eth-26498-02

DISS. ETH NO. 15078

Property Predictions for

Short Fiber and Platelet Filled Materials by

Finite Element Calculations

A dissertation submitted to the

SWISS FEDERAL INSTITUTE OF TECHNOLOGY ZURICH

for the degree of

Doctor of Sciences

presented by

HANS RUDOLF LUSTI

Dipl. Werkstoff-Ing. ETH

born September 7, 1973

citizen of Nesslau, SG

accepted on the recommendation of

PD Dr. A.A. Gusev, examiner

Prof. Dr. U.W. Suter, co-examiner

Prof. Dr. P. Smith, co-examiner

2003

Diese Arbeit widme ich

meinen Eltern

Rösli & Christian

sowie meiner Frau

Natacha

DANKSAGUNG

Ich möchte mich bei PD Dr. Andrei Gusev ganz herzlich für die grossartige

persönliche und fachliche Unterstützung bedanken. Er hat mich während meiner

Doktorarbeit mit Rat und Tat unterstützt und ich konnte durch zahlreiche

Diskussionen von seinem grossen Wissen und seiner langjährigen

Wissenschafts-Erfahrung profitieren.

Ein besonderer Dank geht auch an Prof. Dr. Ulrich W. Suter, der in seiner

Forschungsgruppe ein hervorragendes Arbeitsklima aufgebaut hat und

bereitwillig auf Wünsche und Probleme eingegangen ist.

Ich möchte mich ebenfalls bei Dr. Peter Hine von der Universität Leeds, UK,

für die gute Zusammenarbeit auf dem Gebiet von kurzfaserverstärkten

Kompositen bedanken, die schon viele Früchte getragen hat.

Ein Dankeschön geht auch an:

• Dr. Chantal Oberson, für ihre Hilfeleistungen, wenn ich mit meiner Mathematik

am Ende war und für die anregenden Diskussionen,

• Ilya Karmilov für die gute Zusammenarbeit und die feinen Spezialitäten, die er

mir regelmässig aus Moskau mitgebracht hat,

• Martin Heggli für die gute Zusammenarbeit und die hilfreichen Ratschläge

bezüglich Mathematica,

• Albrecht Külpmann für die interessanten und anregenden Diskussionen,

• Dr. Marc Petitmermet für seinen prompten und kompetenten Computersup-

port und seinen grossartigen Einsatz bei der Wiederinbetriebnahme der HP-

Workstation,

• Sylvia Turner und Christina Graf für ihre Hilfe bei administrativen Angelegen-

heiten,

• alle anderen Mitarbeiter der Forschungsgruppe für das angenehme Arbeits-

klima.

ZUSAMMENFASSUNG

Eine neue, mächtige Finite-Elemente (FE) Simulationsmethode wurde kürz-

lich von Gusev entwickelt, die es erlaubt, die linear-elastischen, elektrischen,

thermischen und Transport-Eigenschaften von mehrphasigen Werkstoffen, ba-

sierend auf realistischen 3D-Computermodellen, zu studieren. Im ersten Teil die-

ser Doktorarbeit wurde dieses neue Verfahren validiert, indem gemessene

thermoelastische Eigenschaften mit den numerischen Voraussagen von FE-Mo-

dellen verglichen wurden, die aufgrund von mikrostrukturellen Daten von spritz-

gegossenen, kurzfaserverstärkten Zugproben generiert wurden. Die

numerischen Voraussagen zeigten eine ausgezeichnete Übereinstimmung mit

allen gemessenen Eigenschaften. Die erfolgreiche Validierung erlaubte es dann,

die Genauigkeit sowohl von den in der Praxis am weitesten verbreiteten mikro-

mechanischen Modellen (Halpin-Tsai und Tandon-Weng) zur Voraussage der

elastischen Eigenschaften von unidirektional kurzfaserverstärkten Kompositen

als auch des Orientierungsmittelungs-Verfahrens zu beurteilen. Die Untersu-

chungen haben gezeigt, dass das Modell von Tandon-Weng wesentlich genauer

ist als dasjenige von Halpin-Tsai. Trotzdem sind die Abweichungen zu gross, als

dass es für Auslegungszwecke im Engineering taugen würde. Der Vergleich zwi-

schen den Voraussagen von numerischen Berechnungen und dem Orientie-

rungsmittelungs-Verfahren haben ergeben, dass die Orientierungsmittelung sehr

geeignet ist, um die thermoelastischen Eigenschaftstensoren von jeglichen Fa-

ser- und Plättchen-Orientierungszuständen zu bestimmen. Das unter der Bedin-

gung, dass die Orientierungsmittelung mit zuverlässigen Eigenschaftsdaten von

unidirektionalen Kompositen durchgeführt wird. Mit dem numerischen Verfahren,

das in dieser Arbeit verwendet wurde, können die Eigenschaften von unidirektio-

nalen Kompositen problemlos bestimmt werden.

Numerische Berechnungen zu den thermoelastischen und Barriere-Eigen-

schaften von Polymer-Schichtsilikat-Nanokompositen mit perfekt ausgerichteten

Silikatplättchen haben gezeigt, dass der Abfall, sowohl der Gaspermeabilität als

auch des thermischen Ausdehnungskoeffizienten, durch eine gestreckte Expo-

nentialfunktion beschrieben werden kann, die von x = af abhängt, wobei a das

Achsenverhältnis und f die Volumenfraktion der Plättchen ist. Diese Masterkur-

ven erlauben eine rationale Auslegung der Barriere- und der thermischen Aus-

dehnungs-Eigenschaften von Nanokompositen mit perfekt ausgerichteten

Plättchen. Es wurde ausserdem demonstriert, wie die thermische Ausdehnung

von Nanokompositen mit Auslegungsdiagrammen, die von der Masterkurve ab-

geleitet wurden, massgeschneidert werden kann. Der Minderungseffekt von

Fehlausrichtungen der Plättchen auf die Barriereeigenschaften wurde auch un-

tersucht. Die Voraussage von Fredrickson et. al. dass verdünnte Konzentrationen

von zufällig orientierten Plättchen hohen Achsenverhältnisses ein Drittel so effek-

tiv sind wie entsprechende Nanokomposite mit perfekt ausgerichteten Plättchen,

wurde durch numerische Berechnungen bestätigt. Es war allerdings nicht be-

kannt, dass dieser Minderungseffekt im halbverdünnten Konzentrationsregime

abnimmt, weil die fehlgerichteten Plättchen gemeinsam anfangen, die Diffusions-

wege der penetrierenden Moleküle zu vergrössern. Für typische Achsenverhält-

nisse und Volumenfraktionen der Plättchen in gegenwärtig existierenden

Nanokompositen bewegt sich der Minderungseffekt von zufällig orientierten Plätt-

chen im Rahmen von 40-50%.

ABSTRACT

Recently, a new powerful finite element (FE) based simulation technique

has been developed by Gusev, which allows to study the linear-elastic, electric,

thermal and transport properties of multi-phase materials based on realistic 3D

multi-inclusion computer models. In the first part of this thesis this new procedure

has been validated by comparing measured thermoelastic properties with

numerical predictions obtained with FE-models, which were generated based on

microstructural data of real injection molded short fiber reinforced dumbbells.

Numerical predictions showed excellent agreement with all the measured

properties. The successful validation then allowed to assess the accuracy of most

widely used in practice micromechanics-based models (Halpin-Tsai and Tandon-

Weng) which predict the elastic properties of unidirectional short fiber

composites, and also the accuracy of the orientation averaging scheme. It was

found that the Tandon-Weng model is considerably more accurate than the

Halpin-Tsai equations, but nonetheless deviations are too large to make this

model appropriate for engineering design purposes. Comparison of direct

numerical and orientation averaging predictions revealed that the orientation

averaging scheme is highly suitable to determine the thermoelastic property

tensors of any fiber and platelet orientation state. This under the condition that

orientation averaging is done based on reliable property data of unidirectional

composites. With the numerical approach employed in this work one can readily

determine the properties of unidirectional composites.

Numerical calculations of the barrier and thermoelastic properties of

polymer-layered silicate nanocomposites comprising perfectly aligned silicate

platelets elucidated that the decline both of the gas permeability and of the

thermal expansion coefficient can be described by a stretched exponential

function which depends on x = af, the product of the platelet aspect ratio a and

the platelet volume fraction f. These mastercurves allow to rationally design the

barrier and thermal expansion properties of nanocomposites with perfectly

aligned platelets. Furthermore, it has been demonstrated how the thermal

expansion coefficient of nanocomposites can be tailored by using design

diagrams adapted from the mastercurve. The degrading effect of platelet

misalignments on the barrier properties has also been investigated numerically.

The prediction of Fredrickson et al. that dilute concentrations of randomly oriented

high-aspect-ratio platelets are 1/3 as effective compared to a corresponding

nanocomposite with perfectly aligned platelets was confirmed by numerical

calculations. It has, however, not been known that the degrading effect decreases

in the semidilute concentration regime due to the fact that the misaligned platelets

start to collectively increase the tortuosity of the penetrant’s diffusion path. For

platelet aspect ratios and volume fractions which are typical of currently existing

nanocomposites the expected degradation effect of randomly oriented platelets

is in the range of 40-50%.

PUBLICATIONS AND PRESENTATIONS IN CONNECTION WITH THIS THESIS

Articles

• A.A. Gusev, H.R. Lusti, Rational Design of Nanocomposites for Barrier Appli-

cations, Adv. Mater. 2001, 13, 1641-1643.

• H.R. Lusti, P.J. Hine, A.A. Gusev, Direct Numerical Predictions for the Elastic

and Thermoelastic Properties of Short Fibre Composites, Compos. Sci. Tech.

2002, 62, 1927-1934.

• P.J. Hine, H.R. Lusti, A.A. Gusev, Numerical Simulation of the Effects of

Volume Fraction, Aspect Ratio and Fibre Length Distribution on the Elastic

and Thermoelastic Properties of Short Fiber Composites, Compos. Sci. Tech.

2002, 62, 1445-1453.

• A.A. Gusev, H.R. Lusti, P.J. Hine, Stiffness and Thermal Expansion of Short

Fiber Composites with Fully Aligned Fibers, Adv. Eng. Mater. 2002, 4, 927-

931.

• A.A. Gusev, M. Heggli, H.R. Lusti, P.J. Hine, Orientation Averaging for Stiff-

ness and Thermal Expansion of Short Fiber Composites, Adv. Eng. Mater.

2002, 4, 931-933.

• H.R. Lusti, I.A. Karmilov, A.A. Gusev, Effect of Particle Agglomeration on the

Elastic Properties of Filled Polymers, Soft Materials 2003, 1, 115-120.

• M. Wissler, H.R. Lusti, C. Oberson, A.H. Widmann-Schupak, G. Zappini, A.A.

Gusev, Non-Additive Effects in the Elastic Behavior of Dental Composites,

Adv. Eng. Mater. 2003, 3, 113-116.

• P.J. Hine, H.R. Lusti, A.A. Gusev, The Numerical Prediction of the Elastic and

Thermoelastic Properties of Multiphase Materials, in preparation.

• H.R. Lusti, O. Guseva, A.A. Gusev, Matching the Thermal Expansion of Mica-

Polymer Nanocomposites and Metals, in preparation.

Poster presentations

• Materials Workshop, Crêt Berard, Switzerland, September 9-12, 2001:

H.R. Lusti, A.A. Gusev, “Rational Design of Nanocomposites for Barrier Appli-

cations”

• Top Nano 21 Third Annual Meeting, Bern, Switzerland, October 1, 2002:

H.R. Lusti, V. Mittal, A.A. Gusev, “Numerical Permeability Predictions for

Nanocomposites comprising Morphological Imperfections”

Oral presentations

• Materials Science Seminar, Department of Materials, ETH Zürich, November

14, 2001

• C4-Workshop, ETH Zürich, November 22, 2001

• Workshop in Analysis Techniques for Polymer Nanostructures, St. Anne’s Col-

lege, Oxford, UK, April 8-10, 2002

• CAD-FEM User's Meeting, Friedrichshafen, Germany, October 9-11, 2002

• 5. Werkstofftechnisches Kolloquium, Chemnitz, Germany, October 24-25,

2002

TABLE OF CONTENTS

Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

Acronyms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.1 Importance of Composite Materials . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.2 Short Fiber Reinforced Parts Made by Injection Molding . . . . . . . . . . 6

1.3 Polymer Nanocomposites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2. Analytical and Numerical Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.1 Micromechanical Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.2 Orientation Averaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.3 Gusev’s Finite-Element Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3. Short Fiber Reinforced Composites . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.1 Fiber Length Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.1.1 Numerical . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.1.2 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.2 Comparison between Micromechanical Models, Numerical Predictions

and Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.2.1 Micromechanical Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.2.2 Experimental . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.2.3 Numerical . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38


3.3 Stiffness and Thermal Expansion of Short Fiber Composites with Fully

Aligned Fibers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.3.1 Numerical . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

Table of Contents


3.4 Prediction of Stiffness and Thermal Expansion by the Orientation

Averaging Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

3.4.1 Numerical . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50


3.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

4. Polymer-Layered Silicate Nanocomposites . . . . . . . . . . . . . . . . . . . . 59

4.1 Permeability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

4.1.1 Morphologies with Perfectly Aligned Platelets . . . . . . . . . . . . . . 60

4.1.2 Morphologies with Misaligned Platelets . . . . . . . . . . . . . . . . . . . 64

4.2 Thermoelastic Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

4.2.1 Morphologies with Perfectly Aligned Platelets . . . . . . . . . . . . . . 74

4.2.2 Morphologies with Misaligned Platelets . . . . . . . . . . . . . . . . . . . 83

4.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

5. Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

Curriculum Vitae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

Table of Contents

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NOTATION

a Aspect ratio

aN Number average of a ARD

aS Skewed number average of a ARD

aRMS Root-Mean-Square average of a ARD

aW Weight average of a ARD

aij 2nd order orientation tensor

aijkl 4th order orientation tensor

Cijkl Stiffness tensor GPa

Cij Stiffness tensor in Voigt notation (6x6 Matrix) GPa

d Fiber diameter µm

Ef Young’s modulus of isotropic fibers GPa

Em Young’s modulus of isotropic matrix GPa

f Fiber or platelet volume fraction %

Gf Shear modulus of isotropic fibers GPa

Gm Shear modulus of isotropic matrix GPa

Km Bulk modulus of isotropic matrix GPa

L Fiber length µm

LN Number average of a FLD µm

LS Skewed number average of a FLD µm

LRMS Root-Mean-Square average of a FLD µm

LW Weight average of a FLD µm

N Number of inclusion in a computer model

Vf Fiber volume fraction

Vm Matrix volume fraction

P Permeability Barrer

p Unit vector pointing along the symmetry axis of

a fiber or platelet

Sijkl Compliance tensor GPa

αm Thermal expansion coefficient of isotropic matrix K-1

αL Thermal expansion coefficient lamellar composite K-1

Notation & Acronyms

- 2 -

αkl Thermal expansion tensor K-1

δij Unit tensor

εkl Effective mechanical strain

ζ Empirical Halpin-Tsai parameter

φ, θ Angles which define the orientation of fibers and platelets deg

µf , λf Lamé constants of isotropic fibers GPa

µm , λm Lamé constants of isotropic matrix GPa

νf Poisson’s ratio of isotropic fibers

νm Poisson’s ratio of isotropic matrix

σij Effective mechanical stress GPa

σTij Effective thermal stress GPa

ψ(p) Normalized probability density function of

a fiber orientation state

Notation & Acronyms

- 3 -

ACRONYMS

ARD Aspect Ratio Distribution

CTE Coefficient of Thermal Expansion

FE Finite Element

FEA Finite Element Analysis

FEM Finite Element Method

FLD Fiber Length Distribution

MC Monte Carlo

PBC Periodic Boundary Conditions

PDF Probability Density Function

RMS Root Mean Square

RVE Representative Volume Element

SCORIM Shear Controlled Orientation Injection Molding

Notation & Acronyms

- 4 -

Notation & Acronyms

- 5 -

1. INTRODUCTION

1.1 IMPORTANCE OF COMPOSITE MATERIALS

Heterogeneous and composite materials, like hardened steel, bronze or

wood were valued since ancient times because they provide a better performance

compared to the individual phases or components which they consist of.

Nowadays the idea of combining eligible materials to form a composite material

with new and superior properties compared to its individual components is still

subject of ongoing research. For example, polymers, which by nature have a low

density, can be reinforced by highly stiff and strong carbon fibers, both continuous

and discontinuous. Such fiber reinforced composites excel in high specific

mechanical properties. For lightweight structures high specific stiffness and

strength are crucial requirements. The higher the specific mechanical properties

are the lighter a part or construction can be designed. This is of great importance

for moving components, especially in the automotive and airplane industry, where

reductions in weight result in greater efficiency and reduced energy consumption.

The expression “fiber reinforced composites” already says that the focus for these

materials is on the mechanical performance. There exist, however, a variety of

other composites where the functionality is more important. For example carbon/

polyethylene composites which suddenly increase electrical resistivity by several

order of magnitudes upon heating because the carbon particles get separated

due the thermal expansion of the surrounding polymer matrix.

In former times people only had an empirical knowledge about the property

changes taking place when combining different types of materials in a composite.

It was in the 20th century that research on composite materials started and got

increasingly important also due to the need of new lightweight and high

performance materials in armor and astronautics. By the time composite

materials also found their way into civil applications e.g in passenger aircrafts,

cars, boats and sports equipment. But still the fraction of composite materials

employed in industrial goods is rather small because often traditionally used

lightweight materials are preferred to composite parts. The reasons for that are

manyfold. On the one hand there is a lack of experience in designing and

Chapter 1 - Introduction

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constructing with this class of materials and on the other hand there are the higher

cost of composite parts caused by expensive raw materials, i.e. carbon fibers,

and costly production processes. To overcome these inhibitions one of the

measures is to reduce the production cost in order to promote a broader

application of these materials. However, it is not only about highly automated and

fast production processes but also about the development of new predictive tools

which can be used to reliably design composite parts. The design stage decides

if a composite part can show its merits in a specific application and if it will perform

successfully in operation.

