VOLUME 37 NUMBER 1
`JANUARY/FEBRUARY 2001
`
`(ISSN 0021 —955X)
`
`Page 1 of 27
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`BOREALIS EXHIBIT 1073
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`Page 1 of 27
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`BOREALIS EXHIBIT 1073
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`

`
`EDITOR
`
`Sidney H. Metzger, Jr.
`Consultant, Pittsburgh, PA, USA
`EDITORIAL ADVISORY BOARD
`
`—Daniel Klempner
`Polymer Institute
`University ofDetroit Mercy
`USA
`—Lynn M. Martynowicz
`NOVA Chemwals’ Inc" USA
`——Ken G. McDaniel
`Bayer Corporation, USA
`
`EDITORIAL POLICY
`
`—Don Mente
`BASF Corporation, USA
`__Andrew N Paquet
`Dow Chemical Co., USA
`__Chu1 B Park
`University of Toronto,
`Canada
`
`—Fyodor A. Shutov
`Tennessee Technological
`University, USA
`——Keith Spitler
`Bayer Corporation, USA
`——Robert L. Zimmerman
`Hunstman Corporation, USA
`
`The primary purpose of the Journal ofCellular Plastics is to provide a permanent record of
`achievements in the science, technology, and economics of cellular plastics. Implicit in this
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`The Journal ofCellular Plastics is included in the following indexing and ab-
`stracting services:
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`Engineering Index
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`,.<,~'sr‘
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`Copyright © 2001
`Technomic Publishing Co., Inc.
`Lancaster, Pennsylvania USA
`ISSN 0021-955X
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`Printed in the United States of America
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`Page 4 of 27
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`Page 4 of 27
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`

`
`
`
`Patents
`
`21
`
`43
`
`58
`
`72
`
`The Thermal Conductivity of a Polyethylene Foam
`Block Produced by a Compression Molding Process
`J. A. Mart/’nez-Diez, M. A. Rodr/’guez—Pérez, J. A. De Saja,
`L. O. Arcos YF?abago and O. A. Almanza
`’
`
`’
`
`Methods of Minimising Density Gradients in Rigid
`Polyurethane Foams
`.
`
`Dale R. Harbron, Christopher J. Page and R. Keith Scarrow
`
`HFC-245fa Spray Polyurethane Foam Systems if.’
`Co-Blown with Water: A Quality, Cost Effective‘;
`Safe Substitute for HCFC-141b
`5;;
`
`"
`
`Mary Bogdan, David Williams and Paul pverbiest
`
`New Polyisocyanurate Catalysts Which Exhibitflligh
`:3
`Activity at Low Temperature
`_.A
`Shuichi Okuzono, Katsumi Tokumoto, Yutaka Tamano aide
`Donald W. Lowe
`r
`.
`‘
`
`5-..\;;j..‘1;.a¥‘
`
`JOURNAL OF CELLULAR PLASTICS Volume 37—— January 2001
`Page 5 of 27
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`Page 5 of 27
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`

