Thermal expansion coefficient and bulk modulus of polyethylene closed‐cell foams
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
Thermal Expansion Coefficient and Bulk Modulus of
Polyethylene Closed-Cell Foams
N. J. MILLS,
M. A. RODRÍGUEZ-PE
Departamento Fı´sica, Universidad Nacional de Colombia, Santafe´ de Bogota´, Colombia
Metallurgy and Materials, University of Birmingham, Birmingham, United Kingdom
Departamento Fı´sica de la Materia Condensada, Universidad de Valladolid, 47011 Valladolid, Spain
Received 18 September 2003; revised 14 June 2004; accepted 22 June 2004
Published online in Wiley InterScience (www.interscience.wiley.com).
ABSTRACT: A regular Kelvin foam model was used to predict the linear thermal
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.
Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 42: 3741–3749, 2004
Keywords: bulk modulus; foams; modeling; polyethylene (PE); thermal expansion
The uses of crosslinked, closed-cell polyoleﬁn
include 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 moduli
and process expansion
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.
However, to obtain rea-
sonable predictions, faces were assumed to buckle
if compressed in plane. Mahapatro et al.
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
varies almost linearly with R for
R ⬎ 0.15. In previous articles,
Correspondence to: N. J. Mills (E-mail: firstname.lastname@example.org)
Journal of Polymer Science: Part B: Polymer Physics, Vol. 42, 3741–3749 (2004)
© 2004 Wiley Periodicals, Inc.
expansion coefﬁcient of PE foams has been
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/m
but to increase at lower densities.
3. To decrease if Young’s modulus (E)ofPE
4. To be anisotropic, with lower values in the
direction in which cells are elongated.
These trends have been explained qualita-
in 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 Saja
that cell faces contain oriented crystalline struc-
tures; consequently, the face properties must be
for oriented low-density polyethylene (LDPE).
Mills and Gilchrist
showed 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 (K
). Kraynik et al.
predicted, using Abaqus ﬁnite element analysis
for a material with E and a Poisson’s ratio (
0.49, that K
of R could be obtained as follows:
⫽ 0.435ER (1)
In general, the same model should be used to
predict the bulk modulus and thermal expansion
coefﬁcient. Both properties involve volume
changes, with isotropic expansion of the foam.
KELVIN FOAM MODEL
In the model, the cell faces are assumed to be ﬂat
and to remain ﬂat, although there is scanning
Figure 2. (a) Kelvin closed-cell foam unit cell with
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
Figure 1. Packing of three tetrakaidecahedra with
planar faces in a body-centered-cubic Kelvin foam
3742 ALMANZA ET AL.
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
Ⰶ 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
共1 ⫹ 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
. A biaxial tensile stress (
) acts in a hex
agonal cell face of width 公3L and thickness
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 (p
) also acts on the area 2公2L
(ignoring the correction for the face cross-sec-
tional area as R Ⰶ 1). The total compressive
3 ⫹ 0.5兲L
⫺ P兲 (4)
Combining eqs 2 and 4 gives
Equation 6 in Mahapatro et al.
is 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 (
⫽ 共1 ⫺
The cell face material is anisotropic; E is for
stresses acting in the plane of the face, and the
linear thermal expansion coefﬁcient (
the expansion in the face length and width.
also appropriate for the anisotropic material.
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
, we obtain
For small volume strains, the foam volume strain
) is equal to 3
, and K
is deﬁned by
The bulk modulus of the air (K
) in the cells is
is the absolute air pressure in the cells.
Because the foam was stored at atmospheric pres-
) for several months before the tests, this
was assumed to be p
. Therefore, we obtain
is 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
with Kraynik et al.’s
eq 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.
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.
The linear thermal expansion coefﬁcient (
deﬁned as follows:
is 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
) is the sum of that of the polymer
structure and the excess air volume expansion,
constrained by the foam structure:
The appendix gives the details of the derivation.
When eq 12 is divided by 3, the foam linear ex-
pansion coefﬁcient (
) is yielded:
The general form of this equation allows its use
either with the theoretical K
value of eq 10 or
with the experimental value. As air behaves as an
ideal gas, its volume thermal expansion coefﬁ-
) is equal to 1/T, where T is the absolute
temperature; consequently, its linear thermal ex-
pansion coefﬁcient (
) is 11.4 ⫻ 10
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 modeling
of 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
), 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. 1D representations of (a) the bulk modulus and (b) the thermal expansion
experiments, with the cell-air and polymer structure acting as springs. The thermal
expansion is by (
3744 ALMANZA ET AL.
have not been made on PE foam faces. Conse-
quently, E and
, values measured in the plane
of a biaxially oriented LDPE ﬁlm, which has a
similar microstructure, are used for the modeling.
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.
After 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.
