The Morphology of Polymer foams - Materials … Introduction Today solid polymers are often used and can be found in all kinds of applications. Much is already know about these polymers

UNIVERSITY OF TECHNOLOGY EINDHOVEN

Mentors: MSc. J.G.F. Wismans dr.ir. J.A.W.v. Dommelen dr.ir. B.v. Rietbergen Date: 14th July 2009

The Morphology of

Polymer foams R.H.M. Titulaer

MT09.23

1

Table of Contents

Introduction ....................................................................................................................................................... 3

1. Polymer foams .............................................................................................................................................. 4

1.1 Introduction ............................................................................................................................................................. 4

1.2 Properties.................................................................................................................................................................. 4

1.3 Structure .................................................................................................................................................................... 4

1.4 Characterization ..................................................................................................................................................... 5

1.4.1 Geometric properties ................................................................................................................................... 5

1.4.2 Intrinsic properties ....................................................................................................................................... 7

2. X-Ray Tomography ..................................................................................................................................... 8

2.1 NanoCT-scan ............................................................................................................................................................ 8

2.2 Methodology ............................................................................................................................................................ 9

2.2.1 Sample preparation ...................................................................................................................................... 9

2.2.2 Measurement settings ................................................................................................................................. 9

2.2.3 Calibration ........................................................................................................................................................ 9

2.3 Results ..................................................................................................................................................................... 10

2.3.1 Reconstruction ............................................................................................................................................ 10

2.3.2 3D volume...................................................................................................................................................... 10

3. Image processing ...................................................................................................................................... 11

3.1 Image processing ................................................................................................................................................ 11

3.2 Script file................................................................................................................................................................. 11

3.3 Test cases ............................................................................................................................................................... 15

3.3.1 Cell size ........................................................................................................................................................... 15

3.3.2 Results cell size ........................................................................................................................................... 15

3.3.3 Strut thickness ............................................................................................................................................. 16

3.3.4 Orientation .................................................................................................................................................... 17

3.4 Remarks .................................................................................................................................................................. 17

4. Analysis of polymer foam ...................................................................................................................... 18

4.1 Closed structure .................................................................................................................................................. 18

4.2 Methodology ......................................................................................................................................................... 18

4.2.1 Import ............................................................................................................................................................. 18

4.2.2 Segmentation ............................................................................................................................................... 18

4.3 Results ..................................................................................................................................................................... 19

4.3.1 Strut thickness ............................................................................................................................................. 19

4.4 Remarks .................................................................................................................................................................. 19

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Conclusion ....................................................................................................................................................... 20

References ....................................................................................................................................................... 21

Appendix A ...................................................................................................................................................... 22

Appendix B ...................................................................................................................................................... 23

Appendix C ....................................................................................................................................................... 27

3

Introduction

Today solid polymers are often used and can be found in all kinds of applications. Much is

already know about these polymers. More interesting are the polymer foams, which are more

often being used. Because they extend the range of properties for particular requirements

compared to the solid polymers. These foams are already used in all kinds of applications like,

the sandwich panels of an aircraft, the packaging material of products, in the isolation of a house

and in the seating of a car. It is important that before a polymer foam is used, it is analyzed

whether it meets the requirements of a certain application. Therefore these foams have to be

characterized to gain information about their properties (Chapter one).

Several of these properties like mean cell strut thickness, mean cell size and their standard

deviations are obtained with the use of X-ray tomography. During this project the Phoenix

Nanotom has been used. The nanotom produces X-rays by accelerating electrons which hit a

target material. During a measurement, a sample (in this case a polymer foam) rotates while it is

exposed to X-rays. The projections made, are combined to generate a reconstruction of the

volume of the specimen, which results in a high 3D resolution image of the sample (Chapter

two).

The high resolution image is then analyzed with image processing software to obtain properties

like the mean cell size, the mean cell strut thickness and their standard deviations. Within this

project, the software developed by Scanco Medical AG is used. Mainly this software is used by

the department of biomedical engineering to analyze bone structures. However the structure of

bones is very similar to the structure of polymer foams. Therefore the Scanco software is in this

case also used to analyze the structure of foams. To determine if the image processing software

is reliable and how it works, test cases will be made to test this method (Chapter three).

If the software developed by Scanco Medical AG proves to be accurate and reliable, any polymer

foam could be characterized. Last the structure of a closed-celled foam will be analyzed.

Parameters like the mean strut thickness and their standard deviations are obtained (Chapter

4).

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1. Polymer foams

1.1 Introduction

Today the use of polymer foams is widespread. They can be found for example in sports and

luxury products, vehicles, aircraft and everybody’s home. That’s why people encounter polymer

foams every day, whether it is in packaging, furniture, their car, the insulation of their house or

as disposable coffee cups and in many more applications.

1.2 Properties

Polymer foams are increasingly used because they extend the range of properties for particular

requirements. Polymer foams have physical, mechanical and thermal properties measured by

the same methods as those used for solid polymers. Figure 1.1 shows the range of four of these

properties: density, thermal conductivity, Young’s modulus and compression strength. The bar

with dotted shading shows the range of property spanned by conventional solids, where the

solid bar shows the extension of this range made possible by polymer foams. This increase in

available materials creates applications for foams where conventional solid polymers cannot

meet the particular requirements. The low densities permit the design of light and stiff

components such as sandwich panels in modern aircrafts. The low thermal conductivity allows

cheap, reliable thermal insulation which is used in a product like the disposable coffee cup but

also as insulation of the booster rockets for the space shuttle. The low stiffness makes polymer

foams ideal for a wide range of cushioning applications, for example the seating in cars. And

because of the low compression strength polymer foams are often used in energy-absorbing

applications, like packaging or body protection.

1.3 Structure

The cells of a polymer foam are polyhedral which pack in three dimensions to fill space [1].

There are two kinds of typical structure. The first one is called open-celled when the solid

material is in the cell edges only, so that the cells connect through open faces (figure 1.2). The

second structure is said to be closed-celled when the faces are solid too; each cell is sealed off

from its surrounding cells (figure 1.3).

Figure 1.1 The range of properties available to the engineer through foaming: (a) density; (b) thermal conductivity; (c)Young’s modulus; (d) compressive strength. [1]

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Figure 1.2 Open celled. Figure 1.3 Closed celled.