`
`The Thermal Conductivity of a
`Polyethylene Foam Block
`Produced by a Compression
`Molding Process
`
`J. A. MARTiNEz-DiEz, M. A. RoDRiGUEz-PEREz* and J. A. DE SAJA
`Polymeric Foams Group
`Departamento de Fisica de la Materia Condensada
`Facultad de Ciencias
`Universidad de Valladolid
`
`47011 Valladolid, Spain
`
`L. O. ARCOS Y RABAGO
`
`Facultad de Ingenieria
`Uniuersidad Auzfonoma de Querétaro
`76010 Santiago de Querétaro
`Querétaro, Mexico
`
`0. A. ALMANZA
`
`Departamento de Fisica
`Universidad Nacional de Colombia
`
`Santa Fé de Bogota, D.C.
`Colombia
`
`ABSTRACT: The thermal conductivity of 10 mm thick low density polyethyl-
`ene foam sheets cut from a block produced by a compression molding process has
`been studied in the temperature range between 24°C and 50°C. The cellular
`structure and the matrix polymer morphology have also been characterized to
`find out the main microscopic characteristics that influence on the foam proper-
`ties. A previously developed theoretical model has been applied to compute the
`
`*Author to whom correspondence should be addressed.
`
`JOURNAL OF CELLULAR PLASTICS Volume 37-January 2001
`
`21
`
`0021-955X/01/01 0021-22 $10.00/0 DOI: 10.1106/DOMJ-HJH8-5YDQ-H5VB
`© 2001 Technomic Publishing Co., Inc.
`
`Page 6 of 27
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`Page 6 of 27
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`
`
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`22
`
`J. A. MARTiNEZ-DiEZ ET AL.
`
`thermal conductivity of the foams under study. This model was successful in the
`considered temperature range. Moreover, the evolution of the thermal conduc-
`tivity along the thickness of the foam block has also been considered and ex-
`plained in terms of the structure of the materials.
`
`INTRODUCTION
`
`ne of the primary applications of foams is as thermal insulators. As
`a consequence, considerable effort has been made to find out the
`conditions to minimize the thermal conductivity and, therefore, to re-
`duce insulation costs and energy requirements [1—6]. Moreover, from a
`scientific point of view, the understanding of the heat transfer mecha-
`nisms in a two phase material has been a challenge for scientists all over
`the world.
`
`One approach to materials design is to develop suitable theoretical
`models (without adjustable parameters), which allow predicting the
`physical properties of interest in terms of the main microscopic charac-
`teristics of the materials. By using these models, not only the properties
`could be estimated without producing such materials, but also the best
`combination of microscopic characteristics to improve the tested proper-
`ties could be deduced. Following this approach, in a previous work [7],
`we proposed a theoretical equation to compute the thermal conductivity
`of foams as a function of density, cell size, cell shape, fraction of material
`in the struts, cell wall thickness and properties of the base material as
`the density, refractive index and absorption coefficient. This model was
`developed and successfully applied to predict the thermal conductivity at
`24°C of polyethylene foams produced by a high pressure nitrogen solu-
`tion process [7]. These foams were desirable materials for scientific stud-
`ies due to some interesting characteristics, such as, lack of residual
`foaming agent in the final foam, almost isotropic cellular structure and
`same cell shape for different densities. Other interesting characteristics
`would be the similar properties of the solid polymer that comprises the
`cell walls and ofthe solid sheet from which the foam was produced. These
`materials can be considered as a foam-model system.
`On the other hand, it is well known that different technologies are
`widely used to manufacture polyolefin foams [8—10]. Each industrial
`method provides materials with different cellular structure and, there-
`fore, with different physical properties [8]. For example, polyethylene
`foams produced by a semi—continuous process, in which a foaming agent
`1S used, show an anisotropic cellular structure that gives different prop-
`erties, depending on the measurement direction. The previous foams
`and those produced by compression molding present some residue of
`foaming agent in the final structure [8], and different cell shapes
`
`
`
` iage7 of27
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`Page 7 of 27
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`

`
`The Thermal Conductivity of a Polyethylene Foam Block
`
`23
`
`depending on the foam density. Consequently, the characterization of
`the cellular structure for these last foams is more complicated and, in
`principle, it should be more difficult to establish the structure property
`relationships for these materials.
`Taking into account the previous notions, the main object of this work
`is to adapt the previous model to predict the thermal conductivity of
`polyethylene foams with a more complicated structure. For this purpose,
`foams produced by a compression molding process were selected. An-
`other target pursued by this investigation is to extend the range of appli-
`cability of the model at different
`temperatures. Moreover,
`the
`compression molding technology is used to manufacture thick foam
`blocks. Due to the foaming process, these blocks could have non-homoge-
`neous properties along the thickness of the block. Therefore, foam
`sheets cut from different positions would show different thermal proper-
`ties. In order to check this possibility, the evolution of the properties
`along the block has also been analyzed.
`
`MATERIALS AND FOAMING
`
`The foams under study were made of low—density polyethylene by us-
`ing a compression molding process and were generously provided by
`Microcel (Burgos, Spain).
`This foaming process started with the mixture of the base polymer
`(LDPE), foaming agent (azocarbonamide) and crosslinking agent (per-
`oxides). A solid sheet was obtained from the previous mixture. In a first
`processing stage, the mixed compound was put into a prism geometry
`mould which was closed under high—pressure conditions (>5000 kPa).
`Both top and bottom surfaces of the mould were heated until equal tem-
`peratures. As the temperature was rising, the crosslinking agent was ac-
`tivated and the chemical crosslinking started. In the meantime, the
`foaming reaction (decomposition of the foaming agent) started. After a
`period of time, the mould was opened. In a second step, the material was
`heated in a second mould to complete the expansion. Then, the material
`was cooled.
`
`A foam thick block of 1 In X 1 m X 0.09 m was thus obtained [Figure
`1(a)] and processed so that two 5 mm thick sheets were cut from the up-
`per and lower surfaces [Figure 1(b)]. Finally, the foam block was cut in
`eight 10 mm thick sheets [Figure 1 (c)]. The tested foam samples were cut
`from the central part of each sheet.
`.
`The acronym for the foams used in this investigation was PE3O which
`denotes the base polymer (LDPE) and the grade of expansion (thirty),
`which is the ratio between the final foam volume and the initial solid
`
`Page 8 of 27
`
`
`Page 8 of 27
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`