1.2 SHORT FIBER REINFORCED PARTS MADE BY INJECTION MOLDING

In contrary to time-consuming, elaborate and expensive winding or

laminating techniques used to manufacture long fiber reinforced parts short fiber

composites can be fabricated into complex shapes using automated mass

production methods, such as injection molding, compression molding or

extrusion. One process, already widely used in the production of unreinforced

thermoplastic parts, which is also predestinated in order to manufacture short

fiber reinforced parts with complex shapes, is injection molding. This process is

efficient and offers due to the high degree of automation the possibility of making

complex shaped structural parts in large quantities at reasonable cost. Therefore

injection molded short fiber reinforced thermoplasts are increasingly finding their

way into industrial applications where high specific mechanical properties,

durability and corrosion resistance are required but where cost are a decisive

factor e.g. in car industry.

Figure 1: Injection molded glass fiber reinforced nylon 6 acceleration pedal which was developed for the Ford Focus. Picture taken from [1].


- 7 -

Because the overall effective properties of a short fiber reinforced composite

can vary between isotropic (3D randomly oriented fibers) and highly anisotropic

(aligned fibers) it is of great importance to design the mold and to control the

process parameters, e.g in a way that everywhere locally across the finished part

the short fibers act along the axes of principal stresses. There exist commercial

software packages (e.g. Moldflow, Sigmasoft,...) which are used to simulate the

mold filling process and at the same time to determine the local fiber orientation

states in a finished injection molded part after cooling. It was shown that

simulation results agree remarkably well with measured microstructural data.[1]

Based on the local fiber orientation states one can calculate the local

thermoelastic properties which serve as input for structural FEM packages (e.g

Abaqus, Ansys,...). Structural FEA reveals if a particular part’s design is

appropriate to perform well under the expected loads during operation and if the

degree of shrinkage and warpage during cooling from the process temperature is

acceptable. At present, all mold filling simulation programs use one of the

micromechanical models (either Tandon-Weng or Halpin-Tsai model) to calculate

the thermoelastic properties of a unidirectional short fiber reinforced unit. The

elastic tensor for the unidirectional reinforced unit is subsequently used in

orientation averaging to determine the elastic tensor for all the actual fiber

orientation states which are present in the simulated injection molded part. The

procedure of using mold filling simulations followed by local property calculations

and structural FEA provides a very efficient way for product design with short fiber

composites. It is, however, unclear if this approach is accurate enough in order to

be used in practice for designing injection molded load bearing short fiber

reinforced structures. Therefore, one of the goals of this thesis was to investigate

if the combination of micromechanical models and orientation averaging is

accurate and reliable enough to successfully predict the local thermoelastic

properties based on the local fiber orientation states calculated by mold filling

simulations.


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1.3 POLYMER NANOCOMPOSITES

An interesting class of materials are polymer nanocomposites whose matrix

properties are promoted with the dispersion of high-aspect ratio, submicron-sized

particles, such as intercalated or exfoliated atomic-thickness sheets of layered

silicates, carbon nanotubes[2] and cellulose whiskers[3,4]. Especially polymer-

layered silicate nanocomposites have attracted a lot of attention in the last

decade.[5-15] An even dispersion of just a few weight percent of such 1nm thick,

high aspect ratio silicate platelets in a polymer can already significantly enhance

modulus, thermal stability, flame retardance, dimensional stability, heat distortion

temperature as well as the barrier properties and corrosion resistance.[16,17] For

example, a doubling of the tensile modulus and strength is achieved for a nylon 6

matrix comprising 2 vol% of montmorillonite.[18-20] In addition, the heat distortion

temperature increases by up to 100°C which opens new possibilities of

applications, for example for automotive under-the-hood parts.[18] In comparison

to conventional, highly filled microcomposites one can save up to 25% weight with

nanocomposites due to the low weight fraction of layered silicates, which have

about a 3 times larger density than polymers. At the same time one can benefit

from an improved and broadened property portfolio. Therefore such materials are

especially attractive for the automotive and the mobility sector in general. It is

crucial, however, that the cost/performance ratio for this class of materials

becomes attractive enough for this highly competitive industry sector. Since the

raw materials of nanocomposites are cheap it is mainly the processing cost which

are decisive for the final cost of these materials. There are already a few existing

applications of polymer nanocomposites e.g. automotive timing-belt covers[21]

and fascia. It is expected that this class of materials can provide property

portfolios which are not only interesting for the automotive[22] but also for the food

packaging industry where one can take advantage of the decreased gas

permeability while retaining flexibility and optical clarity of the pure polymer.

Research on polymer nanocomposites comprising layered silicates started

only 10 years ago and focused mostly on the synthesis of such materials by

elaborate and expensive processing routes revealing outstanding property


- 9 -

enhancements. More recently researchers focused more on simpler and cost-

effective processing routes like melt compounding in order to promote this class

of materials for industrial applications. In parallel researchers also began to build

up a better theoretical understanding about the reasons which lead to the

outstanding property enhancements. An extensive and interesting review article

on the preparation, the properties and the use of polymer-layered silicate

nanocomposites has been published by Alexandre and Dubois.[23]

In this thesis the thermoelastic and barrier properties of polymer-layered

silicate nanocomposites were investigated by direct numerical FE-simulations.

The goal was to develop a more profound understanding of how the morphology

(aspect ratio, volume fraction and orientation of platelets) affects the overall

thermoelastic and barrier properties of nanocomposites and to predict the

possible property improvements both for ideal and imperfect morphologies.

Concretely, the geometry dependent enhancements of the thermoelastic and

barrier properties for nanocomposites comprising perfectly aligned mineral

platelets and the barrier losses due to platelet misalignments were investigated.

The findings of this thesis enable experimentalists to rationally choose suitable

morphologies for certain property requirements before any experiment is done in

the lab. This can obviously accelerate the development of nanocomposites for

barrier, load bearing and other applications.


- 10 -


- 11 -

2. ANALYTICAL AND NUMERICAL METHODS

2.1 MICROMECHANICAL MODELS

In addition to the experimental techniques of processing and measuring the

overall effective properties of composites their theoretical prediction starting from

the intrinsic properties of the constituents and the composite’s morphology has

been the subject of extensive studies.[24] The first theoretical considerations of

two phase systems go back to James Clerk Maxwell who derived an expression

for the specific resistance of a dilute suspension of spheres in an infinite isotropic

conductor.[25] Subsequently, accurate rational models of two-phase composites

with spherical or infinitely long cylindrical inclusions have been developed for

predicting elastic, thermal, transport and other properties.[24]

There exist numerous micromechanics-based models which were

developed to predict a complete set of elastic constants for aligned short-fiber

composites. One of the most popular ones is the Halpin-Tsai model which was

initially developed for continuous fiber composites and which was derived from

the self-consistent models of Hermans[26] and Hill[27]. The Halpin-Tsai

equations can be expressed in a short and easily usable form which might be one

of the reasons why they have found a broad usage:

(1)

Vf is the fiber volume fraction and stands for any of the moduli listed in

Table 1. E11 and E22 are the longitudinal and the transverse Young’s modulus,

G12 and G23 the in-plane and out-plane shear modulus, respectively. K23 is the

plane-strain bulk modulus and ν21 the longitudinal Poisson’s ratio of the

unidirectional transversely isotropic short fiber composite. The corresponding

values of the empirical parameter ζ are also listed in Table 1.

MMm--------

1 ζηVf+1 ηVf–

--------------------- with ηMr 1–Mr ζ+----------------==

MrMfMm--------=

Chapter 2 - Analytical and Numerical Methods

- 12 -

Table 1: Halpin-Tsai parameter ζ is listed for the different substitutions of Mf and Mmused in Eq. (1)

ζ is correlated with the geometry of the reinforcement and it was found empirically

that predictions for E11, the Young’s modulus in fiber direction, are best if ζ=2a,

where a is the fiber aspect ratio, defined as:

(2)

L is the fiber length and d the fiber diameter. It can be shown that for the

Halpin-Tsai equations become the rule of mixtures (Voigt bound) where fiber and

matrix experience the same, uniform strain:

(3)

The rule of mixtures is also applied to calculate the longitudinal Poisson’s ratio ν21

although predictions are not accurate when matrix and fibers have considerably

different Poisson’s ratios.

(4)

The Halpin-Tsai model can deal both with isotropic and transversely

isotropic fibers e.g carbon fibers because the underlying self-consistent theories

of Hermans and Hill apply also to transversely isotropic fibers.

M Mf Mm ζ

E11 Ef Em 2a

E22 Ef Em 2

G12 Gf Gm 1

G23 Gf GmKmGm-------

KmGm------- 2+ ⁄

a Ld---=

ζ ∞→

M VfMf VmMm+=

ν21 Vfνf Vmνm+=


- 13 -

Empirical and semi-empirical equations like the treatments of Halpin and

Tsai [28,29], which are widely used in industry can only be useful in reproducing

available experimental data.[30-32] They always reflect the existing technological

level and can therefore not be helpful in deciding if in principle the performance

of a certain composite can be improved further or not. To make this decision it is

necessary to quantify the degradation effects of imperfections like fiber or platelet

agglomerations and their poor adhesion to the polymer. Based on these

quantifications one could decide about the potential of further improvements of

the composite’s properties by controlling the degree of imperfections. Empirical

equations can not fulfill this task and therefore it is necessary to have another

method at hand which can predict the in principal achievable effective properties

of fiber- and platelet-reinforced composites.

A well established and theoretically well founded micromechanical model is

the one of Tandon and Weng[33] which is based on the Eshelby’s solution of an

ellipsoidal inclusion in an infinite matrix[34] and Mori-Tanaka’s average

stress[35]. This model is applicable to spherical, fiber- as well as to disk-shaped

particles, which are called “platelets” throughout this work. The Tandon-Weng

model predicts the five independent effective elastic constants of a transversely

isotropic composite for any fiber aspect ratio from zero to infinity by the following

analytical equations:

(5)

(6)

(7)

E11Em-------- 1

1 VfA1 2νmA2+( )

A6---------------------------------+

--------------------------------------------------=

E22Em-------- 1

1 Vf2νmA3– 1 νm–( )A4 1 νm+( )A5A6+ +[ ]

2A6---------------------------------------------------------------------------------------------------+

-------------------------------------------------------------------------------------------------------------------=

G12Gm--------- 1

Vfµm

µf µm–----------------- 2VmS1212+-----------------------------------------------+=


- 14 -

(8)

(9)

(10)

Eq. (9) was derived by Tucker[32] and is not the original equation for ν21 which

was found by Tandon and Weng because the original equation is coupled with Eq.

(10) and could therefore only be solved iteratively. The parameters A1,...,A6,

B1,...,B5 and D1,...,D3 are defined as following:

(11)

(12)

G23Gm--------- 1

Vfµm

µf µm–----------------- 2VmS2323+-----------------------------------------------+=

ν21νmA6 Vf A3 νmA4–( )–A6 Vf A1 2νmA2+( )+------------------------------------------------------=

K23Km--------

1 νm+( ) 1 2νm–( )

1 νm 1 2ν21+( )– Vf2 ν21 νm–( )A3 1 νm 1 2ν21+( )–[ ]A4+{ }

A6----------------------------------------------------------------------------------------------------+

-----------------------------------------------------------------------------------------------------------------------------------------------------------=

A1 D1 B4 B5+( ) 2B2–=

A2 1 D1+( )B2 B4 B5+( )–=

A3 B1 D1B3–=

A4 1 D1+( )B1 2B3–=

A51 D1–( )B4 B5–( )-----------------------=

A6 2B2B3 B1 B4 B5+( )–=

B1 VfD1 D2 Vm D1S1111 2S2211+( )+ +=

B2 Vf D3 Vm D1S1122 S2222 S2233+ +( )+ +=

B3 Vf D3 Vm S1111 1 D1+( )S2211+( )+ +=

B4 VfD1 D2 Vm S1122 D1S2222 S2233+ +( )+ +=

B5 Vf D3 Vm S1122 S2222 D1S2233+ +( )+ +=


- 15 -

(13)

λm, µm and λf, µf are the Lamé constants of the matrix and the fibers,

respectively. Sijkl are the non-vanishing components of the Eshelby’s tensor

which depend themselves on the Poisson’s ratio of the matrix νm and the fiber

aspect ratio a. The expressions for the Eshelby’s tensor components Sijkl can be

found in [33]. The Tandon-Weng model was developed for isotropic inclusions but

Qiu and Weng extended it to transversely isotropic inclusions[36]. Therefore it is

also possible to predict the effective elastic constants of carbon fiber reinforced

composites. Although the Tandon-Weng model is widely perceived to give the

best predictions for fiber and platelet filled composites it has never been shown

by direct comparison with experimental results of unidirectional short fiber

composites that predictions are accurate. This due to the fact that it is almost

impossible to produce samples of short fiber composites with perfectly aligned

fibers.

Therefore, another concept, namely the one of the rigorous upper and lower

bounds was developed which is one of the most firmly established. Bounds are

clearly preferable to the use of uncertain micromechanical models because they

deliver rigorous upper and lower margins on the effective properties of a

composite. The most popular bounds are the Hashin-Shtrikman variational

bounds which were developed in order to predict both the elastic[37,38] and the

dielectric constants[39-41] if no morphological information apart from the volume

fractions of the phases is available. The bounds for the dielectric constant are

equally applicable in order to predict properties like the electric and thermal

conductivity as well as the diffusion coefficient. The main drawback of the

rigorous upper and lower bounds is that if the ratio of the constituent’s properties,

e.g , is rising, the bounds become increasingly widely separated and thus

D1 1 2µf µm–( )λ f λm–

----------------------+=

D2λm 2µm+( )λ f λm–( )

----------------------------=

D3λm

λ f λm–( )----------------------=

GfGm-------


- 16 -

practically useless if one wants to predict the effective properties of a two-phase-

composite. Often one is interested in mixing two constituents with a preferably

large difference in their intrinsic properties because then the most attractive

property enhancements can be expected. In this case, however, neither

micromechanics-based models nor rigorous upper and lower bounds can make

firm predictions about the overall effective properties of the composite.

Furthermore both micromechanical models and rigorous upper and lower bounds

are only capable of dealing with two-phase composites. As soon as more than two

phases are present one is supposed to use a series of two-phase homogenization

steps as it is described in classical textbook guidelines. It has been found, though,

that using this additivity premise does not deliver reliable property predictions,

e.g. for composites which are highly filled with ceramic particles.[42]

Another approach to determine the effective properties of composites is FE-

modeling. The problem of this method is that the models are often rudimentary,

e.g. consisting of one or two aligned fibers with regular spatial symmetries, which

are hardly found in real composites. As a consequence numerical results are not

representative of real composites and therefore useless for practical design

purposes. At the end of the 1990’s, however, Gusev developed a FEM with which

it is possible to generate sophisticated multi-inclusion Monte-Carlo (MC) models

for a great variety of composite morphologies. By consistently using periodic

boundary conditions (PBC) throughout model and mesh generation as well during

the numerical solution for the overall, effective properties it has been shown that

this FEM delivers remarkably accurate predictions from surprisingly small

computer models. In chapter 2.3 the Gusev’s FEM is described in detail.

2.2 ORIENTATION AVERAGING

As soon as the inclusions have an anisotropic shape we observe anisotropic

overall properties of the composite. Maximal anisotropy is achieved when all

inclusions e.g. fibers or platelets are unidirectional aligned. In this case we

observe a maximum reinforcement for fibers in the longitudinal direction and for

platelets in the transverse directions. If the inclusions are randomly oriented

throughout the matrix, the composite shows macroscopically isotropic behavior.


- 17 -

Between these two extremes the degree of anisotropy gradually decreases until

it disappears for randomly oriented inclusions. It was found that the anisotropy of

several material properties (elastic stiffness, thermal conductivity, viscosity) can

be directly related to the orientation state of the inclusions. As a consequence,

different methods have been developed which can be used to determine the

property tensors of anisotropic materials based on the orientation state of the

inclusions in a composite (equivalent to the treatment for the degree of crystalline

orientation in an unreinforced pure polymer). The orientation averaging scheme

is one of the methods to predict the overall properties of a known orientation state

of e.g. fibers1 by averaging the unidirectional property tensor T(p) over all

directions weighted by the orientation distribution function ψ(p).[43] The

orientation of a fiber is defined by a direction unit vector p with components p1,

p2, p3 in a cartesian coordinate system (see Figure 2).

Figure 2: The orientation of a fiber can be defined by a unit vector p whose components p1, p2, p3 depend on the two angles θ and φ depicted in this figure.

1. From now on fibers are considered although the orientation averaging scheme is equally applica-ble to any other anisotropic shaped inclusions like platelets, spheroids, ellipsoids etc.

φ

θ

y

x

z

2

3

1

P


- 18 -

The components can be expressed by the angles φ and θ as following:

(14)

Thus, the orientation averaging scheme can be expressed as:

(15)

The probability distribution function ψ(p) indicating the probabilities of finding

fibers with a certain orientation p in the composite is the most accurate form to

describe the fiber orientation state. It is, however, too cumbersome for numerical

calculations and therefore efforts have been made to find alternative ways of

describing orientation states.[45,46] One of the most general but nevertheless

most concise descriptions can be made by using orientation tensors. The

orientation state of a set of fibers, for example, can be defined by an infinite series

of even order orientation tensors. The 2nd order orientation tensor is determined

by forming dyadic products with all possible direction unit vectors p and

integrating the product of the resulting tensors with the distribution function ψ(p)

over all possible directions of p.[43]

(16)

(17)

The indices i, j, k, l run from 1 to 3. All orientation tensors are symmetric and the

2nd and 4th order tensors consist of 6 and 15 independent components,

respectively. If the laboratory frame coincides with the principal axes then all non-

diagonal components become zero and the number of non-zero components is

reduced to 3 and 6, respectively. In this case the components are defined as

follows:

p1 θcos=

p2 θ φcossin=

p3 θ φsinsin=

T⟨ ⟩ T p( )ψ p( )dp∫°=

aij pipj⟨ ⟩ pi pj ψ p( ) dp∫°= =

aijkl pipjpkpl⟨ ⟩ pi pj pk pl ψ p( ) dp∫°= =


- 19 -

(18)

(19)

Any tensor property T(p) of a unidirectional microstructure aligned in the

direction of p must be transversely isotropic, with p as its axis of symmetry. To be

transversely isotropic a 2nd order property tensor Tij(p) must have the form

(20)

where δij is the unit tensor.

