# Thermal expansion coefficient and bulk modulus of polyethylene closed‐cell foams

**ABSTRACT** A regular Kelvin foam model was used to predict the linear thermal expansion coefficient and bulk modulus of crosslinked, closed-cell, low-density polyethylene (LDPE) foams from the polymer and gas properties. The materials used for the experimental measurements were crosslinked, had a uniform cell size, and were nearly isotropic. Young's modulus of biaxially oriented polyethylene was used for modeling the cell faces. The model underestimated the foam linear thermal expansion coefficient because it assumed that the cell faces were flat. However, scanning electron microscopy showed that some cell faces were crumpled as a result of foam processing. The measured bulk modulus, which was considerably smaller than the theoretical value, was used to estimate the linear thermal expansion coefficient of the LDPE foams. © 2004 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 42: 3741–3749, 2004

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**ABSTRACT:**The cellular structure, physical properties, and structure–property relationships of novel open-cell polyolefin foams produced by compression molding and based on blends of an ethylene/vinyl acetate copolymer and a low-density polyethylene have been studied and compared with those of closed-cell polyolefin foams of similar chemical compositions and densities and with those of open-cell polyurethane foams. Properties such as the elastic modulus, collapse stress, energy absorbed in mechanical tests, thermal expansion, dynamic mechanical response, and acoustic absorption have been measured. The experimental results show that the cellular structure of the analyzed materials has interconnected cells due to the presence of large and small holes in the cell walls, and this structure is clearly different from the typical structure of open-cell polyurethane foams. The open-cell polyolefin foams under study, in comparison with closed-cell foams of similar densities and chemical compositions, are good acoustic absorbers; they have a significant loss factor and lower compressive strength and thermal stability. The physical reasons for this macroscopic behavior are analyzed. © 2009 Wiley Periodicals, Inc. J Appl Polym Sci, 2009Journal of Applied Polymer Science 06/2009; 114(2):1176 - 1186. · 1.40 Impact Factor - [show abstract] [hide abstract]

**ABSTRACT:**Finite element analysis, of regular Kelvin foam models with all the material in uniform-thickness faces, was used to predict the compressive impact response of low-density closed-cell polyethylene and polystyrene foams. Cell air compression was analysed, treating cells as surface-based fluid cavities. For a typical 1 mm cell size and 50 s-1 impact strain rate, the elastic buckling of cell faces, and pop-in shape inversion of some buckled square faces, caused a non-linear stress strain response before yield. Pairs of plastic hinges formed across hexagonal faces, then yield occurred when trios of faces concertinaed. The predicted compressive yield stresses were close to experimental data, for a range of foam densities. Air compression was the hardening mechanism for engineering strains < 0.6, with face- to-face contact also contributing for strains > 0.7. Predictions of lateral expansion and residual strains after impact were reasonable. There were no significant changes in the predicted behavior at a compressive strain rate of 500 s-1.01/2009; -
##### Article: Analysis of Space Shuttle External Tank Spray-On Foam Insulation With Internal Pore Pressure

Journal of Engineering Materials and Technology-transactions of The Asme - J ENG MATER TECHNOL. 01/2008; 130(4).

Page 1

Thermal Expansion Coefficient and Bulk Modulus of

Polyethylene Closed-Cell Foams

O. ALMANZA,1Y. MASSO-MOREU,2N. J. MILLS,2M. A. RODRÍGUEZ-PE´REZ3

1Departamento Fı ´sica, Universidad Nacional de Colombia, Santafe ´ de Bogota ´, Colombia

2Metallurgy and Materials, University of Birmingham, Birmingham, United Kingdom

3Departamento Fı ´sica de la Materia Condensada, Universidad de Valladolid, 47011 Valladolid, Spain

Received 18 September 2003; revised 14 June 2004; accepted 22 June 2004

DOI: 10.1002/polb.20230

Published online in Wiley InterScience (www.interscience.wiley.com).

