mass transfer modeling in gas barrier envelopes for vacuum insulation panels- a review

R

MA

Ma

b

c

a

ARA

KVMPGM

C

0h

Energy and Buildings 55 (2012) 903–920

Contents lists available at SciVerse ScienceDirect

Energy and Buildings

j ourna l ho me p age: www.elsev ier .com/ locate /enbui ld

eview

ass transfer modeling in gas barrier envelopes for vacuum insulation panels: review

athias Bouquerela,b,∗, Thierry Duforestela, Dominique Baillis c, Gilles Rusaouenb

EDF R&D, Dpt EnerBAT, Av. des Renardières, Ecuelles, 77818 Moret-sur-Loing Cedex, FranceCETHIL (Centre de Thermique de Lyon), UMR 5008, INSA de Lyon, Bât. Sadi CARNOT, 9 rue de la physique, 69621 Villeurbanne Cedex, FranceLaMCoS (Laboratoire de Mécanique des Contacts et des Structures), UMR 5259, INSA de Lyon, Bât. Jean d’Alembert, 18-20 rue des Sciences, 69621 Villeurbanne Cedex, France

r t i c l e i n f o

rticle history:eceived 4 July 2012ccepted 4 September 2012

eywords:acuum insulation panel (VIP)ass transfer modeling

a b s t r a c t

A vacuum insulation panel (VIP) is a very efficient thermal insulation system for buildings. It is consti-tuted of an evacuated porous core material, enveloped in a gas barrier membrane. The total conductivitymeasured is as low as 5 mW/(m K). The high performance is due to the low pressure inside the panel andthe gas barrier envelope plays a key role in maintaining the vacuum during the whole VIP service life.Indeed, due to the permeation of atmospheric gases through the envelope, a slow increase of pressure andhumidity occurs over time, which involves a thermal conductivity increase in the mean time. This review

ermeabilityas barrier envelopeultilayer membrane with metalized films

paper details the mass transfer models used to predict the permeation rates of gases through the VIPenvelope. The sorption–diffusion model for gas permeation through polymer membranes is presentedas well as alternative permeation models. The parameters which play a key role for mass transfer aredetailed. The adaptation of the permeation models from homogeneous polymer membranes to multilayergas barrier membranes is then presented, including an important section about metal-coated polymerfilms. The conclusions of the works based on several approaches are reported.

© 2012 Elsevier B.V. All rights reserved.

ontents

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9041.1. Vacuum insulation panels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9041.2. Membranes used as gas barrier envelope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 905

2. Mass transfer modeling: the linear sorption–diffusion model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9052.1. Introduction to the theory of gas permeation through membranes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9052.2. Linear sorption–diffusion model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 906

2.2.1. Fick’s law of diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9062.2.2. Henry’s law of sorption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9062.2.3. Spatial integration and permeability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 906

2.3. Thermodynamics justification of the sorption–diffusion model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9072.3.1. Chemical potential as a driving force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9072.3.2. Interface gas/membrane and sorption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9072.3.3. Total mass transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 908

2.4. Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9082.4.1. Diffusion and solubility coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9082.4.2. Permeation measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 908

2.5. Synthesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9083. Alternative mass transfer modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 908

3.1. Dual mode model in glassy polymers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.1.1. Dual mode sorption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.1.2. Diffusion coefficient and permeability . . . . . . . . . . . . . . . . . . .

∗ Corresponding author. Tel.: +33 1 60 73 69 57; fax: +33 1 60 73 65 39.E-mail address: [email protected] (M. Bouquerel).

378-7788/$ – see front matter © 2012 Elsevier B.V. All rights reserved.ttp://dx.doi.org/10.1016/j.enbuild.2012.09.004

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 908 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 908. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 909

dx.doi.org/10.1016/j.enbuild.2012.09.004

http://www.sciencedirect.com/science/journal/03787788

http://www.elsevier.com/locate/enbuild

mailto:[email protected]

dx.doi.org/10.1016/j.enbuild.2012.09.004

904 M. Bouquerel et al. / Energy and Buildings 55 (2012) 903–920

3.2. Brief review on other modeling concepts for mass transfer in polymers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9093.2.1. Flory–Huggins mode sorption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9093.2.2. BET mode sorption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9103.2.3. Gas–polymer matrix model in glassy polymers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9103.2.4. Site distribution model for sorption in glassy polymers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9103.2.5. Molecular models of transport in rubbery polymers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9103.2.6. Free Volume theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 910

3.3. Synthesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9104. Parameter’s influence on transport coefficients for polymers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 910

4.1. Influence of the temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9104.1.1. Arrhenius law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9104.1.2. Experimental results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 910

4.2. Influence of the pressure and the concentration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9114.2.1. Diffusion coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9114.2.2. Solubility coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9114.2.3. Experimental results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 911

4.3. Influence of the crystallinity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9114.4. Influence of the orientation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9114.5. Synthesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 911

5. From polymer permeability to multilayer gas barrier membrane permeability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9115.1. Multilayer gas barrier membrane morphology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9115.2. Analytical modeling for mass transfer through the barrier membrane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 912

5.2.1. Permeation properties of the whole barrier membrane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9125.2.2. Ideal laminate theory for a multilayer membrane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 913

5.3. Apparent permeability of a coated polymer film . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9135.3.1. Transport mechanisms in a coated polymer film . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9135.3.2. Influence of the coating layer and polymer substrate thicknesses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9135.3.3. Roberts model for the permeability of a coated film . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9145.3.4. Analytical modeling of the micro-defects permeability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9155.3.5. Musgrave model for the permeability of a coated film . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9155.3.6. Thorsell model for the permeability of a coated film . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9165.3.7. Garnier model for the permeability of a coated film . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9165.3.8. Influence of temperature and transport mechanisms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 916

5.4. Synthesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9176. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 917

Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 918References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 918

Nomenclature

Indexes/exponentsalu aluminumamor amorphousapp apparentcoat coatingcop center of panelcrit criticalD Henry’s modedef defectdry dryelem elementaryf free volumeg gasglass glass transitionH Langmuir’s modei gas iin insideILT ideal laminate theorylin linearmemb membranemin minimumout outsidepoly polymer substrate

surf surfacetot total

Symbols Naito equation parameter

� enthalpic interaction parameter� layer thickness� activity coefficient� gas–polymer matrix model parameter� thermal conductivity� chemical potential˘ lin linear permeance˘surf surface permeance� thermal bridge linear coefficient tortuosity parameter time lag� volume fraction� chain immobilization factorA areaa chemical activityATR air transmission rateB water conductivity coefficientb Langmuir affinity coefficientBIF barrier improvement factor

rad radiativesat saturationsol solid

c mass concentrationD diffusion coefficientEa activation energy

and Buildings 55 (2012) 903–920 905

1

1

daitiiwArgarc

tsah�TnneTEo

�

Fig. 1. Constitution of a vacuum insulation panel [3].

humidity increases over time. These increases are hardly measur-able on a short time scale (few months) but of great importanceover the total service life of the panel. Indeed, the pressure and

M. Bouquerel et al. / Energy

f free volume fractiong mass flow rateGTR gas transmission rateGTRlin linear gas transmission rateGTRsurf surface gas transmission rateHs sorption enthalpyK ad hoc coefficientkD Henry’s mode solubilityl membrane thicknessM molar massm massn0 pinhole densityP perimeterp pressurePesurf surface permeabilityR ideal gas constantr0 pinhole radiusRth thermal resistanceS solubility coefficientT temperaturet timeu water contentV volumeVm molar volumeWVTR water vapor transmission rateX molar fractionx coordinate in the orthogonal direction

. Introduction

.1. Vacuum insulation panels

Super insulation materials are materials with a thermal con-uctivity below the conductivity of standing still air (25 mW/(m K)t standard conditions of temperature and pressure). They arentended to enable the design of very thin thermal insulation sys-ems for buildings, as efficient as systems based on conventionalnsulation materials, but with a much lower thickness. A vacuumnsulation panel (VIP) is one kind of super insulation material,

hich was the topic of the Annex 39 of the International Energygency [1,2]. It is constituted of an evacuated open porous mate-ial, most of the time a nanoporous silica, wrapped into a multilayeras barrier envelope (see Fig. 1). An apparent thermal conductivitys low as 5 mW/(m K) has been measured [1]. For the same thermalesistance, the thickness can be six to ten times smaller than foronventional insulation materials (see Fig. 2).

Heat transfer studies have been conducted on VIPs [1,3–6],he total heat transfer through a vacuum insulation panel is theum of four contributions and each contribution is associated ton elementary thermal conductivity �elem (see Eq. (1)): radiativeeat transfer �rad, solid conduction through the porous matrixsol, gaseous heat transfer �g, and envelope thermal bridge �memb.he gaseous heat transfer appears to be a function of the inter-al pressure (see Eq. (2): ε is the core material porosity, �0

g is theon-confined gas conductivity at atmospheric pressure, p1/2 is anmpirical pressure that decreases the gaseous conductivity by 50%).he total conductivity is also a function of the water content u (see
q. (3), B is an empirical coefficient), whose major part is adsorbedn the silica matrix when nanoporous silica is used as core material.
tot(p, u) = �rad + �sol(u) + �g(p) + �memb (1)

Fig. 2. Thermal performance of insulation materials: thermal conductivity and insu-lation thickness for a thermal resistance Rth = 5 m2 K/W.

�g(p) = ε�0

g

1 + (p1/2/p)(2)

�sol(u) = �sol,dry + Bu (3)

The initial low pressure and low humidity inside a VIP areresponsible for the very low thermal conductivity. However dry airand water vapor can migrate through the envelope, which is notan absolute barrier to atmospheric gases (see Fig. 3). Mass trans-fer is very low but never null, through the panel faces (membrane)and through the edges (heat-sealed joint) [1]. These mass trans-fers – dry air permeation, quantified by the air transmission rate(ATR), and water vapor permeation, quantified by the water vaportransmission rate (WVTR) – are responsible for the pressure and

Fig. 3. Gas transfer in a VIP.

9 and Buildings 55 (2012) 903–920

ht

tfhhm

emtmasdmfielwcmtoert

1

vc

nPhmmp

malhdabl

v

Table 1Comparison of thermal bridges due to MF and AF membranes for a 1 m × 1 m VIP.

Type � memb (mW/(m K)) �memb (mW/(m K)) �memb/�cop (%)

MFmin 1.3 0.1 2.5max 24 2.1 38

min 41 3.3 66

06 M. Bouquerel et al. / Energy

umidity increases are responsible for the total thermal conduc-ivity increase, which is the main mechanism of VIP aging [7–9].