`
`24
`
`J. A. MARTiNEZ-DiEZ ET AL.
`
`Foam samples area
`
`
`
`a) block of foam
`
`b) surface cut
`
`c) sheet cut
`
`Figure 1. Schematic representation of the method used to cut the foam sheets from the
`initial foam block.
`
`sheet volume. It was necessary to add another index to distinguish
`among the eight sheets cut from the same block of foam. This sheet index
`numbers the PE3O foams from 1 (upper sheet) to 8 (lower sheet)
`Foamed materials are usually anisotropic [1 1]. For this reason, it was
`necessary to characterize the different directions of the block. Thickness
`and longitudinal directions were denoted as T and X,Y, respectively, as
`shown in [Figure 1(c)].
`
`EXPERIMENTAL
`
`Density
`
`Density was measured by Archimedes’ principle using the density de-
`termination kit for the AT261 Mettler Toledo balance. All the tests were
`carried out at a constant temperature of 21°C. The experiment was per-
`formed in five samples cut from each sheet. The 95% confidence interval
`of these measurements was about 3%.
`
`Differential Scanning Calorimetry (DSC)
`
`A Mettler DSC30 differential-scanning calorimeter was used to study
`several thermal properties. A previous calibration with Indium was nec-
`eS5a1‘y- Samples weighed approximately 2.5 mg and the experiment
`worked under the temperature range between —40°C and 200°C at
`10i’C/min. Two characteristic properties of the base polymer, melting
`P011“? (Tm) and Crystallinity (Xe), were obtained. The melting point was
`taken as the minimum of the melt peak in the enthalpy curve. The
`Crystalllnity Was calculated from the DSC curve by dividing the
`
`5a
`‘9.
`
`
`
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`
`Page 9 of 27
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`

`
`The Thermal Conductivity of a Polyethylene Foam Block
`
`25
`
`measured heat of fusion by the heat of fusion of a 100% crystalline mate-
`rial (288 J/g for LDPE) [12]. The experiment was done three times for
`each kind of sample in order to obtain the average values of melting point
`and crystallinity. The 95% confidence interval of these three measure-
`ments was approximately : 1% of the average value for the melting tem-
`perature and :8% of the average value for the crystallinity.
`
`Scanning Electron Microscopy (SEM)
`
`Quantitative image analysis was used to assess the type of cellular
`structure, apparent mean cell diameter, anisotropy, mean cell wall
`thickness, as well as the relative fraction of polymer in the struts. For
`this purpose, cross—sections of samples were microtomed at low tempera-
`ture to provide a smooth surface which, after vacuum coating with gold,
`was examined by SEM using a JEOL JSM-820 microscope.
`Several micrographs, on the parallel direction to the X and Y axes,
`were taken from the microtomed samples. Each micrograph was ana-
`lyzed by obtaining data from 10 parallel and equidistant reference lines
`following each principal direction. Apparent mean cell size along each di-
`rection [¢(X),<[)(Y),¢(T)] was estimated by calculating the number of cells
`that intersected each reference line, so that the appropriate reference
`length was divided by the number of cells [13]. The previous result was
`multiplied by 1.62 because there is a relationship between the measured
`values, average length of cells which were randomly truncated, and the
`real diameter of the cells [14]. Three micrographs for each pair of direc-
`tions (T/X and T/Y) were studied. The 95% confidence interval for the
`
`mean cell size in each direction was estimated as approximately : 7% of
`the average value. From the previous considerations, the apparent mean
`cell size ((1)) was obtained as the mean value of the cell sizes along each di-
`rection.
`
`The coefficient of anisotropy was estimated from the values of cell
`sizes along the principal directions [11]. This magnitude was evaluated
`as the ratio between the cell size along longitudinal directions and the
`thickness direction [Equation ( 1)].
`
`[¢<X>+¢<y>]
`
`2
`A=————i
`<l>(T)
`
`[
`
`K1
`
`\
`
`-
`
`(1)
`
`The thickness of 30 cell walls, randomly chosen along the samples, was
`measured. The average value of the performed measurements was con-
`sidered as the mean cell wall thickness (E) of each foam. The 95%
`
`Page 10 of 27
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`Page 10 of 27
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`