ABSTRACT:

expansion coefficient and bulk modulus of crosslinked, closed-cell, low-density polyeth-

ylene (LDPE) foams from the polymer and gas properties. The materials used for the

experimental measurements were crosslinked, had a uniform cell size, and were nearly

isotropic. Young’s modulus of biaxially oriented polyethylene was used for modeling the

cell faces. The model underestimated the foam linear thermal expansion coefficient

because it assumed that the cell faces were flat. However, scanning electron microscopy

showed that some cell faces were crumpled as a result of foam processing. The mea-

sured bulk modulus, which was considerably smaller than the theoretical value, was

used to estimate the linear thermal expansion coefficient of the LDPE foams. © 2004

Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 42: 3741–3749, 2004

Keywords:

bulk modulus; foams; modeling; polyethylene (PE); thermal expansion

A regular Kelvin foam model was used to predict the linear thermal

INTRODUCTION

The uses of crosslinked, closed-cell polyolefin

foams1include packaging, automotive applica-

tions, and thermal insulation for pipes. The prop-

erties required for these applications include low

thermal expansion, high thermal stability, and

good mechanical properties. In the shoe industry,

polyethylene (PE) foams are bonded to textile

substrates; differences between the thermal ex-

pansion of the foam and that of the substrate

could cause tensile stresses at the material inter-

face, possibly leading to adhesive failure.

The elastic moduli2,3and process expansion4of

such foams have been modeled in terms of their

cellular microstructure. A regular lattice of tetra-

kaidecahedral cells, known as the Kelvin foam

(Fig. 1), was used to compute the foam-compres-

sive Young’s modulus.2However, to obtain rea-

sonable predictions, faces were assumed to buckle

if compressed in plane. Mahapatro et al.4used the

Kelvin foam model, in which the cell faces re-

mained flat under biaxial tension, to compute the

equilibrium density of crosslinked PE foams from

the gas content and the modulus of the PE melt.

However, only qualitative models exist for the

thermal expansion coefficient and thermal con-

ductivity. These properties vary with the foam

relative density (R), which is defined as the foam

density divided by the polymer density. They tend

to the gaseous value when R tends to 0 and to the

polymer value when R tends to 1. The thermal

conductivity5–7varies almost linearly with R for

R ? 0.15. In previous articles,8–10the thermal

Correspondence to: N. J. Mills (E-mail: n.j.mills@bham.ac.uk)

Journal of Polymer Science: Part B: Polymer Physics, Vol. 42, 3741–3749 (2004)

© 2004 Wiley Periodicals, Inc.

3741

Page 2

expansion coefficient of PE foams has been

reported

1. To increase slightly between 0 and 40 °C.

2. To be approximately equal to the PE value

for foam densities greater than 80 kg/m3

but to increase at lower densities.

3. To decrease if Young’s modulus (E) of PE

increases.

4. To be anisotropic, with lower values in the

direction in which cells are elongated.

These trends have been explained qualita-

tively8,9in terms of the gas and polymer contri-

butions to the total thermal expansion coefficient.

However, the goal has been to develop a model

predicting the thermal expansion coefficient from

the polyhedral cell structure and the material

properties. Rodriguez-Perez and de Saja11showed

that cell faces contain oriented crystalline struc-

tures; consequently, the face properties must be

for oriented low-density polyethylene (LDPE).

Mills and Gilchrist12showed that air diffuses

slowly out of LDPE foams when compressive

stresses are applied. For a cylindrical sample

10 mm in diameter and 10 mm high, it will take

more than 10 h for significant gas loss and several

weeks for the foam to reach equilibrium. There-

fore, the long-term storage temperature of the

foam affects the equilibrium absolute air pressure

in the cells, but there is no gas loss during the

thermal expansion measurements, which typi-

cally last 30 min. Consequently, the model can

ignore gas diffusion.

The Kelvin model can be used to predict the

bulk modulus of the foam (KF). Kraynik et al.13

predicted, using Abaqus finite element analysis

for a material with E and a Poisson’s ratio (?) of

0.49, that KFof R could be obtained as follows:

KF? 0.435ER

(1)

In general, the same model should be used to

predict the bulk modulus and thermal expansion

coefficient.Bothproperties

changes, with isotropic expansion of the foam.

involvevolume

KELVIN FOAM MODEL

General

In the model, the cell faces are assumed to be flat

and to remain flat, although there is scanning

Figure 2.

mirror symmetry planes at the boundaries (faces meet-

ing at G have been omitted for clarity) and (b) left side

surface of the unit cell, on which a force balance is

performed.

(a) Kelvin closed-cell foam unit cell with

Figure 1.

planar faces in a body-centered-cubic Kelvin foam

lattice.

