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Introduction Introduction to to Plastic Foams Plastic Foams Andr André Moreira Moreira Polymer Polymer Physics Physics , BASF AG , BASF AG Ludwigshafen, Germany Ludwigshafen, Germany Les Les Houches Houches , Jan. 2006 , Jan. 2006 Acknowlegments Acknowlegments: H. H. Wei Weiß V. V. Sch Schädler dler P. P. Müller ller M. M. Rüllmann llmann E. E. Wa Waßner ner H. H. Schuch Schuch K. Hahn K. Hahn C. C. Exner Exner M. M. Allmendinger Allmendinger P. Lopes P. Lopes B. B. Sowart Sowart A. A. Alteheld Alteheld D. Longo D. Longo K. K.- H. H. Wa Waßmer mer T. Francis T. Francis F. F. Heilmann Heilmann M. M. Rüllmann llmann M. M. Bothe Bothe

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Page 1: Introduction to Plastic Foams - Trinity College, Dublinfoams/PRESENTATIONS/Friday13/Friday13_Lecture… · Introduction to Plastic Foams André Moreira Polymer Physics, BASF AG Ludwigshafen,

Introduction Introduction to to Plastic FoamsPlastic Foams

AndrAndréé MoreiraMoreiraPolymer Polymer PhysicsPhysics, BASF AG, BASF AG

Ludwigshafen, GermanyLudwigshafen, Germany

Les Les HouchesHouches, Jan. 2006, Jan. 2006

AcknowlegmentsAcknowlegments::

H.H. WeiWeißß V. V. SchSchäädlerdlerP. P. MMüüllerller M. M. RRüüllmannllmannE. E. WaWaßßnerner H. H. SchuchSchuchK. HahnK. Hahn C. C. ExnerExnerM. M. AllmendingerAllmendinger P. LopesP. LopesB. B. SowartSowart A. A. AlteheldAlteheldD. LongoD. Longo K.K.--H. H. WaWaßßmermerT. FrancisT. Francis F. F. HeilmannHeilmannM. M. RRüüllmannllmann M.M. BotheBothe

Page 2: Introduction to Plastic Foams - Trinity College, Dublinfoams/PRESENTATIONS/Friday13/Friday13_Lecture… · Introduction to Plastic Foams André Moreira Polymer Physics, BASF AG Ludwigshafen,

Dr. AndrDr. Andréé MoreiraMoreira, BASF AG, BASF AG

Global summaryGlobal summary

Introduction

Plastic foams: Physics and Chemistry

Plastic foams: industrial processes

Mechanical properties

Acoustic properties

Thermal properties

Nanoporous foams: the future?

Part I

Part II

Page 3: Introduction to Plastic Foams - Trinity College, Dublinfoams/PRESENTATIONS/Friday13/Friday13_Lecture… · Introduction to Plastic Foams André Moreira Polymer Physics, BASF AG Ludwigshafen,

Dr. AndrDr. Andréé MoreiraMoreira, BASF AG, BASF AG

Summary: Part IISummary: Part II

Mechanical propertiesStress strain behaviorBendingCrash simulation of foams

Acoustic propertiesSound and sound managementAcoustic impedance and sound reflectionSound dissipation in porous media

Thermal propertiesThermal conductivity: definitionConductivity through the matrixRadiative contributionConductivity through the gas

Nanoporous foams: the future?

Page 4: Introduction to Plastic Foams - Trinity College, Dublinfoams/PRESENTATIONS/Friday13/Friday13_Lecture… · Introduction to Plastic Foams André Moreira Polymer Physics, BASF AG Ludwigshafen,

Mechanical propertiesMechanical properties

Page 5: Introduction to Plastic Foams - Trinity College, Dublinfoams/PRESENTATIONS/Friday13/Friday13_Lecture… · Introduction to Plastic Foams André Moreira Polymer Physics, BASF AG Ludwigshafen,