`
`26
`
`J. A. MARTENEZ-Diez ET AL.
`
`
`
`confidence interval of this magnitude was estimated as approximately
`*5 10% of the average value.
`_
`.
`Finally, the fraction of mass in the struts (1%) was obtained using the
`method proposed by Kuhn [15]. Eight micrographs taken randomly
`along the foam were studied by this method. The 95% confidence inter-
`val was estimated as approximately 1 10% of the average value.
`
`Thermal Conductivity
`
`A Rapid K Heat Flow Meter from Holometrix was used for thermal
`measurements. Heat flow (q) through the sample results from having a
`temperature gradient (AT) across the material. Thermal conductivity K
`is defined according to Fourier’s Equation:
`
`AT
`=?tA———
`d
`
`q
`
`2
`
`(
`
`)
`
`where A and d are the sample surface area and the sample thickness,
`respectively.
`The complete face area is a square of 30 cm side where the heat flow
`meter occupies a 10 cm side square in the central portion of the cold face
`of the equipment. This heat flow meter is a thermopile which gives an
`output of 40 uV for a temperature drop of 1°C. As the tested sample is a
`30 cm side square, the remaining portion acts as a shield that keeps the
`heat flow uniform in the measuring central section. The method is not
`absolute and, therefore, needs to be calibrated using a standard sample.
`Once this has been completed, the heat flow per unit area can be found
`from the reading of the heat flow transducer. Thus, thermal conductiv-
`ity can be calculated by Fourier’s Equation.
`Measurements were made under steady state heat flow conditions
`through the sample, in accordance with ASTM C518 and ISO DIS 8301
`methods. A dispersion ofless than 1% between consecutive readings was
`taken as the criterion to ensure steady state conditions. The interval be-
`tween readings was 15 minutes. These measurements were performed
`at 24°C, 30°C, 40°C and 50°C. The precision of the equipment was evalu-
`ated as approximately 1.5%.
`
`RESULTS
`
`Density
`
`Data obtained from each ofthe eight PE30 foams are shown in Figure
`2- A Parabolic shape is estimated for the six central sheets of the block.
`
`Page 11 of 27
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`I
`
`-.
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`Page 11 of 27
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`