Packing of three tetrakaidecahedra with

3742

ALMANZA ET AL.

Page 3

electron microscopy (SEM) evidence for some cell

faces being wrinkled or buckled. Although PE

foams have irregular cell shapes, rather than the

uniform-size and -shape cells of the Kelvin model,

it is unlikely that the cell shape irregularity

greatly affects the thermal expansion coefficient.

The model is for foams with isotropic cell shapes.

Figure 2(a) shows the chosen unit cell, which is

stacked (repeated by translational symmetry op-

erations) to form the complete foam structure. It

contains four halves of hexagonal faces, four

quarters of horizontal square faces (because these

are shared with two structural units, they con-

tribute half a square face), and two halves of

vertical square faces. Thus, the 4:3 ratio of hex-

agonal faces to square faces is the same in the

unit cell as that in a single isolated tetrakaideca-

hedral cell. For ? ? L (where ? is the face thick-

ness and L is the edge length), if there is negligi-

ble polymer in the edges, R is given by

R ?3?2

16

?1 ? 2?3?

?

L

(2)

To generalize the analysis either for an external

application of relative pressure P or for thermal

expansion when P is 0, we consider the total com-

pressive force acting perpendicularly to a vertical

boundary of the unit cell. This boundary has area

2?2L2. A biaxial tensile stress (?f) acts in a hex-

agonal cell face of width ?3L and thickness ? and

in two shared, sectioned, half-square faces of total

width 0.5L and thickness ?/2, both of which are

perpendicular to the boundary plane. The cell-air

relative pressure (pC) also acts on the area 2?2L2

(ignoring the correction for the face cross-sec-

tional area as R ? 1). The total compressive

force is

P2?2L2? pC2?2L2? ?f??3 ? 0.5?L?

(3)

so

?f?

2?2

0.5 ? ?3

L

??pC? P?

(4)

Combining eqs 2 and 4 gives

?f?3?pC? P?

2R

(5)

Equation 6 in Mahapatro et al.4is incorrect be-

cause of an error in evaluating the area of the

polymer crossing the left boundary of the unit cell

[Fig. 2(b)]. The term in brackets in eq 3 was

incorrectly taken to be 3, and this led to a face

stress that was 74.4% of that in eq 5.

The polymer is assumed to be linearly elastic,

so the tensile elastic strain in each face (?E) is

?E? ?1 ? ???f

E

(6)

The cell face material is anisotropic; E is for

stresses acting in the plane of the face, and the

linear thermal expansion coefficient (?lP) is for

the expansion in the face length and width. ? is

also appropriate for the anisotropic material.

Bulk Modulus

If eq 5 is divided by the tensile strain in the face

(which is also the tensile strain in the foam), and

eq 6 used to replace ?f/?E, we obtain

ER

1 ? ??3?pC? P?

2?E

(7)

For small volume strains, the foam volume strain

(?V) is equal to 3?E, and KFis defined by

KF? ?P

?V

(8)

The bulk modulus of the air (KA) in the cells is

KA? ?pC

?F? p0C

(9)

where p0Cis the absolute air pressure in the cells.

Because the foam was stored at atmospheric pres-

sure (pa) for several months before the tests, this

was assumed to be pa. Therefore, we obtain

KF? Kp? KA?

2ER

9?1 ? ??? pa

(10)

where Kpis the polymer bulk modulus.

Figure 3(a) shows a one-dimensional (1D) ver-

sion of the bulk modulus experiment. The poly-

mer and cell-air springs act in parallel, so their

bulk moduli are summed, as expressed by eq 10.

The polymer contribution to eq 10 is in agreement

POLYETHYLENE CLOSED-CELL FOAMS

3743

Page 4

with Kraynik et al.’s13eq 1. Abaqus would only

consider the stresses in the solids, so the effect of

the compressed cell air would be ignored. Equa-

tion 10 contrasts with the bulk modulus of ER/9

for an open-cell Kelvin foam.14Open-cell foams

are connected by edges running in one dimension,

whereas closed-cell foams are connected by faces

running in two dimensions. The doubled connec-

tivity causes a doubling of the polymer contribu-

tion to the bulk modulus, with an extra effect of ?.

As LDPE is viscoelastic rather than linearly

elastic, E in eq 10 can be replaced by the creep

modulus [E(t)] on timescale t of the experiments

to obtain a time-dependent bulk modulus.