1.4 Characterization

1.4.1 Geometric properties

Nowadays polymer foams are used in very different applications with their own requirements,

as mentioned before. Before a foam is used it is important to analyze whether it is suitable for a

certain application. Therefore polymer foams are characterized to gain information about their

properties. The first set of parameters describes its geometric structure like the size and shape

of the cells, the way the material is distributed between cell edges and faces, and the relative

density. Table 1.1 shows the characterization chart for a polymer foam.

Table 1.1 Characterization chart for polymer foam

Material - Open or closed cells - Density 𝜌∗ (kg/m3) Relative density 𝜌∗/𝜌s Volume fraction ∅ Mean cell strut thickness t (mm) Mean cell size D (mm) Standard deviation of cell size sd (mm) Edge connectivity Z𝑒 Face connectivity Zf Mean edges per face 𝑛 Mean faces per cell 𝑓 Cell shape - Symmetry of structure - Largest principal cell dimension 𝐿1 (mm) Smallest principal cell dimension 𝐿3 (mm) Intermediate principal cell dimension 𝐿2 (mm) Shape anisotropy ratios 𝑅12=𝐿1/𝐿2 and 𝑅13=𝐿1/𝐿3

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The material description should be as full as possible, to allow the solid material properties to be

identified [1]. A distinction has to be made whether the foam is open or closed celled. The density

is usually obtained by cutting a block of foam, measure its dimensions and weigh it. A very

important parameter for the characterization of a polymer foam is its relative density, 𝜌∗/𝜌s (the

density, 𝜌∗, of the foam divided by that of the solid of which it is made, 𝜌s ) [1]. If the cell edge-

length is l and the cell-wall thickness is t, and t << l, then the relative density of a polymer foam

always scales as follows :

closed-celled, 𝜌∗/𝜌s = 𝐶1(t/l) (1.1)

open-celled , 𝜌∗/𝜌s = 𝐶2(𝑡/𝑙)2 (1.2)

where the Cs are numerical constants, near unity, that depend on the details of the cell shape.

Generally speaking, polymer foams have relative densities which are less than about 0.3; most of

them are much less, as low as 0.003 [1]. The volume fraction ∅ is the volume of solid material

divided by total volume of foam. This parameter gives information about the distribution of solid

material in the edges and is important in characterizing foams. The mean cell strut thickness (t),

mean cell size (D) and their standard deviations (sd) are important parameters which are

obtained by using X-ray tomography (chapter 2) and 3d image analyzing software (chapter 3).

The edge-connectivity 𝑍𝑒 , is the average number of edges which meet at a vertex (𝑍𝑒 , usually 4)

and the face-connectivity 𝑍𝑓 , is the average number of faces which meet at an edge (Zf , usually

3) (figure 1.4) [1]. These properties together with the mean number of edges per face, 𝑛 are

measured from micrographs using X-ray tomography. The corresponding number of faces per

cell, 𝑓, is then given by Euler’s law [1]:

𝑛 =Z𝑒Zf

Z𝑒−𝟐 1 −

2

𝑓 (1.3)

The cell shape can be found from the connectivity and the number of faces per cell. In appendix A

figure A1 a list of the characteristics of simpler shapes are given. Although most foams are not

regular, the cells often very in shape, using the mean values of 𝑍𝑒 , 𝑍𝑓 , 𝑛 and 𝑓, the cells still can

have a ‘typical’ shape (figure 1.4) [1]. More ‘typical’ shapes are shown in appendix A figure A2.

Most polymer foams have a high symmetry structure. Asymmetry does occur but only when the

cells are elongated or flattened [1]. The principal cell dimensions are obtained from a micrograph

plane section (figure 1.5), by counting the number of cells per unit length (𝑁𝑐) of strait lines

drawn parallel to the principal directions (𝐿1, 𝐿2 and 𝐿3). The mean cell dimension is [1]:

𝐿1 = 1.5

𝑁𝑐 (1.4)

And so on for 𝐿2 and 𝐿3. At last the shape anisotropy ratios 𝑅12=𝐿1/𝐿2 and 𝑅13=𝐿1/𝐿3 are

calculated from the principal cell dimensions [1]. After obtaining all of the above parameters the

geometric structure of all polymer foams can be characterized.

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Figure 1.4 An example of typical shape polymer Figure 1.5 Micrograph plane section

foam: tetrakaidecahedra. of closed celled polymer foam.

1.4.2 Intrinsic properties

The second set of parameters describe the relation between the mechanical and thermal

properties of the solid material and those of the foam. These properties will only be mentioned

in this chapter. The most important parameters for the cell wall properties are the density of the

solid material 𝜌s , the Young’s modulus 𝐸𝑠 , the plastic yield strength 𝜎𝑠 , the fracture strength 𝜎𝑓𝑠 ,

the thermal conductivity 𝜆𝑠 , the thermal expansion coefficient 𝛼𝑠 and the specific heat 𝐶𝑝𝑠 . The

subscript ‘s’ indicates a property of the solid cell wall material.

𝜌∗

𝜌𝑠= 1.18

𝑡

𝑙

Z𝑒= 4

Zf= 3

𝑛= 4

𝑓=12

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2. X-Ray Tomography

2.1 NanoCT-scan

Properties to characterize polymer foams were discussed in chapter one. Several properties like

mean cell strut thickness, mean cell size and their standard deviations are obtained with the use

of X-ray tomography. This method can create a high 3D resolution image of any specific polymer

foam which has to be analyzed. Therefore the Phoenix Nanotom (figure 2.1) has been used. The

Nantom works as follows (figure 2.2). First X-rays are produced when accelerated electrons hit

a target material, molybdenum in this case. During a measurement, the sample rotates 360

degrees, while the projections are recorded in the 5-megapixel detector of the nanoct-scanner.

The attenuation of each projection is adjustable. These projections contain information about

the difference in X-ray rate coefficients in the examined specimen. Finally, when the scan is

finished, the set of projections is used for the reconstruction of the 3D volume.

Figure 2.1 Phoenix Nanotom. Figure 2.2 Working principle of a micro ct-scanner.