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`
`
`The Thermal Conductivity of a Polyethylene Foam Block
`
`27
`
`23
`
`27
`
`,.,»~
`§26
`
`y = 034;’ - 3.01; + 30.79
`
`R‘ = 0.991
`
`0
`
`1
`
`2
`
`'
`
`3
`
`T
`
`5
`4
`Sheet index
`
`6
`
`7
`
`8
`
`Figure 2. Density of the foam sheets as a function of their position in the block.
`
`Densities from the six central sheets were compared against a second or-
`der linear regression. The curve obtained, as well as the equation and
`correlation index, are also shown in Figure 2. A similar behavior has
`been previously observed for materials produced by compression mold-
`ing [16].
`'
`T
`A possible explanation would lie in the foaming process. Once the ex-
`pansion begins and after a while, the foam material gets in touch with
`the surface of the mould. Therefore, there is a bound effect against the
`' "mould that results in a lower expansion. Consequently, a higher density
`is observed as foam sheets were tested from the center towards the
`surface of the block.
`
`. Microscopic Characterization
`
`The values of the crystallinity and melting point are summarized in
`Table 1. It can be deduced, from the low dispersion estimated in both
`kinds of results (3.25% for xc and 0.25% for Tm), that the -morphology of
`the base polymer present in all the tested foams is very similar.
`Two typical micrographs ofthe foams under study are shown in Figure
`3 (PE30[1] and PE30[4]). Both materials present a closed cell cellular
`structure with some residue of foaming agent in the final foam. No dif-
`ferences between the structure of both foams can be inferred from a
`qualitative observation of the micrographs. Therefore, a" more detailed
`analysis is needed. The analysis was performed in terms of the mean cell
`size, coefficient of anisotropy, mean cell wall thickness and fraction of
`mass in the struts.
`
`Page 12 of 27
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`Page 12 of 27
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`

`
`l
`
`
`
`28
`
`J. A. MAFiTlNEZ-DlEZ ET AL.
`
`Table 1. Melting point and crystallinity
`of the foams under study.
`
`Crystallinity xc
`Melting Point Tm
`
`Samples
`(%)
`(°C)
`
`PE30[1]
`PE30[2]
`pE3o[3]
`PE30[4]
`PE30[5]
`PE30[6]
`PE30[7]
`PE3o[a]
`
`40.4
`42.8
`42.4
`4 40.9
`40.6
`43.2
`43.7
`40.7
`
`107.4
`1079
`107.6
`107-6
`108.2
`107.8
`107.6
`107.9
`
`
`
`The numerical data obtained for the mean cell size ((1)) are included in
`Table 2 and in Figure 4. It can be seen that the mean cell size is slightly
`higher for the central samples of the foam block (from PE30[3] to
`PE30[6]). Two groups of foams may be distinguished: central samples
`(from PE30[3] to PE30[6]) show cells of slightly higher size, and foams
`close to the surface (PE30[1], PE30[2], PE30[7] and PE30[8]) present a
`lower cell size. The explanation of this behavior should be the same that
`explains the evolution of the density.
`It can be deduced from the values of CA (Table 2) that the cell size is
`
`ism‘:
`
`
`
`
`
`I‘/at£L_}___;.€§_,,._4T§..........._AA‘.'.?__....
`
`Page 13 of 27
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`

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`
`
`The Thermal Conductivity of a Polyethylene Foam Block
`
`29
`
`
`
`Figure 3 (continued). Micrographs of:
`
`(b) PE30[4] foam.
`
`longer along the thickness direction than along any other direction.
`Therefore, these materials are anisotropic with cells slightly elongated
`along the thickness direction. However, there is not a clear trend of this
`magnitude as a function of the sheet index.‘
`The data for the mean cell Wall thickness (<3) have been presented in
`Table 2. Once again, there is not a clear trend with sheet index.
`The values measured for the fraction of mass in the struts (fs) are also
`given in Table 2. A low dispersion is observed as all the values are around
`0.2. This value is similar to that obtained for polyolefin foams manufac-
`
`Table 2. Value of the main parameters that
`characterize the cellular structure.
`.
`Mean Cell Wall
`Fraction of
`Coefficient of
`Thickness ’
`Mass in the
`Mean Cell Size
`Anisotropy CA
`é (um)
`’ Struts (fs)
`¢ (um)
`Samples
`0.940
`1.3 : 0.1
`0.19 : 0.02
`346 : 23
`PE30[1]
`1.002
`1.2 : 0.1
`0.23 : 0.02
`348 : 23
`PE30[2]
`0.917
`1.1 : 0.1
`0.27 : 0.03
`414 : 28
`PE30[3]
`0.928
`1.2 : 0.1
`0.20 : 0.02
`386 1‘ 26
`PE30[4]
`0.932
`. 1.4 : 0.1
`0.20 : 0.02
`349 : 23
`PE30[5]
`1.003
`1.4 : 0.1
`0.29 i 0.03
`376 1 25
`PE30[6]
`0.899
`1.3 : 0.1
`0.29 :2 0.03
`347 i 23
`PE30[7]
`
`
`
`0.976 1.4 i 0.1325 : 22PE30[8] 0.24 : 0.02
`
`
`'
`
`Page 14 of 27
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`Page 14 of 27
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`