Dr. AndrDr. Andréé MoreiraMoreira, BASF AG, BASF AG

Mechanical mapMechanical map

Source: Gibson & Ashby

Page 6: Introduction to Plastic Foams - Trinity College, Dublinfoams/PRESENTATIONS/Friday13/Friday13_Lecture… · Introduction to Plastic Foams André Moreira Polymer Physics, BASF AG Ludwigshafen,

Dr. AndrDr. Andréé MoreiraMoreira, BASF AG, BASF AG

Bending work, bending strength

Compressive strength

Testing materials (some examples)Testing materials (some examples)

Tensile strength

Source: Gibson & Ashby, Cellular solids , 2nd Ed., Cambridge (1997)

Page 7: Introduction to Plastic Foams - Trinity College, Dublinfoams/PRESENTATIONS/Friday13/Friday13_Lecture… · Introduction to Plastic Foams André Moreira Polymer Physics, BASF AG Ludwigshafen,

Dr. AndrDr. Andréé MoreiraMoreira, BASF AG, BASF AG

OpenOpen-- and and closedclosed--cell foamscell foams

Open-cell foams are generally weaker than closed-cell foams

Page 8: Introduction to Plastic Foams - Trinity College, Dublinfoams/PRESENTATIONS/Friday13/Friday13_Lecture… · Introduction to Plastic Foams André Moreira Polymer Physics, BASF AG Ludwigshafen,

Dr. AndrDr. Andréé MoreiraMoreira, BASF AG, BASF AG

Model foam: HoneycombsModel foam: Honeycombs

Page 9: Introduction to Plastic Foams - Trinity College, Dublinfoams/PRESENTATIONS/Friday13/Friday13_Lecture… · Introduction to Plastic Foams André Moreira Polymer Physics, BASF AG Ludwigshafen,

Dr. AndrDr. Andréé MoreiraMoreira, BASF AG, BASF AG

Example: stressExample: stress--strain curve strain curve of a rigid foam (experimental)of a rigid foam (experimental)

Source: Gibson & Ashby

σpl

( ) ( )termpressuregas+−+

S

F

S

F

S

F

EE

ρρ

φρρ

φ 12

φ: fraction of material in the edges(=1 in open-cell foams)

Open- and Closed-cell foams (G & A)

( ) ( )termpressuregas+−+

S

F

S

F

ys

pl

ρρ

φρρ

φσ

σ1

2/3

Page 10: Introduction to Plastic Foams - Trinity College, Dublinfoams/PRESENTATIONS/Friday13/Friday13_Lecture… · Introduction to Plastic Foams André Moreira Polymer Physics, BASF AG Ludwigshafen,

Dr. AndrDr. Andréé MoreiraMoreira, BASF AG, BASF AG

0 0.2 0.4 0.6 0.8 1rel. density

0

0.2

0.4

0.6

0.8

1

ler.

gnuoYdo

m.

0 0.2 0.4 0.6 0.8 1rel. density

0

0.2

0.4

0.6

0.8

1

ler.

gnuoYdo

m.

Comparison between Gibson & Ashby and simulations (linear regime )

2

S

F

S

F

EE

ρρ

( )S

F

S

F

S

F

EE

ρρ

φρρ

φ −+

≈ 1

2

2

φ: fraction of material in the edges

Cubic closed cells (G & A)

Cubic open cells (G & A)

ES=3.4 GPa

Cubic foamsCubic foams: linear : linear elasticityelasticity

Simulation

G&A

Simulation

G&A

Horst Weiss, Peter Müller, Frank Heilmann, BASF

Page 11: Introduction to Plastic Foams - Trinity College, Dublinfoams/PRESENTATIONS/Friday13/Friday13_Lecture… · Introduction to Plastic Foams André Moreira Polymer Physics, BASF AG Ludwigshafen,

Dr. AndrDr. Andréé MoreiraMoreira, BASF AG, BASF AG

§ Polymer matrix:

§ Molecular weight distribution

§ Linear, branched

§ Additives and fillers

§ Homogeneity of foam structure

§ Cell density

§ EPS: fusion of the beads

Properties that influence the Properties that influence the mechanical behavior of foamsmechanical behavior of foams

Page 12: Introduction to Plastic Foams - Trinity College, Dublinfoams/PRESENTATIONS/Friday13/Friday13_Lecture… · Introduction to Plastic Foams André Moreira Polymer Physics, BASF AG Ludwigshafen,

Dr. AndrDr. Andréé MoreiraMoreira, BASF AG, BASF AG

On On passingpassing: : Why isWhy is an an inhomogeneous inhomogeneous foam weakerfoam weaker??