Thermal Expansion

The linear thermal expansion coefficient (?l) is

defined as follows:

?l?1

l0

dl

dT

(11)

where l0is the sample length at the reference

temperature of 25 °C and l is the length at tem-

perature T. Figure 3(b) shows a 1D version of the

thermal expansion experiment; there is no exter-

nal pressure change, and the polymer and cell-air

springs act in series. The foam volume expansion

coefficient (?VF) is the sum of that of the polymer

structure and the excess air volume expansion,

constrained by the foam structure:

?VF? ?VP? ??VA? ?VP?KA

KF

(12)

The appendix gives the details of the derivation.

When eq 12 is divided by 3, the foam linear ex-

pansion coefficient (?lF) is yielded:

?IF? ?IP? ??IA? ?IP?p0

KF

(13)

The general form of this equation allows its use

either with the theoretical KFvalue of eq 10 or

with the experimental value. As air behaves as an

ideal gas, its volume thermal expansion coeffi-

cient (?VA) is equal to 1/T, where T is the absolute

temperature; consequently, its linear thermal ex-

pansion coefficient (?lA) is 11.4 ? 10?4K?1at

293 K.

LDPE FOAM CHARACTERIZATION

The assumptions in the model about the foam

geometry need to be verified; in particular, it is

necessary to determine whether the cells are

equiaxed, the cell faces are flat, and the fraction of

the polymer in the edges is low. Furthermore, the

appropriate polymer properties must be used in

the model. The modeling4of cell expansion during

foam processing suggests that the crystals have a

preferred orientation in the cell faces. Conse-

quently, the face properties are anisotropic; the

in-plane E value and linear thermal expansion

coefficient differ from those in the direction nor-

mal to the cell faces. Hence, polymer properties (E

and ?lP), measured on bulk LDPE with a spheru-

litic microstructure, should not be used in the

thermal expansion model. Ideally, the properties

should be measured in the plane of the foam cell

faces, but such micromechanic measurements

Figure 3.

experiments, with the cell-air and polymer structure acting as springs. The thermal

expansion is by (?VA? ?VP)?T.

1D representations of (a) the bulk modulus and (b) the thermal expansion

3744

ALMANZA ET AL.

Page 5

have not been made on PE foam faces. Conse-

quently, E and ?lP, values measured in the plane

of a biaxially oriented LDPE film, which has a

similar microstructure, are used for the modeling.

Foams

Crosslinked, closed-cell LDPE foams, manufac-

tured by Zotefoams (Croydon, United Kingdom),

were used to check the validity of the theoretical

model because they do not contain foaming agent

residues, they have nearly isotropic cellular

shapes, and the polymer crystallinity is indepen-

dent of the foam density. The product codes and

densities of the foams are given in Table 1. The

black foam (LD70B) contains approximately

2 wt % carbon black. The crystallization charac-

teristics of the solid LDPE sheet, used in the

Zotefoams process, are included in the table. Dur-

ing the process, the LDPE is compounded with a

peroxide crosslinking agent and is extruded as a

thick sheet, which is passed through a hot oven to

effect crosslinking to a gel content of approxi-

mately 40%. Slabs, cut from the extruded sheet,

are subjected to several hundred bars of nitrogen

gas pressure in an autoclave for several hours, at

a temperature above the PE crystal melting point,

so that nitrogen dissolves.15After the slabs cool,

the pressure is reduced to zero, and the slabs are

placed in a larger autoclave and reheated above

the polymer melting point under a lower pres-

sure. When the pressure is released, the foam

expands to its final low density. It is taken out of

the autoclave and cooled to room temperature.

Foam Characterization

Density measurements were carried out with the

density determination kit for a Mettler AT261

balance according to Archimedes’ principle.

SEM was used to assess the cellular structure.

Foam samples were microtomed at a low temper-

ature and, after being vacuum-coated with gold,

were examined in a JEOL JSM-820 microscope.

The foam needed to be kept in vacuo for more

than the time for 90% air loss by diffusion, which

was estimated16to be 10 h for a 5-mm cubic

sample of LD19 foam, for the artificial expansion

of cells on the vacuum application to disappear.