Applying orientation averaging to Tij(p) gives:

(21)

Eq. (21) proves that the orientation average of a material property which can be

represented by a 2nd order tensor, e.g the permeability, is completely determined

by the 2nd order orientation tensor aij and by the underlying unidirectional

permeability tensor which determines the scalar constants A1 and A2 as following:

(22)

a11 θ2cos⟨ ⟩=

a22 θ φ2cos2

sin⟨ ⟩=

a33 θ φ2sin2

sin⟨ ⟩=

a1111 θ4cos⟨ ⟩=

a1122 θ2 θ φ2cos2

sincos⟨ ⟩=

a1133 θ θ φ2sin2

sin2

cos⟨ ⟩=

a2233 θ φ φ2sin2

cos2

sin⟨ ⟩=

a2222 θ φ4cos4

sin⟨ ⟩=

a3333 θ φ4sin4

sin⟨ ⟩=

Tij p( ) A1pipj A2δij+=

T⟨ ⟩ ij A1 pipj⟨ ⟩ A2 δij⟨ ⟩ A1aij A2δij+=+=

A1 P1 P2 and A2 P2=–=


- 20 -

Therefore to calculate the permeability one only needs to know the 2nd order

orientation tensor aij of the actual composite morphology and the longitudinal and

transverse permeability coefficient P1 and P2 of the corresponding unidirectional

composite. The linear-elastic and the thermoelastic properties, however, require

knowledge of both the 2nd and the 4th order orientation tensor because the elastic

properties are characterized by a 4th order tensor. The orientation averaged

elastic tensor , is defined as:

(23)

are five scalar constants related to the elastic constants of a

transversely isotropic orientation state with fully aligned fibers[43, 44]

(24)

Although the thermal expansion is characterized by a 2nd order tensor the

orientation averaging of the thermal expansion tensor also requires the 4th

order orientation tensor. The reason is that the thermal expansion is directly

related to the elastic properties of a material. The orientation averaged thermal

expansion tensor is given by:

(25)

Cijkl⟨ ⟩

Cijkl⟨ ⟩ B1aijkl B2 aijδkl aklδij+( ) B3 aikδjl ailδjk ajkδil ajlδik+ + +( )+ + +=

B4 δijδkl( ) B+ 5 δikδjl δilδjk+( )

B1 … B5, , Cijkl

B1 C11 C22 2C12– 4C66–+=

B2 C12 C23–=

B3 C6612--- C23 C22–( )+=

B4 C23=

B512--- C22 C23–( )=

αkl

αkl⟨ ⟩

αkl⟨ ⟩ Cijklαkl⟨ ⟩ Cijkl⟨ ⟩ 1– D1aij D2δij+( ) Sijkl⟨ ⟩= =


- 21 -

where D1 and D2 are again two invariants which depend on the elastic and

thermal expansion tensor of the unidirectional composite.[44]

(26)

2.3 GUSEV’S FINITE-ELEMENT METHOD

A new FEM for predicting the properties of multi-phase materials based on

3D periodic multi-inclusion computer models has been developed by

Gusev.[47,48] This FEM excels that with remarkably small computer models one

can accurately determine the overall effective properties of ‘real’ composites with

complex morphologies comprising any desired number of anisotropic phases. In

the last few years the problem of obtaining accurate predictions from small

computer models has extensively been studied. It has been demonstrated that

PBC are most appropriate to predict the behavior and properties of multi-phase

materials from very small computer models. Numerical calculations have shown

that a unit cell comprising 25 spheres is already representative of a particle filled

composite with a random microstructure.[47] The same was done for short fibers

for which the minimal representative volume element (RVE) size is somewhat

larger. It was demonstrated that computer models comprising 100 parallel fibers

are appropriate to get accurate predictions for the longitudinal Young’s modulus

E11 (see chapter 3.1).

Often composite materials have a complex microstructure containing

inclusions of different size and shape featuring non-uniform orientation

distributions. Based on measured microstructural data a computer model

representative of a real composite morphology can be generated. Then the

computer model is meshed into an unstructured, morphology-adaptive FE-mesh

which is fully periodic.[48] In the first step of mesh construction a set of nodal

points is placed onto the inclusions’ surfaces. In the following step an additional

set of nodes is inserted on a regular grid inside the unit cell. A sequential Bowyer-

Watson algorithm[49] is used to uniquely connect both the surface and grid nodes

to a periodic, morphology-adaptive 3D-mesh consisting of tetrahedra following

D1 A1 B1 B2 4B3 2B5++ +( ) A2 B1 3B2 4B3+ +( )+=

D2 A1 B2 B4+( ) A2 B2 3B4 2B5+ +( )+=


- 22 -

the Delaunay triangulation[50,51] scheme. The initial 3D-mesh normally contains

a large number of ill-shaped tetrahedra (sliver-, cap-, needle- and wedge-like

tetrahedra) which influence the speed of convergence and the accuracy of the

numerical results. The same problems occur for bridging elements which directly

connect two or even more inclusions. To get rid of these tetrahedra types the FE-

mesh is locally refined by inserting new nodes at the centers of the ill-shaped

tetrahedra circumspheres.[48]

When the FE-mesh is finished material properties are assigned to the

individual inclusions and the matrix. Like this each tetrahedron acquires certain

material properties depending on which phase it belongs to. One of the

outstanding possibilities of this FEM is that one can assign anisotropic properties

of crystalline materials belonging to any of the 7 crystal systems (triclinic,

monoclinic, orthorhombic, tetragonal, trigonal/rhombohedral, hexagonal and

cubic) both to matrix and inclusions.

To numerically calculate the overall, effective properties of the modelled

composites, a perturbation of certain type is applied to the computer model and

the material’s response on the perturbation is calculated numerically. For

example, to calculate the effective dielectric properties, one applies an external

electric field and solves the Laplace’s equation for the unknown local nodal

potentials by minimizing the total electric energy of the system. At the minimum

the nodal potentials can be determined and the local polarization fields inside

each tetrahedron are uniquely defined. The overall, effective dielectric tensor of

the multi-phase material is finally calculated based on the linear-response relation

between the effective induction and the external electric field. By successively

applying the external electric field in the 1-, 2- and 3-directions of the computer

model’s coordinate system one can calculate the complete dielectric tensor of an

anisotropic composite material. Analogous by numerically solving the Laplace’s

equation, also the overall, effective permeability as well as the electric and

thermal conductance can be determined.

A displacement-based, linear-elastostatic solver is used to numerically

compute the elastic constants and thermal expansion coefficients of multi-phase

materials. The effective elastic properties can again be calculated from the


- 23 -

response to an applied perturbation in the form of a constant effective mechanical

strain εkl. The solver finds iteratively a set of nodal degrees of freedom that

minimize the total strain energy of the system. Conjugate gradient

minimization[52] is used to find this unique energy minimum in the space of

system’s degrees of freedom which is defined by a certain set of nodal

displacements. The knowledge of the nodal displacements allows to determine

the local strains in each tetrahedron and consequently the effective stress σij of

the system. The effective elastic constants Cijkl can then be calculated from the

linear response equation:

(27)

Six independent strain energy minimizations conducted under 6 different

effective mechanical strains (tensile strains in each of the 3 directions and shear

strains in each of the 3 planes of the coordinate system) are necessary to

determine all the 21 independent components of the stiffness matrix. To obtain

the effective thermal expansion coefficient local non-mechanical strains

corresponding to a temperature change of one Kelvin are applied, assuming a

zero effective mechanical strain εkl. One last strain energy minimization is

necessary in order to calculate the effective thermal stress σΤij at the energy

minimum. Using the previously calculated effective stiffness matrix Cijkl of the

composite it is possible to determine the 6 independent components of the

effective thermal expansion tensor αij:

(28)

In a similar way, one can evaluate the effective swelling coefficients and the

effective shrinkage caused by chemical reactions or the relaxation of residual

stresses.

It has already been shown that the FEM of Gusev delivers remarkably

accurate predictions for the overall, effective properties of multi-phase

materials.[53-55] Therefore this FEM has also been applied to identify the

technological potential of sphere and platelet filled polymers.[42,56-58]

σij Cijklεkl=

α ij Cijkl1– σkl

T Sijkl σklT= =


- 24 -


- 25 -

3. SHORT FIBER REINFORCED COMPOSITES

In injection molded short fiber composites microstructural variations like

polydispersed fiber lengths and arbitrary fiber orientation states are unavoidable,

and influence the overall elastic properties of the composite. For structural design

of short fiber reinforced parts one would like to be in the position to reliably predict

the thermoelastic properties either by micromechanical or numerical models.

Many micromechanical have been developed[32] but they are often based on

idealized composite morphologies, e.g. a matrix comprising aligned fibers of

equal size[28,33], or a single ellipsoid in an infinite matrix[34]. FEMs often deal

with rudimentary models comprising one or two fibers with regular spatial

symmetries which are rarely if ever found in real composites. In order to predict

the properties of realistic composite morphologies it is, however, necessary to

use models which take into account the ‘real’ composite morphology with all its

imperfections. In this chapter it is shown that with Gusev’s FEM one can make

accurate and precise predictions of the thermoelastic properties of short fiber

composites with complex morphologies which excellently agree with

experimental data. Furthermore it is demonstrated that the property prediction for

composites comprising morphological imperfections, like polydispersed fiber

lengths or arbitrary fiber orientation states can be simplified by eligible averaging

methods.

3.1 FIBER LENGTH DISTRIBUTIONS

One aspect of short fiber composites which can be difficult to address

analytically is the distribution of fiber lengths that are normally present in a real

material. The most popular approach is to replace the fiber length distribution

(FLD) with a single length, normally the number average length LN.

(29)LNNiLi∑Ni∑

-----------------=

Chapter 3 - Short Fiber Reinforced Composites

- 26 -

A number of proposals for this “mean length” have been published for special fiber

orientation states. Takao and Taya [59] and Halpin et al.[60] concluded that the

number average length LN of a distribution was an appropriate value. Eduljee and

McCullough [61] suggested a different average LS to take into account the

skewed nature of real FLDs, in particular to give a heavier weighting to shorter

fibers.

(30)

The Root-Mean-Square (RMS) average LRMS has also been suggested as a

possible descriptor of the FLD.

(31)

For completeness the weight average LW was also taken into account in this

study.

(32)

For fibers of constant diameter one can express Eq. (32) again by using Ni, the

frequency of fibers in a certain length interval.

(33)

It would seem, therefore, that there is merit in being able to model an assembly

of fibers with a ‘real’ FLD, in order to establish whether the distribution can be

replaced by one of the above mean values in order to establish what

LSNi∑NiLi-----∑

------------=

LRMSNiLi

2∑

Ni∑------------------=

LWWiLi∑Wi∑

------------------=

LWNiLi

2∑NiLi∑

------------------=


- 27 -

McCullough[61] describes as ‘the appropriate statistical parameters to represent

the microstructural features of the composite’. The FEM of Gusev offers the

chance to establish which type of mean length is appropriate in order to replace

a length distribution. For this purpose a fiber length distribution measured by

image analysis of a typical injection molded plate was sampled in a MC-run,

producing computer models with an equivalent FLDs. Results from models with

polydispersed fibers were compared to models comprising assemblies of

monodispersed fibers to assess whether the length distribution could be replaced

by a single length.

3.1.1 NUMERICAL

Direct FE-calculations with 3D multi-inclusion computer models were done

under periodic boundary conditions in an elongated unit cell of orthorhombic

shape. All computer models comprised fibers perfectly aligned along the x-axis of

the unit cell and placed on random positions using a MC-algorithm[47]. A typical

example is shown in Figure 5A. For monodispersed fibers the length-to-width

ratio was set to 7.5. The computer models comprising fibers with a distribution of

lengths, however, were generated in a more elongated unit cell with a length-to-

width ratio of 25 (see Figure 5A). This due to the fact that the fibers must not be

longer than the box itself because this would imply self-overlaps under periodic

boundary conditions. Previously, it was checked with monodispersed fibers that

numerical predictions are not influenced by increasing the box aspect ratio from

7.5 to 25.

In order to assure that the computer models are representative of large

laboratory samples the minimal RVE size was investigated. Five computer

models were built with unit cells of different size comprising random dispersions

of 1, 8, 27, 64 and 125 aligned fibers of aspect ratio 30 at volume fraction 15%.

For each unit cell size three MC-runs were performed which delivered three

different fiber arrangements. The elastic properties of each set of three computer

models with a particular size were calculated numerically. From the results the

arithmetic mean and the 95% confidence interval of the longitudinal Young’s


- 28 -

modulus E11 were determined. Figure 3 shows that with 27 fibers one can already

get predictions for E11 deviating only a few percent from the true value provided

that one averages the results of three individual calculations. In case of larger

computer models comprising 125 fibers the individual predictions for the different

MC-configurations show hardly any scatter.

Figure 3: Predictions for the longitudinal Young’s modulus E11 depending on the size ofthe computer models (number of fibers N). The filled circles indicate the arithmetic meanof three individual estimates and the error bars depict the 95% confidence interval.

Consequently, the effective elastic properties for composites with a

monodispersed fiber length were obtained from one single MC-configuration of a

computer model comprising 100 fibers. The short fibers were assigned the

isotropic elastic properties of glass fibers and the matrix the ones of a typical

thermoplast (see Table 2).

Table 2: Isotropic phase properties for glass fibers and a model matrix.

Glass fibres Model matrix

E (GPa) 72.5 2.28

ν 0.2 0.335

α (x 10-6/°C) 4.9 117


- 29 -

Based on these isotropic phase properties the Young’s modulus E11 in fiber

direction was calculated for 15 models each comprising monodispersed fibers

with an aspect ratio between 5 and 50 at a volume fraction of 15%.

The next stage was to generate computer models using a distribution of fiber

lengths. In order to be representative, a real data set, measured with image

analysis facilities developed at Leeds, was used as the basis for the computer

model generation. The measured data for 27,500 fibers, collected by Bubb from

an injection molded short glass fiber filled plate[62], is shown in Figure 4. For this

non-symmetrical distribution the number average length was determined as

388µm and the weight average length as 454µm. Assuming a common glass fiber

diameter of 10µm gives aspect ratios of 38.8 and 45.4 for the length and weight

averages, respectively.

Figure 4: Experimentally measured fiber length distribution (FLD) collected by Bubb froman injection molded short glass fiber filled plate.[62]

As described earlier, the measured FLD was used to bias the MC-runs. For

this purpose the measured frequency distribution of the fiber lengths was

transformed into the cumulative PDF. The cumulative PDF was then sampled by

generating 100 random numbers in the interval [0,1], whence each random

number corresponds to a particular fiber length. The 100 fibers with the previously

0

1000

2000

3000

4000

5000

0 400 800 1200

L (µm)

frequ

ency


- 30 -

sampled fiber lengths were randomly placed in parallel in the unit cell without

overlaps at a volume fraction of 15% (see Figure 5A). The fiber length distribution

was sampled in 3 different MC-runs in order to better approximate the measured

distribution. In each of the 3 MC-runs a different seed was used for the random

number generator, which consequently delivered computer models with three

different FLDs. Averaging these three fiber length distributions excellently

approximated the experimentally measured FLD (see Figure 5B). The computer

models with the polydispersed fibers, were meshed and solved numerically in

order to determine the longitudinal modulus, E11.

Figure 5: A: Orthorhombic unit cell containing 100 randomly situated, perfectly alignedfibers of different length at volume fraction 15%. The fiber lengths were determined bysampling the measured fiber length distribution (see Figure 4) during a MC-run. All fiberswere assumed to have a diameter of 10µm. B: Measured fiber length distribution (solidline) and the average fiber length distribution of 3 different MC-runs (bars).

3.1.2 RESULTS AND DISCUSSION

The numerical results of E11 are shown in Figure 6. The diamond symbols

represent the results for different aspect ratios of monodispersed fibers, and the

solid line the best fit through all the data. The random nature of the generated

microstructures is reflected by the scatter of the points around the best fit line. It

is typical of short fiber reinforced composites that E11 is levelling off towards

larger aspect ratios. Above a certain critical fiber aspect ratio no substantial gains

in E11 can be achieved.

A

12

3

B


- 31 -

Figure 6: Numerical results for E11 are depicted as filled symbols for compositescomprising monodispersed fibers. The solid line fits the simulation data best.

The question to be answered in this chapter is: “What is the length of a

monodispersed distribution, which would have the same longitudinal modulus as

the ‘real’ distribution?” Figure 7 shows a comparison of the numerical results from

simulations with monodispersed and polydispersed fiber lengths.

Figure 7: A comparison of numerical results for E11 calculated with computer modelswhich comprised either of monodispersed or of polydispersed fibers. The solid, horizontalline shows the average E11 calculated from three different MC-configurations ofpolydispersed fibers. The triangles symbolize E11 which was predicted from severalcomputer models with monodispersed fibers of different aspect ratio.

4

6

8

10

12

0 10 20 30 40 50a

E11

(GP

a)

7

9

11

13

20 30 40 50

a

E11

(GP

a)


- 32 -

The horizontal lines show the band of predictions made with the three

computer models comprising polydispersed fibers, in this case 10.9 ± 0.12 GPa.

The triangles represent the predictions from the simulations with monodispersed

fibers for six particular aspect ratios. The crossing point between the best line fit

through the diamonds and the horizontal lines, determines the monodispersed

aspect ratio which matches E11 of the composite with the real distribution. For this

set of data the equivalent monodispersed aspect ratio was 36.6 ± 2.5.

To explore different regions of the modulus versus aspect ratio curve shown

in Figure 6, fiber aspect ratio distributions (ARD) were generated by using the

FLD of Figure 5B assuming different fiber diameters of 15, 20 and 25µm. Figure

8 shows the ARD for fiber diameters of 10µm (as used so far) 15µm and 20µm.

One can see that as the fiber diameter is increased the distribution is pushed to

lower aspect ratios. As above, the monodispersed length needed to match the

modulus of the ‘real’ distribution was determined for each distribution.

Figure 8: ARD for fibers with diameter d of 10, 15 and 20 microns generated by usingthe measured FLD of Figure 5B.

Results are shown in Figure 9 and Table 3. Although the monodispersed

fiber aspect ratio that matches the properties of composites with polydispersed

fibers does not fit exactly with one of the four considered averages, the number

average LN appears to be the best choice to cover the whole range of likely aspect

0

1000

2000

3000

4000

5000

0 20 40 60 80 100 120

a

frequ

ency

10 microns 15 microns 20 microns


- 33 -

ratios. This result explains why the number average LN has proved so successful

in substituting FLDs, although until this point there has been little justification for

its use.