Before starting a X-ray scan of the specimen, it is important to optimize the settings in such a

way to achieve the smallest voxelsize (highest resolution) and the highest image quality

(sharpness and contrast). The resolution is dependent on the distance between the source and

the sample (FOD), and the distance between the source and the detector (FDD). And the

resolution of the detector itself (figure2.3). The highest resolution is achieved by decreasing FOD

and increasing FDD, while the whole sample is kept in the detector view and is exposed

sufficiently. The sharpness of the image largely depends on the

spot size. To achieve maximum sharpness, the spot size should

be as small as possible. The sharpness will also be effected by

thermal fluctuations in the system caused by the temperature

increase of the X-ray source. The contrast of the image depends,

apart from the sample characteristics, on the X-ray spectrum

and timing. Generally low energy X-rays are used to improve

the contrast, because low energy X-rays are attenuated more

easily (figure 2.4). But by decreasing the voltage the intensity

also decreases, so the current has to be maximized.

Figure 2.4 Attenuation of X-rays.

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Figure 2.3 The resolution (voxelsize) of reconstruction image depends on the magnification (FDD/FOD) and the detector pixel size (P).

2.2 Methodology

2.2.1 Sample preparation

A small specimen of a polymer foam is used to obtain a high resolution and sharp image.

Therefore a cylinder-shaped sample with a height of 6 mm and a cross sectional area of

approximately 7 mm2 is used. The sample is mounted on a quartz glass rod, which is barely

sensitive too thermal fluctuations.

2.2.2 Measurement settings

For each measurement, the settings that lead to the highest resolution image are used. At each

measurement the distance between source and sample (FOD) is 12 mm. And the distance

between the source and the detector (FFD) is 200 mm (maximum position). This leads to a

voxelsize of 0.003 x 0.003 x 0.003 mm3. To achieve maximum sharpness, beam mode 2 has been

used throughout the measurements and an exposure time of 2000 milliseconds per each

projection. During the measurements two types of specimens were used; a closed-celled foam

and an open-celled foam. This leads to an optimum setting for each material to the highest image

quality. For the open structure a voltage of 40 kVolt and an current of 325 𝜇𝐴 were used. And for

the closed structure a voltage of 50 kVolt and an current of 270 𝜇𝐴. To make sure that these

settings give the best result, a live image gray value histogram is used, to investigate the

intensity and contrast. In both cases the histogram showed a wider peak in the middle of the

histogram which results in good contrast in the reconstruction image.

2.2.3 Calibration

To get the best high resolution images of the sample, it is important that before each

measurement a calibration is done. First the number of steps of rotation was adjusted to 1400

steps. Then number of steps which were skipped was adjusted to 1000 steps. This is done to

make sure the specimen is up to temperature (temperature of the X-ray source increases and

warms the specimen up) when the scan starts. After the skipped steps which takes about 20

minutes, the sample will rotate 360 degrees in 1400 steps. The next calibrations steps are very

important before a measurement can start. The X-ray is turned off and the filament is adjusted

(additional functions > adjust filament). Secondly the X-ray is turned on and the X-ray source is

centered (additional functions > centuring1). Next an image calibration is performed to get the

highest quality image. First the sample is lowered to the zero point (z direction). Then the X-ray

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is turned off and an offset scan is started. Secondly the x-ray is turned on and a multiple gain

scan is started. Finally the sample is restored to its old position and a measurement can start.

2.3 Results

2.3.1 Reconstruction

After the scan, a reconstruction has to be made of the set of projections. The program datosloc-

reconstruction is used to reconstruct the volume. The program uses a *.pca file to combine the

1400 (2D) projection images of the sample. But before a reconstruction can start the data from

the scan have to be optimized. The function automatic geometry calibration calculates the

deviation of the first and last projection, preferable under 10. Secondly the beam hardening

correction has to be adjusted to the value 2. Then a scan optimization with the first and last image

of the projections is performed. This is necessary because during the scan the sample moves

which causes a small difference between the first and last image. With the scan optimization this

can be corrected, by moving the first and last image in any direction to let them fit. Then a

reconstruction of the volume can be made (maximum volume 790x790x790 pixels).

2.3.2 3D volume

After the reconstruction, the new volume can be analyzed with the program called VGStudiomax.

It opens a *.vgi file produced by the reconstruction program. With the tool classification a

histogram is used to segment the image. The histogram shows two peaks, the left peak

represents all the air in the volume and the right peak the solid polymer material. Next a good

threshold has to be found to make sure that only the solid material is shown (figure 2.5 and

figure 2.6). Finally the tool filter is used to run a Gauss filter, which reduces the noise in the 3D

image. The results are shown above (figure 2.5 and figure 2.6).

Figure 2.5 Open-celled polymer foam. Figure 2.6 Closed-celled polymer foam.

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3. Image processing

3.1 Image processing

The process of the reconstruction of 2D projections to a high resolution 3D image of the volume

of the specimen is described in chapter 2. The next step is to analyze the volume with image

processing software to obtain properties like mean cell strut thickness, mean cell size and their

standard deviations. In this chapter image processing software is used developed by Scanco

Medical AG. Mainly this software is used by the department of biomedical engineering to analyze

bone structures. Because the structure of bones (figure 3.2) resemblance the structure of

polymer foams (figure 3.2) the image processing software can also be used to analyze polymer

foams.

Figure 3.1 Polymer foam. Figure 3.2 Trabecular bone structure [3].

3.2 Script file

A script file has been made which renders all the important functions to obtain the mean cell

size, mean strut thickness and their standard deviations. Below the structure of the script is

given and all its functions are described [2].

1. Read: reads an object from the disk by giving it an internal name (org). The file type has

to be *.aim . ipl/read

-name org

-filename ORG_FILE

2. Sup_divide: this command is used to subdivide an object into smaller object. This is

helpful in examining only a small part of the complete object or for segmenting the whole

object using small subobjects to save memory. There are two ways to define the

subvolume(s). The first one is by entering the number of subvolumes (supdim_numbers).