Initial 10% compression

Extreme bending of struts near irregular cells!

Homogeneous compression In homogeneous Foams!

Density ~ 40 g / l

Horst Weiss, Peter Müller, Frank Heilmann, BASF

Page 13: Introduction to Plastic Foams - Trinity College, Dublinfoams/PRESENTATIONS/Friday13/Friday13_Lecture… · Introduction to Plastic Foams André Moreira Polymer Physics, BASF AG Ludwigshafen,

Acoustic propertiesAcoustic properties

Page 14: Introduction to Plastic Foams - Trinity College, Dublinfoams/PRESENTATIONS/Friday13/Friday13_Lecture… · Introduction to Plastic Foams André Moreira Polymer Physics, BASF AG Ludwigshafen,

Dr. AndrDr. Andréé MoreiraMoreira, BASF AG, BASF AG

8 Hz – 20 kHz

25 – 80 kHz

40 – 50 kHz

1 – 5 Hz

Fre

quen

cy (

Hz)

1

10

102

103

104

105

343

34.3

3.4

0.34

0.034

0.0034

Wav

elen

gth

(m)

Air: c = 343 m/sρ = 1.3 kg/m³

SoundSound

Page 15: Introduction to Plastic Foams - Trinity College, Dublinfoams/PRESENTATIONS/Friday13/Friday13_Lecture… · Introduction to Plastic Foams André Moreira Polymer Physics, BASF AG Ludwigshafen,

Dr. AndrDr. Andréé MoreiraMoreira, BASF AG, BASF AG

Threshold of hearing measured at 1 kHz:

Lowest Intensity: I0 = 10-12 Watt / m²Highest Intensity: 10 Watt / m²

Lowest Pressure: 2 x 10-5 Pa = 2 x 10-10 bar Highest pressure: 60 Pa = 0.6 mbar

Decibel system:

I(db) = 10 Log10(I / I0)

Lowest value: 0 db (I = I0)

Highest value (threshold of pain): 130 db

20 50 100 200 500 1000 2000 5000

20

40

60

80

100

120

Conversation

Music

Threshold of pain

Threshold ofhearing

Frequency (Hz)

Inte

nsity

(dB

)

Source: Veit, Teschnische Akustik

SoundSound

Page 16: Introduction to Plastic Foams - Trinity College, Dublinfoams/PRESENTATIONS/Friday13/Friday13_Lecture… · Introduction to Plastic Foams André Moreira Polymer Physics, BASF AG Ludwigshafen,

Dr. AndrDr. Andréé MoreiraMoreira, BASF AG, BASF AG

Absorption:

Isolation:

Insulation:

Absorb and dissipate sound generated in a room. Low sound reflection.

Keep sound out of a room through reflection.

Avoid sound propagation though solids by coupling rigid parts to resilient materials.

Foams OK

Foams OK

Foams KO

Sound Sound management management in in buildingsbuildings

Page 17: Introduction to Plastic Foams - Trinity College, Dublinfoams/PRESENTATIONS/Friday13/Friday13_Lecture… · Introduction to Plastic Foams André Moreira Polymer Physics, BASF AG Ludwigshafen,

Dr. AndrDr. Andréé MoreiraMoreira, BASF AG, BASF AG

cZ ρ=Acoustic impedance: Reflection factor:

12

12)(

)(

ZZZZ

pp

r i

r

+−

=≡1 2

incident

absorbed

reflectedp(r) reflected pressurep(i) incident pressure

The higher is the modulus of r, the better is the reflexion…Bulky materials reflect better (higher contrast with the air)