Figure 9: For different fiber diameters d the monodispersed fiber aspect ratios a (filledcircles) are depicted which match the E11 predictions of computer models comprisingpolydispersed fibers. The four lines represent the different averages that wereconsidered.

Table 3: For different fiber diameters the monodispersed fiber aspect ratio a is listedwhich matches the E11 predictions from computer models comprising polydispersedfibers. Also the numerical values of the four considered average types are listed.

3.2 COMPARISON BETWEEN MICROMECHANICAL MODELS, NUMERICAL PREDICTIONS AND MEASUREMENTS

In this subchapter the focus is on another type of morphological

imperfection, namely the one of fiber misalignments. The goal was to reproduce

10

20

30

40

50

5 10 15 20 25 30d (µm)

a

Monodispersed fibers

Number average

Weight average

RMS average

Skewed average

d (µm) 10 15 20 25 a 36.6 ± 2.5 24.3 ± 1.4 20.7 ± 0.5 15.8 ± 0.5

aN 38.8 25.9 19.4 15.6 aW 45.4 30.2 22.7 18.1

aRMS 41.9 28.0 21.0 16.8 aS 33.2 22.1 16.6 13.3


- 34 -

measured fiber orientation distributions in 3D-multi-inclusion computer models, to

numerically calculate their thermoelastic properties and to compare the results

with experimental measurements and micromechanical models. For this purpose,

the fiber orientation distributions of two differently processed short glass fiber

reinforced composites were determined experimentally and subsequently

reproduced in 3D multi-inclusion computer models. In analogy to the previous

subchapter, the two measured fiber orientation distributions were sampled during

a MC-run.

3.2.1 MICROMECHANICAL MODELS

Micromechanical models combined with the orientation averaging scheme

can be used to predict the elastic properties of composites with misaligned fibers.

For this purpose the composite is considered as an aggregate of elastic units

comprising perfectly aligned fibers, whose properties can be calculated by a

micromechanical model. The properties of the aggregate are predicted by

orientation averaging the unit properties according to the measured orientation

distribution via the tensor averaging scheme described in chapter 2.2. Crucially,

the averaging can be done either assuming constant strain between the units

(averaging the stiffness constants of the units) which leads to an upper bound

prediction, or by assuming constant stress between the units (averaging the

compliance constants of the units) which leads to a lower bound prediction. The

advantage of the numerical approach of Gusev employed here is that only a

single estimate is produced, with no assumptions of constant strain or stress

being imposed.

In terms of the unit predictions, the micromechanical models chosen here

were those accepted as the most appropriate in literature[32,53]. For isotropic

fibers (i.e. glass) the approach of Tandon and Weng [33] is widely accepted as

giving the best unit predictions. The Halpin-Tsai model was chosen because it is

the most widely used micromechanical model in industry.

With respect to the thermal expansion, the overall CTEs αi of two phase

composites with arbitrarily shaped phases are uniquely related to the overall


- 35 -

elastic compliances , and one can use the explicit formula of

Levin:[24,71]

(34)

The superscripts 1 and 2 stand for the fiber and the matrix phases,

respectively, and the general summation convention is used for the indices

occurring twice in a product. Thus, for any composite with a single type, fully

aligned but not necessarily equal length fibers, the overall thermal expansion

coefficients αi are not truly independent entities and one can always use Eq. (34)

to obtain the αi in a simple calculation from the accurate in principle numerical Cik.

If both fibers and matrix are isotropic the Levin formula can also be applied to

calculate the CTE of composites with misaligned fibers because still it can be

viewed as a two phase composite. However, for anisotropic fibers e.g carbon

fibers Eq. (34) is not valid any more because differently oriented fibers have

generally different laboratory-frame elastic constants. Since in this chapter we

deal with composites where both matrix and misaligned fibers are isotropic the

Levin formula was used to compute the CTEs of composites with misaligned

fibers.

3.2.2 EXPERIMENTAL

Circular dumbbells (see Figure 10) were injection molded by conventional

and shear controlled orientation injection molding (SCORIM) using a mold gated

at both ends. During processing, the flow from the larger to the smaller dumbbell

cross section produces preferred fiber alignment in the smaller central section

due to elongational flow.[64] The first set of samples was produced by

conventional injection molding where the polymer/glass-fiber melt was injected

into the mold through one gate of the mold before the sample was cooled down.

For the second set of samples the SCORIM process developed at the University

of Brunel was used. Again the polymer/glass-fiber melt was injected through one

gate but during cooling of the sample, the polymer melt containing the glass fibers

was forced back and forth through the mold cavity using both gates of the mold.

Sik Cik1–=

α i αk1( ) αk

2( )–( ) Skl1( ) Skl

2( )–( )1–Sli Sli

2( )–( ) α i2( )+=


- 36 -

Due to the additional shear forces experienced by the melt during the SCORIM

process, the fibers are more aligned along the dumbbell axis than in

conventionally injection molded samples.[65]

Figure 10: Picture of a circular dumbbell manufactured by injection molding from a glass-fiber-polypropylene granulate.

The material used was a glass-fiber-polypropylene granulate from Hoechst,

Grade G2U02, containing 20 wt% of short fibers. The polypropylene was an easy

flowing injection molding grade with a melt flow index of 55. Specifications of the

thermoelastic properties of the polypropylene matrix and the glass fibers are

listed in Table 4.

Table 4: Thermoelastic properties of polypropylene and glass fibers that were used tocalculate the overall properties of short glass fiber reinforced composites bothnumerically and by the use of micromechanical models.

The degree of fiber orientation in each type of injection molded samples was

measured on a two-dimensional longitudinal cut through the axis of the central

gauge length section, using an image analysis system[66] developed in-house at

the University of Leeds. This image-analysis system, whose reliability and

accuracy has already been validated,[67] was used to measure the angular

deviations θ (see Figure 2) of the glass fibers from the ideal orientation along the

80mm

5mm 8.5mm

25mm

Polypropylene Glass fibres E [GPa] 1.57 72.5 ν 0.335 0.2 α [x 10−6 Κ−1] 108.3 4.9


- 37 -

dumbbell axis. The orientation in both samples was found to be non-uniform with

a well aligned shell region around a central, less well aligned, core. This pattern

of fiber orientation was found to be symmetric about the centre line of the section

and consistent along the gauge length.

Typical image frames (700µm x 530µm) taken from the shell region of each

sample type are shown in Figure 11 with the injection axis in horizontal direction.

It is clear that in the SCORIM sample the fibers are more highly aligned along the

1-axis compared to the conventionally molded sample, which itself has a high

preferential alignment. To compare with mechanical measurements, the fiber

orientation distributions for each gauge length cross section was required. To

produce this distribution, the 2D image analysis data was divided into 10 strips

across the sample diameter and then normalized in terms of the appropriate

angular area. As the distributions were found to be transversely isotropic, for

orientation averaging purposes they can be described by only two orientation

averages, <cos2θ> and <cos4θ>. The measured values of these two averages

were 0.872 and 0.769 for the conventionally and 0.967 and 0.936 for the SCORIM

molded samples, for the second and fourth orders respectively.

Figure 11: Figure 11A and Figure 11B show longitudinal cuts through the gauge sectionof a conventionally and a SCORIM injection molded dumbbell, respectively. Typicalimage frames (700 x 530 µm) from the shell region of the samples’ gauge section aredepicted.

In order to measure the fiber length distribution of the two samples the

polypropylene matrix was first burnt off at a temperature of 450°C in a furnace.

The remaining glass fibers were spread onto a glass dish and their length

BA


- 38 -

distribution was determined by image analysis. The burn off technique was also

used to confirm that the weight fraction of the glass fibers was 20%, which is

equivalent to a volume fraction of 8%.

The Young’s modulus E11, of the glass fiber reinforced samples was

measured in a tensile test at a constant strain rate of 10-3 s-1. The sample strain

was measured using a Messphysik video extensometer and 10 samples were

measured for both conventionally and SCORIM processed dumbbells. To

determine the properties of the matrix phase, compression molded plates were

made from pellets of the unreinforced polymer. The matrix Young’s modulus was

measured using the same technique as above, while the Poisson’s ratio was

determined using an ultrasonic immersion method.

CTEs were determined for both short fiber reinforced samples and

unreinforced polypropylene using a dilatometer by measuring the length change

of the samples for a temperature change from +10 to +30 °C both in the

longitudinal and the transverse direction of the dumbbells.

3.2.3 NUMERICAL

Computer models comprised 150 misaligned fibers of equal aspect ratio

randomly positioned in a cubic unit cell at a volume fraction of 8%. For both a

conventionally and a SCORIM molded sample the length distribution of the fibers

was measured. In the previous subchapter it was found that the number average

LN is the best choice to substitute a fiber length distribution by a single fiber

length. As a consequence in the computer models for the conventionally molded

composite the number average LN = 448µm was assigned to all fibers whereas

for the SCORIM molded composites a slightly smaller number average

LN = 427µm was employed. The diameter of the glass fibers was measured as

well and found to be 12µm in both samples. The specification of number N,

length L and diameter d of the fibers determines the total fiber volume. Since we

know that the fiber volume fraction is 8% the length of the cubic unit cell is

determined. In a MC-run the cumulative PDF for each of the two measured θ -

distributions was sampled with 150 random numbers in the interval [0,1]. Each


- 39 -

random number assigns an angle θ to one of the 150 fibers in the computer

model. Measurements elucidated that the angle φ is homogeneously distributed

in the interval [0°, 360°] which means that the gauge section of the dumbbell was

transversely isotropic. Therefore another 150 random numbers were used to

randomly determine the second angle φ in the interval [0°, 360°].

Figure 12: The average θ-distribution (grey bars) of 450 fibers in three different MC-snapshots which were generated by sampling the PDF (black curve) in three MC-runs isshown for the conventionally (left) and for the SCORIM molded dumbbell (right). Theangle θ characterizes the fibers’ misalignments in a transversely isotropic composite.

After having specified length, diameter and orientation of all 150 fibers they

were successively placed in the unit cell on random positions while a subroutine

checked for overlaps with already positioned fibers. If overlaps occurred the

position was rejected and the MC-algorithm repeated the procedure until the fiber

could be placed without overlaps and until all fibers were inserted into the unit cell.

Because the fibers were misoriented it was impossible to randomly place the

fibers without overlaps even at the relatively small volume fraction of 8%. This

problem was solved by increasing the box size and inserting the fibers at a dilute

volume fraction of 0.1%. The box was then compressed during a variable-box-

size MC-run to the desired volume fraction of 8% keeping the fiber orientations

constant while repeatedly displacing each fiber in the unit cell. In Figure 14 a

computer model for both the conventional and the SCORIM morphology is shown

together with a cut through the FE-mesh. In order to obtain information about the


- 40 -

scatter of the numerical predictions three different MC-snapshots were generated

for both composite morphologies by sampling the cumulative PDFs with three

different seeds for the random number generator. By averaging the individual

orientation distributions of the three computer models the measured distribution

was better approximated (see Figure 12).

Figure 13: On the left side 3D multi-fiber computer models are shown for both theconventional (top) and the SCORIM (bottom) morphology. On the right side longitudinalcuts through the FE-mesh of both computer models are depicted.

The 6 computer models (3 for the conventional and 3 for the SCORIM

morphology) were meshed into unstructured, morphology-adaptive FE-meshes

and numerically solved for the overall, effective thermoelastic properties. The FE-

meshes of all 6 computer models consisted of about 2.4 x 106 nodes and 15 x 106

tetrahedra. Calculations were done on a HP Visualize J6700 Workstation with two

PA-RISC 8700 processors and took about 25 hours for 7 strain energy

1

2

3


- 41 -

minimizations (6 minimizations to determine the elastic properties and 1

minimization to determine the CTEs) on a single processor.


In this section we present both experimental and numerical results for glass

fiber reinforced polypropylene composites and compare them with values which

were computed by two micromechanical models, namely the ones of Tandon-

Weng[33-36,69] and Halpin-Tsai[28,70], together with the orientation averaging

scheme. Experimental, numerical and micromechanical results of the Young’s

modulus E11 in the longitudinal direction of the glass-fiber/polypropylene

dumbbells are listed in Table 5.

Table 5: The Young’s modulus E11 in the longitudinal direction of both conventionallyand SCORIM injection molded glass-fiber/polypropylene dumbbells. Measured andnumerical results are listed together with micromechanical model predictions.

For the longitudinal Young’s modulus E11 there is an excellent agreement

between the numerically calculated and measured values. The numerically

calculated E11 is nominally higher than the measured value for both the

conventional and the SCORIM sample but the difference is less than 1% and is

well inside the error range of the measurements. The Tandon-Weng model

combined with orientation averaging for determining the aggregate properties

Young’s Modulus E11 [GPa]

Conventional SCORIM

Measured 5.09 ± 0.25 5.99 ± 0.31

Numerical 5.14 ± 0.1 6.04 ± 0.02

Tandon-Weng + Orientation Averaging

Upper bound 5.13 5.91 Lower bound 3.94 5.51 Halpin-Tsai + Orientation Averaging

Upper bound 4.42 5.02 Lower bound 3.63 4.78


- 42 -

gives an upper and a lower bound. Both the experimental and the numerical

values are close to the upper bound predictions of the Tandon-Weng/Orientation-

Averaging approach, confirming the well held belief that a state of constant strain

is the most appropriate for well aligned glass fiber reinforced polymers. The upper

bound deviates by ~1% from the measured E11 for both the conventionally and

the SCORIM molded samples, while the lower bound underestimates E11 by

22.6% and 8%, respectively. The upper bound predictions of the Halpin-Tsai

model combined with the orientation averaging scheme underestimates the

measured E11 by 13% for the conventionally molded and by 16% for the SCORIM

molded composite. Therefore the Halpin-Tsai/Orientation-Averaging approach is

not appropriate to accurately predict the Young’s modulus E11 of short fiber

composites.

Table 6: Longitudinal and transverse CTEs of conventionally and SCORIM moldedglass-fiber/polypropylene dumbbells. Measured and numerical results are comparedwith micromechanical models. The Levin formula (Eq. (34)) was employed together withthe upper bound prediction of the stiffness tensor and the orientation averaging schemein order to predict the CTEs.

Longitudinal thermal expansion α1 [x 10-6 K-1]

Conventional SCORIM

Measured 32.9 ± 1.5 27.7 ± 1.7

Numerical 30.6 ± 1 29.3 ± 0.1

Tandon-Weng + Orient. Av. + Levin 30.1 29.7

Halpin-Tsai + Orient. Av. + Levin 36.1 35.3

Transverse thermal expansion α2 [x 10-6 K-1]

Conventional SCORIM

Measured 121 ± 2 121 ± 1

Numerical 115 ± 1 119 ± 0.1

Tandon-Weng + Orient. Av. + Levin 116 120

Halpin-Tsai + Orient. Av. + Levin 106 109


- 43 -

Table 6 lists for the two investigated composite morphologies the measured,

numerical and micromechanical results for the longitudinal and the transverse

CTE α1 and α2. The micromechanical results were determined with Eq. (34) using

the upper bound prediction of the elastic tensor Cik from the combined

micromechanics/orientation-averaging approach. For α1 and α2 the agreement

between numerical predictions and measurements is good. The measured

longitudinal CTE α1 for the conventional sample is slightly larger than what was

predicted numerically whereas for the SCORIM sample the measured value is

lower than the numerically calculated one. For the transverse CTE α2 resulted the

same CTE from the dilatometer measurement. Numerical calculations, however,

predict a lower α2 for the conventional compared to the SCORIM composite,

which is in harmony with both micromechanical models. The Tandon-Weng/

Orientation-Averaging approach agrees very well with the numerical predictions

for α1 and α2 while the Halpin-Tsai approach overestimates α1 and at the same

time underestimates α2. It has already been shown that the Halpin-Tsai model

underestimated the longitudinal Young’s modulus E11. Because the elastic

properties are directly connected to the CTE this explains why α1 is overestimated

by the Halpin-Tsai approach.

3.3 STIFFNESS AND THERMAL EXPANSION OF SHORT FIBER COMPOSITES WITH FULLY ALIGNED FIBERS

Predicting the overall, effective properties of short fiber composites with fully

aligned fibers from the properties of the individual phases and the composite’s

morphology has attracted a great deal of attention during the last decades, and a

variety of micromechanics-based and empirical models have been

proposed.[24,28,32,33,34,69-70] It has however been difficult to objectively

assess the predictive capability of the models, as it is rather hard in practice to

fabricate well-controlled samples of composites with fully aligned short fibers.

Accordingly, it has also been difficult to validate the adequacy of the underlying

assumptions made upon formulation of micromechanics-based models, as well

as to decide on the significance of parameters obtained by fitting a plausible

empirical form against a particular set of experimental data. In this work, we focus


- 44 -

on the overall elastic constants and use the Gusev’s FEM[47,48] to assess the

adequacy of two of the most widely used models, namely that of the

micromechanics-based model of Tandon-Weng[33-36,69] and the semi-empirical

model of Halpin-Tsai[28,70]. In chapter 3.2 a comparison with measurements on

well-characterized laboratory samples, showed that the Gusev’s FEM gave

excellent predictions for injection molded short fiber composites. Therefore the

approach taken here is to use accurate in principle numerical predictions in place

of measured values.

3.3.1 NUMERICAL

In this study periodic computer models comprised of a polymer matrix

reinforced by 100 fully-aligned randomly-positioned non-overlapping identical

fibers were studied (Figure 14).

Figure 14: Periodic computer model comprised of 100 fully aligned randomly placed non-overlapping identical fibers of aspect ratio a = 20. The volume fraction is f = 20%.Orthorhombic PBC are imposed.

Using both glass and carbon fibers, calculations with three different polymer

matrices typical of industrial short fiber composites were done (Table 7). Perfect

adhesion was imposed at the fiber-matrix interfaces. A total of nine different

combinations of fiber aspect ratio, a = 10, 20, and 30, and fiber volume fraction,

f = 10%, 20% and 30%, were studied. This parameter range is representative of

most industrial injection molded short fiber reinforced composites.