Or by entering the size of the volume (subdim_pixels). Suppos is the offset of the object in

pixels in local coordinates. If you leave the value at -1, the object will be centered. Testoff

is used for overlapping the small subobjects in case of a procedure such as a gauss-filter. ipl/sup_divide

-input org -supdim_numbers 4 4 1

-testoff_pixels 0 0 0 -suppos_pixels_local -1 -1 -1

-subdim_pixels -1 -1 -1

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3. Threshold: this command is used to binarize an object. All voxels below lower_in_perm

and above upper_in_perm will be set to value 0. The rest of the object will be given a

value of value_in_range usually 127. The values for the lower and upper threshold are

given in 1/1000, covering the whole range of data values. ipl/threshold

-input org

-output seg

-lower_in_perm_aut_al 300.000000

-upper_in_perm_aut_al 100000.000000

-value_in_range 127

-unit 0

4. Write: writes an object to the disk. The file type should be *.aim. It is possible to

compress segemented files using a binary compression. ipl /write_v020

-name seg

-filename SEG_FILE

-compress_type bin

-version_020 true

5. Vox_scanco_param: this command evaluates a segmented image for the fraction of voxels

sets and writes the result to the database. For gray scale images, the mean attenuation

coefficient is written to the database. ipl/vox_scanco_param

-input seg

-region_number 0

6. Tri_da_metric_db: this command triangulates a segmented object (figure 3.3 white lines)

and calculates the object volume as well as the structure model index (SMI). Using the

plate model (2D slices), the trabecular number, thickness and separation of the

segmented object are derived. However the results of this function are not reliable

because the outcome of the parameters is dependent on which slice is used for

calculations. ipl/tri_da_metric_db -input seg -output tri

-peel_iter -1 -ip_sigma 0.000000

-ip_support 0 -ip_threshold 50

-nr_ave_iter 2 -t_dir_radius 2

-epsilon 1.200000 -size_image 512 512

-scale_image 0.700000 -edges false -nr_views 0

Figure 3.3 Function tri_da_metric uses 2D skeleton method to calculate parameters.

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7. Copy: this command is used to copy files to the disk. In this case the segmentation file

(seg) is copied and saved as temporal file (tmp). ipl/copy

-in seg

-out tmp

8. Delete: this command is used to delete files from the disk. In this case the temporal file

(tmp) is deleted. ipl/delete

-name tmp

9. Dt_object_param: calculates the mean separation of the structure with the distance

transformation (DT) method by filling largest spheres into the object and calculating

their mean diameter, t (figure 3.4). Mean thickness and standard deviation are written to

the database. The output object shows the spheres fitting inside the structure with the

voxel values being their diameter in voxel units (voxel size = 1x1x1 mm). The

Roi_radius_factor controls whether only a sphere is evaluated or the whole region. The

ridge_epsilon and assign_epsilon are used for suppressing artefacts due to rough surfaces.

Finally Histofile_or_screen, computes a histogram of the thickness distribution and writes

it to a text file (_th.txt). ipl/dt_object_param

-input seg -output out

-peel_iter -1 -roi_radius_factor 10000.000000

-ridge_epsilon 0.900000 -assign_epsilon 1.800000

-version2 true -histofile_or_screen _th.txt

Figure 3.4 Function dt_object_param calculates strut thickness (t).

10. Dt_background_param: calculates the mean cell size of the structure with the distance

transformation (DT) method by filling largest spheres into the background of the object

and calculating their mean diameter, D (figure 3.5). Mean cell size and standard

deviation are written to the database. The output object shows the spheres fitting inside

the background with the voxel values being their diameter in voxel units (voxel size =

1x1x1 mm). The Roi_radius_factor controls whether only a sphere is evaluated or the

whole region. The ridge_epsilon and assign_epsilon are used for suppressing artefacts

due to rough surfaces. Finally Histofile_or_screen, computes a histogram of the cell size

distribution and writes it to a text file (_sp.txt).

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ipl/dt_background_param




-version2 true -histofile_or_screen _sp.txt

Figure 3.5 Function dt_background_param calculates cell size (D).

11. Dt_mat_param: calculates the mean trabecular number of the structure with the distance

transformation (DT) method by filling largest spheres into the background of the mid

axis transformed object and calculating their mean diameter, l (figure 3.6). Mean number

and standard deviation are written to the database. The output object shows the spheres

fitting inbetween the mid axis structure with the voxel values being their diameter in

voxel units (voxel size = 1 1 1 mm). The Roi_radius_factor controls whether only a

sphere is evaluated or the whole region. The ridge_epsilon and assign_epsilon are used for

suppressing artefacts due to rough surfaces. Finally Histofile_or_screen, computes a

histogram of trabecular number distribution and writes it to a text file (_1-over-n.txt). ipl/dt_mat_param




-version2 true -histofile_or_screen _1-over-n.txt

Figure 3.6 Function dt_mat_param calculates the distance between mid axis structure (l).

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3.3 Test cases

Before a polymer foam is analyzed with the above method, the image analyzing software has

been tested for its accuracy. Therefore several test cases were defined to compare the known

input with the computed output of the script file. The test cases were made using Matlab, see

appendix B for the m-files. The m-files produces a signed 8-bit binary file (extension *.raw),

which will be imported in the image processing software . Then the script file was used to

analyze the test case.

3.3.1 Cell size

First the function dt_background_param, which calculates cell size, was tested using the eight

test cases below. Each case figures a cubic of 100x100x100 voxels. The red colored voxels

represent the material, where the blue voxels represent the air. Figure 3.7a till 3.7f show the

cross sectional area of the test volume.

Figure 3.7a Closed structure: Figure 3.7b Closed structure: Figure 3.7c Closed structure: Sphere. Ellipse. two connecting spheres.

Figure 3.7d Closed structure: Figure 3.7e Open structure: Figure 3.7f Open structure: edge diffusion. six half spheres. six half ellipse and one sphere.

3.3.2 Results cell size

All the results of the above test cases, analyzed by Scanco Medical AG software, are shown in

appendix B. The Scanco software outputs a text file which figures the mean cell diameter, the

standard deviation and a list with three columns. The first column figures the cell diameters

measured, the second column show their corresponding number of voxels, and the third column

gives these voxels as a percentage of the total number of voxels. The test cases lead to some

interesting results. The first and second test case (figure 3.7a, 3.7b ) are as expected, one

diameter is measured. But the third case is unexpected(3.7c). Instead of one, the Scanco

software measures two diameters, although the two spheres are connected. The fourth case

shows an enclosed sphere with diffusion on the edge (figure 3.7d). Due to the threshold (all the

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Figure 3.8 Sphere defined by Matlab.

material gets one gray value), only one diameter is measured.

Interesting are test cases five and six, the analyzing software cannot

measure the open structures; it does not measure the diameter of the

six half spheres (figure 3.7e) and the six ellipses (figure 3.7f). Contrary

it only measures the diameter of the enclosed sphere (figure 3.7f).