`
`30
`
`J. A. MAnTiNEz—DiEz ET AL.
`
`LIIO9
`
`-J=-U\C
`
`3
`
`La.)U)-ROU!GO0
`
`
`
`Meancellsize(pm)
`
`Sheet index
`
`Figure 4. Mean cell size as a function of the sheet index.
`
`tured by a high-pressure nitrogen gas solution process [17], and lower
`than the value measured in those foams produced by a semi-continuous
`foaming process [18].
`Some results can be inferred from the previous microscopic character-
`ization that will be helpful for the understanding of the thermal conduc-
`tivity results. These foams have an anisotropic closed cell structure with
`cells that are slightly elongated in the thickness direction. Within our ex-
`perimental deviation, the mean cell size is the only microscopic parame-
`ter which shows a slight variation along the thickness of the block; mean
`cell wall thickness, coefficient of anisotropy and fraction of mass in the
`struts seem to be approximately constant. Moreover, the crystallinity
`and melting point are also constant.
`
`Thermal Conductivity (A)
`
`There are two kinds ofdata obtained from the experiments performed.
`The behavior ofthe thermal conductivity as a function ofthe sheet index
`is shown in Figure 5(a). Two groups of foams from the same block are
`found; the thermal conductivity is higher for the central foams than for
`the superficial ones. It is also a fact that the differences among the nu-
`merical values are very small; they are estimated under 3.0%. Another
`behavior of the thermal conductivity is the one versus the temperature-
`The experiments were performed at 24°C, 30°C, 40°C and 50°C; the val-
`ues obtained for the foams PE30[1] and PE30[4] are shown in Figure
`5(b.). There is a clear linear trend which can be fitted by a linear curve-
`This result was obtained for all the foams from the PE30 block.
`
` aige:15 of27
`
`Page 15 of 27
`
`

`
`
`
`The Thermal Conductivity of a Polyethylene Foam Block
`
`31
`
`0,043
`
`0 24°C El 30°C
`A 40°C X 50°C
`
`0,037 0,036
`
`0
`
`1
`
`2
`
`3
`
`5
`4
`Sheet index
`
`6
`
`7
`
`8
`
`(a)
`
`0.043
`
`0.042
`
`0.041
`
`0.038
`
`X PE30[1]
`0 PE30[4]
`
`0.036
`
`0.04
`
`A(W/IIIK) 0.039
`
`0.037
`
`20
`
`25
`
`30
`
`35
`
`40
`
`45
`
`50
`
`55
`
`Temperature (°C)
`
`(b)
`
`Figure 5. (a) Experimental thermal conductivity as a function of the sheet index. (b)
`Thermal conductivity for the PE30[1] and PE30[4] foams as a function of the tempera-
`ture.
`’
`
`Page 16 of 27
`
`Page 16 of 27
`
`