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
to 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 Saja
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 (D
) of the 2D spherulites varies
with the foam density, but it is a nearly constant
multiple (6) of
. The mean cell diameter (D
Table 1 is the mean of the average cell diameters
in the three directions. The cell shape anisotropy
is deﬁned as the ratio of the largest D
value to the smallest D
value 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 (f
was obtained with the method of Kuhn et al.,
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 D
, the edge
, the volumes of the polymer
in the edges and faces are obtained as follows:
Table 1. Foam Densities, Cell Size Parameters, and Crystallinity for Zotefoams LDPE Foams
LD15 16.7 313.5 1.00 1.4 0.22 105.9 40.6
LD18 22.5 879.7 1.01 5.8 0.21 108.4 41.8
LD24 24.6 311.9 1.01 1.9 0.16 108.6 43.8
LD29 30.7 528.1 1.02 4.2 0.24 108.5 42.9
LD33 32.0 424.4 1.00 3.6 0.28 105.8 43.9
LD33(1) 32.5 396.9 0.99 2.5 0.36 108.6 41.6
LD60 58.5 773.4 1.02 10.3 0.24 109.0 42.1
LD70B 69.5 528.1 1.04 6.0 0.35 106.6 40.8
LD Solid 910 — — — — 105.9 40.6
POLYETHYLENE CLOSED-CELL FOAMS 3745
Edge cross-sectional areas were measured at four
randomly chosen positions on each micrograph.
These and the average cell diameter were used to
. 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
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,
the 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.
Thermal Expansion Coefficient
The linear thermal expansion coefﬁcient of
the LD solid sheet from Zotefoams was 1.3
. 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.
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
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. Linear thermal expansion coefﬁcient, in
the direction perpendicular to the foam sheet, versus
the temperature for Zotefoams LD24.
Figure 5. Experimental linear expansion coefﬁcient
data for Zotefoams foams versus the density, compared
with theoretical predictions for
⫽ 3.3 ⫻ 10
and foam bulk moduli proportional to R.
ALMANZA ET AL.
Linear Thermal Expansion Coefficient of the
Oriented LDPE Films
Mills and Zhu
measured the tensile response of a
biaxially stretched LDPE packaging ﬁlm, with a
density of 910 kg m
and a thickness of 45
which had a biaxial draw ratio of 6.5. Its E value
at a strain rate of 7 ⫻ 10
, 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
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.
The 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.
For uniaxially oriented LDPE ﬁlms, the linear
expansion coefﬁcient decreases with drawing, to
values close to 2 ⫻ 10
in the parallel direc
tion and to 2 ⫻ 10
in the perpendicular
COMPARISON WITH THEORY
Masso-Moreu and Mills
showed 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
in an LD19 foam,
whereas a 20-kPa decrease in pressure causes a
linear strain of 15.2 ⫻ 10
. 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
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
cussed later), that is, higher than the 1.3
value for bulk LDPE of the same
crystallinity percentage. If this is used with E
⫽ 202 MPa and the theoretical K
value in eq 12,
the predicted linear expansion coefﬁcient for
LD19 foam is 3.8 ⫻ 10
, much lower than
the experimental value of 6.7 ⫻ 10
. 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
It is known that K
is 440 kPa for an LDPE
foam with a density of 19 kg m
. To estimate the
bulk moduli of LDPE foams with densities
greater than 19 kg m
, 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. Micrographs of LD19 foam (a) soon after insertion into the SEM instrument
and (b) after 64 h in vacuo. Cut faces A–C and complete face D are (a) ﬂat and (b) buckled.
POLYETHYLENE CLOSED-CELL FOAMS 3747
linear thermal expansion coefﬁcient versus the
density were ﬁtted with a range of
best ﬁt at high densities was for a cell face ther-
mal expansion coefﬁcient of
⫽ 3.3 ⫻ 10
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.
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 Zhu
predicted 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,
in 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.
However, 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-
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.
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.
The theoretical model predicts K
and 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 K
value 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 (
) in each
cell face, as a result of a temperature increase (⌬T
⫽ T ⫺ T
), is the sum of the elastic and thermal
strains (for any closed-cell foam model):
Figure 7. Experimental E data for Zotefoams LDPE
foams versus the density, compared with the theoreti-
cal models of Mills and Zhu
and Kraynik et al.
3748 ALMANZA ET AL.
⌬T ⫹ 共1 ⫺
This can be rearranged to yield
For the Kelvin foam model,
is given by eq 5 for
P ⫽ 0. When this is substituted, we obtain
In terms of the volume thermal expansion coefﬁ-
cients, we obtain
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:
⫽ 3共1 ⫺
Rearranging this and substituting for
eq 5, we obtain
Substituting for p
/⌬T from eq A6 into eq A4 gives
which is eq 12.
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