Absorption factor:21 r−=α

Source: Gibson and Ashby, Cellular Solids

Interfaces and Interfaces and sound absorptionsound absorption

Page 18: Introduction to Plastic Foams - Trinity College, Dublinfoams/PRESENTATIONS/Friday13/Friday13_Lecture… · Introduction to Plastic Foams André Moreira Polymer Physics, BASF AG Ludwigshafen,

Dr. AndrDr. Andréé MoreiraMoreira, BASF AG, BASF AG Source:K.W. Suh et al., Adv. Mat. 12, 1779 (2000)

Sound Sound absorption absorption ((open cell foamsopen cell foams))

(effect of moisture)

Page 19: Introduction to Plastic Foams - Trinity College, Dublinfoams/PRESENTATIONS/Friday13/Friday13_Lecture… · Introduction to Plastic Foams André Moreira Polymer Physics, BASF AG Ludwigshafen,

Dr. AndrDr. Andréé MoreiraMoreira, BASF AG, BASF AG

tp

pctp

gas ∂∂Ξ

−∇=∂∂

ρ22

2

2

Combining (1) Navier-Stokes with dissipation, (2) mass conservation and (3) the equation-of-state for the medium (gas), one obtains:

Traveling 1d plane wave solution (damped!):

)(0

tzkiz eepp ωξ −−−=

Dispersion relation: 04 22

222242 =

Ξ−−

gasckkc

ρω

ω

c: velocity of soundΞ: dissipative factor

ckc gas

/

2

ω

ρξ

Ξ=

Dispersion relation (c=343 m/s)

0

1000

2000

3000

4000

5000

6000

0 5 10 15 20 25 30

kr

om

ega low diss.

high diss.

air

Penetration length:

Sound Sound dissipationdissipation: : the wave equation the wave equation in in porous porous mediamedia

penetration length (m)

0

0.1

0.2

0.3

0.4

0.5

0 200 400 600 800 1000 1200

frequency (Hz)

len

gth

(m

)

low diss.

high diss.

What do we know about Ξ ?It has a maximum at some porosity

It decreases the amount of “blind ends”(connection to the Rayleigh model)

Ξ = 5 x 103 Ns/m4

Ξ = 100 x 103 Ns/m4

Page 20: Introduction to Plastic Foams - Trinity College, Dublinfoams/PRESENTATIONS/Friday13/Friday13_Lecture… · Introduction to Plastic Foams André Moreira Polymer Physics, BASF AG Ludwigshafen,

Thermal Thermal propertiesproperties

Page 21: Introduction to Plastic Foams - Trinity College, Dublinfoams/PRESENTATIONS/Friday13/Friday13_Lecture… · Introduction to Plastic Foams André Moreira Polymer Physics, BASF AG Ludwigshafen,

Dr. AndrDr. Andréé MoreiraMoreira, BASF AG, BASF AG

25Air

800 – 1200Concrete

41Glass wool

38Cork (low dens.)

46Balsa

25 – 35PUR

15Aerogel

400 – 800Brick

4VIP

30 – 35Styropor

35 – 40Styrodur

150Polystyrene

700 – 800Glass

TC (mW / K m)TC (mW / K m)MaterialMaterial

Source: Handbook of Chemistry and Physics, 56th Ed.

Thermal Thermal conductivityconductivity

Page 22: Introduction to Plastic Foams - Trinity College, Dublinfoams/PRESENTATIONS/Friday13/Friday13_Lecture… · Introduction to Plastic Foams André Moreira Polymer Physics, BASF AG Ludwigshafen,

Dr. AndrDr. Andréé MoreiraMoreira, BASF AG, BASF AG

Transport propertiesTransport properties

φα∇−=Jr

tJ

∂∂

−=⋅∇φr

(1) The flux of φ is against its gradient

Pos

itive

gra

dien

t

(2) Sources of flux responsible for change in time

Combining (1) and (2) leads to: φαφ 2∇=

∂∂

t

ρρ 2∇=

∂∂

Dt

Transport of matter: Transport of heat: Tct

T 2∇=∂∂

ρλ

ρ: Density field

D: Diffusion coefficient

T: Temperature fieldλ: thermal conductivity (W / K m)c: specific (mass) heat capacityρ: (mass) density of the material