1

2

3


- 45 -

Table 7: Matrix and fiber elastic properties. E denotes the Young’s modulus and ν thePoisson’s ratio.The elastic parameters of matrix M3 are typical of a glassy polymer, M1of a solid semicrystalline polymer at room temperature, and M01 is representative of asemicrystalline polymer at elevated temperature. For glass fibers we took elasticparameters of E-glass while for carbon fibers those of highly anisotropic fiber-symmetryCourtaulds H370 fibers. The index numbering system for carbon fibers follows Figure14. For example, G12 denotes the shear modulus in the 12-plane.

Computer models were meshed into periodic unstructured morphology-

adaptive tetrahedra-based quality meshes (Figure 15).[48,63] The external strain

was applied by changing the size and shape of the unit cell. A preconditioned

conjugate-gradient solver was employed for minimizing the total strain energy as

a function of the nodal displacements.[47,48] The overall elastic constants Cik

were obtained on the basis of a linear response relation between the average

volume stress and the effective strain applied. More than a hundred different

computer models were studied.1

M3 E = 3 GPa ν = 0.35

M1 E = 1 GPa ν = 0.40

M01 E = 0.1 GPa ν = 0.45

Glass fibers E = 70 GPa ν = 0.20

Carbon fibersE11 = 370 GPa E22 = 12 GPa

G12 = 17.5 GPa ν21 = 0.35 ν23 = 0.48

1. Minimization runs were conducted on the mainframe stardust.ethz.ch cluster of PA8600 (550 MHz) processors administrated by the Department of Mathematics at the ETH-Zürich. Typically, it took several single processor CPU hours to calculate the Cik of a particular computer model.


- 46 -

Figure 15: A: 2D cut through the periodic computer model shown in Figure 14. One cannotice that the fiber sections sketched as spreading outside the simulation box in Figure14 are now entering the simulation box from the opposite side. Both visualization modesare equally consistent under periodic boundary conditions, as in the first case oneperforms visualization based on the fiber center-of-mass positions while in the secondcase on the tetrahedra center-of-gravity coordinates. B: 2D cut through the periodictetrahedra-based mesh employed for predicting the overall elastic constants Cik.numerically. The same fragment as shown in A. The meshes typically consisted of amillion nodes and several million tetrahedra.


Figure 16 presents results on the predictive capability of the Tandon-Weng

model for the overall longitudinal elastic constant, C11, of short glass fiber

composites with a semicrystalline polymer matrix. One can see that the Tandon-

Weng model is very accurate for the cases of high fiber aspect ratios and low fiber

loadings, which was already observed in chapter 3.2 for glass fibers of aspect

ratio 37 at volume fraction 8%. This observation is consistent with the model’s

underlying assumptions, as the model was derived under a dilute condition and it

was shown that it delivered the proper asymptotic predictions for the limiting case

of infinitely long fibers.[24,33] The model’s predictions become, however,

progressively less reliable for composites comprised of shorter fibers dispersed

at higher loadings.

A B


- 47 -

Figure 16: Comparison between numerical and Tandon-Weng predictions for the overalllongitudinal elastic constant C11 of glass fiber composites with matrix M1. In the figure, adenotes the fiber aspect ratio and f the fiber volume fraction. The relative error is definedas . The elastic parameters of matrix M1 and the range of a and fstudied are typical of most widely used in industry short glass fiber composites withcommodity polymer matrices (polypropylene, polyethylene, etc.).

Figure 17: Comparison between numerical and Tandon-Weng predictions for the overallelastic constants of glass fiber composites. Each bar is an average over nine individualerror estimates obtained with a particular matrix reinforced by fibers with aspect ratios 10,20, and 30 dispersed at volume fractions 10%, 20% and 30%. For example, the bar withlabels M1 and C11 is an average over the nine error estimates shown in Figure 16.

10

20

30

[VOL%]

C11num C11

TW–( ) C11num⁄


- 48 -

Figure 17 shows further results on the predictive capability of the Tandon-

Weng model, for a variety of glass fiber reinforced composites. It is seen that the

model gives quite accurate predictions for the C12 and C23 elastic constants,

satisfactory predictions for the transverse C22 elastic constant, but delivers less

reliable predictions for the longitudinal C11 and shear C44 elastic constants,

especially with increasing difference between the fiber and matrix stiffness.

Table 8 provides an overview of the predictive capability of the Tandon-

Weng (the Qiu-Weng version for anisotropic carbon fibers) and Halpin-Tsai

models.

Table 8: Accuracy assessment for the Tandon-Weng and Halpin-Tsai predictions. Thestandard matrix norm errors, , between numerical and model predictions areanalyzed.1 All error estimates are given in percents relative to the numerical results.Each individual error estimate is an average of those obtained with nine combinations offiber aspect ratios a and volume fractions f studied in this chapter, see Figure 16. Wealso show, in the parentheses, the maximal error among these nine individual standardmatrix norm error estimates. For the adjustable parameters of the Halpin-Tsai equations,we used literature recommended values[32,53]. The Halpin-Tsai predictions areobviously unsatisfactory, mostly due to poor predictions for the transverse Poisson’sratio ν23.[54]

One can see that for glass fiber composites with a glassy polymer matrix (M3) the

Tandon-Weng model delivers reliable results, with an accuracy appropriate for

engineering design purposes. However, the model predictions become less

accurate for the most widely industrially used glass fiber composites with

semicrystalline matrices (M1 and M01). For carbon fiber composites, the Qiu-

1. The standard matrix norm is defined as with the summation carried out over

indices i and k running from 1 to 6. As the standard matrix norm tends to smooth all extreme deviations in the individual Cik components, a low error is a necessary rather than a sufficient condition for the all round adequacy.

Tandon-Weng Qiu-Weng Halpin-Tsai

glass carbon glass carbon

M3 4 (7) 13 (21) 40 (103) 28 (35)

M1 11 (17) 26 (33) 45 (127) 42 (57)

M01 32 (46) 40 (61) 50 (89) 60 (76)

∆C C⁄

C Cik Cik⋅∑=

∆C∆C C⁄


- 49 -

Weng predictions are considerably less accurate than the corresponding Tandon-

Weng ones for glass fiber composites. One can also see that the predictions of

the semi-empirical Halpin-Tsai model are systematically less accurate than those

of the Tandon-Weng model.

Regarding the relation to ‘real’ materials, as neither of the two models

considered nor the numerical FEM include the interfacial layers, one can express

concern for the adequacy of both of the routes for the stiffness predictions. To this

important point, the results of the extensive validation program in chapter 3.2 and

in [54] have indicated that for all ‘real’ continuous fiber and short fiber composites

studied at the University of Leeds the FEM gave excellent stiffness predictions,

without any explicit account for the presence of interface layers. A further issue to

be addressed is whether there is any molecular orientation in the matrix phase,

as a consequence of the processing route particularly when injection molding is

used. Detailed experimental studies of injection molded glass-fiber

polypropylene-matrix samples, conducted in Leeds by using Wide Angle X-Ray

Diffraction (WAXS),[72] showed that while molecular orientation was seen for

some unfilled polypropylene samples, this was never prevalent in glass filled

samples, under the same industrially employed processing conditions. The

validation study in chapter 3.2 showed excellent agreement between

experimental measurements and numerical predictions assuming an isotropic

matrix. We also considered the effect of the fiber length distribution,[53] and

demonstrated that one can obtain accurate stiffness and thermal expansion

predictions by replacing the ‘real’, measured fiber length distribution with a

monodispersed, number average fiber length LN.

3.4 PREDICTION OF STIFFNESS AND THERMAL EXPANSION BY THE ORIENTATION AVERAGING SCHEME

It is fairly common in practice that during injection molding, the mold filling

process results in non-uniform fiber orientation distributions in the final injection

molded short fiber reinforced composite part.[73] Consequently, one needs to be

able to deal with spatially non-uniform elastic constants in order to describe the

structural performance of the part. This is, in principle, no problem for the FEM of


- 50 -

structural analysis[74,75], provided that all the elastic constants for all the mesh

elements across the part are known. Nearly two decades ago, Advani and

Tucker[43] proposed an extension of the laminate analogy of Halpin-Pagano[76],

such that the elastic constants of a short fiber composite with any given fiber

orientation distribution can be obtained by averaging the elastic constants of a

composite with fully aligned fibers, weighted by the fiber orientation distribution. It

would be very attractive to employ this orientation averaging scheme to form

predictions for the design of short fiber reinforced composite parts, should this

scheme be accurate enough. Here, Gusev’s FEM is applied for the first time to

directly predict the stiffness and thermal expansion of several hundred multi-fiber

computer models with a variety of different predefined fiber orientation states. The

direct predictions are compared with those obtained by using the orientation

averaging scheme and demonstrate that the orientation averaging scheme

delivers reliable, engineering accuracy predictions. Orientation averaging of the

elastic properties described in Eq. (23) involves fast arithmetic operations and

allows for a quick evaluation of for all the mesh elements during the FE-

assembly stage.[74,43]

3.4.1 NUMERICAL

Periodic computer models comprising 150 non-overlapping fibers with

predefined second order orientation tensors aij were studied, see Figure 18. In

each model the fiber orientation state was adjusted to a specific second order

orientation tensor aij of diagonal form following a regular grid with a spacing of

∆a = 0.1 (see Table 9). Since and all , only 13 grid points

are in fact symmetry independent. To generate a model with a given tensor aij, a

MC-run was conducted with a set of 150 isolated fibers, by changing the Euler’s

angles of the fibers and accepting new configurations whenever they progressed

the system towards the desired orientation state. Then, the fibers were placed

with the so-assigned Euler’s angles inside a large unit cell, and during a variable-

box-size MC-run the fibers were displaced and the box size steadily decreased

Cijkl⟨ ⟩

a1 a2 a3+ + 1= ai 0≥


- 51 -

towards the desired fiber volume fraction. At this stage, fiber orientations were

kept constant and all configurations with fiber overlaps were rejected.

Figure 18: Four different orientation states of short fiber composites. Periodic computermodels with 150 non-overlapping identical fibers of aspect ratio a = 20 are shown. Thefiber volume fraction is f = 15%. In the coordinate frame shown, the models havediagonal second order orientation tensors aij defined by three eigenvalues {a1, a2, a3}.The model with fully aligned fibers has eigenvalues {0, 0, 1}, the 2D-random model {0,0.5, 0.5}, the 3D-random model {0.33, 0.33, 0.33}, and the arbitrary model {0.6, 0.3, 0.1}.

Table 9: Diagonal components a1, a2, a3 of the 13 symmetry independent, diagonal-form second order orientation tensors aij following a regular grid with a spacing of∆a = 0.1.

a1 a2 a3

1 0.9 0.1 0.0

2 0.8 0.2 0.0

3 0.8 0.1 0.1

4 0.7 0.3 0.0

Fully aligned 2D random

3D random Arbitrary

1

2

3


- 52 -

Computer models were then meshed into periodic tetrahedra-based

morphology-adaptive quality meshes,[48,63] see Figure 19. Assuming typical

elastic properties for the fibers and the matrix, see Table 7, the elastic constants

of a total of several hundred computer models were predicted.[53]

Figure 19: A: 2D cut through a computer model with a fiber orientation state describedby the second order orientation tensor aij with eigenvalues {0.6, 0.3, 0.1}, see Figure 18.In the cutting plane, circular fibers appear as ellipses, with the two semi-axes defined bythe fibers’ orientation relative to the cutting plane. This sort of information is typically usedfor experimental characterization of fiber orientation states, based on digitized imageframes obtained from polished sections of laboratory short fiber composite samples.B: magnified fragment of the 2D cut through the three dimensional morphology-adaptivetetrahedra-based mesh used for predicting the properties of this computer modelnumerically. The fragment is marked accordingly in part A. Periodic meshes consist of afew million nodes and several million tetrahedra.

5 0.7 0.2 0.1

6 0.6 0.4 0.0

7 0.6 0.3 0.1

8 0.6 0.2 0.2

9 0.5 0.5 0.0

10 0.5 0.4 0.1

11 0.5 0.3 0.2

12 0.4 0.4 0.2

13 0.4 0.3 0.3

a1 a2 a3

1

2

3

A B


- 53 -


Direct numerical predictions were compared with the obtained by

using the orientation averaging scheme of Eq. (23). The needed invariants

of Eq. (23) were calculated from the Cijkl of the computer models with

fully aligned fibers (see Figure 18) using Eq. (24).

Figure 20 presents results for glass fiber composites with a typical

commodity polymer matrix (M1). One can see that the orientation averaging

predictions agree remarkably well with direct numerical results, with a largest

error of about 3% seen for a 2D-random planar orientation state.

Figure 20: Relative standard matrix norm error between direct and orientationaveraging predictions. Results for a composite with glass fibers of aspect ratio a = 20dispersed in matrix M1 at a volume fraction of f = 15% are shown. The error assessmentis carried out based on computer models with predefined second order orientationtensors. We consider diagonal form tensors and use a regular grid with a spacing of∆a = 0.1. Only 13 grid points are in fact symmetry independent. All of them are shown inthe figure, together with 7 additional symmetry equivalent points included for visualexpediency.

Table 10 provides a summary of the error assessment, over all the

orientation states studied. One can see that the orientation averaging is highly

suitable for predicting the of short glass fiber composites. Although short

glass fiber reinforced composites are the most widely used in industry today,

short carbon fiber reinforcement is also used. Similar orientation averaging

Cijkl⟨ ⟩

B1 … B5, ,

∆C C⁄

Cijkl⟨ ⟩


- 54 -

predictions have been produced for carbon fiber composites. The results are also

shown in Table 10, and although less accurate than the glass fiber predictions,

they still suggest that the orientation averaging scheme is satisfactory for most

practical purposes.

Table 10: Accuracy assessment for the orientation averaging predictions. We show thestandard matrix norm errors, and , between direct and orientationaveraging predictions for the elastic constants and thermal expansion coefficients. Hereresults for composites with fibers of aspect ratio a = 20 dispersed at a volume fraction off = 15% are listed. Each error estimate shown in this table is an average over theestimates obtained with computer models of all the predefined second order orientationtensors studied, see Figure 20. All error estimates are given in percents relative to thedirect numerical predictions. The accuracy assessment for composites with a = 30 andf = 10% and a = 10 and f = 30% has shown results which are very similar to thosepresented in this table.

As an alternative, one can predict the elastic constants by the

inversion of the elastic compliances obtained by averaging the Sijkl of a

composite with fully aligned fibers. The same Eq. (23) can readily be used for this

purpose, with the scalar constants calculated from the Sijkl

components.[24] We checked this calculation and found that this alternative

averaging scheme resulted in very poor predictions, typically with 20-60%

standard matrix norm errors, depending on the particular orientation state and

fiber fraction.

For glass fibers, one commonly assumes isotropic elastic behavior. As a

result, any glass fiber composite with a uniform matrix can be viewed as a two

phase composite and one can therefore use the explicit formula of Levin (see Eq.

(34)) to predict the overall thermal expansion coefficients α ij.[71] The situation is

however different for carbon fiber composites, as carbon fibers are anisotropic

and differently oriented fibers have generally different laboratory-frame elastic

constants. As thermal expansion of three and more phase composites is no

Glass fibers Carbon fibers

Matrix M3 M1 M01 M3 M1 M01

[%] 1.1 1.9 3.3 6.1 8.4 5.6

[%] 3.4 2.5 1.2 5.4 7.6 8.7

∆C C⁄ ∆α α⁄

∆C C⁄

∆α α⁄

Cijkl⟨ ⟩

Sijkl⟨ ⟩

B1 … B5, ,


- 55 -

longer uniquely determined by the overall elastic constants,[24,71] the accuracy

of orientation averaging predictions was assessed.[44] For this purpose, directly

predicted tensors αij were compared with those obtained by using the orientation

averaging scheme the thermal expansion described by Eq. (25). Table 10

provides a summary of the relative standard matrix norm error for all

fiber and matrix types. For completeness, we also give predictions for glass fiber

composites. Overall, the results for the thermal expansion are in concert with

those seen for the elastic constants: the orientation averaging gives excellent

predictions for glass fiber composites and relatively less accurate but still

satisfactory results for carbon fiber composites.

3.5 CONCLUSIONS

It has been shown that the direct FE-based procedure of Gusev has been

developed to a degree of sophistication where it becomes possible to predict the

thermoelastic properties of short fiber reinforced composites more accurately

than with any micromechanical model, and often with a better precision than from

measurements. Composites with polydispersed and misaligned fibers have been

studied and it was demonstrated that the numerical property predictions for

composites comprising fibers with non-uniform length and orientation

distributions can be simplified by eligible averaging methods.

3D multi-inclusion computer models comprised of 100 fibers whose lengths

reproduced a measured fiber length distribution were generated. Assuming

different fiber diameters, it was possible to investigate a variety of ARDs based

on one and the same fiber length distribution. It has been found that the Young’s

modulus E11 in fiber direction of a composite with polydispersed fibers is best

matched by the E11 of a composite with monodispersed fibers when the fiber

length distribution is replaced by the number average fiber length. Based on this

finding the microstructures of the central gauge section of 2 transversely isotropic,

glass fiber reinforced dumbbells produced by two different injection molding

procedures were accurately reproduced in 3D multi-fiber computer models.

Numerical predictions from these computer models were compared with

experimental data of the Young’s modulus E11 and the CTEs α1 and α2. For all of

∆α α⁄


- 56 -

them, it was found that agreement between measurements and numerical

simulations is excellent thus validating Gusev’s numerical approach and

demonstrating that it accurately predicts the elastic constants and the CTEs not

only for idealized morphologies of aligned fibers but also for morphologies

containing imperfections such as polydispersed fibers and fiber misalignments.

Comparison with micromechanical models showed that the Tandon-Weng model

together with the upper bound from orientation averaging is very accurate in

predicting the Young’s modulus E11, confirming the current published literature.

Because the overall thermal expansion coefficients αi of two phase composites

are uniquely related to the overall elastic compliances Sik of the composite it is no

surprise that the Tandon-Weng approach was also in excellent agreement with

the numerically calculated CTEs. The Halpin-Tsai model, however, makes

predictions which deviate considerably from the measured values both for the

conventional and the SCORIM composite.