Last, in all cases the measured diameters (output) differs from the

input, the spheres defined in Matlab. The difference for small spheres

is about three voxels and one voxel for larger spheres. The cause is not

the inaccuracy of the Scanco software but the way Matlab defines a

sphere (equation 2.1). Where r is the radius, l the length of the volume

(in this case a cubic of 100x100x100 voxels) and x0, y0, z0 coordinates to position the sphere.

(l − x0)2 + (l − y0)2 + (l − z0)2 ≦ r2 (2.1)

The sphere (figure 3.8) has a diameter of 16 voxels, though Scanco measures a diameter of 14

voxels. The difference can be explained by the black line being shorter than the white line. The

Scanco software uses the shorter black line as the diameter for the sphere instead of the white

line. Taken all the above comments in to account, two new test cases were defined which

resembles a polymer foam more and are analyzed by Scanco (figure 3.9a and 3.9b). The results,

shown in appendix B, are as expected. Namely only one diameter is measured for the first case

and four different diameters are measured for the second case.

Figure 3.9a Much spheres. Figure 3.9b Different spheres

3.3.3 Strut thickness

Also the function dt_object_param, which calculates the strut thickness, was tested using the two

test cases below. In both test cases rectangular shapes are defined, so the strut thickness

measured by the Scanco software can be compared with the known strut thickness. The results

shown in appendix B, are as expected.

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Figure3.10a Four rectangular. Figure 3.10b Different rectangular.

3.3.4 Orientation

It is important to know if the orientation of the volume has influence on the image analysis.

Figure 3.11a and 3.11b show two test cases which have the same volume. The difference

between them is that the second volume is rotated 90 degrees anti-clockwise. Both test cases are

analyzed by the Scanco Medical AG software. The results are shown in appendix B, and they

show that the orientation has nearly no influence.

Figure 3.0.11a Different spheres. Figure 3.11b Different spheres Other orientation.

3.4 Remarks

The image analyzing software developed by Scanco Mediacal AG looks like a very useful tool for

obtaining morphology parameters of polymer foams. As tested above it calculates the cell size

and strut thickness of a foam quite accurate. But as the results also show in appendix B, the

software only works well for enclosed structures, where it runs into problems with open

structures (figure 3.7e and 3.7f). Therefore more research is needed to understand how the

software works. Also more research is needed on the Scanco software, which features more

functions to calculate other morphology parameters as the connectivity density, mean intercept

length and the degree of anisotropy.

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4. Analysis of polymer foam

4.1 Closed structure

From the test cases defined in chapter 3, it is appropriate to conclude that the image analyzing

software developed by Scanco Medical AG is reasonable accurate and reliable (for closed

structures). The next step will be the analysis of a real polymer foam. The results from the

previous chapter also show that the Scanco software gives problems when analyzing an open

structure. Therefore in this chapter the analysis results of a closed-celled polymer foam (figure

4.1) will be presented.

Figure 4.1 Closed-celled polymer foam.

4.2 Methodology

4.2.1 Import

First the reconstruction file*.vig has to be opened with VGStudiomax (Chapter two). The volume

must be remained unfiltered and not segmented otherwise the image analysis program runs into

problems. Next the volume is exported as a headerless *.raw file. This headerless RAW file

contains 16 bit unsigned data. The data file is then imported, with function /import into the

Sanco Medical AG software. The headerless RAW file has to be converted to a signed 16 bit data

file with the function /convert_signed , or else the Scanco software runs into problems. Then the

signed RAW data file is saved as an *.AIM file, with function /write, so the reconstructed volume

can be analyzed by the image analysis program developed by Scanco Medical AG.

4.2.2 Segmentation

The binary AIM file is yet unfiltered, which means it contains noise. Next a Gauss filter is used,

with function /Gauss, to reduce this noise (figure 4.2). The final step is to segment the AIM file.

This is done by applying a threshold on to the filtered data file. Because the AIM is a signed 16

bit data file, the histogram has gray values with a lower limit of -32768 and an upper limit of

32768. If the threshold is chosen too low the data file contains too much noise (gray values

which represent the air in a volume). Otherwise if the threshold is chosen too high than there

will be no noise but too much material is gone (gray values which represent the polymer

structure). The best threshold for the segmentation of the closed structure has a gray value with

19

a lower limit of 1950 and an upper limit of 32767. But to study the influence of the threshold on

the analysis, two more threshold are being used. One threshold has gray values with a lower

limit of 1930 and an upper limit of 32767, which implies more solid material in the structure of

the foam and more noise. The second other threshold has gray values with a lower limit of 1970

and an upper limit of 32767, which implies less noise and less material in the structure of the

polymer foam .

4.3 Results

4.3.1 Strut thickness

The closed-celled polymer foam (figure 4.1) is analyzed for each threshold mentioned above.

The complete results are shown in Appendix C. First the function Dt_object_param is used to

calculate the strut thickness. The mean strut thickness, Standard deviation and Median are shown

in table 4.1. The results show that if the lower limit of the threshold decreases the mean strut

thickness increases, and if the lower limit increases the strut thickness will decrease. To

conclude it is very important to chose the right threshold before analyzing a foam.

Table 4.1 Results strut thickness

Threshold Mean strut thickness Standard deviation Median 1930 - 32767 0.0337 [mm] 0.0126 [mm] 0.0330 [mm]

1950 - 32767 0.0332 [mm] 0.0124 [mm] 0.0330 [mm]

1970 - 32767 0.0328 [mm] 0.0122 [mm] 0.0330 [mm]

4.4 Remarks

The results above can be conclude the Scanco software is an useful method to study the

morphology of polymer foams. But in the future more research is needed into the impact of the

threshold on the results. And into the other functions to calculate other morphology parameters

as the cell size, the connectivity density, mean intercept length and the degree of anisotropy.