Page 23: Introduction to Plastic Foams - Trinity College, Dublinfoams/PRESENTATIONS/Friday13/Friday13_Lecture… · Introduction to Plastic Foams André Moreira Polymer Physics, BASF AG Ludwigshafen,

Dr. AndrDr. Andréé MoreiraMoreira, BASF AG, BASF AG

Heat transport across materials:Heat transport across materials:oneone--dimensional casedimensional case

nrr JJJ +=

Two contributions to the heat flux: radiative and non-radiative (convection plays no role for air at length scale below a few mm)

xT

J nr ∂∂

−= λ

( ) xKr eTTJ −−= 4

04

1σβT1 T0

β: constant (below 1 for real surfaces, 1 for black body)

σ: Stefan’s constant

Kp: extinction coefficient of polymer

λ: thermal conductivity (W / K m)

Jr

Jnr

Page 24: Introduction to Plastic Foams - Trinity College, Dublinfoams/PRESENTATIONS/Friday13/Friday13_Lecture… · Introduction to Plastic Foams André Moreira Polymer Physics, BASF AG Ludwigshafen,

Dr. AndrDr. Andréé MoreiraMoreira, BASF AG, BASF AG

NonNon--radiativeradiative heat transport across heat transport across composites: onecomposites: one--dimensional casedimensional case

Two limiting cases:

xT

J nr ∂∂

−= λ

T1 T0

T1 T0

2

1

1

1 11λ

ϕλϕ

λ−

+=

1

1

2

22111 )1( λϕλϕλ −+=

(a) Serial (eg. coextrusion)

(b) Parallel

REMEMBER! conductivity = 1 / Resistivity

ϕ1: vol. frac. of mat. 1

λ1: thermal cond. of mat. 1

Parallel X Serial

01234567

0 0.2 0.4 0.6 0.8 1

Vol. frac. of component 1λ

/ λ2 parallel

21 6 λλ =

serial

Page 25: Introduction to Plastic Foams - Trinity College, Dublinfoams/PRESENTATIONS/Friday13/Friday13_Lecture… · Introduction to Plastic Foams André Moreira Polymer Physics, BASF AG Ludwigshafen,

Dr. AndrDr. Andréé MoreiraMoreira, BASF AG, BASF AG

Heat transport across foams:Heat transport across foams:oneone--dimensional casedimensional case

( )LK

pp

Fg

P

F p

Fp

eTTxT

J ρρ

σβλρρ

τλρρ −

−+∂∂

+

−−= 4

04

11

ρF: density of the foam (usually <300 g/l)

ρp: density of the polymer (~1 kg/l)

τ: tortuosity factor (empirically found to be ~ 2/3)

β: constant (below 1 for real surfaces, 1 for black body)

σ: Stefan’s constant

Kp: extinction coefficient of polymer

L: thickness of foam across x-axis

T1 T0

L x

Page 26: Introduction to Plastic Foams - Trinity College, Dublinfoams/PRESENTATIONS/Friday13/Friday13_Lecture… · Introduction to Plastic Foams André Moreira Polymer Physics, BASF AG Ludwigshafen,

Dr. AndrDr. Andréé MoreiraMoreira, BASF AG, BASF AG

Heat transport across foams:Heat transport across foams:oneone--dimensional casedimensional case

( )radiationmatrixgaseff LTT

Jλλλλ ++≈

−−≡

10

Effective Thermal Conductivity

0

10

20

30

40

50

60

0 20 40 60 80 100

Foam density (g/l)

TC (

mW

/ K

m)

polymer

gas

radiation

total

T1 T0

L x

( )LK

pp

Fg

P

F p

Fp

eTTxT

J ρρ

σβλρρ

τλρρ −

−+∂∂

+

−−= 4

04

11

Page 27: Introduction to Plastic Foams - Trinity College, Dublinfoams/PRESENTATIONS/Friday13/Friday13_Lecture… · Introduction to Plastic Foams André Moreira Polymer Physics, BASF AG Ludwigshafen,