This review paper is focused on mass transfer modeling throughhe gas barrier envelope, at the scale of the membrane only, as a sur-ace phenomenon. The issues of the permeation rates through theeat-sealed joint and at the scale of a whole VIP are not addressedere, they are in the scope of another review paper about agingodeling in VIPs [10].This paper details the background theory of mass transfer mod-

ling through polymer membranes and the application of theseass transfer models for multilayer barrier envelopes in VIPs. Mass

ransfer modeling can be used to predict the atmospheric gases per-eation through the envelope and thus the evolution of pressure

nd humidity inside the panel. The usual approach is based on aorption–diffusion model, a linear mass transfer model originallyeveloped to evaluate permeation through homogeneous polymerembranes [11]. The permeance to a particular gas, which quanti-

es the gas transmission rate, is a crucial property for a gas barriernvelope. Several models have been proposed, in order to calcu-ate the permeances to atmospheric gases of multilayer membranes

ith metalized polymer films. They may help the design of very effi-ient barrier envelopes. The influence of external parameters on theembrane permeances, mainly the climatic conditions (tempera-

ure and humidity), is also studied: depending on the climate andn the building implementation, the climatic conditions experi-nced by several VIPs can be very different and thus the permeationates would vary a lot if the permeance is highly dependent on theemperature and humidity.

.2. Membranes used as gas barrier envelope

Two types of gas barrier membrane can satisfy the air and waterapor tightness needed to ensure a long term performance [1]. Theonstitution of each type is illustrated in Fig. 4.

AF membranes are made with a laminated aluminum foil (thick-ess 5–10 �m). A cover layer is added on the outside (≈50 �m ofET) and a weld layer on the inside (≈50 �m of PE). AF membranesave very high gas barrier properties thanks to the continuousetal barrier. However, the thermal bridge produced by an AFembrane is very high and decreases dramatically the VIP thermal

erformance [12–14].MF membranes are multilayer membranes made with several

etal-coated polymer films (≈12 �m of PET with 20–100 nm ofluminum) used as gas barrier layers. A cover layer and a weldayer are also added. Unfortunately, the metallic layers contain aigh density of defects (mainly pinholes) [15–17]. Because of thisefect distribution, barrier performance to atmospheric gases is nots good as laminated aluminum foils [17]. This effect is attenuatedy the superposition of several metalized layers, so that the path

ength and tortuosity are increased for the permeant molecules.The polymers (PE, PET, PP, etc.) and thicknesses of the layers may

ary from one product to another. In both laminated aluminum foils

Fig. 4. Ultra gas barrier membranes: AF and MF design.

AF max 105 8.4 210

Data from [12–14,18–20].

and metalized polymer membranes, polyurethane layers are usedto glue the layers [15,16].

Aluminum has a thermal conductivity �alu ≈ 200 W/(m K) fiveorders of magnitude higher than the evacuated core materialconductivity �cop ≈ 0.004 W/(m K). The conventional polymer con-ductivity (PE, PET, PP, etc.) �poly ≈ 0.1–0.5 W/(m K) is only twoorders of magnitude above �cop. This difference explains why MFmembranes have a better thermal performance: the aluminumlayer thickness is much lower than for AF membranes. The com-parison of thermal performance for several AF and MF membranesis reported in Table 1. These values have been obtained by numeri-cal simulations [13,18] and measurements [12,14,19,20]. � memb isthe thermal bridge linear coefficient, �memb the elementary thermalconductivity due to the membrane thermal bridge, calculated asthe difference of apparent thermal conductivity between a squaremeter VIP with its gas barrier envelope and a without this envelope.

2. Mass transfer modeling: the linear sorption–diffusionmodel

2.1. Introduction to the theory of gas permeation throughmembranes

Laminated aluminum foils (AFs) and multilayer membraneswith metalized films (MFs) are the main gas barrier membranesable to satisfy the thermal and gas barrier requirements [1]. MFmembranes have better thermal properties, but the dispersion oftheir gas barrier performances is much higher, between low andhigh quality membranes [21]. The usual way to evaluate the gasbarrier performance is to use the permeability or permeance of themembrane [21]. It is based on the sorption–diffusion model, a lin-ear mass transfer model established for polymer membranes, inparticular rubbery polymers [11,22].

This model is applied to non-porous and homogeneous polymermembranes, at a given temperature. The transport mechanism isdivided in three main steps (see Fig. 5): condensation and sorp-tion at one surface of the membrane (Sorption 1), diffusion of thedissolved permeant, driven by a concentration gradient, and thenevaporation at the other surface (Sorption 2).

It is common to use volumetric units for gases: amount of gasin cm3(STP), concentrations in cm3(STP)/cm3, GTR (gas transmis-sion rate, i.e. mass flow rate) in cm3(STP)/(m2 s). 1 cm3(STP) is theamount of gas that has a volume of 1 cm3 in standard conditions oftemperature and pressure (IUPAC definition of STP: 273.15 K and1 bar [23]). For water vapor, however, mass units are often used(respectively kg, kg/cm3, and kg/(m2 s)). This unit’s change is asource of complexity and the use of STP volumes is only arbitrary.Moreover, about fifteen definitions of Standard conditions can befound in several publishing or establishing entities [24].

In order to increase the consistency of mass transfer theory andexperimental data, it might be smart to use the same unit for allgases, as it has already been pointed out in the literature [25]. Mass
units are somehow more “objective” units. Equations in this sectionare all written with mass units, nevertheless properties found in theliterature may be written in volume units as it is the most conve-nient way to rewrite them here. In Table 2, some conversions are

M. Bouquerel et al. / Energy and Buildings 55 (2012) 903–920 907

Table 2Conversion between volume and mass units for dry gases.

N2 O2 Dry air

Volume cm3(STP) 1 1 1Mass kg 1.23 × 10−6 1.41 × 10−6 1.28 × 10−6

Flow rate (GTR)

cm3(STP)/(cm2 d) 1 1 1cm3(STP)/(m2 s) 0.116 0.116 0.116g/(cm2 d) 1.23 × 10−3 1.41 × 10−3 1.28 × 10−3

kg/(m2 s) 1.43 × 10−7 1.68 × 10−7 1.48 × 10−7

cm3(STP)/(cm2 d bar) 1 1 13 2 −6 −6 −6

pSu

2

2

t

g

srtth

flc

g

tar

Fb

Permeabilitycm (STP)/(m s Pa)g/(cm2 d bar)

kg/(m2 s Pa)

roposed between the most commonly used “technical” units andI units. The amount of experimental data expressed in technicalnits prevents to get rid of them in the technical literature.

.2. Linear sorption–diffusion model

.2.1. Fick’s law of diffusionThe mass flow rate gi is defined as the amount of permeant dmi

hat crosses a surface of area dA during one unit of time dt:

i = dmi

dA dt(4)

It is defined at a microscopic scale. At a macroscopic scale, theurface gas transmission rate (GTRsurf,i) is defined as the mass flowate that is transferred through the whole membrane, from one sideo the other one, per surface and time unit (see Fig. 5). It is equalo the mass flow rate gi in steady state, when gi is constant andomogeneous through the membrane thickness.

The Fick’s first law sets the linear relation between the massow rate of a molecule diffusing through a membrane and theoncentration gradient of this molecule in the membrane [26]:

i = −Di∇ci (5)

Di is called the diffusion coefficient, ci is the molecule concen-
ration in the membrane, in kg/m3. This is a steady state equation,pplicable for a constant concentration distribution and mass flowate. For a unidirectional case, when the diffusion is only important
ig. 5. Sorption–diffusion mechanism for mass transfer through a polymer mem-rane.

1.16 × 10 1.16 × 10 1.16 × 101.23 × 10−3 1.41 × 10−3 1.28 × 10−3

1.43 × 10−12 1.68 × 10−12 1.48 × 10−12

in the x-direction (orthogonal to the membrane plane), the equa-tion is simplified:

gi = −Di∂ci

∂x(6)

This unidirectional equation is valid for membranes where thethickness is much smaller than the other dimensions. That is thecase for thin VIP barrier membranes. For transient state, Fick’s sec-ond law of diffusion is applicable, where the mass fluxes and theconcentrations are function of time and position (written here inunidirectional case):

∂ci(x, t)∂t

= −∂gi(x, t)∂x

= ∂(Di∇ci)∂x

= ∂Di

∂x

∂ci

∂x+ Di(ci)

∂2ci

∂x2(7)

For several polymer–permeant systems, Di can be considered asa constant in the whole membrane, so that the equation is simpli-fied:

∂ci

∂t= Di

∂2ci

∂x2(8)

This equation is formally equivalent to Fourier heat equationand thus has the same kind of solutions based on Fourier series andFourier transform [27].

2.2.2. Henry’s law of sorptionThe boundary conditions of the diffusion problem depend on the

sorption at the polymer/gas interface. The sorption is a general termfor the dissolution of the transported molecule in the polymer. Itincludes absorption, adsorption, and trapping in micro-voids [22].The simplest law is Henry’s law, which considers a linear relationat fixed temperature between the dissolved molecule concentra-tion at the interface and the partial pressure of the molecule in thegaseous phase:

ci = Sipi (9)

Si is called the solubility coefficient and is a constant at giventemperature if Henry’s law is respected. Henry’s law is more likelyto be respected for ideal gases and low pressure [22].

2.2.3. Spatial integration and permeabilityFor an homogeneous and non-porous membrane at given

temperature, with the assumption of an homogeneous diffusioncoefficient and at steady state, the steady state Fick’s law (Eq. (6))can be integrated on the whole membrane thickness l, between theinterfaces 1 and 2. One gets an homogeneous mass flow rate which
is thus the gas transmission rate:
GTRsurf ,i = gi = −Dici,2 − ci,1

l(10)

9 and Bu

is

G

P

p

G

iap

G

a

˘

2m

tBe

2

faa

Hp

g

w

d

Xta

fw

�

a

�

He

g


From Henry’s law (Eq. (9)), the permeant concentration at thenterfaces is a function of the partial pressure. Assuming that theolubility coefficient is the same on both sides:

TRsurf ,i = −DiSi(pi,2 − pi,1)

l(11)

It is then possible to introduce the surface permeability Pesurf,i:

esurf ,i = DiSi (12)

So that the GTR is only a function of the permeability and theartial pressure gradient over the membrane:

TRsurf ,i = −Pesurf ,i

l�pi (13)

For an heterogeneous material, the permeability has no mean-ng. It is however possible to define the surface permeance ˘surf,i,

global membrane property, so that the GTR is a function of theermeance and the partial pressure difference over the membrane:

TRsurf ,i = −˘surf ,i �pi (14)

For an homogeneous membrane, the relation between perme-nce, permeability, diffusion and solubility coefficients is written:

surf ,i = Pesurf ,i

l= DiSi

l(15)

.3. Thermodynamics justification of the sorption–diffusionodel

In this section, the sorption–diffusion model is obtained thankso thermodynamics equations, from the review by Wijmans andaker [11]. At each step, the hypotheses used to progress are put inmphasis.

.3.1. Chemical potential as a driving forceThe demonstration starts with the proposition that the driving

orces (pressure, temperature, concentration, electromotive forces)re interrelated and that the overall driving force can be expresseds a function of the chemical potential �i of the permeant species.

ypothesis 1. The mass flow rate gi is proportional to the chemicalotential gradient.

i = −Kid�i

dx(16)

For pressure and concentration gradients only, it is possible torite:

�i = RT d ln(�iXi) + Vm dp (17)

i is the molar fraction (mol/mol), � i the activity coefficient so thathe chemical activity ai is written ai = � iXi, p is the total pressurend Vm the molar volume.