20

Conclusion

Polymer foams are being used increasingly, because they extend the range of properties for

particular requirements compared to solid polymers. To study the morphology, these foams are

characterized. The most important parameters are the mean cell size, mean strut thickness and

relative density. In this case the mean strut thickness of a closed-celled foam is obtained by using

X-ray tomography. It is important to make a high resolution reconstruction of the polymer foam

because this image is used by an image processing software to obtain the mean cell strut

thickness, standard deviation and distribution. If for example the resolution is too low, the

structure of the specimen will not be reconstructed well, and information about the structure is

lost. The image processing software is developed by Scanco Medical AG. To test its accuracy,

several test cases were defined to compare the known input with the computed output of the

Scanco software. They show the software only works well for enclosed structures (which figures

a closed-celled foam), where it runs into problems with open structures. This requires more

research. When using the Scanco software it is very important to find the right threshold. If it is

chosen too low the reconstructed volume contains too much noise and results of the Scanco

software are unreliable. Otherwise if the threshold is chosen too high than there will be no noise

but too much material is gone, which means information about the structure is lost. Then the

results by the image analysis software are also useless. To make sure they are reliable, the

structure has to stay intact and the noise has to be minimal. Therefore a Gauss filter is used,

which reduces the noise. After the segmentation, the Scanco software calculates the mean strut

thickness with function dt_object_param. The results show a lower threshold leads to a mean

strut thickness of 0.0337 mm, where a higher threshold leads to a mean strut thickness of

0.0328 mm. It is not clear how big the impact of the threshold has on the results. Therefore more

research is needed to understand how the software works and to control other functions to

calculate morphology parameters, such as the mean cell size, connectivity density, mean intercept

length (MIL) and the degree of anisotropy.

21

References

[1] Lorna J.Gibson and Michael F.Ashby (1997) Cellular solids structure and properties, 2nd

edition. editors D.R Clarke, S. Suresh and I.M. Ward FRS. Cambridge University Press,

Cambridge.

[2] Scanco Medical AG (1999) MicroCT 80 user’s guide. Scanco Medical, Bassersdorf.

[3] J.H. Kinney, J.S. Stolken, T.S. Smith, J.T. Ryaby, and N.E. Lane (2005) An orientation distribution function for trabecular bone, Bone 36(2), 193-201.

22

Appendix A

Figure A1 The relative densities of cell aggregates [1].

Figure A2 The packing of polyhedra to fill space: (a) triangular prisms, (b) rectangular prisms, (c) hexagonal prisms, (d) rhombic dodecahedra, (e) tetrakaidecahedra [1].

23

Appendix B

Results Cell size

DT Histogram: one sphere

(closed structure)

Patient Name: Imported_Sample

Patient/Sample Number: 0

Measurement Number: 0

Dim: 100 100 100

Off: 2 2 2

Pos: 0 0 0

El_size_mm: 1.0000 1.0000 1.0000

Segmentation with Unit: 0.00 / 0

/300.000000 / 6

Total Number of voxels 113081

Mean = 59.00000 [mm] sd = 0.00000 [mm]

Median = 59.00000 [mm]

---------------------------------------

60 Bins with Size 1.00000 [mm] each

Binstart(incl)[mm] Counts in % of

Total Nr

42.0000 0 0.000

43.0000 0 0.000

44.0000 0 0.000

45.0000 0 0.000

46.0000 0 0.000

47.0000 0 0.000

48.0000 0 0.000

49.0000 0 0.000

50.0000 0 0.000

51.0000 0 0.000

52.0000 0 0.000

53.0000 0 0.000

54.0000 0 0.000

55.0000 0 0.000

56.0000 0 0.000

57.0000 0 0.000

58.0000 0 0.000

59.0000 113081 100.000

DT Histogram: two connecting spheres

(closed structure)




Dim: 100 100 100

Off: 2 2 2

Pos: 0 0 0

El_size_mm: 1.0000 1.0000 1.0000


/300.000000 / 6


Mean = 67.26280 [mm] sd = 8.25880 [mm]

Median = 69.00000 [mm]

----------------------------------------



Total Nr

25.0000 0 0.000

26.0000 0 0.000

27.0000 0 0.000

28.0000 7986 4.237

29.0000 0 0.000

30.0000 0 0.000

31.0000 0 0.000

32.0000 0 0.000

38.0000 0 0.000

39.0000 0 0.000

65.0000 0 0.000

68.0000 0 0.000

69.0000 180493 95.763

DT Histogram: ellipse

(closed structure)




Dim: 100 100 100

Off: 2 2 2

Pos: 0 0 0

El_size_mm: 1.0000 1.0000 1.0000


/300.000000 / 6


Mean = 44.80386 [mm] sd = 0.59483 [mm]

Median = 45.00000 [mm]

---------------------------------------



Total Nr

28.0000 0 0.000

29.0000 0 0.000

30.0000 0 0.000

31.0000 0 0.000

32.0000 0 0.000

33.0000 0 0.000

34.0000 0 0.000

35.0000 0 0.000

36.0000 0 0.000

37.0000 0 0.000

38.0000 0 0.000

39.0000 0 0.000

40.0000 0 0.000

41.0000 0 0.000

42.0000 0 0.000

43.0000 7310 9.807

44.0000 0 0.000

45.0000 67227 90.193

DT Histogram: edge diffusion

(closed structure)




Dim: 100 100 100

Off: 2 2 2

Pos: 0 0 0

El_size_mm: 1.0000 1.0000 1.0000


/300.000000 / 6


Mean = 79.00000 [mm] sd = 0.00000 [mm]

Median = 79.00000 [mm]

----------------------------------------



Total Nr

66.0000 0 0.000

67.0000 0 0.000

68.0000 0 0.000

69.0000 0 0.000

70.0000 0 0.000

71.0000 0 0.000

72.0000 0 0.000

73.0000 0 0.000

74.0000 0 0.000

75.0000 0 0.000

76.0000 0 0.000

77.0000 0 0.000

78.0000 0 0.000

79.0000 267761 100.000

24

DT Histogram: six half spheres (open

structure)




Dim: 100 100 100

Off: 2 2 2

Pos: 0 0 0

El_size_mm: 1.0000 1.0000 1.0000


/300.000000 / 6


Mean = 0.00000 [mm] sd = 0.00000 [mm]

Median = 0.00000 [mm]

-----------------------------------------



Total Nr

0.0000 0 0.000

1.0000 0 0.000

DT Histogram: much spheres




Dim: 100 100 100

Off: 2 2 2

Pos: 0 0 0

El_size_mm: 1.0000 1.0000 1.0000


/300.000000 / 6


Mean = 9.00000 [mm] sd = 0.00000 [mm]

Median = 9.00000 [mm]