Rodriguez-Perez and de Saja11described the

determination of the cell diameter, cell shape an-

isotropy, crystallinity, and crystal orientation for

these foams. They showed, by SEM of etched

foam, that two-dimensional (2D) spherulites nu-

cleated on both cell face surfaces. The average

surface diameter (Ds) of the 2D spherulites varies

with the foam density, but it is a nearly constant

multiple (6) of ?. The mean cell diameter (Dc) in

Table 1 is the mean of the average cell diameters

in the three directions. The cell shape anisotropy

ratio17is defined as the ratio of the largest Dc

value to the smallest Dcvalue for the three direc-

tions of measurement.

? of thirty cell faces, chosen randomly, was

measured directly from the SEM screen, and the

mean value was calculated. The 95% confidence

interval of these measurements was ?8% of the

mean. Finally, the mass fraction in the edges (fs)

was obtained with the method of Kuhn et al.,18

which assumes that the cells are regular dodeca-

hedra with pentagonal faces and that the faces

have uniform thickness where they meet the

edges. From the average values of Dc, the edge

diameter (De), and ?, the volumes of the polymer

in the edges and faces are obtained as follows:

Ve??(1.3Dc

2? 5.4DcDe? 1.7Dc

2) (14)

Vf? 2.8DcDe

2? 3.9De

3

(15)

Table 1.

Foam Densities, Cell Size Parameters, and Crystallinity for Zotefoams LDPE Foams

Product

Code

Density

(kg/m3)

Dc

(?m)

Cell

Anisotropy

Ratio

?

(?m)

Edge

Fraction

Melting

Point

(°C)

Crystallinity

(%)

LD15

LD18

LD24

LD29

LD33

LD33(1)

LD60

LD70B

LD Solid

16.7

22.5

24.6

30.7

32.0

32.5

58.5

69.5

910

313.5

879.7

311.9

528.1

424.4

396.9

773.4

528.1

—

1.00

1.01

1.01

1.02

1.00

0.99

1.02

1.04

—

1.4

5.8

1.9

4.2

3.6

2.5

10.3

6.0

—

0.22

0.21

0.16

0.24

0.28

0.36

0.24

0.35

—

105.9

108.4

108.6

108.5

105.8

108.6

109.0

106.6

105.9

40.6

41.8

43.8

42.9

43.9

41.6

42.1

40.8

40.6

POLYETHYLENE CLOSED-CELL FOAMS

3745

Page 6

Edge cross-sectional areas were measured at four

randomly chosen positions on each micrograph.

These and the average cell diameter were used to

calculate fs. The 95% confidence error of these

measurements was ?8% of the mean.

The linear thermal expansion coefficient was

measured with a PerkinElmer TMA7 thermome-

chanical analyzer. The test specimens were cylin-

ders 10 mm in diameter and 5–10 mm high. These

were placed between parallel metal plates 10 mm

in diameter. An applied compressive stress of

130 Pa kept the plates in contact with the sample;

it caused a 0.06% elastic strain at room temper-

ature for a low-density foam. As the thermal ex-

pansion strain between 5 and 25 °C was 20 times

higher than the elastic strain, the latter could be

neglected. The sample height direction was the

direction perpendicular to the foam sheet for all

foams.

Two types of measurements were made:

1. The temperature was raised from 20 to

130 °C at 5 °C/min to characterize the over-

all thermal expansion response.

2. To measure the linear thermal expansion

coefficient, away from any thermal transi-

tions of the polymer,8the foam was cooled

from room temperature to 5 °C, at which it

was kept for 15 min. It was then heated

from 5 to 25 °C at 1 °C/min and kept at

25 °C for 15 min. Each material was mea-

sured three times with new samples. The

95% confidence limits of the measurements

were ?6% of the mean.

RESULTS

Thermal Expansion Coefficient

The linear thermal expansion coefficient of

the LD solid sheet from Zotefoams was 1.3

? 10?4K?1. Figure 4 shows typical data for the

variation of the linear thermal expansion coeffi-

cient of the isotropic foam with the temperature.

The coefficient increases slowly from 20 to 30 °C

and then decreases at higher temperatures as the

melting process of the crystals starts. Figure 5

shows experimental values from 5 to 25 °C as a

function of the density.