Dr. AndrDr. Andréé MoreiraMoreira, BASF AG, BASF AG

Side note: RSide note: R--values in architecturevalues in architecture

RLTTJ eff 1

10

≡=−

−λ

Inne

r wal

l

Ther

mal

isol

atio

n (fo

am)

Out

er w

all

LTT

Jeff

10 −−≡λ

outerfoaminnertotal RRRR ++=

outer

outer

foam

foam

inner

inner LLLλλλ

++=

JTT

JTT

JTT

JTT

Rofi

total01011111 −

−=−′′

−′′−′

−′−

−=

ofi JJJ ==

Proof:

Conservation of energy:

Page 28: Introduction to Plastic Foams - Trinity College, Dublinfoams/PRESENTATIONS/Friday13/Friday13_Lecture… · Introduction to Plastic Foams André Moreira Polymer Physics, BASF AG Ludwigshafen,

Dr. AndrDr. Andréé MoreiraMoreira, BASF AG, BASF AG

Effective Thermal Conductivity

0

10

20

30

40

50

60

0 20 40 60 80 100

Foam density (g/l)

TC

(m

W /

K m

)

polymer

gas

radiation

total

Thermal conductivity of foams:Thermal conductivity of foams:low density systemslow density systems

TC dominated by radiative contribution!

Page 29: Introduction to Plastic Foams - Trinity College, Dublinfoams/PRESENTATIONS/Friday13/Friday13_Lecture… · Introduction to Plastic Foams André Moreira Polymer Physics, BASF AG Ludwigshafen,

Dr. AndrDr. Andréé MoreiraMoreira, BASF AG, BASF AG

Styropor®

IR-Absorber:Graphite

l 25% lower thermal conductivity (at density of 10 g/l)l 50% lower amount of Material needed (for thermal conductivity of 33 mW / K m)

NeoporNeopor®®

Increasing the KIncreasing the K--factorfactor

Page 30: Introduction to Plastic Foams - Trinity College, Dublinfoams/PRESENTATIONS/Friday13/Friday13_Lecture… · Introduction to Plastic Foams André Moreira Polymer Physics, BASF AG Ludwigshafen,

Dr. AndrDr. Andréé MoreiraMoreira, BASF AG, BASF AG

NeoporNeopor®® versusversus StyroporStyropor®®

Styropor®Neopor®

IR Camera

Page 31: Introduction to Plastic Foams - Trinity College, Dublinfoams/PRESENTATIONS/Friday13/Friday13_Lecture… · Introduction to Plastic Foams André Moreira Polymer Physics, BASF AG Ludwigshafen,

Dr. AndrDr. Andréé MoreiraMoreira, BASF AG, BASF AG

Thermal conductivity of foams:Thermal conductivity of foams:intermediate density systemsintermediate density systems

Effective Thermal Conductivity

0

10

20

30

40

50

60

0 20 40 60 80 100

Foam density (g/l)

TC

(m

W /

K m

)

polymer

gas

radiation

total

TC dominated by gas contribution!

Page 32: Introduction to Plastic Foams - Trinity College, Dublinfoams/PRESENTATIONS/Friday13/Friday13_Lecture… · Introduction to Plastic Foams André Moreira Polymer Physics, BASF AG Ludwigshafen,

Nanoporous foamsNanoporous foams::The futureThe future??

Page 33: Introduction to Plastic Foams - Trinity College, Dublinfoams/PRESENTATIONS/Friday13/Friday13_Lecture… · Introduction to Plastic Foams André Moreira Polymer Physics, BASF AG Ludwigshafen,

Dr. AndrDr. Andréé MoreiraMoreira, BASF AG, BASF AG

25Air

800 – 1200Concrete

41Glass wool

38Cork (low dens.)