Further investigations on the accuracy of the Tandon and Weng and the

Halpin-Tsai model in terms of predicting the properties of fully aligned short fiber

composites have shown a more detailed picture. In chapter 3.3 it has been

demonstrated that the Tandon-Weng model is very accurate for glass fiber

reinforced polymers of glassy type (M3) with a Poisson’s ratio around 0.35. But

as soon as one goes to less stiff polymer matrices with a Poisson’s ratio of 0.4 or

more the Tandon-Weng model starts to deviate increasingly from the numerical

predictions. The same trends were observed for the Halpin-Tsai model, only that

its deviations are consistently larger than for the Tandon-Weng model. The

properties of composites with fully aligned short fibers constitute the basis for the

orientation averaging scheme in order to predict the properties of all the vastly

different local fiber orientation states occurring in injection molded composite

parts.[73] Therefore, one would like to have the upmost confidence in these basic

property predictions in order to avoid any possible interference with the

unavoidable uncertainties of orientation averaging schemes commonly employed

for predicting the properties of general fiber orientation states.[73] It appeared

though that neither the Tandon-Weng nor the Halpin-Tsai model provides the sort

of robust accuracy needed for the reliable general purpose structural design of


- 57 -

injection molded short fiber composite parts under all conditions, for example, for

all the intermediate temperatures occurring upon cooling from the processing to

the end use temperature.

On the basis of the numerical predictions for unidirectional short fiber

composites, the numerical procedure of Gusev was used to predict the stiffness

and thermal expansion of periodic multi-fiber computer models with various

predefined fiber orientation states. The numerical predictions were compared with

those obtained by combining predictions of the composites with fully aligned

fibers with the orientation averaging scheme both for the elastic constants and

thermal expansion coefficients. It was demonstrated, on the basis of several

hundred computer models, that the orientation averaging scheme is highly

suitable for the reliable general purpose design of short fiber composite parts. The

only prerequisite is dependable elastic constants for the composites with fully

aligned fibers, and they can now be reliably obtained numerically as described in

chapter 3.3.

Putting things in perspective, direct accurate in principle numerical

predictions provide an appealing alternative to the traditional model route. It was

demonstrated that one can readily use the numerical route for the first principle

predictions of the overall elastic constants and CTE of composites with fully

aligned fibers as well as with arbitrary fiber orientation states, both isotropic

(glass) and anisotropic (carbon). Furthermore, the same numerical procedure can

also be employed for predicting the overall thermal expansion and transport

coefficients, such as thermal and electric conductivity, dielectric constants, etc., of

composites with two and more different sort and varying aspect ratio

fibers.[48,56,57,63]


- 58 -


- 59 -

4. POLYMER-LAYERED SILICATE NANOCOMPOSITES

4.1 PERMEABILITY

Sheets of layered minerals like mica or smectic clays are impermeable for

molecular species. By putting just a few percent of exfoliated high-aspect-ratio

atomic-thickness sheets of a layered mineral into a polymer, one can significantly

improve barrier properties[77-80] but still retain flexibility and optical clarity of the

pure polymer. The problem is to understand the permeability levels that can be

achieved with a particular nanocomposite. For this, two factors should be

considered, namely a geometric factor that favors permeability reduction by

forcing diffusing molecules to make long detours around the platelets and

changes in the local permeability due to molecular-level transformations in the

polymer matrix caused by the presence of mineral’s sheets. As laboratory

measurements deliver only a combined effect of the two factors, one would like

to separate them to enable the reliable design. Here, for the first time direct FE

permeability calculations have been conducted with multi-inclusion computer

models comprising random dispersions of non-overlapping platelets. Both

morphologies with perfectly aligned and with randomly oriented platelets were

investigated. For morphologies of perfectly aligned platelets a design master-

curve for the overall permeability reduction was identified assuming no

permeability change in the matrix, thus not only establishing the role of the

geometric factor alone but also providing a rational reference point for the

understanding of the contribution of molecular-level transformations occurring in

the matrices of nanocomposites. The presence of high-aspect-ratio atomic-

thickness nanoplatelets can lead to molecular level transformations of the

polymer matrix in the vicinity of the platelets which may cause changes in the local

gas permeability coefficient.[81] It would be interesting to understand the effect of

these molecular changes on the overall barrier properties of nanocomposites.

This can be achieved, for example, by a combined use of numerical predictions

and experimental data. Indeed, as numerical predictions directly establish the role

of geometric factor, the difference between numerically predicted and measured

Chapter 4 - Polymer-Layered Silicate Nanocomposites

- 60 -

barrier properties of nanocomposites should elucidate the contribution of

molecular level transformations.

4.1.1 MORPHOLOGIES WITH PERFECTLY ALIGNED PLATELETS

Periodic 3D computer models comprised of a random dispersion of perfectly

aligned impermeable round platelets in an isotropic matrix were studied, see

Figure 21. Based on unstructured morphology-adaptive meshes[48], the

Laplace’s equation

(35)

for the local chemical potential µ was solved, with position-dependent local

permeability coefficients P(r) taken to be zero inside the platelets and Pm

everywhere in the matrix.1The overall, effective permeability coefficients were

calculated numerically2 based on a linear-response relation between the overall

flux and the external chemical potential gradient applied.[48,82] In this work the

overall permeability coefficient P1 relating the overall flux and the chemical

potential gradient in the 1-direction was studied, see Figure 21. It is exactly this

permeability coefficient P1 that is of interest in most barrier applications including

advanced coatings, food packaging, and beverage bottling because in the 1-

direction a penetrant’s diffusion is maximally hindered by the impermeable

mineral platelets therefore causing the largest permeability reduction.

1. Rigorously speaking, gas permeability through a material is described by a 3×3 sym-metric permeability tensor. Here we consider isotropic matrices so the local permeabil-ity tensor can be written as P(r)·δik, where P(r) is the local permeability coefficient, δik the unit tensor with indices i and k running from 1 to 3.

2. An in-house iterative conjugate-gradient solver with a diagonal preconditioner was used in calculations.[48,52] The stopping criterion was that the first residual norm be reduced by a factor of 105 relative to its initial value. Calculations were carried out on a DEC AXP 8400 5/300 workstation. On a single processor, a single permeability calcula-tion typically took 1 CPU hour.

divP r( )gradµ 0=


- 61 -

Figure 21: A: a sketch of a periodic multi-inclusion computer model comprised of 25identical perfectly aligned non-overlapping platelets of aspect ratio 50. The plateletvolume fraction is 2%. Orthorhombic periodic boundary conditions were imposed.B: a cut through the periodic morphology-adaptive quality mesh of the computer modelshown on the left used for predicting the effective permeability numerically. Thistetrahedra-based Delaunay mesh was made up of about 106 nodes and 5·106 tetrahedra.The cutting plane was situated somewhat behind the central 12-plane. For meshgeneration and visualization, we used a commercial preprocessor.[63]

Numerical results are presented in Figure 22. One can see that the

permeability reduction is governed by the product x = a·f, where a is the platelets’

aspect ratio defined as the ratio of diameter to thickness, and f the volume

fraction. Moreover, for the practical design1, one can readily use the stretched

exponential function

(36)

with least-square parameters β = 0.71 and xo = 347. One can also see from

Figure 22B that it is platelets with aspect ratios on the order of 103 that are

1. Eq. (36) does not reduce to the Maxwell equation for spheres whereas direct numerical predic-tions accurately follow this equation.

2

31

A B

P1Pm------- x xo⁄( )β–[ ]exp=


- 62 -

remarkably efficient in improving the barrier properties of nanocomposites while

platelets with smaller aspect ratios are much less efficient.

Figure 22: Permeability reduction for a random dispersion of identical perfectly alignednon-overlapping platelets. A: Each square stands for a numerical prediction obtainedwith a particular computer model comprised of 25 identical platelets of a certain aspectratio a from 1 to 400 dispersed at a specific volume fraction f varying between 1 and 5%.The solid line shows the least-square fit of Eq. (36). B: Design diagram generated basedon the results of least-square fitting. The domain of aspect ratios and volume fractionsconsidered is typical of currently existing nanocomposites for barrier applications.

Table 11: Comparison of predictions obtained assuming selected platelet volumefractions f and aspect ratios a. We compare Eq. (36) of this work and three literatureformulas[83,85,86] currently used for permeability predictions: the Nielsen formula

, the modified Cussler-Aris formula with thegeometric factor advocated in [86], and the Fredrickson-Biceranocomposite formula with ,

, and .

Table 11 compares numerical predictions obtained in this work with those of

three selected literature models.[83-86] Interestingly, the Nielsen formula gives

f a this work Nielsen Cussler-Aris

Fredrickson-Bicerano

1 % 200 0.51 0.50 0.97 0.59

2 % 500 0.12 0.16 0.66 0.25

3 % 1000 0.0097 0.061 0.22 0.092

4 % 1000 0.0013 0.037 0.091 0.047

A B

P1 Pm⁄ 1 f–( ) 1 x 2⁄+( )⁄= P1 Pm⁄ 1 1 µx2+( )⁄=µ π2 8 a 2⁄( )ln[ ] 2⁄=

P1 Pm⁄ 1 2 a1κx+( )⁄ 1 2 a2κx+( )⁄+[ ] 2= a1 2 2–( ) 4⁄=a2 2 2+( ) 4⁄= κ π a 2⁄( )ln⁄=


- 63 -

quite accurate predictions for . Nonetheless, for technologically attractive

values of , all three models systematically underestimate the improvement

attainable in barrier properties. To rationalize the discrepancies, one should

realize that the currently existing models were either derived based on some

approximations or tuned to reproduce available laboratory data. For example, the

composite formula[86] is the second order approximant in the formalism of

multiple scattering expansion. For large x, the asymptotic predictions of the first

and second order approximants still differ by a factor of 4 so the convergence of

this expansion is obviously not rapid.

To check to see if the computer models with N = 25 platelets studied were

large enough to allow representative numerical predictions, we generated

computer models comprising 1, 8, 27, 64 and 125 platelets of aspect ratio 50 at a

volume fraction of 5%. For each model the permeability of 3 MC-snapshots was

calculated. Results are shown in Figure 23.

Figure 23: Investigation of the minimal RVE size. Increasing the number of platelets N inthe computer model reduces the scatter of the predicted permeability P1. The filled circlesindicate the arithmetic mean of 3 numerical calculations with three different MC-snapshots and the error bars indicate the 95% confidence interval.

The scatter of individual permeability estimates steadily decreases with

increasing N but no significant change in the average values is observed, except

from the prediction with one platelet. Figure 23 clearly shows that the minimal

x 10<

x 10>


- 64 -

RVE size for the periodic random composite studied was very small and that

computer models with 25 platelets were already representative. Similar behavior

was already established, both theoretically[87] and numerically,[47,48,56] for the

elastic constants and thermal expansion coefficients of periodic random-

microstructure two-phase composites.

Here the barrier properties of a nanocomposite comprised of perfectly

aligned randomly dispersed platelets were studied. This “nematic-phase”

morphology should apparently maximize the barrier properties in the direction of

the platelets’ symmetry axes and is therefore of most interest to applications

where a polymer film is supposed to build a barrier between two media.

Nonetheless, Gusev’s FEM is generic and one can readily use it to identify the

role of various morphological imperfections[16,88], such as incomplete

exfoliation, platelet misorientation and agglomeration, which are typically present

in nanocomposites. The presence of such defects is common in practice but their

precise degradation influence on the barrier properties of nanocomposites is not

always clear today. Consequentially, in the next subchapter the influence of

platelet misalignments on the barrier performance is addressed.

4.1.2 MORPHOLOGIES WITH MISALIGNED PLATELETS

In chapter 4.1.1 the barrier enhancements which can be achieved by

aligned, impermeable platelets randomly dispersed in a polymer matrix were

quantified. Perfect alignment of the platelets’ symmetry axes parallel to the

external concentration gradient is more effective in hindering the penetrant’s

diffusion in the direction of the external concentration gradient than any other

platelet orientation state. In real nanocomposites, however, the microstructure is

more complex and not ideal like presumed previously. The platelets can have

different size and shape and it is unavoidable that exfoliated silicate sheets are

disoriented which is going to influence the barrier properties. Using a multiple

scattering expansion approach[86,89] Fredrickson and Bicerano addressed the

issue of misalignments for dilute concentrations where platelets are able to orient

independently of each other (see Figure 24A). They showed that with randomly


- 65 -

oriented platelets only 1/3 of the barrier performance can be achieved as with

perfectly aligned platelets whose symmetry axes are parallel to the external

concentration gradient. The multiple scattering approach, however, is only

applicable in the dilute regime where platelets have their full rotational freedom.

Therefore one can not employ it to make predictions about the barrier properties

of nanocomposites comprising semidilute platelet concentrations. With high

aspect ratio platelets one enters the semidilute regime already at relatively low

volume loadings. This is due to their large shape anisotropy which lets them

experience rotational constraints at loadings of less than 1 vol%. In the previous

chapter it was shown that technologically interesting barrier properties are

achieved for high aspect ratio platelets (a > 100) dispersed at moderate volume

fractions (3-7%) which is already well inside the semidilute concentration regime.

In the semidilute concentration regime due to rotational constraints the platelets

start to form liquid crystalline like domains. A domain is built up of a bunch of

slightly misaligned mineral sheets (see Figure 24B). It must be assumed, though,

that the microstructure of a nanocomposite consists of many different domains

which can be arbitrarily oriented (see Figure 24C). Depending on the particular

orientation of the domains compared to the external concentration gradient the

penetrating molecules have to diffuse on more or less tortuous paths.

In chapter 3.4 it has been shown for short fiber reinforced composites that

the orientation averaging scheme can be readily used to accurately predict the

elastic tensor of composites comprising any fiber orientation state. Orientation

averaging, however, is in principal applicable to calculate any tensor property

based on the unit prediction for a unidirectional composite. Therefore also the

permeability tensor of a nanocomposite comprising platelets with an arbitrary

orientation state can be calculated. It is, though, not clear if this scheme is

accurate enough to deliver reliable predictions. Thus, in this section the errors of

the permeability predictions made by orientation averaging have been assessed

in comparison with direct numerical predictions.


- 66 -

Figure 24: Sketches depicting a platelet filled composite in the dilute (top) and thesemidilute regime (middle). In the dilute regime the platelets have the full rotationalfreedom whereas in the semidilute regime they start to form liquid crystalline like domainswith preferred orientations. The bottom sketch depicts different domains with arbitraryorientations. Each domain consists of a liquid crystalline like morphology of mineralplatelets.

4.1.2.1 NUMERICAL

Computer models were generated for three different platelet aspect ratios,

namely 3, 10 and 100, and for a wide range of concentrations starting in the dilute

and extending deep into the semidilute concentration regime (see Table 12).

Figure 24C shows that in order to reproduce the morphology of a nanocomposite

in the semidilute regime with a FE-model would require a very large number of

platelets which is currently out of reach with the available computational

resources. It was therefore necessary to adjust the size of the computer models

to a reasonable number of 50 platelets. It is a justifiable assumption that a

nanocomposite with a morphology similar to the one shown in Figure 24C but in

Dilute regime

Semidilute regime

Arbitrarily oriented domains

A

B

C


- 67 -

3 dimensions features a 3D random orientation state of the platelets. Based on

this assumption 20 computer models, each one with a different combination of

aspect ratio a and volume fraction f of the platelets (see Table 12), were

generated using 50 platelets of equal shape and size.

Table 12: The investigated platelet volume fractions f are listed for each of the 3considered aspect ratios a. Volume fractions are given in percent.

In the first step of model generation 50 platelets were randomly placed at a

dilute concentration of 0.001 vol% throughout a cubic unit cell. In a MC-run their

orientations were adjusted to a 3D random orientation state where all 3

eigenvalues a1, a2 and a3 of the second order orientation tensor aij equal 1/3. The

unit cell of each computer model was then compressed to one of the 20

investigated platelet volume fractions during a variable-box-size MC-run keeping

platelet orientations fixed and accepting only configurations without platelet

intersections.

In order to predict the permeability by using the orientation averaging

scheme the permeability tensors of nanocomposites with perfectly aligned

platelets are required. Hence, for each of the 20 investigated combinations of

aspect ratio a and volume fraction f the corresponding computer model with

perfectly aligned platelets was generated using a cubic unit cell. In most cases it

was possible to directly put the 50 platelets at the desired volume fraction into the

unit cell. Otherwise the platelets were placed at a somewhat lower volume

a = 3 a = 10 a = 100

0.0333 0.01 0.001

0.333 0.1 0.01

3.33 1.0 0.1

16.6 10.0 0.5

33.3 25.0 2.5

50.0 30.0 4.5

52.8 - 5.0


- 68 -

fraction and subsequently their volume fraction was increased during a variable-

box-size MC-run.

In Figure 25 two computer models each comprising 50 platelets featuring a

3D random orientation state are shown both for a dilute and a semidilute

concentration of 0.001 and 5%, respectively. Remarkably, such small computer

models already confirm the assumption that at semidilute concentrations the

platelets of similar orientation start to build liquid-crystalline-like domains (see

Figure 25B).

Figure 25: Two computer models containing 50 platelets of aspect ratio 100 in a cubicunit cell: A) at a dilute concentration of 0.001 vol% B) at a semidilute concentration of5 vol%. The entirety of platelets features a 3D random orientation state with a diagonalform 2nd order orientation tensor aij where all 3 diagonal components are 1/3.

All 40 computer models were meshed into periodic, unstructured,

morphology-adaptive FE-meshes. The meshes were rather large and consisted

of up to 1.5 x 106 nodes and 9 x 106 tetrahedra which is due to the large surface

area of the platelets. Because our interest is in the relative decrease of the

permeability due to the presence of the platelets a permeability Pm was assigned

to the matrix and zero permeability to the impermeable mineral platelets. A

Laplace solver was employed to numerically determine the overall, effective

permeability tensor.

1

23

A B


- 69 -

4.1.2.2 RESULTS AND DISCUSSION

Perfectly aligned platelets provide the maximal possible permeability

reduction in the direction of the platelets’ symmetry axes. In this direction the

permeability reduction ∆PII is defined as the difference between the permeability

Pm of the pure polymer matrix and the longitudinal permeability P1 of a nano-

composite with parallel aligned platelets:

(37)

For nanocomposites consisting of arbitrarily oriented domains (see Figure

24, bottom) the barrier performance is lower because the diffusion paths for the

permeating molecules get less tortuous due to the presence of domains where

the preferred orientation of the platelets’ symmetry axes deviates from the

direction of the external concentration gradient. For this kind of morphologies one

can express the permeability reduction ∆PX as:

(38)

P1, P2 and P3 are the eigenvalues of the permeability tensor of a

nanocomposite comprising arbitrarily oriented domains. Due to the fact that we

assumed a 3D random orientation state for the platelets the eigenvalues must

approach identity as the size of the models is increased, resulting in

macroscopically isotropic transport properties. For the relatively small model size

employed here, however, there were slight differences between the eigenvalues

P1, P2 and P3 of maximally 3%. Therefore the arithmetic average of the 3

eigenvalues was used to calculate the permeability reduction ∆PX.