Cell Geometry

Figure 6 shows SEM images of an LD19 foam

immediately upon insertion into the SEM instru-

ment and after 64 h of storage in vacuo. The cut

faces labeled A, B, and C were initially flat,

whereas they appear buckled after 64 h. The com-

plete face D appeared flat initially, whereas it was

wrinkled after 64 h. The sudden exposure to a

vacuum caused the stretching of the faces of com-

plete cells at the surface of the foam, but, after

64 h, the majority of the air diffused from the

foam, leaving the faces in the same wrinkled state

in which they had before being placed into the

instrument.

The melting points and crystallinity (ca. 110 °C

and 40%, respectively) of the foams, given in

Table 1, are typical of LDPE. The cell size did not

depend on the density, whereas the edge mass

fraction had a mean value of 0.25.

Figure 4.

the direction perpendicular to the foam sheet, versus

the temperature for Zotefoams LD24.

Linear thermal expansion coefficient, in

Figure 5.

data for Zotefoams foams versus the density, compared

with theoretical predictions for ?lP? 3.3 ? 10?4K?1

and foam bulk moduli proportional to R.

Experimental linear expansion coefficient

3746

ALMANZA ET AL.

Page 7

Linear Thermal Expansion Coefficient of the

Oriented LDPE Films

Mills and Zhu2measured the tensile response of a

biaxially stretched LDPE packaging film, with a

density of 910 kg m?3and a thickness of 45 ?m,

which had a biaxial draw ratio of 6.5. Its E value

at a strain rate of 7 ? 10?4s?1, up to 1.5% strain,

was 202 MPa. The E value was consistent with

the range of values (175–225 MPa) measured for

other LDPE blown films of different degrees of

orientation.19

No published values for the linear thermal ex-

pansion coefficient of biaxially oriented LDPE

film could be found. The anisotropic thermal ex-

pansion of 2D PE spherulites was discussed by

Barham and Keller.20The crystal b axis appears

to be similar to that of the whole 2D spherulite

and to be very small. For both amorphous and

semicrystalline polymers, the thermal expansion

coefficient along the draw direction decreases

with increasing deformation and increases in the

direction perpendicular to the draw direction.21

For uniaxially oriented LDPE films, the linear

expansion coefficient decreases with drawing, to

values close to 2 ? 10?5K?1in the parallel direc-

tion and to 2 ? 10?4K?1in the perpendicular

direction.22

COMPARISON WITH THEORY

Masso-Moreu and Mills16showed that the slightly

wrinkled cell faces of LDPE foams affect their re-

sponse to pressure changes. They found that the

bulk modulus of an LD19 foam was 440 kPa for

relative pressures of 0 to ?20 kPa, but it increased

to 820 kPa for relative pressures of ?40 to ?80 kPa.

However, the theoretical value for a Kelvin foam

with flat cell faces (E ? 202 MPa and ? ? 0.4) is

1550 kPa. Hence, the bulk modulus is lower than

the theoretical value, even for large volumetric

expansions. A temperature increase of 10° causes

a linear strain of 6.7 ? 10?3in an LD19 foam,

whereas a 20-kPa decrease in pressure causes a

linear strain of 15.2 ? 10?3. The strains are com-

parable in magnitude, so the bulk modulus data is

at an appropriate strain level for predicting the

thermal expansion coefficient.

Given the difficulty of finding values for ?lPof

biaxially oriented LDPE cell faces, the asymptotic

value of the foam linear thermal expansion coef-

ficient, when the density became large, was used.

This was estimated to be 3.3 ? 10?4K?1(dis-

cussed later), that is, higher than the 1.3

? 10?4K?1value for bulk LDPE of the same

crystallinity percentage. If this is used with E

? 202 MPa and the theoretical KFvalue in eq 12,

the predicted linear expansion coefficient for

LD19 foam is 3.8 ? 10?4K?1, much lower than

the experimental value of 6.7 ? 10?4K?1. The too

small experimental bulk modulus and the too

large experimental linear thermal expansion co-

efficient result from the foam cells having buckled

faces. Cells with heavily buckled faces would re-

spond like a bellows and so would have a linear

thermal expansion coefficient almost that of air,

11.4 ? 10?4K?1.

It is known that KFis 440 kPa for an LDPE

foam with a density of 19 kg m?3. To estimate the

bulk moduli of LDPE foams with densities

greater than 19 kg m?3, we assumed that the

bulk modulus was proportional to R, as predicted

by eq 10, being 440 kPa when R was 0.018. With

this information, the data in Figure 5 for the foam

Figure 6.

and (b) after 64 h in vacuo. Cut faces A–C and complete face D are (a) flat and (b) buckled.