For a compressible gas, the saturation pressure pi,sat can be takenor the reference potential. In the liquid phase, the potential can beritten:

i = �0i + RT ln(�iXi) + Vm(p − pi,sat) (18)

nd in the gaseous phase:

i = �0i + RT ln(�iXi) + RT ln

p

pi,sat(19)

ypothesis 2. The pressure in the membrane is homogeneous,qual to the high pressure side.

Then Eqs. (16) and (18) can be combined:

i = −RTKi

�iXi

d(�iXi)dx

(20)

ildings 55 (2012) 903–920

The molar fraction Xi and the concentration ci (kg/m3) arerelated thanks to the following equation:

Xi = ci

MiVmemb

m (21)

where Vmembm is the average molar volume of the

membrane–permeant system and Mi the molar mass of thepermeant i. Then:

gi = − RTKi

�ici(Vmembm /Mi)

d((�ici)(Vmembm /Mi))

dx(22)

Hypothesis 3. The activity coefficient and molar volume arehomogeneous in the membrane

Mi is independent of x, so are � i and Vmembm from Hypothesis 3,

they vanish in Eq. (22).

gi = −RTKi

ci

dci

dx(23)

It should be noted that this equation is formally identical toFick’s law (Eq. (6)), the diffusion coefficient is:

Di = RTKi

ci(24)

Hypothesis 4. The membrane is isothermal, the concentrationsvary slightly.

It is possible to integrate Eq. (23) and to get the expression ofthe GTR as a function of the permeant concentrations ci,1 and ci,2 atinterfaces 1 and 2:

gi = Dici,1 − ci,2

l(25)

2.3.2. Interface gas/membrane and sorptionThe thermodynamics equilibrium at the “liquid”/gas interface is

written:

�0i + RT ln(�i,membXi,memb) + Vm(p − pi,sat)

= �0i + RT ln(�i,gXi,g) + RT ln

p

pi,sat(26)

It is possible to rewrite this equation:

Xi,memb = �i,g

�i,memb

p

pi,satXi,g exp

(−Vm(p − pi,sat)

RT

)(27)

The expression of molar fractions is:

Xi,memb = ci

MiVmemb

m (28)

Xi,g = pi

p(29)

We get:

ci = Mi

Vmembm

�i,g

�i,memb

pi

pi,satexp

(−Vm(p − pi,sat)

RT

)(30)

It is reasonable to assume that exp((− Vm(p − pi,sat))/RT) ≈ 1 inmost cases. Indeed, the deviation from this assumption is below 1%for �p < 10 bar and below 10% for �p < 100 bar.

Hypothesis 5. �p is small, so that exp((− Vm(p − pi,sat))/RT) = 1.

The equilibrium is then written:

ci = Mi

Vmembm

�i,g

�i,memb

pi

pi,sat(31)

and Bu

wic

S

2

vaa

H

G

wE

D

S

2

2

iebett

D

iosig

2

tmv(TpaaHdtp

2

mts


here pi = p × Xi,g is the partial pressure in the gaseous phase. Its formally equivalent to Henry’s law for sorption, the solubilityoefficient is written:

i = Mi

Vmembm

�i,g

�i,membpi,sat(32)

.3.3. Total mass transferFrom Hypothesis 3, activity coefficients and membrane molar

olume are homogeneous. The solubility coefficients on both sidesre equal as soon as the activity coefficients in both gaseous phasesre equal.

ypothesis 6. � i,g is equal on both sides of the membrane.

Combining Eqs. (25) and (31), one gets:

TRsurf ,i = gi = DiSi

l�pi (33)

hich is the global sorption–diffusion transport model, identical toq. (13), where:

i = RTKi

ci(34)

i = Mi

Vmembm

�i,g

�i,membpi,sat(35)

.4. Measurements

.4.1. Diffusion and solubility coefficientsThe value of Di can be measured by the “time lag” method. It

s based on the transient state, at the beginning of a permeationxperiment. The quantity of permeants that has crossed the mem-rane has to be plotted as a function of time. The interception of thextrapolated linear steady state line (asymptotic curve) with theime axis is the time lag . If Di is independent of the concentration,ime lag and diffusion are linked through [22]:

i = l2

6(36)

Volumetric or gravimetric methods are available for measur-ng the sorption isotherm – and thus the solubility coefficients –f polymers. These methods are independent of permeation mea-urements. The principle of the gravimetric method, for instance,s to measure the mass variation of a sample when the surroundingas atmosphere is modified.

.4.2. Permeation measurementsSeveral methods are available to measure the permeation rate

hrough a sample, for gases or water vapor: the manometricethod (any gas), the cup method or gravimetric method (water

apor), the electrolytic process (water vapor), the calcium testwater vapor), and the tritiated water method (water vapor) [1,28].he principle is to put a membrane sample between two gaseoushases at different pressures, the upstream side at high pressurend the downstream side at low pressure, and to measure themount of permeant that crosses the sample as a function of time.igh pressure and low pressure are assumed to remain constanturing measurements, until the steady state is reached. The gasransmission rate is calculated from the definition, Eq. (4), and theermeability from Eq. (13).

.5. Synthesis

The sorption–diffusion model has been widely used for gas per-eation through polymer membranes [11]. It assumes that the

ransmission rate of a species is proportional to its partial pres-ure gradient over the membrane. The proportionality coefficient

ildings 55 (2012) 903–920 909

is the product of a solubility coefficient, governing the equilibriumbetween the gas phase and the sorbed phase inside the mem-brane, and a diffusion coefficient, governing the transport fromthe high concentration area toward the low concentration area.The simplicity of the set of equations is one of its great advan-tage and the consistency with experimental measurements as well[11]. However, the demonstration from thermodynamics needsa lot of hypotheses, which may not be true in all cases, espe-cially hypotheses concerning the properties homogeneity in themembrane. Measurements of the solubility coefficient, diffusioncoefficient and permeability on the same sample with independentmethods should be a good way to separate diffusion and sorptionphenomena and to test the validity of the sorption–diffusion theoryfor a determined couple polymer/permeant.

3. Alternative mass transfer modeling

The sorption–diffusion was developed for rubbery polymers, butmechanisms of transport in glassy polymers differ totally becauseof the difference in polymer structure [22]. Other analytical modelsare thus available, for glassy polymers and rubbery polymers aswell.

3.1. Dual mode model in glassy polymers

3.1.1. Dual mode sorptionIn semi-crystalline polymers with a glassy part and a crystalline

part, the molecular configuration is different from the rubberystate. To explain the deviation from the linear sorption, the dualmode sorption model was developed [29].

Let us introduce the Langmuir-mode sorption. Contrary to thelinear Henry’s law of sorption (Eq. (9)), this mode is based onthe assumption that the permeant molecules occupy specific sites(micro-voids, etc.) and that a saturation may occur while all theLangmuir sorption sites are occupied [22]. The Langmuir concen-tration cH is expressed as:

cH = c′Hbpi

1 + bpi(37)

c′H is the Langmuir capacity (or “hole saturation” constant) and b

is the Langmuir affinity coefficient (or “hole affinity” constant). Atlow pressure (bpi � 1) the behavior is linear and at high pressure(bpi � 1) the saturation is reached and the concentration of thispopulation is very close to the constant value c′

H .In a semi-crystalline polymer below the glass transition tem-

perature, the dual mode sorption model postulates that twopopulations of sorbed molecules coexist in the polymer matrix. Inthe dense area, ordinary “dissolution” occurs and linear Henry’s lawis respected. In this area, Henry’s sorption population has a concen-tration written cD. The idea that micro-voids are present becauseof the glassy phase was suggested by Barrer et al. [30] and thesemicro-voids are responsible for a Langmuir sorption population, ata concentration cH. The total concentration is thus the sum of Henryand Langmuir populations:

ci = cD + cH = kDpi + c′Hbpi

1 + bpi(38)

kD is equal to the solubility coefficient Si in the case of pureHenry’s mode sorption. Otherwise, the solubility coefficient definedby Eq. (9) is equal to:

Si = kD + c′Hb

1 + bpi(39)

The solubility coefficient is thus a function of the gas pressure.


ion is

id

KT(tsfa

3

mmtm

g

fi

l[mei

whpo

g

Fig. 6. Typical sorpt

A typical Langmuir and dual mode sorption isotherm is plottedn Fig. 6, together with a Henry sorption isotherm to illustrate theifference.

One analysis of the dual-mode sorption model was conducted byanehashi and Nagai [31], based on 250 types of glassy polymers.hey tried to find correlation parameters for dual mode parameterskD, c′

H and b) on various factors (glass transition temperature, frac-ional free volume, etc.). Good correlations were observed wheneveral analyses were conducted with the focus on one polymeramily at a time. However, a wide variation appears when lookingt the whole polymer panel.

.1.2. Diffusion coefficient and permeabilityThe first mass transfer models based on the dual mode sorption

odel postulate that the “adsorbed” molecules population of Lang-uir’s mode is immobilized and that it doesn’t participate to the

otal transfer [29]. Fick’s law is thus respected only for the Henry’sode population. For a unidirectional flux:

i = −Di∂cD

∂x(40)

The second Fick’s law in the case of homogeneous diffusion coef-cient (Eq. (8)) is then written:

∂(cD + cH)∂t

= Di∂2

cD

∂x2(41)

This unsteady state equation has analytical solution for theimiting cases bpi � 1 and bpi � 1, but not for the general case29]. And for steady state, it is equivalent to the sorption–diffusion

odel, since Langmuir’s mode population vanishes from thequation, once in equilibrium with the gas partial pressure andmmobilized.

Other authors developed a model of “partial immobilization”,here Langmuir’s mode population is not totally immobilized butas a diffusion coefficient different from the one of Henry’s modeopulation [32–34]. Fick’s first law is thus written for a unidirecti-
nal flux:
i = −DD∂cD

∂x− DH

∂cH

∂x(42)

otherm curves [22].

For a steady state and homogeneous membrane, considering apressure that vanishes at the downstream side (pi,2 = 0), the massflow rate becomes after integration:

GTRsurf ,i = gi = 1l

(DDkDpi,1 + DHc′

Hbpi,1

1 + bpi,1

)(43)

Diffusion, solubility and permeability coefficients are thus writ-ten for this case:

Si = ci

pi= kD + c′

Hb

1 + bpi,1(44)

Pesurf ,i = GTRil

�pi,1= 1

l

(DDkD + DHc′

Hb

1 + bpi,1

)(45)

Di = lPesurf ,i

Si= DDkD + ((DHc′

Hb)/(1 + bpi,1))kD + ((c′

Hb)/(1 + bpi,1))(46)

All three parameters are thus dependent on the pressure. Evolu-tions of S, D and Pe as a function of the pressure have been observedexperimentally. This model would be able to explain these behav-iors [22]. However, there is no clear evidence of the existence ofboth populations of Henry’s mode and Langmuir’s mode sorption,which would have different solubility and diffusion coefficients.The nature and distribution of the micro-voids responsible for Lang-muir’s mode sorption remain unknown.

Islam and Buschatz [35] have developed a theoretical expla-nation of the Dual Mode formalism, based on a flux equationexpressed in term of chemical potential gradient. In this model,the formalism is identical but the meaning is different, since theLangmuir’s mode population is assumed to be fully immobilized.