The effectiveness of a composite comprising arbitrarily oriented domains of

misaligned platelets relative to the ideal morphology of aligned platelets can now

be expressed by the ratio . So the permeability reduction ∆PX is set in relation

with the maximal possible permeability reduction ∆PII of perfectly aligned

platelets. A ratio means that a morphology of arbitrarily oriented domains

∆P|| Pm P1–=

∆PX PmP1 P2 P3+ +( )

3------------------------------------–=

∆PX∆PII-----------

∆PX∆PII----------- 1=


- 70 -

is equally effective as one with perfectly aligned platelets. It represents the upper

bound of what can be achieved for composites comprising misaligned,

impermeable platelets. As mentioned before for the dilute concentration regime it

has theoretically been shown that arbitrarily oriented platelets are one third as

effective in hindering a penetrant’s diffusion as perfectly aligned ones.[86]

Therefore, if this finding is correct, there exists, at least in the dilute regime, a

lower bound below which the ratio never drops.

In this work both the dilute and the semidilute concentration regimes were

addressed using the FEM of Gusev. From the numerical results the ratio has

been calculated for all 20 a-f-combinations which have been considered. Results

are shown in Figure 26 again as a function of the product af.

Figure 26: The ratio between the permeability reduction ∆PX achieved with amorphology of arbitrarily oriented domains of misaligned platelets and the maximalpossible permeability reduction ∆PII of a morphology with perfectly aligned platelets isdepicted against the product af. The dashed line stands for composites with sphericalinclusions where misalignments don’t affect the permeability. The dotted line shows theone-third-rule of Fredrickson and Bicerano.[86]

Unlike as in chapter 4.1.1 there is no universal relation between the ratio

and the product af. Anyway, this scaling is sensible because the results are

brought on the same scale, although considerably different volume fractions f

were investigated for the 3 distinct aspect ratios a (see Table 12). Like this the

∆PX∆PII-----------

∆PX∆PII-----------

∆PX∆PII-----------


- 71 -

effect of different a-f-combinations on the ratio can be easily compared. In

Figure 26 one can see that for dilute concentrations of platelets with aspect ratio

100 one approaches , which confirms the one-third-rule of Fredrickson

and Bicerano. For platelet aspect ratios of 3 and 10, however, the ratio

converges to larger values at dilute concentrations. As the aspect ratio, which

characterizes the shape anisotropy of the platelets, is decreasing the degrading

effect of platelet misalignments on the barrier properties is diminished until it

totally disappears for spherical particles. In Figure 27 the stationary values of

at dilute platelet concentrations (af = 0.1) are depicted as a function of aspect

ratio. This plot clarifies that the one-third-rule is only valid in case of high aspect

ratio platelets, for a > 70. Considering again Figure 26 at larger af one can

observe for all 3 aspect ratios that the ratio starts to deviate considerably from

the stationary value. For a = 3 and a = 10 the ratio begins to rise for af > 10

whereas for a = 100 this takes place already for af > 1. This effect is caused by

the transition from the dilute to the semidilute concentration regime which is

characterized by the fact that platelets loose their full rotational freedom and that

the penetrant’s diffusion is collectively hindered. The increasing ratio in the

technologically interesting semidilute concentration regime means that the

degrading effect of platelet misalignments is less pronounced than in the dilute

regime. For example for a nanocomposite comprising platelets of aspect ratio 100

at volume fraction 5% one must expect only a 40% loss in the barrier performance

due to platelet misalignments compared with a perfectly aligned morphology.

For platelets of aspect ratio 3 and 10 the largest values of correspond

to volume fractions of 30% and 53%, respectively, which were actually the largest

volume fractions which could be achieved in the variable-box-size MC-run by

keeping the platelets’ orientations fixed. For such large concentrations

(concentrated regime) the role of misalignments is getting secondary and it is

mainly the volume fraction which determines the permeability of the composite.

Larger volume fractions can be obtained if one allows the platelets to perform

rotations during the variable-box-size MC-run. In this case, however, the domains

would not be arbitrarily oriented any more and the permeability would become

anisotropic with a maximal permeability reduction ∆PX in one single direction. It

∆PX∆PII-----------

∆PX∆PII----------- 1

3---=∆PX∆PII-----------

∆PX∆PII-----------

∆PX∆PII-----------

∆PX∆PII-----------

∆PX∆PII-----------

∆PX∆PII-----------


- 72 -

can be assumed that the permeability reduction ∆PX in this particular direction

would reach the maximal possible permeability reduction ∆PII at further increased

platelet concentrations.

Figure 27: Dependence of the ratio on the aspect ratio a in the dilute concentration

regime (af=0.1). Towards smaller aspect ratios effect of platelet misalignments

diminishes. For spheres the permeability does not depend on the orientation and

therefore the ratio is one for any concentration.

As described in chapter 3.4 for the elastic properties of fiber reinforced

composites it is also possible to apply the orientation averaging scheme to

calculate the permeability of a composite comprising platelets with a defined

orientation state. For each of the 20 computer models with perfectly aligned

platelets the two independent permeabilities P1 and P2 which characterize the

permeability of a transversely isotropic material were determined. With both the

longitudinal and the transverse coefficient P1 and P2 the permeability tensor of a

nanocomposite with misaligned platelets was calculated using the orientation

averaging scheme which is defined as:

(39)

∆PX∆PII-----------

∆PX∆PII-----------

Pij⟨ ⟩ P1 P2–( )aij P2δij+=


- 73 -

Nanocomposites with a 3D random platelet orientation state feature

isotropic effective properties. Therefore the permeability is characterized by one

single coefficient. Here, the prediction from orientation averaging is denoted as

, the one from direct numerical calculations as . The deviations of the

orientation averaged permeabilities from the direct numerically calculated

are depicted in Figure 28. One can see that the error of the predictions made by

orientation averaging rises more or less linearly with increasing af. For the largest

investigated af = 500 (a = 100, f = 5 vol%) a deviation of 33% was found. In order

to prevent errors larger than 5% one should therefore not employ the orientation

averaging scheme for af > 100. For example, with platelets of a = 100 one should

not use the orientation averaging scheme for concentrations above 1 vol%.

Figure 28: Deviations of the permeability coefficients calculated by orientationaveraging from the direct numerically predicted permeability coefficients .

4.2 THERMOELASTIC PROPERTIES

Layered silicates like mica are much stiffer than polymers due to the strong

covalent Si-O bonds and therefore a few volume percent of exfoliated high aspect

ratio silicate sheets can considerably change the stiffness and thermal expansion

of a polymer matrix. It has already been found experimentally that the increase in

the Young’s modulus depends on the average platelet aspect ratio as well as on

P⟨ ⟩ P

P⟨ ⟩ P

P⟨ ⟩P


- 74 -

the extent of exfoliation.[6] Only for exfoliated and well dispersed silicate platelets

one can fully profit from the nano-size-effect. With models based on the Halpin-

Tsai equations it has been shown that intercalated silicate platelets, in contrast,

have a much lower impact on the modulus because the effective aspect ratio of

such stacked structures is smaller.[23, 90]

The CTE of silicate minerals is with 10-5 K-1 about the same as for metals

and roughly one order of magnitude lower than for semicrystalline polymers.

Consequently, the overall CTE of polymer-layered silicate nanocomposites

decreases due to the presence of the mineral filler.[91] Nanocomposites with a

tailored CTE could be interesting for advanced hybrid polymer-metal structures in

microelectronic, automotive and airplane industry.

It is, however, unclear what can be gained in principal for the thermoelastic

properties of nanocomposites due the presence of exfoliated, atomic-thickness

mineral platelets because most of the micromechanical models can only deal with

isotropic or transversely isotropic inclusions. In contrary to the micromechanical

models Gusev’s FEM can deal with anisotropic phase properties of any crystal

symmetry. Here, for the first time this FEM was applied to quantify how the

effective thermomechanical properties of polymer nanocomposites are affected

by different aspect ratios and volume loadings of mineral platelets. For the

present numerical studies muscovite mica which has monoclinic crystal symmetry

was chosen as mineral filler because its 13 independent elastic tensor

components have been measured by means of Brioullin scattering.[92]

4.2.1 MORPHOLOGIES WITH PERFECTLY ALIGNED PLATELETS

3D computer models each comprising 25 round and perfectly aligned

platelets randomly placed throughout a cubic unit cell without intersecting each

other were generated. As in chapter 4.1.1 it was checked first if the computer

models with N = 25 platelets studied were large enough to allow representative

numerical predictions. Computer models comprising 1, 8, 27, 64 and 125 platelets

of aspect ratio 50 at a volume fraction of 5% were generated and the transverse

Young’s modulus E22 of 3 different MC-snapshots was calculated numerically.


- 75 -

Figure 29 indicates that the minimal RVE size is small and that models comprising

25 platelets are large enough to deliver reliable predictions.

Figure 29: Investigation of the minimal RVE size. Increasing N, the number of platelets,in the computer model reduces the scatter of the predicted transverse Young’s modulusE22. The filled circles indicate the arithmetic mean of 3 numerical calculations with threedifferent MC-snapshots and the error bars indicate the 95% confidence interval.

For numerical calculations three different isotropic polymers were taken into

consideration: a glassy polymer, a semicrystalline polymer at room temperature

and a semicrystalline polymer at elevated temperature denoted M3, M1 and M01,

respectively. Typical thermoelastic properties of these three polymer types are

given in Table 13.

Table 13: Typical thermoelastic properties of three polymer matrices. M3 and M1represent a glassy and a semicrystalline polymer at room temperature, respectively.M01 has properties of a semicrystalline polymer at elevated temperature.

The elastic properties of monoclinic muscovite mica (see Table 14) were

assigned to the in-plane arbitrarily oriented platelets in the computer models. The

E [GPa] ν α [K-1]

M01 0.1 0.45 2 x 10-4

M1 1.0 0.4 1 x 10-4

M3 3.0 0.35 0.7 x 10-4


- 76 -

CTE of mica was assumed to be isotropic with a value of 10-5 K-1.[93] The

adhesion between mica platelets and matrix was assumed to be perfect.

Table 14: 13 independent elastic constants of monoclinic muscovite mica measured byBrioullin scattering[92]

Periodic unstructured morphology-adaptive meshes built up of tetrahedra

were generated and a displacement-based linear-elastic solver was used to

determine the thermomechanical properties iteratively (see chapter 3.2.3).

24 different computer models delivered both the stiffness and the CTE

tensor of nanocomposites comprising muscovite mica platelets of aspect ratio 1,

3, 10, 31, 100 or 316 at volume fractions of 2.5%, 5%, 7.5% or 10%. Due to the

random in-plane orientation of the perfectly aligned platelets the composite

revealed transversely isotropic properties with 5 independent elastic constants

C11, C22, C12, C23, C66 and two independent CTEs α1 and α2. Because of the

shape and property anisotropy of the aligned mica platelets the largest property

changes are observed in the transverse directions of the nanocomposite. Hence,

Elastic constant [GPa]

C11 60.9

C22 179.5

C33 176.5

C44 70.7

C55 13.1

C66 15.0

C12 23.0

C13 20.0

C15 -0.7

C23 47.7

C25 11.1

C35 -1.2

C46 0.7


- 77 -

here we focus on the transverse Young’s modulus E22 and on the transverse CTE

α2.

As a limiting case we also considered mica composites with a lamellar

morphology. It is clear that a composite built up by polymer and mica lamellae

(platelets with infinitely large aspect ratio) is most effective in terms of increasing

E22 and decreasing α2.

Figure 30: Lamellar composite comprised of 25 vol% muscovite mica and 75 vol%polymer matrix. Unit cell comprises 8x8x80 nodes on a regular grid which are connectedto a periodic morphology-adaptive mesh of equally sized tetrahedra. Mica propertieswere assigned to 20 of the 80 cross-sectional layers and matrix properties to theremaining ones.

In an orthorhombic unit cell a regular grid of 8x8x80 nodes was placed and

connected to a periodic mesh of equally sized tetrahedra (see Figure 30).

According to the desired mica volume fraction mica properties were assigned to

an appropriate number of cross-sectional layers (1 layer corresponds to a volume

fraction of 1/80 = 0.125 vol%). Matrix properties were assigned to the remaining

layers. The transverse Young’s modulus EL and the transverse CTE αL of a

lamellar mica composite was calculated for mica volume fractions of 2.5%, 5%,

7.5% or 10%.

Results for the transverse Young’s modulus E22 of the nanocomposite are

expressed relative to the Young’s modulus Em of the pure matrix. This allows to

compare the stiffening effect of mineral platelets for different polymer matrices. In

Figure 31 a graph shows how the relative transverse modulus E22/Em increases

depending on the aspect ratio a for each of the three considered polymer

matrices. Compared to the pure polymer matrices E22 is very much enhanced by

the presence of, in-plane, 2-3 orders of magnitude stiffer mica platelets. 10 vol%

mica platelets of aspect ratio 300 in M3, M1 and M01 can increase E22 by a factor

3

12


- 78 -

Figure 31: The relative transverse Young’s modulus E22/Em as a function of the aspectratio a of the mica platelets. Each graph consists of 4 curves, one for each of the fourinvestigated mica volume loadings. The dashed lines in two of the three graphs depictthe relative transverse modulus EL/Em of a lamellar morphology with 10.0 vol% mica.

M3

M1

M01


- 79 -

of 5, 12 and 58, respectively, presumed that the platelets are perfectly aligned and

well dispersed throughout the polymer matrix. Furthermore, the graphs for M1

and M3 in Figure 31 show that a nanocomposite containing 10 vol% platelets of

a = 300 already approaches the maximal possible E22 = EL of a lamellar

composite with equal mica loading. For example, a nanocomposite comprising

10 vol% mica platelets of aspect ratio 300 dispersed in M3 is only 12% less stiff

than the corresponding lamellar morphology, whereas for the low-modulus matrix

M01 the nanocomposite is still 63% less stiff. This in analogy to short fiber

reinforced composites where the longitudinal Young’s modulus E11 is also

levelling off towards larger fiber aspect ratios. This observation can be justified by

a critical aspect ratio above which the modulus stays practically constant. This

critical aspect ratio depends on the constituents’ modulus ratio. The critical aspect

ratio for the Mica/M01 nanocomposite is considerably larger because the

modulus ratio is 10 and 30 times larger than for Mica/M1 and Mica/M3,

respectively. The graph for matrix M01 in Figure 31 shows that with platelet

aspect ratios around 300 the nanocomposite does not yet approach the maximal

possible E22 = EL of a lamellar morphology.

Numerical calculations deliver the complete stiffness matrix of the

composite. Therefore one can compute the moduli in any direction other than the

one of the applied stress. This was done both for the relative Young’s modulus

Ec/Em and the shear modulus Gc/Gm as a function of θ, which is the angle

between the symmetry axis of the platelets and the applied normal and shear

stress, respectively. Figure 32 shows that the enhancement both of the Young’s

and the shear modulus is very much direction dependent. As already discussed

before the maximal enhancement of the Young’s modulus occurs in the

transverse direction perpendicular to the platelets’ symmetry axes (θ = 90°). In

the direction of the platelets’ symmetry axes (θ = 0°) a moderate increase of the

Young’s modulus takes place, too. However, at an angle θ = 40° almost no

change occurs for any platelet aspect ratio. Exactly the same direction

dependence of the Young’s modulus has also been found by Brune and

Bicerano[90] who employed the Halpin-Tsai equations and adjusted them using

an interpolation approach in order to bring them in harmony with the analytical


- 80 -

equation for infinitely thin disks. In contrary to the Young’s modulus Ec the shear

modulus Gc is maximally increased when the shear stress acts in a plane which

is 45° tilted away from the symmetry axis of the platelets.

Figure 32: Young’s and shear modulus as a function of θ, the angle between thesymmetry axis of perfectly aligned platelets and the applied normal and shear stress,respectively. Both plots were generated from the numerically calculated elastic tensor ofcomputer models comprising 2.5 vol% platelets in matrix M1.

The decrease in the transverse CTE due to the presence of the mica

platelets can be expressed by the ratio φ. The ratio φ sets , the difference

between the transverse CTE of the nanocomposite, α2, and the lamellar

morphology, αL, into relation with the maximal CTE reduction ,

which is the difference between the CTE of the matrix, αm, and the lamellar

morphology, αL.

(40)

The ratio φ can take values in the interval [0,1]. φ=1 corresponds to the pure

polymer matrix with no change in the CTE and φ=0 to the lamellar morphology

with the maximal CTE decrease.

Depicting all numerical results in form of the ratio φ versus x = af, the product

between platelet aspect ratio a and volume fraction f (see Figure 33), clarifies that

the CTE of nanocomposites shows equal behavior for all three considered

A B

α2 αL–

∆αmax α= m αL–

φα2 αL–∆αmax------------------

α2 αL–αm αL–-------------------= =


- 81 -

matrices and that the ratio φ can be described by one and the same stretched

exponential function1 with least-square parameters xo = 278 and β = 0.586.

(41)

Figure 33: Exponential decrease of the ratio φ = (α2 – αL)/(αm– αL), where α2 is thetransverse CTE of the nanocomposite, αL the transverse CTE of the respective lamellarmorphology and αm the CTE of the polymer matrix. Each data point denotes a numericalprediction obtained with a particular computer model comprising 25 round platelets ofcertain aspect ratio a dispersed at a specific volume fraction f varying between 2.5% and10%.

For every combination of aspect ratio a and volume fraction f the present

response-function φ(x) can predict by what fraction, based on the maximal

possible CTE reduction , the transverse CTE α2 of a

nanocomposite drops. For example, let us assume we have a nanocomposite

with mica platelets of an average aspect ratio of 100 at a volume fraction of 1%.

According to the response-function φ(x) one can expect a CTE decrease which is

about 40% of the maximal possible CTE reduction of a lamellar

morphology. Remarkably, this 40% decrease is the same for any polymer matrix.

1. In chapter 4.1.1 the same type of curve described the overall permeability of nanocom-posites in the longitudinal direction as a function of the product x = af.