Micrographs of LD19 foam (a) soon after insertion into the SEM instrument

POLYETHYLENE CLOSED-CELL FOAMS

3747

Page 8

linear thermal expansion coefficient versus the

density were fitted with a range of ?Pvalues. The

best fit at high densities was for a cell face ther-

mal expansion coefficient of ?lP? 3.3 ? 10?4K?1.

Figure 5 shows that the fit was good for relative

densities of 0.035 or greater, but it was an under-

estimate for the lowest relative density foams.

DISCUSSION

The model, in which the cell faces are flat, overes-

timates the bulk modulus and therefore underesti-

mates the linear thermal expansion coefficient of

LDPE foams. Its predictions should be better at

foam densities higher than those tested. However,

at high densities (Fig. 5), the predicted foam ther-

mal expansion coefficient approaches that of LDPE,

so the test of the model is not demanding.

Mills and Zhu14predicted the E values of

closed-cell foams with the Kelvin foam model, in

which cell faces could not support in-plane com-

pressive forces. This underestimated E of LDPE

foams, but an alternative theory,13in which the

faces remained flat, overestimated E. The trend

line of E versus the density (Fig. 7) is steeper than

that of either theory, suggesting that face buck-

ling becomes easier at lower densities. The data

in this figure were measured for the Zotefoams in

slow compressive tests.23However, although face

buckling influences the experimental E values,

the pattern of stresses in the cell faces differs

from that in the bulk modulus and thermal ex-

pansion experiments.

The foam processing route causes the cell faces

to be slightly wrinkled. It is assumed that the

faces buckle during the cooling of the foam from

the melt, when the cell gas relative pressure be-

comes negative. In extruded LDPE foams, for

which pentane is used as a blowing agent, it ap-

pears necessary to add permeability modifiers to

the LDPE to prevent the collapse of cells after

extrusion because the diffusion rate of pentane

out of the cells is faster than the diffusion rate of

air into the cells.24

In the future, efforts should be made to directly

measure the linear thermal expansion coefficients

of polymer foam faces. If such data and bulk mod-

ulus measurements for a range of foam densities

were available, it would be possible to check the

Kelvin model predictions more rigorously.

CONCLUSIONS

The theoretical model predicts KFand linear ther-

mal expansion coefficients from the polymer and

air thermal expansion coefficients and E of the

polymer in the cell faces. This model assumes

that the cell faces are all flat, but SEM shows that

many cell faces are buckled. Experimental bulk

moduli and linear thermal expansion coefficients

of LDPE foams are consistent with the presence

of wrinkled and buckled cell faces, which signifi-

cantly change the values.

If the experimental KFvalue is used, and the

polymer linear thermal expansion coefficient is

estimated from the high-density asymptotic be-

havior of the foam thermal expansion coefficients,

reasonable predictions of the foam linear thermal

expansion coefficient can be made.

Financial assistance from Junta de Castilla y Leo ´n

(VA026/03), from La Secretaria de Estado de Educacio ´n

y Universidades (Spain) for a postdoctoral grant

(O. Almanza), and from the Engineering and Physical

Science Research Council for Y. Masso-Moreu is grate-

fully acknowledged. The authors thank Zotefoams PLC

for supplying the foams.

APPENDIX: DERIVATION OF THE

THERMAL EXPANSION COEFFICIENT

The total (equibiaxial) tensile strain (?T) in each

cell face, as a result of a temperature increase (?T

? T ? T0), is the sum of the elastic and thermal

strains (for any closed-cell foam model):

Figure 7.

foams versus the density, compared with the theoreti-

cal models of Mills and Zhu14and Kraynik et al.13

Experimental E data for Zotefoams LDPE

3748

ALMANZA ET AL.

Page 9

?T? ?IP?T ? ?1 ? ???f

E

(A1)

This can be rearranged to yield

?IF?

?T

?T? ?IP??f?1 ? ??

E?T

(A2)

For the Kelvin foam model, ?fis given by eq 5 for

P ? 0. When this is substituted, we obtain

?IF? ?IP?3pC?1 ? ??