----------------------------------------



Total Nr

0.0000 0 0.000

1.0000 0 0.000

2.0000 0 0.000

3.0000 0 0.000

4.0000 0 0.000

5.0000 0 0.000

6.0000 0 0.000

7.0000 0 0.000

8.0000 0 0.000

9.0000 263680 100.000

DT Histogram: six half ellipse and one

sphere (open structure)




Dim: 100 100 100

Off: 2 2 2

Pos: 0 0 0

El_size_mm: 1.0000 1.0000 1.0000


/300.000000 / 6


Mean = 38.00000 [mm] sd = 0.00000 [mm]

Median = 38.00000 [mm]

----------------------------------------



Total Nr

31.0000 0 0.000

32.0000 0 0.000

33.0000 0 0.000

34.0000 0 0.000

35.0000 0 0.000

36.0000 0 0.000

37.0000 0 0.000

38.0000 33401 100.000

DT Histogram: different spheres




Dim: 200 200 200

Off: 2 2 2

Pos: 0 0 0

El_size_mm: 1.0000 1.0000 1.0000


/300.000000 / 6


Mean = 22.31272 [mm] sd = 9.52565 [mm]

Median = 19.00000 [mm]

----------------------------------------



Total Nr

3.0000 0 0.000

4.0000 0 0.000

5.0000 0 0.000

6.0000 0 0.000

7.0000 0 0.000

8.0000 0 0.000

9.0000 32960 14.117

10.0000 0 0.000

11.0000 0 0.000

12.0000 0 0.000

13.0000 0 0.000

14.0000 52725 22.582

15.0000 0 0.000

16.0000 0 0.000

17.0000 0 0.000

18.0000 0 0.000

19.0000 50175 21.490

20.0000 0 0.000

21.0000 0 0.000

22.0000 0 0.000

23.0000 0 0.000

24.0000 0 0.000

25.0000 0 0.000

26.0000 0 0.000

27.0000 0 0.000

28.0000 0 0.000

29.0000 0 0.000

30.0000 0 0.000

31.0000 0 0.000

32.0000 0 0.000

33.0000 97620 41.811

25

Results strut thickness

DT Histogram: four rectangular




Dim: 100 100 101

Off: 2 2 2

Pos: 0 0 0

El_size_mm: 1.0000 1.0000 1.0000


/300.000000 / 6


Mean = 8.00000 [mm] sd = 0.00000 [mm]

Median = 8.00000 [mm]

----------------------------------------



Total Nr

0.0000 0 0.000

1.0000 0 0.000

2.0000 0 0.000

3.0000 0 0.000

4.0000 0 0.000

5.0000 0 0.000

6.0000 0 0.000

7.0000 0 0.000

8.0000 208128 100.000

DT Histogram: different rectangular




Dim: 100 100 101

Off: 2 2 2

Pos: 0 0 0

El_size_mm: 1.0000 1.0000 1.0000


/300.000000 / 6


Mean = 12.58786 [mm] sd = 4.63872 [mm]

Median = 12.00000 [mm]

----------------------------------------



Total Nr

0.0000 0 0.000

1.0000 0 0.000

2.0000 0 0.000

3.0000 0 0.000

4.0000 33396 7.329

5.0000 6264 1.375

6.0000 0 0.000

7.0000 0 0.000

8.0000 68474 15.027

9.0000 1854 0.407

10.0000 1610 0.353

11.0000 105724 23.202

12.0000 10952 2.404

13.0000 6156 1.351

14.0000 103964 22.816

15.0000 2576 0.565

16.0000 66904 14.683

17.0000 0 0.000

18.0000 0 0.000

19.0000 0 0.000

20.0000 0 0.000

21.0000 64 0.014

22.0000 47728 10.474

26

Orientation

Cell size histogram: different spheres

Dim: 200 200 200

Off: 2 2 2

Pos: 0 0 0

El_size_mm: 1.0000 1.0000 1.0000


/300.000000 / 6


Mean = 22.31272 [mm]

sd = 9.52565 [mm]

Median = 19.00000 [mm]

----------------------------------------



Total Nr

8.0000 0 0.000

9.0000 32960 14.117

10.0000 0 0.000

11.0000 0 0.000

12.0000 0 0.000

13.0000 0 0.000

14.0000 52725 22.582

15.0000 0 0.000

16.0000 0 0.000

17.0000 0 0.000

18.0000 0 0.000

19.0000 50175 21.490

20.0000 0 0.000

21.0000 0 0.000

22.0000 0 0.000

23.0000 0 0.000

27.0000 0 0.000

28.0000 0 0.000

29.0000 0 0.000

30.0000 0 0.000

31.0000 0 0.000

32.0000 0 0.000

33.0000 97620 41.811

Strut thickness histogram: different

spheres

Dim: 200 200 200

Off: 2 2 2

Pos: 0 0 0

El_size_mm: 1.0000 1.0000 1.0000


/300.000000 / 6


Mean = 10.58505 [mm] sd = 4.11815 [mm]

Median = 11.00000 [mm]

----------------------------------------



Total Nr

1.0000 0 0.000

2.0000 0 0.000

3.0000 16795 5.330

4.0000 13521 4.291

5.0000 15827 5.022

6.0000 14010 4.446

7.0000 31465 9.985

8.0000 7773 2.467

9.0000 26040 8.263

10.0000 5753 1.826

11.0000 45427 14.415

12.0000 18988 6.025

13.0000 35684 11.324

14.0000 8409 2.668

15.0000 60007 19.042

16.0000 98 0.031

17.0000 2047 0.650

18.0000 4412 1.400

19.0000 8875 2.816

Cell size histogram: different spheres

other orentation

Dim: 200 200 200

Off: 2 2 2

Pos: 0 0 0

El_size_mm: 1.0000 1.0000 1.0000


/300.000000 / 6


Mean = 22.31272 [mm] sd = 9.52565 [mm]

Median = 19.00000 [mm]