46Balsa

25 – 35PUR

15Aerogel

400 – 800Brick

4VIP

30 – 35Styropor

35 – 40Styrodur

150Polystyrene

700 – 800Glass

TC (mW / K m)TC (mW / K m)MaterialMaterial

Source: Handbook of Chemistry and Physics, 56th Ed.

Thermal Thermal conductivityconductivity

Page 34: Introduction to Plastic Foams - Trinity College, Dublinfoams/PRESENTATIONS/Friday13/Friday13_Lecture… · Introduction to Plastic Foams André Moreira Polymer Physics, BASF AG Ludwigshafen,

Dr. AndrDr. Andréé MoreiraMoreira, BASF AG, BASF AG

Knudsen effect in PU-foam (pore diameter ~ 400 µm)

8

13

18

23

28

33

-2 -1 0 1 2 3 4

log(pore diameter/mean free path)

Ther

mal

con

duct

ivit

y (m

W/

K m

)Freon 12

CO2

N2

PUR foam, porosity 0.88, pore size ~400 µm

Experimental data from Harper and Sahrigi, I & EC Fundamentals 3, 318 (1964)

Mean free path of N2 at 10 C ~ 100 nm

Why Why nanoscopicnanoscopic pores are good:pores are good:the Knudsen effectthe Knudsen effect

Mean free path(from kinetic

theory):

22 dPTk

l B

π≈

d: diameter of the molecule

Page 35: Introduction to Plastic Foams - Trinity College, Dublinfoams/PRESENTATIONS/Friday13/Friday13_Lecture… · Introduction to Plastic Foams André Moreira Polymer Physics, BASF AG Ludwigshafen,

Dr. AndrDr. Andréé MoreiraMoreira, BASF AG, BASF AG

Gas conductivity in usual foamsGas conductivity in usual foams

Thermal conductivity through diffusion & collision, no conductivity through convection (for air below 1mm typical length scale)

Page 36: Introduction to Plastic Foams - Trinity College, Dublinfoams/PRESENTATIONS/Friday13/Friday13_Lecture… · Introduction to Plastic Foams André Moreira Polymer Physics, BASF AG Ludwigshafen,

Dr. AndrDr. Andréé MoreiraMoreira, BASF AG, BASF AG

Gas conductivity in Gas conductivity in nanoporous nanoporous mediamedia

In words: in a gas, heat is transported either by radiation or by diffusion à reducing diffusion length means reducing the heat flux through the gas!

Page 37: Introduction to Plastic Foams - Trinity College, Dublinfoams/PRESENTATIONS/Friday13/Friday13_Lecture… · Introduction to Plastic Foams André Moreira Polymer Physics, BASF AG Ludwigshafen,

Dr. AndrDr. Andréé MoreiraMoreira, BASF AG, BASF AG

Knudsen made easy:Knudsen made easy:thermal transport in the ideal gasthermal transport in the ideal gas

vl0=τ Frequency of collision = mean free path / mean molecular velocity

In a bulk system (no walls):

τ1

=p Probability of collision per unit time

Ideal gas confined between 2 walls:

wmmm ppp ,, +=Lll111

0

+=

L

l is the effective mean free path of the system, i.e., the distance that a molecule on average travels before collision (either with another molecule or with a wall)

Page 38: Introduction to Plastic Foams - Trinity College, Dublinfoams/PRESENTATIONS/Friday13/Friday13_Lecture… · Introduction to Plastic Foams André Moreira Polymer Physics, BASF AG Ludwigshafen,

Dr. AndrDr. Andréé MoreiraMoreira, BASF AG, BASF AG

Knudsen made easy:Knudsen made easy:thermal transport in the ideal gasthermal transport in the ideal gas

z=0n: number density of molecules

321 n

Molecules move, on average, in the z+ direction…

z

)(61

lnv −…of which cross z=0 per unit time and area without experiencing collision

So the particle flux is:zn

lvlnvlnvJ z ∂∂

−≈+−−=31

)(61

)(61

10

0 131

31 −

+==

Ll

lvlvD

Mass transport:

Page 39: Introduction to Plastic Foams - Trinity College, Dublinfoams/PRESENTATIONS/Friday13/Friday13_Lecture… · Introduction to Plastic Foams André Moreira Polymer Physics, BASF AG Ludwigshafen,

Dr. AndrDr. Andréé MoreiraMoreira, BASF AG, BASF AG

Knudsen made easy:Knudsen made easy:thermal transport in the ideal gasthermal transport in the ideal gas

z=0n: number density of molecules (no gradient!)