φ x( ) e

xxo---- β

–

=

∆αmax α= m αL–

∆αmax


- 82 -

The response-function φ(x) can be used to rationally design

nanocomposites with tailored transverse CTEs. It is only necessary to numerically

calculate αL, the transverse CTE for a particular lamellar morphology, by

specifying the matrix CTE and the platelets’ volume loading. Subsequently, αL

together with the design goal for the transverse CTE α2 of the nanocomposite is

inserted into Eq. (40) which delivers the ratio φ. By rearranging Eq. (41) one can

determine the platelet aspect ratio a which is necessary to reach the targeted α2.

(42)

This procedure was applied to calculate a design map which allows to tailor the

CTE of nanocomposites comprising perfectly aligned mica platelets. For the

polymer matrix M1 the transverse CTE of 20 lamellar morphologies comprising

mica lamellae with volume fractions f between 1.25% and 20% was calculated.

The collected αL allowed then to determine for each mica fraction f the

corresponding platelet aspect ratio a which is required to reach a certain value for

α2.

In Figure 34 a design map for two different design goals is depicted. From this

design map one can extract which combinations of platelet aspect ratio a and

volume fraction f deliver a transverse CTE equal to the CTE of aluminum (α2 = 2.0

· 10-5 K-1) or bisecting the CTE mismatch between matrix M1 and aluminum

(α2 = 6.0 · 10-5 K-1). The graph clearly shows that platelet aspect ratio a and

platelet volume fraction f interdepend when targeting a predefined CTE α2 for the

nanocomposite. For example, in order to reduce the mismatch between the CTEs

of M1 and aluminum by 50% one can either disperse mica platelets of aspect ratio

20 at a volume fraction of 6.5 vol% or of aspect ratio 100 at 2.3 vol% (see Figure

34). To match the CTEs of polymer and aluminium completely it is necessary to

disperse mica platelets of aspect ratio 200-300 at rather large volume fractions of

7-10%. Due to the susceptibility to damage during and the reagglomeration of

mineral sheets after compounding it is, however, unclear if exfoliated

nanocomposites with such large mica loadings can be produced in practice.

axo φ( )ln–( )

1β---

f------------------------------=


- 83 -

Figure 34: Design map applicable to match the CTE of a mica containing nanocompositewith the CTE of aluminum (filled circles) or to reduce the CTE-mismatch between thematrix M1 (α = 10-4 K-1) and aluminum by 50% (open circles). The two dashed verticallines stand for the volume fractions which are asymptotically approached when ,which corresponds to a lamellar morphology.

4.2.2 MORPHOLOGIES WITH MISALIGNED PLATELETS

As in short fiber composites also in nanocomposites the overall effective

thermoelastic properties are depending on the actual orientation state of the

inclusions. In chapter 3.4 it was demonstrated that the thermoelastic properties of

short fiber composites with any fiber orientation state can be calculated from the

thermoelastic properties of a unidirectional composite using the orientation

averaging scheme. In the present chapter it will be shown that the orientation

averaging scheme is also suitable to predict the full stiffness tensor of any platelet

orientation state from the elastic properties of a composite with perfectly aligned

platelets.

4.2.2.1 NUMERICAL

3D computer models comprising 150 platelets in a cubic unit cell were

created for two combinations of platelet aspect ratio a and volume fraction f:

a ∞→


- 84 -

• a = 50, f = 5.0%

• a = 100, f = 2.5%

In each of the computer models, except the one with aligned platelets, the

platelet orientation state was adjusted to one of the 13 symmetry independent 2nd

order tensors aij which were already studied in chapter 3.4 (see Table 9). A choice

of four computer models with distinct platelet orientation states is depicted in

Figure 35. After meshing the thermoelastic properties were assigned to matrix

and platelets. The same polymer matrices M01, M1 and M3 as in the previous

subchapter (see Table 13) were used. The elastic properties of muscovite mica

(see Table 14) were assigned to the platelets. In contrary to the previous

subchapter the CTE of the mica platelets was not assumed to be isotropic but to

consist of two coefficients, the CTE perpendicular and the

CTE parallel to the cleavage plane of mica.[92] With a

displacement-based linear-elastic solver both the stiffness and the CTE tensors

were calculated numerically for a total of 84 different computer models. The

numerical solution of the FE-models was quite expensive both in terms of CPU

time and memory requirements because the FE-mesh for these models usually

consisted of about 106 nodes and 6x106 tetrahedra.1 For such computer models

the linear-elastic solver needed about 1.3 GB of memory and a CPU time of 37

hours to perform seven strain-energy minimizations.

1. The reason for the large meshes is that the surface area of platelets with aspect ratios of 50 or 100 is considerably larger than for spherocylinders (short fibers) of equal aspect ratio. For a mor-phology-adaptive FE-mesh it is crucial to construct first a fine triangular mesh on the surface of the objects which exactly reproduces the shape of the object. Therefore in case of high-aspect ratio platelets a large number of surface nodes is needed for the surface mesh. This number is multiplied by 150, the number of platelets in the computer model.

α⊥ 1.5= 10 5– K 1–⋅

α II 1.0= 10 5– K 1–⋅


- 85 -

Figure 35: Four computer models with four different orientation states comprising 150platelets of aspect ratio 50 at 5 vol%. Top left: Perfectly aligned platelets. The elastictensor calculated with this computer model was subsequently used for orientationaveraging. Top right: Platelets whose normal vectors were randomly tilted in the 1-3-plane. Bottom left: Platelets with an arbitrary chosen orientation state. Bottom right:Platelets with a 3D random orientation state.

4.2.2.2 RESULTS AND DISCUSSION

The orientation averaged elastic tensor of the 13 distinct platelet

orientation states was determined by calculating the 5 invariants B1,...,B5 (see

Eq. (24)) from the elastic tensor of a composite with perfectly aligned platelets.

Subsequently, B1,...,B5 were inserted into Eq. (23) together with the 2nd and 4th

order orientation tensors aij and aijkl in order to calculate the stiffness tensor

a1 = 1, a2 = 0, a3 = 0 a1 = 0.5, a2 = 0.0, a3 = 0.5

a1 = 0.6, a2 = 0.3, a3 = 0.1 a1 = 0.33, a2 = 0.33, a3 = 0.33

1

3

2

Cijkl⟨ ⟩


- 86 -

. The orientation averaging of the CTE tensor was done by employing

Eq. (25) together with two additional invariants D1 and D2 which were computed

according to Eq. (26).

To judge if the orientation averaging scheme is accurate and reliable

enough to predict the thermoelastic properties of platelet reinforced composites,

the deviations of the orientation averaged elastic tensors in Voigt notation

from the respective direct numerically calculated tensors were assessed by

the standard matrix norm error which was already applied in chapter 3.3.2.

The standard matrix norm is defined as

(43)

with the summation carried out over indices i and k running from 1 to 6.

In Figure 36 the standard matrix norm errors for each of the 13 investigated

orientation states of a nanocomposite with matrix M3 are depicted. One can see

that the maximal error is 2.3% for the orientation state a1=0.5, a2=0.0, a3=0.5 in

which the platelets’ symmetry axes are randomly oriented within the 1-3-plane

(see Figure 35 top right). The average standard matrix norm error over all

orientation states amounts to 1.69%.

For the thermal expansion properties the CTEs α1, α2 and α3 in the

directions of the three principal axes were computed both for the numerically

calculated and for the orientation averaged CTE tensor. In Figure 37 the

orientation averaged CTEs of a nanocomposite with matrix M3 are depicted

against those from direct numerical calculations for all 13 investigated

orientation states to show the quality of agreement between the two approaches.

The orientation averaged CTEs deviate by less than 5% from the

numerically calculated ones.

In Table 15 a summary of the average standard matrix norm errors is listed

for both investigated a-f-combinations as well as for all matrix types M01, M1 and

M3. All average standard matrix norm errors are below 4% which proves that the

orientation averaging scheme is accurate enough to calculate both the stiffness

Cijkl⟨ ⟩

Cij⟨ ⟩

Cij∆CC------------

∆CC------------

Cik Cik⟨ ⟩–( )2∑

Cik2

∑---------------------------------------------=

α i⟨ ⟩

α i

α i⟨ ⟩


- 87 -

and the CTE tensors of platelet reinforced composites for every orientation state.

However, the orientation averaging results are only as good as the underlying set

of elastic constants for a composite with perfectly aligned platelets.

Micromechanical models like Halpin-Tsai or Tandon-Weng are not in the position

to deal with the elastic tensors of monoclinic phases and can therefore not provide

the required data for orientation averaging. With the FEM used throughout this

work, however, one can readily deal with phases of any crystal symmetry both for

the inclusions and the matrix.

Figure 36: Relative standard matrix norm error between direct numerical andorientation averaging predictions. Results for a composite with mica platelets of aspectratio a = 50 dispersed in matrix M3 at a volume fraction of f = 5% are shown. The errorassessment is carried out based on computer models with predefined second orderorientation tensors. We consider diagonal form tensors and use a regular grid with aspacing of ∆a = 0.1. Since and all , only 13 grid points are in factsymmetry independent. All of them are shown in this figure, together with 7 additionalsymmetry equivalent points included for visual expediency.

In chapter 3.4 the error assessment revealed deviations of up to 8.7%

between the direct numerical and the orientation averaging route due to the high

anisotropy of the thermoelastic properties of carbon fibers. Mica as well has

anisotropic properties although not as pronounced as for carbon fibers. Because

isotropic material properties have shown better agreement between the direct

∆C C⁄

a1 a2 a3+ + 1= ai 0≥


- 88 -

numerical and the orientation averaging approach one can assume that for

platelets with isotropic properties the orientation averaging predictions would

even better agree with direct numerical ones.

Table 15: Summary of the standard matrix norm errors (in percent) both for the elasticand the CTE properties for the different a-f-combinations and polymer matrices M3, M1and M01 which were considered. The average of the standard matrix norm errors overall considered platelet orientation states was calculated.

a = 50, f = 5% a = 100, f = 2.5%

Matrix M3 M1 M01 M3 M1 M01

[%] 1.69 2.32 1.95 0.83 1.45 1.44

[%] 1.94 3.08 3.73 1.18 2.30 3.29

∆C C⁄

∆α α⁄


- 89 -

Figure 37: CTEs of a nanocomposite comprising 5 vol% mica platelets of aspect ratio 50in an amorphous polymer matrix (M3). Predictions from orientation averaging areplotted against direct numerical predictions . The solid diagonal line symbolizesidentity and the two dashed lines mark ±5% deviation from identity.

α i⟨ ⟩α i


- 90 -

4.3 CONCLUSIONS

Numerical FE-simulations elucidated some important relations between

microstructural parameters, like aspect ratio a, volume fraction f and platelet

orientation state and the overall effective properties of nanocomposites. It

appeared that the product x = af plays an important role for the properties of

nanocomposites with perfect platelet alignment. Numerical investigations have

shown that the longitudinal permeability P1 as well as the transverse CTE α2 is

governed by the product af and that these properties can be described by

mastercurves in the form of a stretched exponential function. This opens new

horizons for experimentalists because the presented results of direct numerical

calculations allow them to estimate the effective transport- and thermoelastic

properties and support them in targeting at the most promising morphologies

before any experiment is done in the lab. Due to the fact that only geometric

parameters were considered in the FE-models the comparison of numerically

calculated and experimentally measured property data could potentially reveal

information about the property changes taking place in the vicinity of the platelet

surfaces caused by molecular level transformations of the polymer matrix. It is

believed that these property changes contribute to the exceptional properties of

nanocomposites. In order to analyze this effect, however, it must be possible first

to produce samples with a controlled morphology. At the moment this is still a

problem mainly because smectites like montmorillonite etc. have a very broad

distribution of particle size and therefore one must use elaborate procedures in

order to separate the technologically interesting large aspect ratio platelets from

the low aspect ratio particles and to measure their size distribution. Furthermore

it is also a challenge to properly modify the mineral sheets’ surfaces to match

them with the surface energy of the polymer matrix which is crucial to get

exfoliated nanocomposites. Finally reliable methods to characterize both the

microstructure need to be established which would then allow to extract the non-

geometric contribution of altered matrix properties.

Investigations about the degradation of the barrier properties due to platelet

misalignments revealed that there exists a lower bound which is exactly one third


- 91 -

of the maximal possible permeability reduction achieved by perfectly aligned

platelets. For technologically interesting a-f-combinations inside the semidilute

concentration regime, however, the degrading effect of platelet misalignments is

even reduced because the impermeable inclusions collectively increase the

tortuosity of the penetrant’s diffusion path. Therefore the barrier performance of

nanocomposites with misaligned platelets is better than predicted by the lower

bound. The large barrier property enhancements predicted in chapter 4.1.1 for

nanocomposites with perfectly aligned platelets are therefore impaired by only

40-50% due to the 3D random orientation distribution of the platelets.

Numerical investigations on the thermoelastic properties of muscovite mica

nanocomposites have shown that with morphologies of perfectly aligned high-

aspect ratio platelets it is possible to increase the transverse Young’s modulus

several times. Especially for low modulus polymer matrices with a large platelet-

to-matrix modulus ratio one can expect improvements up to a factor of 58.

Nanocomposites with aligned platelet morphologies, however, show a

pronounced anisotropy. The longitudinal Young’s modulus is only moderately

changed and at an angle of 40° between the applied normal stress and the

platelets’ symmetry axes almost no improvement can be observed. Therefore

nanocomposite structural parts with perfectly aligned platelets could only prove

their excellent transverse stiffness in a very limited range of applications. Hence,

for structural applications it is desirable to produce nanocomposite parts

comprising 3D randomly oriented platelets. The result is a macroscopically

isotropic material which can prove its excellent stiffness in combination with other

improved properties (thermal stability, flame retardance, dimensional stability,

heat distortion temperature) in a broad range of structural applications, especially

in the automotive sector where high specific properties are needed at the lowest

possible price. It has further been shown that the transverse CTE of

nanocomposites with aligned morphologies can be described by a response

function. This response function can be used to compute design diagrams for

tailoring the transverse CTE of nanocomposites. It has been demonstrated how

the transverse CTE of a nanocomposite can be adjusted to the CTE of aluminum

by choosing the right combination of platelet aspect ratio a and volume fraction f.


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As for the elastic properties also the CTE of perfectly aligned morphologies is

highly anisotropic. As soon as platelets start to disorient the CTE in the transverse

direction is going to rise. In this case, however, the response function can not be

applied any more. For this purpose the accuracy of the orientation averaging

scheme has been validated. It was found that for the investigated range of platelet

aspect ratios and volume fractions orientation averaging can be readily employed

to predict both the elastic and the CTE properties of nanocomposites comprising

any platelet orientation state. This requires, though, that reliable data of the

property tensors of the corresponding aligned morphology are available. The

FEM used throughout this work is highly suitable to calculate these data

numerically.


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5. OUTLOOK

With the rapidly increasing speed of computers it has become feasible to

predict the properties and the behavior of different material classes on various

length and time scales by using increasingly sophisticated and complex computer

models. It is, however, not only about more complex models but also about

eligible combinations of different approaches which are needed in order to

facilitate and shorten the design phase of new products.

For example, FE-based mold filling flow simulations i.e Moldflow, Sigmasoft

are capable of delivering the local fiber orientation states in each element of an

injection molded short fiber reinforced part. Structural FE-packages i.e Ansys,

Abaqus, etc. are, among other things, used to find out if structural parts are

optimally designed to resist the expected loads during operation. Between these

two FE-packages there is, however, a gap which needs to be bridged so that a

fully computer-based design of short fiber reinforced parts becomes possible.

This gap consists of the unknown mechanical properties for the different fiber

orientation states in different sections of an injection molded part. In this work it

has been shown that the Gusev’s FEM, which is available as a commercial

software package called Palmyra1, together with the orientation averaging

scheme is highly suitable to fill this gap because the combination of these two

approaches allows to quickly determine accurate thermoelastic properties for any

fiber orientation state.

Viewing the field of nanocomposites it would be most interesting to include

interfacial layers with properties different from those of the bulk in multi-inclusion

FE-models. The problem, though, is that there exist no well established data on

the matrix properties in the vicinity of platelet interfaces and therefore one is

forced necessary to use reasonable assumptions. Nevertheless, it would be

interesting to see if the wide held believe that it is the changed matrix properties

at the interfaces which cause the extraordinary property enhancements of

nanocomposites is true. Furthermore, to my knowledge, this would be the first

1. PALMYRA is a software product of MatSim GmbH, Zürich (www.matsim.ch)

Chapter 5 - Outlook

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time that altered matrix properties in the vicinity of platelet interfaces would be

included in a FE-modeling approach.

Numerical investigations on morphological imperfections other than platelet

misalignments could bring more insight, too. It is, however important, that such

investigations are based on well established experimental data like it has been

demonstrated for short fiber reinforced composites. Analogous to those

investigations it would be interesting to reveal for instance by what single aspect

ratio a platelet ARD can be substituted in order to get the same effective

properties for the nanocomposite.

Since in this work only the linear elastic properties were considered it would

also be interesting to investigate composite materials under large strain

deformations. For this purpose one should switch, however, to cubic finite

elements which are more appropriate for large strain FE-simulations. Since for

large-strain FE-calculations powerful solvers are already available in Ansys and

Abaqus it would be proximate to export periodic meshes to one of these software

packages in order to use their numerical solvers for large strain calculations.

Chapter 5 - Outlook

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CURRICULUM VITAE

Date of Birth: September 7, 1973

Citizen of: Nesslau, SG

Martial status: Married

Education: 1980 - 1986 Primary School in Nesslau

1986 - 1988 Secondary School in Nesslau

1988 - 1993 High School (Typus C) in Wattwil

1993 - 1999 ETH Zurich, Department of Materials Science

1999 Diploma thesis with PD A.A. Gusev, Institute of

Polymers, ETH Zurich

1999 - 2003 Doctoral studies with PD A.A. Gusev, Institute of


Professional: 1996 3 month internship, Department of High

Performance Ceramics, EMPA, Dübendorf

1998 4 month internship, R&D, Kistler Instruments,

Winterthur

1999 - 2003 Client support and remittance work,

MatSim GmbH, Zurich

2000 - 2001 Teaching assistant in computer science course,

Department of Chemistry, ETH Zurich

2002 Supervision of a diploma student, Institute of


Curriculum Vitae

- 102 -

Curriculum Vitae

eth-26498-02

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