----------------------------------------



Total Nr

8.0000 0 0.000

9.0000 32960 14.117

10.0000 0 0.000

11.0000 0 0.000

12.0000 0 0.000

13.0000 0 0.000

14.0000 52725 22.582

15.0000 0 0.000

16.0000 0 0.000

17.0000 0 0.000

18.0000 0 0.000

19.0000 50175 21.490

20.0000 0 0.000

21.0000 0 0.000

22.0000 0 0.000

23.0000 0 0.000

27.0000 0 0.000

28.0000 0 0.000

29.0000 0 0.000

30.0000 0 0.000

31.0000 0 0.000

32.0000 0 0.000

33.0000 97620 41.811

Strut thickness histogram: different

spheres other orientation

Dim: 200 200 200

Off: 2 2 2

Pos: 0 0 0

El_size_mm: 1.0000 1.0000 1.0000

Segmentation with Unit: 0.00 / 0 \

/300.000000 / 6


Mean = 10.61916 [mm] sd = 4.11698 [mm]

Median = 11.00000 [mm]

----------------------------------------



Total Nr

1.0000 0 0.000

2.0000 0 0.000

3.0000 16856 5.330

4.0000 13728 4.341

5.0000 15248 4.821

6.0000 13885 4.390

7.0000 30788 9.735

8.0000 7732 2.445

9.0000 27782 8.785

10.0000 5867 1.855

11.0000 41236 13.039

12.0000 21411 6.770

13.0000 35767 11.309

14.0000 10520 3.326

15.0000 60008 18.974

16.0000 98 0.031

17.0000 2047 0.647

18.0000 4412 1.395

19.0000 8875 2.806

Appendix C Results strut thickness

Histogram: threshold 1930 - 32767

Dim: 786 786 786

Off: 2 2 2

Pos: 2 2 2

El_size_mm: 0.0030 0.0030 0.0030

Segmentation with Unit: 1.00 / 2 / 1930.000000 / 5


Mean = 11.24

sd = 4.20

Median = 11.00

Mean_unit = 0.0337 [mm]

SD_unit = 0.0126 [mm]

Median_unit = 0.0330 [mm]

---------------------------------------------------


Binstart(incl)[mm] Counts Vol[mm^3] in-%

0.0000 0 0.000000 0.000

0.0030 182540 0.004929 0.151

0.0060 562721 0.015193 0.464

0.0090 874587 0.023614 0.722

0.0120 2280520 0.061574 1.882

0.0150 4681060 0.126389 3.863

0.0180 5524091 0.149150 4.558

0.0210 9377742 0.253199 7.738

0.0240 11109882 0.299967 9.168

0.0270 12056603 0.325528 9.949

0.0300 9842322 0.265743 8.122

0.0330 11526893 0.311226 9.512

0.0360 8570068 0.231392 7.072

0.0390 10975329 0.296334 9.057

0.0420 6430693 0.173629 5.307

0.0450 8000951 0.216026 6.602

0.0480 4236043 0.114373 3.496

0.0510 5449335 0.147132 4.497

0.0540 2720834 0.073463 2.245

0.0570 2642827 0.071356 2.181

0.0600 1369407 0.036974 1.130

0.0630 1341205 0.036213 1.107

0.0660 555331 0.014994 0.458

0.0690 436943 0.011797 0.361

0.0720 193281 0.005219 0.159

0.0750 167707 0.004528 0.138

0.0780 32311 0.000872 0.027

0.0810 6922 0.000187 0.006

0.0840 35351 0.000954 0.029

28


Dim: 786 786 786

Off: 2 2 2

Pos: 2 2 2

El_size_mm: 0.0030 0.0030 0.0030



Mean = 11.08

sd = 4.14

Median = 11.00


SD_unit = 0.0124 [mm]


---------------------------------------------------



0.0000 0 0.000000 0.000

0.0030 139164 0.003757 0.118

0.0060 468408 0.012647 0.396

0.0090 807067 0.021791 0.682

0.0120 2369033 0.063964 2.003

0.0150 4986697 0.134641 4.216

0.0180 5778772 0.156027 4.886

0.0210 9601588 0.259243 8.118

0.0240 11114120 0.300081 9.397

0.0270 11964439 0.323040 10.116

0.0300 9551605 0.257893 8.076

0.0330 11323526 0.305735 9.574

0.0360 8327303 0.224837 7.041

0.0390 10613045 0.286552 8.973

0.0420 6062189 0.163679 5.125

0.0450 7623823 0.205843 6.446

0.0480 4005226 0.108141 3.386

0.0510 5161572 0.139362 4.364

0.0540 2507183 0.067694 2.120

0.0570 2339340 0.063162 1.978

0.0600 1192292 0.032192 1.008

0.0630 1157405 0.031250 0.979

0.0660 472900 0.012768 0.400

0.0690 319833 0.008635 0.270

0.0720 204044 0.005509 0.173

0.0750 126479 0.003415 0.107

0.0780 17447 0.000471 0.015

0.0810 25419 0.000686 0.021

0.0840 16773 0.000453 0.014

29


Dim: 786 786 786

Off: 2 2 2

Pos: 2 2 2

El_size_mm: 0.0030 0.0030 0.0030



Mean = 10.92

sd = 4.08

Median = 11.00


SD_unit = 0.0122 [mm]


---------------------------------------------------



0.0000 0 0.000000 0.000

0.0030 110550 0.002985 0.096

0.0060 403938 0.010906 0.349

0.0090 793869 0.021434 0.686

0.0120 2487697 0.067168 2.151

0.0150 5299797 0.143095 4.582

0.0180 5948445 0.160608 5.142

0.0210 9838666 0.265644 8.506

0.0240 11084408 0.299279 9.583

0.0270 11822145 0.319198 10.220

0.0300 9333108 0.251994 8.069

0.0330 11319144 0.305617 9.786

0.0360 8043242 0.217168 6.953

0.0390 10246887 0.276666 8.859

0.0420 5743796 0.155082 4.966

0.0450 7243258 0.195568 6.262

0.0480 3693081 0.099713 3.193

0.0510 4843110 0.130764 4.187

0.0540 2220937 0.059965 1.920

0.0570 2220126 0.059943 1.919

0.0600 1119061 0.030215 0.967

0.0630 840884 0.022704 0.727

0.0660 444757 0.012008 0.384

0.0690 261169 0.007052 0.226

0.0720 196596 0.005308 0.170

0.0750 76097 0.002055 0.066

0.0780 0 0.000000 0.000

0.0810 37539 0.001014 0.032

Documents

The Morphology of Polymer foams - Materials … Introduction Today solid polymers are often used and can be found in all kinds of applications. Much is already know about these polymers