`
`
`
`32
`
`J. A. MARTiNEz—DiEz ET AL.
`
`DISCUSSION
`
`It is well known that the thermal conductivity 0») of cellular materials
`may
`be
`evaluated
`through
`several
`different mechanisms
`[1—6,15,17,19—25]: conduction through the gas phase (kg), convection
`within the cells (kc), conduction through the solid phase (ks) and thermal
`radiation (Kr). As all these mechanisms contribute to the thermal con-
`ductivity of a foam, the previous evaluation of these terms drives to the
`knowledge of the whole magnitude. Moreover, it has previously been
`shown that thermal conductivity may be expressed by an equation that
`considers the sum of them [1].
`
`)t=?tg+?tc+7ts+7t,
`
`(3)
`
`It is widely accepted that for closed cell materials the convection term
`plays a minor role when cell size is lower than 4 mm [26]. As the foams
`under study in this investigation show lower mean cell sizes, the convec-
`tion term is considered negligible and is not taken into account for the es-
`timation of the thermal conductivity.
`Conduction through solid and gas phases is evaluated, in this work, by
`the equation given by Glicksman [1]. Previous works have demonstrated
`that similar results are obtained if other models are used [3,4,17].
`
`V >.g+i,=;.g,s(1—B":J+x,,é [f,\[C—;+2(1—,g)(aZT]
`
`P
`
`3
`
`1
`
`1
`
`4
`
`1
`
`(4)
`
`where ltgas is the thermal conductivity ofthe gas within the cells, hp is the
`thermal conductivity ofthe matrix polymer, p is the density of the foam,
`p, is the density of the solid polymer, 1% is the fraction of mass in the
`struts and CA is the coefficient of anisotropy. The first term corresponds
`to the thermal conductivity through the gas phase, and the second term
`denotes the contribution given by the thermal conductivity through the
`a solid phase. As a diffusion exchange is given, air was considered as the
`gas within the cells [27,28]. kgas was evaluated from tables where ther-
`mal conductivity for air at several temperatures was shown [29]. The
`values used for hp and p, were KP = 0.214 W/mK and ps = 910 kg/m3 [7]-
`As a first approximation, it is assumed that the thermal conductivity of
`the base polymer does not depend on temperature. It is important to note
`that the previous equation considers the anisotropy ofthe cellular struc-
`ture through the term in which the anisotropy coefficient appears.
`Once the wnduction through the gas and solid phases is evaluated and
`the values ofthe thermal conductivity ofthe samples are experimentally
`
` 17 of27
`
`<
`
`Page 17 of 27
`
`

`
`
`
`The Thermal Conductivity of a Polyethylene Foam Block
`
`33
`
`obtained, the thermal radiation may be calculated by subtracting the
`two first terms from.the experimental values of the thermal conduc-
`tivity.
`The results ofthis term are shown in Figure 6 as a function ofthe sheet
`index. It is evident that the heat transfer by radiation is higher for
`central foams.
`
`From the experimental results, we concluded that the foams close to
`the surface were better insulating materials. One possible way to ana-
`lyze which of the different contributions is mainly responsible for this
`behavior is to obtain the percentages of difference between two foams for
`each mechanism, one cut from the surface (PE30[1]) and the other cut
`from the center (PE30[4]) [Equation (5)].
`
`Experimental =
`
`xexp (PE30[4]) — xexp (PE[1])
`7texp(PE30[4])
`
`100
`
`Gas Weight =
`
`?tg(PE30[4]) — }tg(PE[1]) '
`?texp(PE30[4])
`
`100
`
`Solid Weight =
`
`ls (PE30[4]) — KS (PE[1l) _
`?texp(PE30[4])
`
`100
`
`Radiation Weight =
`
`9., (PE30[4]) — 9., (PE[1]) ,1
`?teXp(PE3O[4])
`
`00
`
`(5)
`
`Table 3 shows the results of this calculation. The difference between
`the experimental thermal conductivity of these two foams at 24°C is
`
`1,200E-02
`1,1 50E-02
`1,100E-02
`1 ,050E-02
`l,OO0E-02
`9,500E-03
`9,000E-O3
`8,500E—03
`8,000E—03
`7,500E-03
`
`x.‘(W/mK)
`
`+ 24°C x 30°C
`
`X 1 A 40°C x 50°C
`
`7,000E—03
`
`0
`
`1
`
`2
`
`3
`
`5
`4
`Sheet index
`
`6
`
`7
`
`8
`
`.
`
`Figure 6. Thermal conductivity by radiation as a function of the position of the foam in
`the block.
`‘
`
`Page 18 of 27
`
`Page 18 of 27
`
`