3.2. Brief review on other modeling concepts for mass transfer inpolymers

3.2.1. Flory–Huggins mode sorptionThis sorption mode was presented by Flory [36]:

ln(ai) = lnpi

pi,sat= ln �i + (1 − �i) + �(1 − �i)

2 (47)

where �i is the volume fraction of the permeant in the polymerand � the enthalpic interaction parameter between the polymer

and Bu

aiptfs

3

LrA

3

fdmamipe

�

S

D

Tpad

S

D

P

3

hgmstt

3

oaaopd

3

la

f


nd the solute. This model is supposed to be adapted when thenteractions between diffusing molecules are stronger than theermeant–polymer interaction [22]. This may be due to plas-icization by sorbed molecules, or the association of clustersor hydrophobic polymer/water systems. A typical Flory–Hugginsorption isotherm is plotted in Fig. 6.

.2.2. BET mode sorptionA BET mode sorption corresponds to the combination of the

angmuir and Flory–Huggins modes [22]. In practice, it can be rep-esentative of the sorption of water in highly hydrophilic polymers.

typical isotherm sorption is plotted in Fig. 6.

.2.3. Gas–polymer matrix model in glassy polymersThis model is presented as an alternative to the dual mode model

or glassy polymers able to explain the dependence of solubility andiffusion coefficients to the permeant concentration [22]. In thisodel, it is assumed that there is only one population of perme-

nt, but that an interaction between the permeant and the polymeratrix exists. The effect of a concentration increase in the polymer

s an increase of the diffusion process (through an increase of theolymer mobility) and a decrease of the sorption rate. The proposedquations are:

= Tglass(0) − Tglass(ci)T

(48)

i = Si,0 exp(−K∗S �) (49)

i = Di,0(1 + K∗D�) exp(K∗

D�) (50)

glass(0) and Tglass(ci) are respectively the glass transition tem-erature of the pure polymer and of the polymer–gas systemt concentration ci. With the additional assumption of a linearependence of Tglass(ci) on ci, one gets:

i = Si,0 exp(−KSci) (51)

i = Di,0(1 + KDci) exp(KDci) (52)

If KDci is small, one gets for the permeability:

ei = Si,0Di,0 exp((KD − KS)ci) (53)

.2.4. Site distribution model for sorption in glassy polymersAn other model is based on the assumption that all sorption sites

ave different energy levels [37,38]. The distribution of site ener-ies can be chosen, the linear Henry’s law and dual mode sorptionodel are two particular cases for this model. The variation of the

olubility coefficient as a function of pressure is well fitted thankso this model, with a lower amount of fitting parameters than forhe dual mode model.

.2.5. Molecular models of transport in rubbery polymersThese models are based on the assumption that a certain amount

f energy is necessary for a molecule to jump from one position tonother [22]. It is consistent with the observations of an energy-ctivated behavior. The estimation of the energy level differs fromne model to another, it is basically related to polymer chain dis-lacements and interaction forces which are involved in molecularisplacement.

.2.6. Free Volume theoryFree volume models are based on the assumption that molecu-

ar transport is due to free volume redistribution and not thermal
ctivation [22]. The free volume fraction f is defined as:
= Vf

Vtot= Vf

Vtot(54)

ildings 55 (2012) 903–920 911

Correlations between f and S or D, able to explain the depend-ence of D to some parameters like concentration, permeant shapeand size, temperature and glass transition temperature, are pro-posed. This theory seems to be well fitted to rubbery polymers/largeorganic vapors systems, but not for small molecules (the depend-ence on concentration is much lower). It is however not predictingan Arrhenius temperature dependence, which is a source of con-troversy [22].

3.3. Synthesis

Several alternative models have been developed, especially forsorption, in order to explain experimental behaviors different fromwhat is predicted by the linear sorption–diffusion model. Thesemodels mainly apply for glassy and semi-crystalline polymers,where it is assumed that the crystalline part and the amorphousone have different properties. For these non-linear behaviors, sev-eral physical explanations coexist, which may be relevant or notdepending on the polymer and permeant species that are studied.This section is just a brief review on these alternative models, theliterature about theory and experimental results is too extensive tobe exhaustively presented here.

4. Parameter’s influence on transport coefficients forpolymers

4.1. Influence of the temperature

4.1.1. Arrhenius lawThe diffusion in polymers is a thermally activated process.

Experimental data suggest that Arrhenius law is an appropriateequation for the thermo-activation, at a given pressure and on anarrow range of temperature [22]:

Di = D0i exp

(−ED

a

RT

)(55)

Si = S0i exp

(−�HS

RT

)(56)

Pei = Pe0i exp

(−EP

a

RT

)(57)

˘i = ˘0i exp

(−EP

a

RT

)(58)

From Eq. (15), it comes:

EPa = ED

a + �HS (59)

For diffusion, the activation energy EDa corresponds to the energy

level necessary to jump from one position to another, it is thusalways positive [22]. The heat of sorption �HS may be positive ornegative. It is generally positive for gases far from their critical tem-perature (H2, N2, O2, etc.) and negative for condensable gases andvapors (CO2, SO2, H2O, etc.). Diffusion has a greater dependence totemperature than sorption, the permeability energy activation ED

ais thus positive.

4.1.2. Experimental resultsFlaconnèche et al. [39] measured the diffusion coefficient, sol-

ubility coefficient and permeability to He, Ar, N2, CH4 and CO2of various polyethylenes (LDPE, MDPE, HDPE, HPDE-RI). Measure-
ments were made from 40 ◦C to 80 ◦C, at a pressure between 4 and10 MPa for He and CH4, and at constant pressure of 10 MPa for othergases. Energy activations are positive for permeabilities, between29 and 51 kJ/mol, for diffusion as well, between 17 and 46 kJ/mol,

9 and Bu

aa

iaPt(lbaIaitepIapl

wfiCgtCtp

4

4

o

•

•

ta

D

Dtt

4

tc

4

1mcpb


nd for sorption between 3 and 20 kJ/mol, except CO2 that has anctivation energy between −2 and −1 kJ/mol.

Gajdos et al. [40] also measured the diffusion coefficient, solubil-ty coefficient and permeability to O2 and N2 of PE and PP (normalnd co-extruded) for three temperatures: 20 ◦C, 40 ◦C and 60 ◦C. ForE, on both thicknesses (0.03 mm and 0.1 mm), a very good correla-ion for Arrhenius thermo-activation is observed, for all coefficientscorrelation coefficient between 0.90 and 0.99). For PP, Arrheniusaw is respected in some cases, but not for all coefficients andoth gases. The diffusion coefficient, in particular, has a temper-ture dependence that deviates from Arrhenius thermo-activation.n these measurements, all activation energies for permeabilityre positive (between 57 and 187 kJ/mol), the permeability is anncreasing function of the temperature. But all enthalpies of sorp-ion are positive (between 57 and 197 kJ/mol) and all activationnergies for diffusion are negative except one (27 kJ/mol for theositive value of O2 in PP, between −4 and −44 kJ/mol for others).

t is in contradiction with the theoretical influence of the temper-ture on the diffusion coefficient. An inversion between D and S isossible, or a bias in the estimation of D, which is based on the time

ag method.Other measurements on a polyimide membrane (6FDA-durene)

ere made by Lin et al. [41], Lin and Chung [42]: the diffusion coef-cient, solubility coefficient and permeability to He, H2, O2, N2,H4 and CO2, from 30 ◦C to 49 ◦C and at 1 MPa. Activation ener-ies are positive for the permeability of O2, N2 and CH4 (from 2.4o 7.25 kJ/mol), and for the diffusion coefficient of O2, N2, CH4 andO2 (from 17.55 to 24.29 kJ/mol). The heats of sorption are nega-ive for O2, N2, CH4 and CO2 (from −19.84 to −15.14 kJ/mol). Theermeability energy activation of CO2 is almost null (0.24 kJ/mol).

.2. Influence of the pressure and the concentration

.2.1. Diffusion coefficientThe effect of high pressure at the upstream side may lead to two

pposite effects [22]:

An increase of the polymer density, that may result in a freevolume reduction and then a transfer velocity decrease.An increase of the permeant concentration, that may result inplasticizing of macromolecular chains and thus an increase of freevolume and transfer velocity.

Below 10 MPa, Naito et al. [43–45] proposed the following equa-ion for rubbery polymers (PE, PP, poly(ethylene-co-vinyl-acetate)nd polybutadienes):

(c, p) = D(0, 0) exp(˛1p + ˛2c) (60)

(0, 0) is the diffusion coefficient at c = 0 and p = 0, exp(˛1p) is theerm for pressure effect (˛1 is a negative term) and exp(˛2c) is theerm for concentration effect.

.2.2. Solubility coefficientAt pressure lower than 10 MPa, the solubility coefficient appears

o be independent on gas pressure and concentration for slightlyondensable gases [22].

.2.3. Experimental resultsOn the measurements of Flaconnèche et al. [39], between 4 and

0 MPa, no influence of total pressure was observed for the per-
eability of PE to He and CH4. Other measurements lead to the
onclusion that CH4 and CO2 solubilities in PE are independent ofressure below 4 MPa [46], as well as CH4 and N2 solubilities in PEelow 15 MPa [47].

ildings 55 (2012) 903–920

4.3. Influence of the crystallinity

Most attempts to take into account the crystallinity are basedon the two-phase model [22,48]. The assumption is that in a semi-crystalline polymer, sorption and diffusion only happen in theamorphous part. Crystalline areas are impermeable barriers fordiffusion and excluded volumes for sorption. The following expres-sions are proposed:

Si = S∗�amor (61)

Di = D∗

�(62)

where S* and D* are the coefficients in an amorphous completelyrelaxed state (degree of crystallinity = 0%), �amor the amorphousphase volume fraction, � the chain immobilization factor (the crys-talline zones might decrease the mobility of polymer chains in theamorphous phase) and the tortuosity factor. It is assumed thatthe nature of the amorphous phase remains the same despite theexistence of crystalline areas. Thus:

S∗ = Samor (63)

D∗ = Damor (64)

From Eq. (61) it appears that a high crystallinity rate has a pos-itive impact on the barrier properties, as it has a linear and directimpact on the solubility coefficient and thus the permeability.

4.4. Influence of the orientation

The influence of the amorphous chains morphology wasobserved, showing that oriented polymers may have differentproperties [22]. The interpretation is based on a free volume frac-tion modification due to the polymer chains reorganization, leadingto an increase or a decrease of transport properties.

4.5. Synthesis

Temperature appears to be the main parameter which influ-ences the transport properties of polymer membranes. Thedependence is close to an Arrhenius activation in most cases. Theinfluence of the pressure and concentration has not been observedon experimental measurements below 1 MPa, which includes intheory the atmospheric pressure (outside VIPs) as well as any vac-uum level (inside VIPs). The crystallinity rate and orientation arealso playing a role in the permeation properties, since they have agreat influence on the microstructure – thus on the sorption sitesshape and density – and on the diffusion properties. In the litera-ture, the influence of the total pressure is almost never reported,the permeant partial pressure on both sides of the polymer mem-brane is the only physical input. The variation of total pressure andthus on the permeant concentration in the gas phases, from onemeasurement to another, should be taken into account in order toensure that this total pressure does not play a role in the permeationproperties.