2ER?T

(A3)

In terms of the volume thermal expansion coeffi-

cients, we obtain

?VF? ?VP?

pC

KP?T

(A4)

Figure 3(b) shows a 1D representation of a ther-

mal expansion experiment, in which the volume

strain in the cell air must be the same as the

volume strain in the polymer when there is a

temperature rise (?T). The air and polymer

springs appear to be in series:

??VA? ?VP??T ?pC

P0? 3?1 ? ???f

E

(A5)

Rearranging this and substituting for ?fwith

eq 5, we obtain

?VA? ?VP?

pC

?T?

1

p0?9?1 ? ??

2E??

pC

?T?

1

KA?

1

KP?

(A6)

Substituting for pC/?T from eq A6 into eq A4 gives

?VF? ?VP?

1

Kp

?VA? ?VP

?

1

KA?

1

Kp?

? ?VP? ??VA? ?VP?KA

KF

(A7)

which is eq 12.

REFERENCES AND NOTES

1. Park, C. P. In Handbook of Polymeric Foams and

Foam Technology; Klempner, D.; Frisch, K. C.,

Eds.; Hanser: Munich, 1991; Chapter 9.

2. Mills, N. J.; Zhu, H. X. J Mech Phys Solids 1999, 47,

669–695.

3. Roberts, A. P.; Garboczi, E. J. Acta Mater 2001, 49,

189–197.

4. Mahapatro, A.; Mills, N. J.; Sims, G. L. S. Cell

Polym 1998, 17, 252–270.

5. Leach, A. G. J Phys D: Appl Phys 1993, 26, 733–

739.

6. Collishaw, P. G.; Evans, J. R. G. J Mater Sci 1994,

29, 486–498.

7. Glicksman, L. R. In Low Density Cellular Plastics:

Physical Basis of Behaviour; Hilyard, N. C.; Cun-

ningham, A., Eds.; Chapman & Hall: London, 1994;

Chapter 5.

8. Rodrı ´guez-Pe ´rez, M. A.; Alonso, O.; Duijsens, A.; de

Saja, J. A. J Polym Sci Part B: Polym Phys 1998,

36, 2587–2596.

9. Rodrı ´guez-Pe ´rez, M. A.; Duijsens, A.; de Saja, J. A.

J Appl Polym Sci 1998, 68, 1237–1244.

10. Rodrı ´guez-Pe ´rez, M. A.; Alonso, O.; Souto, J.; de

Saja, J. A. Polym Test 1997, 16, 287.

11. Rodriguez-Perez, M. A.; de Saja, J. A. J Macromol

Sci Phys 2002, 41, 761–775.

12. Mills, N. J.; Gilchrist, A. J Cell Plast 1997, 33,

264–292.

13. Kraynik, A. M.; Neilsen, M. K.; Reinelt, D. A.;

Warren, W. E. In Foams and Emulsions; Sadoc,

J. F.; Rivier, N., Eds.; NATO ASI Series E 354;

Kluwer: Dordrecht, 1999; pp 259–286.

14. Mills, N. J.; Zhu, H. X. J Mech Phys Solids 1999,

47, 669–695.

15. Eaves, D. E.; Witten, N. Soc Plast Eng ANTEC ’98,

1998, vol. 2, 1842–1849.

16. Masso-Moreu, Y.; Mills, N. J. Polym Test 2004, 23,

313–322.

17. Cowin, S. C. J Mater Sci 1991, 26, 5155–5157.

18. Kuhn, J.; Ebert, H. P.; Arduini-Schuster, M. C.;

Bu ¨ttner, D.; Fricke, J. Int J Heat Mass Transfer

1992, 35, 1795–1801.

19. Patel, R. M.; Butler, T. T.; Walton, K. L.; Knight,

G. W. Polym Eng Sci 1994, 34, 1506–1513.

20. Barham, P. J.; Keller, A. J Mater Sci 1977, 12,

2141–2148.

21. Choy, C. L. In Developments in Oriented Polymers;

Ward, I. M., Ed.; Applied Science: London, 1982; pp

121–151.

22. Kacarevic-Popovic, Z.; Kostoski, D.; Novakovic,

L. J. Radiat Phys Chem 1999, 55, 645–658.

23. Almanza. O. Ph.D. Thesis, University of Vallado-

lid, 1999.

24. Dieckmann, D.; Holtz, B. J Vinyl Addit Technol

2000, 6, 34–38.

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