`
`
`
`
`
`34
`
`J. A. MAi=iTiNEz-DiEz ET AL.
`
`Table 3. Percentages of difference for the experimental
`values, gas conduction, solid conduction and thermal
`radiation between two foams, PE30[1] and PE30[4].
`
`Temperatures
`
`Experimental
`
`Gas Weight
`
`Solid Weight
`
`Radiation
`
`315
`-0.839
`0.167
`2.48
`24°C
`313
`-0.818
`0.166
`253
`30-ac
`3.12
`-0.793
`0.165
`249
`40°C
`
`
`
`
`2_4O 0.162 -0.75850°C 300
`
`
`
`2.48%; this difference comes mainly from the difference between the
`thermal radiation terms (3.15%). Minor changes are due to the conduc-
`tion through the gas and solid phases, 0.167% and —0.839%, respectively.
`Moreover, it is well known that thermal radiation depends strongly on
`the cell size [1,3]. Figure 7 shows the thermal conductivity as a function
`of the cell size. This figure also suggests that the structural reason for
`the evolution of the thermal conductivity along the thickness of the ini-
`tial block is the change in the mean cell size. As it was pointed out in the
`SEM results, cell size is the only microscopic parameter that slightly
`changes along the thickness of the foam block.
`Apart from estimating the radiation term, one reason for this investi-
`gation was to predict this term. Such prediction was performed by using
`the Williams and Aldao model [6]:
`
`i =—»?—«~
`
`(6)
`
`3,800E-02
`
`3,780E-02
`
`3,760E-02
`
`3,7401-:—02
`
`3,7203-02
`
`/mK)
`
`‘< 3,700E-02
`3,680E-02
`
`3,660E-02
`
`300
`
`320
`
`400
`380
`360
`340
`Mean cell size (pm)
`
`420
`
`440
`
`Figure 7. Experimental thermal conductivity as a function of the mean cell size.
`
`Page 19 of 27
`
`

`
`
`
`The Thermal Conductivity of a Polyethylene Foam Block
`
`35
`
`where 6 is the Stefan-Boltzmann constant, T is the temperature, L is the
`foam thickness, q> is the mean cell size and TN is the net fraction of radi-
`ant energy sent forward by a solid membrane of thickness LS, which in
`this work is represented by the mean cell wall thickness (5,).
`TN is given by:
`
`_
`
`TN
`
`_ (1—r){(1—r)t
`
`(1—rt)
`
`(1+rt)
`
`+
`
`(1—t)}
`
`2
`
`(7)
`
`where r is the reflectivity of the gas-solid interface within the cells,
`which is related to the refractive index of the polymer as by:
`
`2
`
`r=(w—1)
`
`w+1
`
`(8)
`
`and t is the transmission coefficient of a solid membrane (thickness L3)
`which is given by the Bouguer’s Law:
`
`t = exp(—aLS)
`
`(9)
`
`where a is the absorption coefficient of the base polymer.
`In order to check the validity of the Williams and Aldao model to
`describe the experimental results, TN was calculated by two different
`ways. On one hand, TN was computed using Equation (6) (TN) and the
`thermal radiation obtained from the experimental thermal conductivity.
`On the other hand, TN was evaluated by Equation (7) (TI§,)using the
`experimental values of the cell wall thickness (<2 = LS) and typical values
`for the refractive index, (0, and the absorption coefficient of the base
`polymer in the infrared region, a. These two values were to = 1.51 (30)
`and a = 660 cm‘1 [7].
`The values of TN estimated by the first method are included in Table 4.
`The differences between the values at different temperatures for each
`foam were very small. Therefore, TN was taken as a constant, and the
`value (TN) at 24°C was chosen as a representative value of this magni-
`tude in the temperature range between 24 and 50°C.
`The values of TN evaluated by the second method (Tfi) , as well as the
`percentage of difference with the value obtained by the first method, are
`included in Table 5. Little differences between both kinds of methods are
`shown.
`
`Once TN is characterized from typical values of co and a, thermal radia-
`tion may be estimated by Equation (6) for each temperature. As a conse-
`quence of the previous results,
`the thermal conductivity may be
`
`Page 20 of 27
`
`Page 20 of 27
`
`

`
`36
`
`J. A. MARTIINEZ-D|'EZ ET AL.
`
`Table 4. Fraction of radiant energy sent forward by a solid
`membrane calculated by using Equation (6).
`
`Samples
`
`Ti} (-24°C)
`
`Till 50°C)
`
`Till (40%))
`
`Til (50°C)
`
`0.880
`0.877
`0.883
`0.883
`PE3o[2]
`0.872
`0.859
`0.87