5. From polymer permeability to multilayer gas barriermembrane permeability

5.1. Multilayer gas barrier membrane morphology

When a moderate gas barrier performance is needed, i.e. therequirements for air or water vapor tightness are not too high and
the service life is not longer than few months, simple polymers canbe used as gas barrier membranes. One of the main applicationsfor barrier polymers is food and beverage packaging [49,50]. Thepermeability of cheap polymers are listed in Table 3.


Table 3Transfer properties of food packaging polymers.

Ref. Sample Temperature (K) Gas S (cm3(STP)/(cm3 Pa)) D (cm2/s) Pe (cm3(STP)/(cm s Pa))

[48]PET 20% cryst. 298 H2O 4 × 10−3 4 × 10−9 1.6 × 10−11

PET amorphous298 N2 5 × 10−7 2 × 10−9 1 × 10−15

298 O2 9 × 10−7 5 × 10−9 4.5 × 10−15

[39]LDPE 30% cryst. 343 N2 2.6 × 10−7 2.0 × 10−6 5.1 × 10−13

HDPE 63% cryst. 342 N2 9.0 × 10−8 1.2 × 10−6 1.0 × 10−13

PE 0.1 mm333 N2 1 × 10−4 6 × 10−8 5 × 10−7

−2 −8 −5

tdltcb

cmthmtmeobbi

ifimcebtdpbrpmf

[40]333 O2

PP 0.03 mm333 N2

333 O2

Simple polymers are really too permeable for VIP. The massransfer has to be as low as possible to get a service life of severalecades and it is only reachable if the barrier properties of the enve-

ope are increased. Multilayer membranes are designed to improvehe gas barrier properties of simple polymer films. The idea is toombine several material properties to get a membrane that hasetter properties than each individual film alone.

As presented in Section 1, two types of multilayer membranesan be used for VIP envelopes (see Fig. 4): laminated aluminumembranes (AF type) and metalized polymer membranes (MF

ype) [1]. Unfortunately, even if laminated aluminum membranesave good gas barrier properties, they are responsible for a ther-al bridge that drastically decreases the thermal performance of

he whole panel (see Section 1 and Ref. [6]). That is why multilayerembranes with metalized polymer films are developed for VIP

nvelopes. They are based on polymer films coated with a thin layerf metal, often aluminum. The low amount of metal in the mem-rane decreases the heat transfer through the membrane thermalridge. An example of such a gas barrier membrane is represented

n Fig. 7.Coating polymer films with a thin metallic or inorganic layer

s a very efficient way to improve the barrier properties of thelm, without losing the good mechanical properties of the poly-er. Barrier properties can be improve by a factor of 100, for a

oating thickness less than 1% of the polymer thickness [50–53]. Forxample, oxygen and water vapor permeabilities were decreasedy respectively 99% and 98.5% on a metalized PET film comparedo a non-metalized PET film [50] and oxygen permeability wasecreased by 99.2% on a SiOx-coated PET [53]. This kind of coatedolymers was extensively developed as oxygen and water vaporarrier envelope for food and medical packaging [51] and more
ecently for electronic devices like solar cells or flat panel dis-lays [54]. Mass transfer modeling is thus focused on multilayerembranes with metalized polymer films, as they have better per-
ormances for vacuum insulation panels.

Fig. 7. Example of a multilayer membran

1 × 10 5 × 10 5 × 108 × 10−5 6 × 10−9 1 × 10−7

7 × 10−5 9 × 10−9 1 × 10−7

5.2. Analytical modeling for mass transfer through the barriermembrane

5.2.1. Permeation properties of the whole barrier membraneMost laboratories working on gas barrier membranes of VIPs

use the model of sorption–diffusion detailed in Section 2 for watervapor and dry air transport [1,8,9,17,21,28,55–58]. It is indeed avery easy model for property identification and for mass transfersimulation.

The assumptions commonly used are listed below:

1. The outer and inner gaseous phases are binary mixtures of watervapor and dry air, considered as a mixture of ideal gases.

2. For each of these gases, the total transmission rate GTRtot is thesum of a faces’ contribution GTRsurf (on the whole surface A) andan edges’ contribution GTRlin (on the whole perimeter P):

GTRtot = A · GTRsurf + P · GTRlin (65)

3. The mass flow rates from outside to inside follow the linear sorp-tion/diffusion model, i.e. each gas transmission rate proportionalto the partial pressure difference (�pg is the positive difference,i.e. �pg = pout − pin which is different from the convention usedin Section 2):

GTRsurf = ˘surf �pg (66)

GTRlin = ˘lin �pg (67)

4. Permeances are independent of the gas composition.5. Permeances are only dependent on the temperature, following

an Arrhenius law.

˘surf (T) = ˘0surf exp

(−EP

a

R

(1T

− 1T0

))(68)

e with metalized polymer films [8].

9 and Buildings 55 (2012) 903–920

sfphofe

5

mag

P

mil

wlPtu

ra

5

5

ap

1

2

ptafgd

t(mt


˘lin(T) = ˘0lin exp

(−EP

a

R

(1T

− 1T0

))(69)

The heat-sealed joint and the envelope are supposed to obey theorption–diffusion model for both dry air and water vapor, with dif-erent transfer coefficients for each case. As stated previously, thisaper does not address the issue of gas permeation through theeat-sealed joint of the envelope, or gas permeation at the scalef a whole VIP. Following paragraphs are thus focused on the sur-ace permeation through barrier membranes, excluding the edgeffects.

.2.2. Ideal laminate theory for a multilayer membraneAs an heterogeneous material, a multilayer coated polymer

embrane has no intrinsic permeability. It is possible to define thepparent permeability, from Eq. (13), as a function of the surfaceas transmission rate and the pressure gradient:

ei,app(T) = GTRi

�pil (70)

It is possible to get the total permeability of a multilayer barrierembrane thanks to the law of “superposition” of the ideal lam-

nate theory, as soon as the property of each layer is known. Thisaw can be written as [25,59]:

l

Peapp=

∑ �elem

Peelem(71)

here �elem and l are respectively the thickness of each elementaryayer and the whole multilayer membrane (

∑�elem = l), Peelem and

eapp are respectively the apparent permeabilities of each elemen-ary layer and of the whole multilayer membrane. It is possible tose permeances, defined by Eq. (59), to rewrite the law [17]:

1˘app

=∑ 1

˘elem(72)

It is actually the same law as for an electric system that containsesistances in series. The “resistance to gas” is defined as Rg = 1/˘g

nd the total resistance is the sum of the individual resistances.

.3. Apparent permeability of a coated polymer film

.3.1. Transport mechanisms in a coated polymer filmAn elementary coated layer is made from a polymer substrate

nd a thin coating barrier layer. The mass transfer through a coatedolymer film is assumed to follow two separate mechanisms [51]:

. In non-damaged bulk materials (the polymer substrate and non-damaged area of the coating layer), mass transfer follows theFickian diffusion model, with the bulk material properties.

. In damaged areas (the thin coating layer containing defects), gastransport can occur through the layer’s defects: pinholes, grainboundaries, micro-cracks, etc.

These mechanisms of mass transfer are different, so that theermeation properties of many coating layers are different fromhe permeation properties of their bulk counterparts [51]. Thisssumption of double transport mechanism was mainly studiedor oxygen and water vapor transport, it can be used for anyas [17,25,28,51,52,54,60–66]. These mechanisms of transport inefect free and imperfect layers are illustrated in Fig. 8.

It should be noted that two kinds of defects may be iden-
ified: nano-defects (between 0.3 and 1 nm) and micro-defects>1 nm, generally about 1 �m) [62,67]. Most studies consider that
icro-defects like pinholes or microcracks are much more effec-ive paths through the coating layer for gas permeation and that

Fig. 8. Mechanism of gas permeation through a defect free substrate coated withbarrier layers containing defects [54].

the permeation rate through the micro-defects free areas is neg-ligible (molecules are blocked) [51,62,67]. Nano-defects may stillbe present, they would result for the micro-defects free area in anapparent permeability greater than the bulk material permeability[62,67].

The improvement of the elementary coated films propertiesleads to the improvement of the multilayer barrier membranemade from these coated films, as explained above (see Eq. (71)). Tocompare the coated film permeability Peapp,coated to the uncoatedfilm permeability Peapp,uncoated and thus the barrier improvement,the barrier improvement factor BIF can be used [51]. It is definedas:

BIF = Peapp,uncoated

Peapp,coated= Pepoly

Peapp,coated(73)

Since the coating thickness is much smaller than the polymersubstrate thickness in most cases (�coat � �poly), the coated polymerthickness is equal to the uncoated one. It is thus possible to write:

BIF = ˘poly

˘coated= GTRpoly

GTRcoated(74)

5.3.2. Influence of the coating layer and polymer substratethicknesses

The permeability of a coated film is experimentally a function ofthe polymer substrate material, coated layer material and coatedlayer geometry: thickness and distribution of defects (size and den-sity) [51]. It has been observed that for many polymer/coatingassociations, the apparent permeability decreases when the coat-ing thickness increases, until a minimum value Pemin obtained for acoating thickness greater than a critical value �coat,crit [51,54,67–69](see Figs. 9, 10 and 11a). It should be noted that for the associationPET/aluminum used in most VIP barrier membrane, �coat,crit is fairlylow, approximately 15 nm [51,68].

Numerical simulations from Hanika [63] showed that a criticalthickness also exists for the polymer substrate. For a coating layerwith a constant defect size and density, the apparent permeance ofa coated polymer decreases when the substrate thickness increases,until a value which remains constant while the thickness is abovea critical thickness �poly,crit (see Fig. 12)

If the coating layer and polymer substrate both have a thick-ness above the critical one (respectively �coat,crit and �poly,crit), theapparent permeance of the coated polymer is independent of thethicknesses. It is then possible to normalize the barrier improve-ment factor, if the reference film thickness (uncoated polymer) isset to 100 �m [25]:

BIF100 = ˘app,uncoated,100

˘app,coated(75)


Fig. 9. OTR of coated PET as a function of coating thickness [51].

F[

5

tpFfRoa

P

mer substrate and the coating layer (�poly + �coat = �tot), Pepoly and

Fi

ig. 10. WVTR of coated PET as a function of coating thickness (T = 28 ◦C and RH=90%)54].

.3.3. Roberts model for the permeability of a coated filmThe intrinsic permeability of a coated film has no meaning, but

he apparent permeability can be measured, following the samerinciple as presented above for multilayer barriers (see Eq. (70)).rom the hypothesis of the coexistence of a bulk diffusive trans-er and a micro defect dominated transfer, a model is proposed byoberts et al. [62] for the total apparent permeability as a functionf the ideal laminate theory permeability PeILT (see Section 5.2.2)nd the defect driven permeability Pedef:

eapp,coated = Pedef +(

1 − Pedef

Pepoly

)PeILT (76)

ig. 11. Water vapor permeance of aluminum-coated PET as a function of (a) aluminum ts the aluminum thickness (T = 25 ◦C, RH=90%) [17].