ε: mean energy per molecule

321 n

Molecules move, on average, in the z+ direction…

z

)(61

lnv −ε… transporting average energy across z=0 per unit time and area without collision

So the energy flux is:zT

TlvJ z ∂

∂∂∂

−≈ε

31

Heat transport (non-radiative):

cT

=∂∂ε Specific heat per

molecule

Dcn=λ

Page 40: Introduction to Plastic Foams - Trinity College, Dublinfoams/PRESENTATIONS/Friday13/Friday13_Lecture… · Introduction to Plastic Foams André Moreira Polymer Physics, BASF AG Ludwigshafen,

Dr. AndrDr. Andréé MoreiraMoreira, BASF AG, BASF AG

Knudsen made easy:Knudsen made easy:thermal transport in the ideal gasthermal transport in the ideal gas

Dcn=λ1

00 1

31 −

+=

Ll

lvD 20 2 dPTk

l B

π=

Knudsen effect in PU-foam (pore diameter ~ 400 µm)

8

13

18

23

28

33

-2 -1 0 1 2 3 4

log(pore diameter/mean free path)

Ther

mal

con

duct

ivit

y (m

W/ K

m)

Freon 12

CO2

N2

0

0.2

0.4

0.6

0.8

1

1.2

-4 -3 -2 -1 0 1 2 3 4

Log(L / l0)

D /

D0

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Dr. AndrDr. Andréé MoreiraMoreira, BASF AG, BASF AG

Small Pores: 10 - 100 nmLow density: 1 - 100 g/literTherm. cond.: 15 - 25 mW/K*m

Discovered in the 1931 by Kistler (Nature 127, 741 (1931))

Silica Silica aerogelsaerogels

Page 42: Introduction to Plastic Foams - Trinity College, Dublinfoams/PRESENTATIONS/Friday13/Friday13_Lecture… · Introduction to Plastic Foams André Moreira Polymer Physics, BASF AG Ludwigshafen,

Dr. AndrDr. Andréé MoreiraMoreira, BASF AG, BASF AG

Si(OR)4 + H2O g Si(OR)3OH + ROH (Hydrolysis)

2 Si(OR)3OH g (RO)3SiOSi(OR)3 + H2O (Condensation)

R: Hydrogen, alkyl or silicate group

0 200 400 600 800 1000radiusHnmL

2

4

6

8

10

rotcaferpni

ecalpaLHrabL

θγ

cos2R

PP lg =−

liqliq

gas gas

Fw

FlFt

Fw

Fl

Ft

-Ft

-Ft

θ

0

20

40

60

80

100

120

140

160

180

200

220

240

50 100 150 200 250 300 350 400

Temperature (C)

Pre

ssur

e (b

ar)

water

isoprop.

CP

CPSupercriticaldrying path

Silica Silica aerogelsaerogels

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Dr. AndrDr. Andréé MoreiraMoreira, BASF AG, BASF AG

Duromers (gels)

Use the right components: Avoid supercritical drying

Thermoplastics

Use the right nucleating agents/processing

NanofoamsNanofoams@BASF@BASF

Page 44: Introduction to Plastic Foams - Trinity College, Dublinfoams/PRESENTATIONS/Friday13/Friday13_Lecture… · Introduction to Plastic Foams André Moreira Polymer Physics, BASF AG Ludwigshafen,

Dr. AndrDr. Andréé MoreiraMoreira, BASF AG, BASF AG

"eierlegende Wollmilchsau“Egg-laying, wool- and milk-producing pig

Plastic foams arePlastic foams are......