Fig. 12. Oxygen permeance of a coated PET film as a function of the polymer sub-strate thickness (defect density n0 = 104 cm−2, AD is the individual defect area) [63].

Pedef is the effective permeability of the coating layer withdefects in the case where this layer is totally impermeable to thepermeant in the non-damaged area. From Eq. (71), the ideal lami-nate theory permeance is equal to:

PeILT = �tot

(�poly

Pepoly+ �coat

Pecoat

)−1

(77)

where �poly and �coat are respectively the thicknesses of the poly-

Pecoat are respectively the permeabilities of the coating layer andthe polymer substrate. The most favorable case is when the coatinglayer is practically impermeable to the permeant gas, then PeILT ≈ 0

hickness and (b) surface fraction of pinholes (defects). The value close to each point

9 and Bu

aP

P

P

P

givd

P

5

dtrdn

P

tt

wn(t

P

tiF

t

P

tn


nd the permeation is only due to the defects driven transport:eapp = Pedef. Otherwise, the apparent permeability is written:


1 − Pedef

Pepoly

) (�poly/�tot

Pepoly+ �coat/�tot

Pecoat

)−1

(78)

For sufficiently good coating layer barrier properties,edef/Pepoly < 0.05, so that Eq. (78) becomes:


�poly/�tot

Pepoly+ �coat/�tot

Pecoat

)−1

(79)

If nano-defects are present in the coating layer and increase theas transport, then the value of Pecoat is different from the bulk coat-ng material permeability. It would be the sum of a Fickian diffusionalue (non-damaged bulk material) Pecoat,bulk and a nano-defectiffusion value Pecoat,nanodef:

ecoat = Pecoat,bulk + Pecoat,nanodef (80)

.3.4. Analytical modeling of the micro-defects permeabilityPrins and Hermans [60] found approximate solutions for the

iffusion problem in a polymer substrate with a coating layer con-aining only cylindrical defects (defects density n0, defects radius0, substrate thickness �poly). The first limiting case is for largeefects, when the defects size is greater than the substrate thick-ess (�poly/r0 < 1):

eapp,coated = Pedef = n0�r20 Pepoly (81)

It means that the coated layer permeability is proportional tohe non-covered area: only the total defects area n0�r2

0 participateo the mass flux.

On the contrary, the second limiting case is for small defects,hen the defects size is much smaller than the substrate thick-ess and the total defects area much smaller than the total area�poly/r0 � 1 and n0�r2

0 � 1). The defect driven permeability canhen be estimated by:

edef = 3.71n0r0�totPepoly (82)

In this case, the permeance ˘def = Pedef/�tot is independent ofhe substrate thickness, which means that the substrate thicknesss above the critical thickness �poly,crit according to Section 5.3.2 (seeig. 12).

An intermediate case, providing �poly/r0 > 1 and n0�r20 � 1, gives

he following equation:

edef = n0�r2(

1 + 1.18�poly

)Pepoly (83)
0 r0
Pedef is from these three equations independent of the coatinghickness, if the total thickness is assimilated to the substrate thick-ess (�coat/�poly � 1). This result is consistent with the experimental

Fig. 13. Micro-defects (pinholes) surface fraction of aluminum-coated PET

ildings 55 (2012) 903–920

observations reported in Section 5.3.2 concerning the coating thick-ness (see Figs. 9, 10 and 11a), as soon as the coating thickness isabove the critical thickness �coat,crit. Nevertheless, the permeabilityis indirectly dependent on the coating thickness since the defectssize distribution (size and density) is dependent on the coatingthickness among other things. Garnier et al. [69] measured thisinfluence (see Fig. 13).

The critical value of the coating thickness �coat,crit is probably theminimal thickness that allows the complete or maximal coverage ofthe polymer, so that the pinhole number as well as size decreases toa minimum situation for �coat,crit and stays constant for thicknessesgreater than �coat,crit [51]. For a determined couple polymer/coating,the critical thickness �coat,crit appears to be the same for oxygen andwater vapor [51].

Rossi and Nulman [70] and Yanaka et al. [71] came to the con-clusion that the defects permeability is directly proportional to thebulk polymer permeability, following a proportionality constantKdef that is function of the defects size distribution:

Pedef = Kdef Pepoly (84)

This result is consistent with Eq. (81). An expression of Kdef isproposed for circular holes by Rossi and Nulman [70] and rect-angular holes by Yanaka et al. [71]. Kdef is only dependent on thegeometry, but is independent of both temperature and permeantspecies.

From the results of different numerical simulations, Hanika [63]proposed the following heuristic equation for evenly distributeddefects of equal area Adef and density n0:

Pedef = n0Adef

1 − exp(−0.507(√

Adef /�poly)) + 0.01n0Adef

Pepoly (85)

If the defects size is not uniform, the defects size distributionis split in different classes. For each class, one elementary defectpermeability is computed from the average defects density and size.The total permeability is the sum of all elementary permeabilities.

Garnier et al. [69] have measured that the water vapor perme-ability linearly increases with the surface fraction of micro-defects(see Fig. 11b). It is consistent with Eq. (81) (and thus Eq. (84))without restriction, with Eqs. (82) and (83) only if �poly/r0 remainsconstant, and with Eq. (85) only if

√Adef �poly remains constant

and above 1 (equivalent to �poly/r0 < 1 for circular micro-defects).Garnier et al. [17] have also measured that the water vapor per-meability is an exponentially decreasing function of the coatingthickness for an aluminized PET (see Fig. 14)

5.3.5. Musgrave model for the permeability of a coated filmMusgrave [72] proposed to use finite element method simula-

tions to estimate the oxygen permeation rate through a polymerlayer between two coating layers containing defects. The geome-try is reduced to a rectangular PET box (2.5 mm × 2.5 mm area and12 �m thick) with two quarters of pinholes in opposite corners.

as a function of aluminum coating thickness (T = 25 ◦C, RH=90%) [69].

M. Bouquerel et al. / Energy and Bu

Fig. 14. WVTR of aluminum-coated PET with one PET protective layer as a functiono

Ftacacsb0

5

batbFiaccfi

rdot

The difference of value of the permeability activation energy, foruncoated and coated polymer, appears to be a good way to detect

Fn

f aluminum coating thickness (T = 25 ◦C, RH=90%) [17].

ick’s law of diffusion is solved for oxygen transport thanks tohe formal analogy with Fourier’s law. The simulation parametersre fitted with experimental results of one and two aluminum-oated layers membrane, permeances used are respectively 0.055nd 0.0005 cm3(STP)/(m2 d bar). Permeance for 3 and 4 layers areomputed, but no experimental data are used to validate the resultsince these ultra high oxygen barrier membranes had permeanceelow the limit of measurement for oxygen permeance in 2005,.0005 cm3(STP)/(m2 d bar). It is still the case in 2011 [66].

.3.6. Thorsell model for the permeability of a coated filmThorsell [65,73] proposed a hybrid model for a polymer film

etween two coating layers containing defects. Based on thessumption that the defect free coating is impermeable and thathe size and distribution of circular defects are known, the dou-le coated film is assimilated to a partial resistance network (seeig. 15). Each defect (considered as an input or an output depend-ng on the side) has a so-called cylinder resistance, calculated fromxisymmetric FEM simulations (COMSOL Multiphysics®). Closeylinders defects are linked through so called field resistances, cal-ulated from a 2D analytical resolution of the dipole problem ineld theory.

Oxygen permeability estimated with this model is coherent withesults from other models and measured data, even though theefect distribution used for simulation and the experimental value
f permeability are not taken on the same sample, which preventso estimate the accuracy of this model.
ig. 15. Hybrid model from Thorsell for gas permeation through a double coated polymetwork between defects in the upper coating (black solid circles) and defects in the low

ildings 55 (2012) 903–920 917

5.3.7. Garnier model for the permeability of a coated filmGarnier et al. [17] proposed an other model based on the analogy

with an electric system, with resistances in parallel and in series.The principle is analog to what is presented in Section 5.2.2. Thepolymer layers and the coating layers are considered as resistancesin series (see Fig. 16a) and the coating layer permeance is calculatedconsidering that the defect-free aluminum coating area and theglue filled holes act as resistances in parallel (see Fig. 16b).

For an aluminum-coated polymer film, where the total surfacefraction of pinholes in the aluminum coating layer is equal to �r2

0 n0,and providing that n0�r2

0 � 1, the apparent permeability Peapp,coatedis from this model:

Peapp,coated = �tot

(�poly

Pepoly+ �coat

n0�r20 Peglue + Pealu

)−1

(86)

Comparison of model and experimental results shows an agree-ment described as surprisingly good by the authors (see Fig. 17).The surface fraction of pinholes n0�r2

0 was measured by scanningelectron microscopy. The good agreement is surprising since it wasobserved by many authors that the permeability is a function ofdefects density and not a function of the total defect surface fraction[53,74].

5.3.8. Influence of temperature and transport mechanismsHenry et al. [52] have measured the activation energies of per-

meability to oxygen and water vapor for normal PET, AlOx coatedPET and indium tin oxide (ITO) coated PET. The activation energy ofthe oxygen permeability is found identical for all films. The inter-pretation is that the transport is diffusive and occurs only on thepolymer, and that the coating layer is an absolute barrier, onlycrossed by the permeant molecules through the micro-defects.Oxygen transport is thus a micro-defect dominated permeation.The activation energy of the water vapor permeability is dependenton the coating, which suggests that diffusive transport might occurthrough the non-damaged part of the coating layer, or at least thatan interaction between water vapor and the coating layer exists.

Fahlteich et al. [54] have observed the same behavior for oxy-gen and water vapor permeation through silicon oxide (SiOx)coated PET and zinc tin oxide (ZTO) coated PET. Oxygen perme-ation through those films is a micro-defect dominated permeationwith constant activation energy, whereas water vapor permeationis also driven by chemical interaction between water vapor and thecoating layer, with a varying activation energy.

the transport mechanism of gas in coated polymer films. Beyondthese two particular examples, other experiments in the literature

er film [65,73]. (a) Three resistances in series between two defects. (b) Resistanceer coating (gray dashed circles).


F coatedc

tnwp

5

fitipptlace

ctdaaom

Fc

ig. 16. Electric resistances analogy for the permeance modeling of an aluminum-oating with micro-defects (pinholes).

end to prove that the oxygen and nitrogen permeation throughon organic coated polymer films is a micro-defect driven process,hereas for water vapor an additional transport mechanism exists,robably involving capillarity condensation [51,67,25].

.4. Synthesis

Barrier properties of simple polymer membranes are not suf-cient for several applications where the atmospheric gasesransmission rates should be extremely low, including vacuumnsulation panels. With a coating layer of inorganic material on theolymer substrate, typically 10–100 nm of silica, metal oxide orure metal, the permeation properties can be improved by one orwo orders of magnitude. The superposition of several coated layerseads to even better barrier properties, the multilayer membranecting like several barriers in series. The apparent permeabilityan be calculated thanks to physical laws analog to those used forlectric circuits.

The transport mechanism through the coated polymer layers isonsidered as a micro-defect driven process. Indeed, the very lowhickness of the coating layers leads to the presence of a micro-efects distribution, depending among other things on the polymer
nd coating materials and on the coating layer thickness. Materi-ls used for coating layers have bulk permeabilities several ordersf magnitude below the permeabilities of polymers or glues (inultilayer membranes) that are directly adjacent to them, so that
ig. 17. Comparison of model and experimental results for the permeability of aoated polymer film (T = 25 ◦C, RH=90%) [17].

polymer film [17]. (a) Aluminum-coated polymer film as a bilayer. (b) Aluminum

the micro-defects represent a much more efficient path through acoating layer than the defect free area.

In order to increase the barrier properties, the number andsize of the micro-defects should be minimized. The apparent per-meability of a coated polymer film, or a multilayer membranecontaining coated polymer films, appears to have the same tem-perature dependence as the polymer alone for oxygen and nitrogenpermeation: the gas flux is totally blocked by the defect freeareas of the coating layers, dry air only diffuses through thepolymer and the empty or glue-filled micro-defects. For watervapor, however, it seems that an interaction between watermolecules and coating layer exists, as the activation energy differ-ence between uncoated and coated polymers permeabilities couldindicate.

6. Conclusion

Mass transfer in VIPs is a crucial issue when looking at the dura-bility of vacuum super insulation materials. A decent service lifefor building applications is only insured if the ability to maintain alow pressure and a low humidity during several decades is proved.The gas barrier envelope thus plays a key role in the durability ofvacuum insulation panels, it is supposed to decrease at a minimumlevel the permeation rates of atmospheric gases. The understand-ing of gas permeation mechanisms through multilayer membraneswith metalized polymer films is needed in the scope of aging mod-eling, especially the dependence of barrier properties to materials,manufacturing, and climatic conditions (temperature and relativehumidity).

Mass transfer theory for the permeation of atmospheric gasesthrough a VIP envelope is based on the sorption–diffusion model,initially designed for permeation through homogeneous poly-mer membranes. This model, very easy to use, has shown acertain consistency with experimental observations on severalpolymers. However, the thermodynamics demonstration of thismodel is not straightforward, a lot of hypotheses is needed. Manyalternative models are also proposed, mainly for glassy and semi-crystalline polymers. They are often based on the assumption that
the crystalline volume fraction of the polymer and the amor-phous one have different behaviors. Sorption is not linear inthese alternative models, but follows other isotherm curves, likeLangmuir of Flory–Huggins modes for instance. Concerning the

and Bu

iptiepoc

pefigsemvcmso

mbitgdvmlhtpTto

geadssptfieve

mfcrcmbp

A

AC

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[


nfluence of external parameters on the permeation properties ofolymers, the thermo-activation, following an Arrhenius law forhe temperature dependence, has been repeatedly observed. Thenfluence of a low pressure (below atmospheric pressure), how-ver, has not been observed. For high pressures, the permeationroperties may vary, and some alternative models have been devel-ped in order to explain this pressure dependence of transportoefficients.

For VIPs, the requirements concerning the atmospheric gasesermeation rates are very constraining, and simple polymernvelopes would not be able to maintain the vacuum over a suf-ciently long period. However, polymer membranes have veryood mechanical properties, particularly in term of flexibility andtrength. In order to benefit from these polymer properties, a veryfficient strategy to increase the barrier properties is to coat poly-er films with a thin inorganic layer, made of a material having

ery good barrier properties to the gases considered. Polymer filmsoated for instance with aluminum can increase by two orders ofagnitude or more their barrier properties. The superposition of

everal coated films is also a way to increase the barrier propertiesf the membrane.

The sorption–diffusion model has been extended to this kind ofultilayer membranes containing aluminum-coated films. It has

een shown that the transport mechanism of permeant moleculesn a coated layer is a micro-defects driven process. The transporthrough the polymer layer substrate is a Fickian diffusion, whereasas molecules can only cross the coating layer through micro-efects. Indeed, the coating layer material is often chosen for itsery good barrier properties, but the low thickness prevents theanufacturing of a homogeneous and perfect layer. In the coating

ayer, the defect free area is totally blocking the gas molecules, theyave to diffuse through the polymer layer to find a micro-defect ando cross the coating layer. The apparent permeance of the coatedolymer film is thus a function of the micro-defects distribution.he total permeance of a multilayer membrane can be calculatedhanks to the ideal laminate theory, based on the electric analogyf resistances in series.

Concerning the temperature influence on the instantaneousas permeation rates through an aluminum-coated polymer film,xperimental results and numerical results based on Arrhenius lawre consistent. For dry air permeation, the energy activation is onlyependent on the polymer properties: the energy activation is theame for coated or uncoated films. For water vapor permeation, iteems that a metal/water interaction exists in the mass transferhenomenon. Because of this interaction, the energy activation ofhe water vapor permeation rates is different for coated or uncoatedlms. For the humidity influence, no explicit model is proposed toxplain the influence of the humidity level on the air and waterapor instantaneous permeation rates that has been observed onxperimental results.

An advanced research work on permeation experiments andodeling is needed to improve the knowledge about mass trans-

er through the multilayer gas barrier membranes, in particular theombined effect of humidity and temperature on the permeationates. Considering the permeation of dry air and water vapor as aoupled mass transfer of several atmospheric gases through theseembranes, instead of independent mass transfers as it has always

een considered so far, might be a first step toward a more realisticermeation model.

cknowledgments

This work was partially supported by the French Nationalgency for Technological Research [ANRT] under the grant-in-aidIFRE 2009/1012.

[[

[

ildings 55 (2012) 903–920 919

References

[1] H. Simmler, S. Brunner, U. Heinemann, H. Schwab, K. Kumaran, P. Mukhopad-hyaya, D. Quénard, H. Sallée, K. Noller, E. Kükükpinar-Niarchos, C. Stramm,M. Tenpierik, H. Cauberg, M. Erb, Vacuum Insulation Panels. Study on VIP-components and panels for service life prediction of VIP in building applications(Subtask A), Technical Report, IEA/ECBCS Annex 39 HiPTI-project (High Perfor-mance Thermal Insulation for Buildings and Building Systems), 2005.

[2] A. Binz, A. Moosmann, G. Steinke, U. Schonhardt, F. Fregnan, H. Simmler, S.Brunner, K. Ghazi Wakili, R. Bundi, U. Heinemann, H. Schwab, H. Cauberg,M.J. Tenpierik, G. Johannesson, T.I. Thorsell, M. Erb, B. Nussbaumer, Vacuuminsulation in the building sector. Systems and applications (subtask B), Tech-nical Report, IEA/ECBCS Annex 39 HiPTI-project (High Performance ThermalInsulation for Buildings and Building Systems), 2005.

[3] M.J. Tenpierik, Vacuum insulation panels applied in building constructions,Ph.D. Thesis, Delft University of Technology, 2009.

[4] R. Baetens, B.P. Jelle, J.V. Thue, M.J. Tenpierik, S. Grynning, S. Uvslokk, A.Gustavsen, Vacuum insulation panels for building applications: a review andbeyond, Energy and Buildings 42 (2010) 147–172.

[5] M. Alam, H. Singh, M.C. Limbachiya, Vacuum insulation panels (VIPs) for build-ing construction industry – a review of the contemporary developments andfuture directions, Applied Energy 88 (2011) 3592–3602.

[6] M. Bouquerel, T. Duforestel, D. Baillis, G. Rusaouen, Heat transfer modeling inVacuum Insulation Panels containing nanoporous silicas – a review, Energy andBuildings 54 (2012) 320–336.

[7] H. Schwab, U. Heinemann, A. Beck, H.-P. Ebert, J. Fricke, Prediction of service lifefor vacuum insulation panels with fumed silica kernel and foil cover, Journal ofThermal Envelope and Building Science 28 (2005) 357–374.

[8] H. Simmler, S. Brunner, Vacuum insulation panels for building applicationsbasic properties, aging mechanisms and service life, Energy and Buildings 37(2005) 1122–1131.

[9] P. Mukhopadhyaya, K. Kumaran, J. Lackey, N. Normandin, D. van Reenen,Methods for evaluating long-term changes in thermal resistance of vacuuminsulation panels, in: Proceedings of the 10th Canadian Conference on BuildingScience and Technology, Ottawa, Canada, 2005.

10] M. Bouquerel, T. Duforestel, D. Baillis, G. Rusaouen, Aging modeling of VacuumInsulation Panels – a review, Energy and Buildings, submitted for publication.

11] J.G. Wijmans, R.W. Baker, The solution–diffusion model: a review, Journal ofMembrane Science 107 (1995) 1–21.

12] K. Ghazi Wakili, R. Bundi, B. Binder, Effective thermal conductivity of vacuuminsulation panels, Building Research & Information 32 (2004) 293–299.

13] H. Schwab, C. Stark, J. Wachtel, H.-P. Ebert, J. Fricke, Thermal bridges in vacuum-insulated building fac ades, Journal of Thermal Envelope and Building Science28 (2005) 345–355.

14] T. Duforestel, B. Yrieix, C. Pompeo, Étude expérimentale et par modélisation del’impact des barrières d’étanchéité sur la performance thermique des isolantssous vide, in: Actes du VIIIème Colloque Interuniversitaire Franco-Québécoissur la Thermique des Systèmes, Montréal, Canada, 2007.

15] S. Brunner, P. Gasser, H. Simmler, K. Ghazi Wakili, Investigation of multilay-ered aluminium-coated polymer laminates by focused ion beam (FIB) etching,Surface & Coatings Technology 200 (2006) 5908–5914.

16] S. Brunner, P.J. Tharian, H. Simmler, K. Ghazi Wakili, Focused ion beam(FIB) etching to investigate aluminium-coated polymer laminates subjectedto heat and moisture loads, Surface & Coatings Technology 202 (2008)6054–6063.

17] G. Garnier, S. Marouani, B. Yrieix, C. Pompeo, M. Chauvois, L. Flandin, Y. Bréchet,Interest and durability of multilayers: from model films to complex films, Poly-mers for Advanced Technology 22 (2011) 847–856.

18] D. Quénard, H. Sallée, From VIP’s to building facades three levels of thermalbridges, in: Proceedings of the 7th International Vacuum Insulation Sympo-sium, EMPA, Duebendorf/Zurich, Switzerland, 2005.

19] R. Bundi, K. Ghazi Wakili, T. Frank, Vacuum insulation panels in buildingapplications, in: Proceedings of the CISBAT 2003 Conference, EPFL, Lausanne,Switzerland, 2003.

20] K. Ghazi Wakili, T. Stahl, S. Brunner, Effective thermal conductivity of a stag-gered double layer of vacuum insulation panels, Energy and Buildings 43 (2011)1241–1246.

21] H. Schwab, U. Heinemann, A. Beck, H.-P. Ebert, J. Fricke, Permeation of differentgases through foils used as envelopes for vacuum insulation panels, Journal ofThermal Envelope and Building Science 28 (2005) 293–317.

22] M.H. Klopffer, B. Flaconnèche, Transport properties of gases in polymers: bib-liographic review, Oil & Gas Science and Technology – Revue d’IFP 56 (2001)223–244.

23] A.D. McNaught, A. Wilkinson, IUPAC Compendium of Chemical Terminology,The Gold Book, 2nd ed., Blackwell Science, Oxford, 1997.

24] http://en.wikipedia.org/wiki/standard conditions for temperature andpressure, 2012.

25] H.-C. Langowski, Plastic Packaging: Interactions with Food and Pharmaceut-icals, 2nd ed., Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim, Germany, 2008,pp. 297–347.

26] A. Fick, Über diffusion, Poggendorff’s Annalen 94 (1855) 59–86.
27] J. Fourier, Théorie analytique de la chaleur, Firmin Didot, père et fils, 1822.28] C. Alaux, Perméance des ultra barrières pour panneaux isolants sous vide, Mas-
ter’s Thesis, INSA-Lyon, 2010.29] W.R. Vieth, J.M. Howell, J.H. Hsieh, Dual sorption theory, Journal of Membrane

Science 1 (1976) 177–220.

http://en.wikipedia.org/wiki/standard_conditions_for_temperature_and_pressure

http://en.wikipedia.org/wiki/standard_conditions_for_temperature_and_pressure

9 and Bu

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[[

[

[

[

[

[

[

[

[


30] R.M. Barrer, J.A. Barrie, J. Slater, Sorption and diffusion in ethyl cellulose. Partiii. Comparison between ethyl cellulose and rubber, Journal of Polymer Science27 (1958) 177–220.

31] S. Kanehashi, K. Nagai, Analysis of dual-mode model parameters for gas sorp-tion in glassy polymers, Journal of Membrane Science 253 (2005) 117–138.

32] J.H. Petropoulos, Quantitative analysis of gaseous diffusion in glassy polymers,Journal of Polymer Science Part A-2 8 (1970) 1797–1801.

33] D.R. Paul, W.J. Koros, Effect of partially immobilizing sorption on permeabilityand the diffusion time lag, Journal of Polymer Science: Polymer Physics Edition14 (1976) 675–685.

34] W.J. Koros, D.R. Paul, Transient and steady-state permeation in poly(ethyleneterephtalate) above and below the glass transition, Journal of Polymer Science:Polymer Physics Edition 16 (1978) 2171–2187.

35] M.A. Islam, H. Buschatz, Gas permeation through a glassy polymer membrane:chemical potential gradient or dual mobility mode? Chemical Engineering Sci-ence 57 (2002) 2089–2099.

36] P.J. Flory, Principles of Polymer Chemistry, Cornell University Press, Ithaca, NY,1953.

37] R. Kirchheim, Sorption and partial molar volume of small molecules in glassypolymers, Macromolecules 25 (1992) 6952–6960.

38] P. Pekarski, R. Kirchheim, On the sorption of gas mixtures by polymer glasses,Journal of Membrane Science 152 (1999) 251–262.

39] B. Flaconnèche, J. Marin, M.H. Klopffer, Permeability diffusion and solubility ofgases in polyethylene, polyamide 11 and poly(vinylidene fluoride), Oil & GasScience and Technology – Revue d’IFP 3 (2001) 261–278.

40] J. Gajdos, K. Galic, Z. Kurtanjek, N. Cikovic, Gas permeability and DSC charac-teristics of polymers used in food packaging, Polymer Testing 20 (2001) 49–57.

41] W.-H. Lin, R.H. Vora, T.-S. Chung, Gas transport properties of 6FDA-Durene/1,4-phenylenediamine (pPDA) copolymides, Journal of Polymer Science Part B:Polymer Physics 38 (2000) 2703–2713.

42] W.-H. Lin, T.-S. Chung, Gas permeability, diffusivity, solubility, and agingcharacteristics of 6FDA-durene polyimide membranes, Journal of MembraneScience 186 (2001) 183–193.

43] Y. Naito, D. Bourbon, K. Terada, Y. Kamiya, Effect of pressure on gas perme-ation through semicrystalline polymers above the glass transition temperature,Journal of Polymer Science Part B: Polymer Physics 29 (1991) 457–462.

44] Y. Naito, D. Bourbon, K. Terada, Y. Kamiya, Permeation of high-pressure gasesin poly(ethylene-co-vinyl-acetate), Journal of Polymer Science Part B: PolymerPhysics 31 (1993) 693–697.

45] Y. Naito, Y. Kamiya, K. Terada, K. Mizoguchi, J.S. Wang, Pressure dependence ofgas permeability in a rubbery polymer, Journal of Applied Polymer Science 61(1996) 945–950.

46] S.S. Kulkarni, S.A. Stern, The diffusion of CO2 CH4 C2H4 and C3H8 in polyethyleneat elevated pressures, Journal of Polymer Science Part B: Polymer Physics 21(1983) 441–465.

47] J.L. Lundberg, Diffusivities and solubilities of methane in linear polyethyl-ene melts, Journal of Polymer Science Part A: Polymer Chemistry 2 (1964)3331–3925.

48] G.E. Serad, B.D. Freeman, M.E. Stewart, A.J. Hill, Gas and vapor sorption anddiffusion in poly(ethylene terephthalate), Polymer 42 (2001) 6929–6943.

49] G. Strupinsky, A.L. Brody, A twenty-year retrospective on plastics: oxygen bar-rier packaging materials, in: Polymers, Laminations & Coatings ConferenceProceedings, San Francisco, CA, USA, 1998.

50] S.N. Dhoot, Sorption and transport of gases and organic vapors in poly(ethyleneterephthalate), Ph.D. Thesis, University of Texas at Austin, 2004.

51] H. Chatham, Oxygen diffusion barrier properties of transparent oxide coatings
on polymeric substrates, Surface & Coatings Technology 78 (1996) 1–9.
52] B.M. Henry, A.G. Erlat, A. McGuigan, C.R.M. Grovenor, G.A.D. Briggs, Y. Tsuka-hara, T. Miyamoto, N. Noguchi, T. Niijima, Characterization of transparentaluminium oxide and indium tin oxide layers on polymer substrates, Thin SolidFilms 328 (2001) 194–201.

[

[

ildings 55 (2012) 903–920

53] G. Rochat, Y. Leterrier, P. Fayet, J.-A.E. Månson, Influence of substrate addi-tives on the mechanical properties of ultrathin oxide coatings on poly(ethyleneterephtalate), Surface & Coatings Technology 200 (2005) 2236–2242.

54] J. Fahlteich, M. Fahland, W. Schönberger, N. Schiller, Permeation barrier prop-erties of thin oxide films on flexible polymer substrates, Thin Solid Films 517(2009) 3075–3080.

55] H. Schwab, U. Heinemann, J. Wachtel, H.-P. Ebert, J. Fricke, Prediction of theincrease in pressure and water content of vacuum insulation panels (VIPs) inte-grated into building constructions using model calculations, Journal of ThermalEnvelope and Building Science 28 (2005) 327–344.

56] M.J. Tenpierik, H. Cauberg, Simplified analytical models for service life pre-diction of a vacuum insulation panel, in: Proceedings of the 8th InternationalVacuum Insulation Symposium, Würzburg, Germany, 2007.

57] J.-S. Kwon, C.H. Jang, H. Jung, S. Tae-Ho, Vacuum maintenance in vacuum insu-lation panels exemplified with a staggered beam VIP, Energy and Buildings 42(2010) 590–597.

58] E. Wegger, B.P. Jelle, E. Sveipe, S. Grynning, A. Gustavsen, R. Baetens, J.V. Thue,Aging effects on thermal properties and service life of vacuum insulation panels,Journal of Building Physics 35 (2011) 128–167.

59] W.J. Schrenk, T.J. Alfred, Some physical properties of multilayered films, Poly-mer Engineering & Science 9 (1969) 393–399.

60] W. Prins, J.J. Hermans, Theory of permeation through metal coated polymerfilms, Journal of Physical Chemistry 63 (1959) 716–720.

61] Y.G. Tropsha, N.G. harvey, Activated rate theory treatment of oxygen andwater transport through silicon oxide/poly(ethylene terephthalate) com-posite barrier structures, Journal of Physical Chemistry B 101 (1997)2259–2266.

62] A.P. Roberts, B.M. Henry, A.P. Sutton, C.R.M. Grovenor, G.A.D. Briggs, T.Miyamoto, M. Kano, Y. Tsukahara, M. Yanaka, Gas permeation in silicon-oxide/polymer (SiOx/PET) barrier films: role of the oxide lattice, nano-defectsand macro-defects, Journal of Membrane Science 208 (2002) 75–88.

63] M. Hanika, Ph.D. Thesis, TU München, Munich, Germany, 2004.64] Y. Carmi, Ultra high barrier VIP laminates – new solutions for tougher require-

ments, in: Proceedings of the 8th International Vacuum Insulation Symposium,Würzburg, Germany, 2007.

65] T.I. Thorsell, A hybrid model for diffusion through barrier films with multiplecoatings, Journal of Building Physics 34 (2011) 351–381.

66] Y. Carmi, Developing 60-year lifetime metallized VIP laminates – challengesand solutions, in: Proceedings of the 10th International Vacuum InsulationSymposium, Ottawa, Canada, 2011.

67] Y. Leterrier, Durability of nanosized oxygen-barrier coatings on polymers,Progress in Materials Science 48 (2003) 1–55.

68] A.S. da Silva Sobrinho, G. Czeremuszkin, M. Latrèche, G. Dennler, M.R.Wertheimer, A study of defects in ultra-thin transparent coatings on polymers,Surface & Coatings Technology 116–119 (1999) 1204–1210.

69] G. Garnier, B. Yrieix, Y. Bréchet, L. Flandin, Influence of structural feature ofaluminium coatings on mechanical and water barrier properties of metallizedPET films, Journal of Applied Polymer Science 115 (2010) 3110–3119.

70] G. Rossi, M. Nulman, Effect of local flaws in polymeric permeation reducingbarriers, Journal of Applied Physics 74 (1993) 5471–5475.

71] M. Yanaka, B.M. Henry, A.P. Roberts, C.R.M. Grovenor, G.A.D. Briggs, A.P.Sutton, T. Miyamoto, Y. Tsukahara, N. Takeda, R.J. Chater, How cracks in SiOx-coated polyester films affect gas permeation, Thin Solid Films 397 (2001)176–185.

72] D.S. Musgrave, Finite element analysis used to model VIP barrier film perfor-mance, in: Proceedings of the 7th International Vacuum Insulation Symposium,
EMPA, Duebendorf/Zurich, Switzerland, 2005.
73] T.I. Thorsell, Vacuum insulation in buildings means to prolong service life,Licentiate Thesis, KTH, Stockholm, 2006.

74] A. Sugiyama, H. Tada, M. Yoshimoto, Gas permeation through the pinholes ofplastic film laminated with aluminium foil, Vuoto XXVIII (1999) 51–54.

mass transfer modeling in gas barrier envelopes for vacuum insulation panels- a review

Documents

gas barrier envelopes

theory of gas permeation

mass transfer models

total mass transfer

permeation measurements

ative mass transfer

vacuum insulation panels

vacuum insulation panel