30
Edited by
Vikas Mittal
Modeling and Prediction of Polymer Nanocomposite Properties
Polymer Nano-, Micro- & Macrocomposite Series
Mittal, V. (ed.)
Surface Modification of Nanotube Fillers
2011
ISBN: 978-3-527-32878-9
Mittal, V. (ed.)
In-situ Synthesis of Polymer Nanocomposites
2012
ISBN: 978-3-527-32879-6
Mittal, V. (ed.)
Characterization Techniques for Polymer Nanocomposites
2012
ISBN: 978-3-527-33148-2
Related Titles
Thomas, S., Joseph, K., Malhotra, S. K., Goda, K., Sreekala, M. S. (eds.)
Polymer Composites
Volume 1
2012
ISBN: 978-3-527-32624-2
Mittal, V. (ed.)
Miniemulsion Polymerization Technology
2010
ISBN: 978-0-470-62596-5
Cosnier, S., Karyakin, A. (eds.)
ElectropolymerizationConcepts, Materials and Applications
2010
ISBN: 978-3-527-32414-9
Mittal, V. (ed.)
Polymer Nanotube NanocompositesSynthesis, Properties, and Applications
2010
ISBN: 978-0-470-62592-7
Leclerc, M., Morin, J.-F. (eds.)
Design and Synthesis of Conjugated Polymers
2010
ISBN: 978-3-527-32474-3
Xanthos, M. (ed.)
Functional Fillers for Plastics
2010
ISBN: 978-3-527-32361-6
Modeling and Prediction of Polymer Nanocomposite Properties
Edited by Vikas Mittal
The Editor
Dr. Vikas MittalThe Petroleum InstituteChemical Engineering DepartmentBu Hasa Building, Room 2204Abu DhabiUAE
All books published by Wiley-VCH are carefully produced. Nevertheless, authors, editors, and publisher do not warrant the information contained in these books, including this book, to be free of errors. Readers are advised to keep in mind that statements, data, illustrations, procedural details or other items may inadvertently be inaccurate.
Library of Congress Card No.: applied for
British Library Cataloguing-in-Publication DataA catalogue record for this book is available from the British Library.
Bibliographic information published by the Deutsche NationalbibliothekThe Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available on the Internet at .
© 2013 Wiley-VCH Verlag & Co. KGaA, Boschstr. 12, 69469 Weinheim, Germany
All rights reserved (including those of translation into other languages). No part of this book may be reproduced in any form – by photoprinting, microfilm, or any other means – nor transmitted or translated into a machine language without written permission from the publishers. Registered names, trademarks, etc. used in this book, even when not specifically marked as such, are not to be considered unprotected by law.
Print ISBN: 978-3-527-33150-5ePDF ISBN: 978-3-527-64436-0ePub ISBN: 978-3-527-64435-3mobi ISBN: 978-3-527-64437-7oBook ISBN: 978-3-527-64434-6ISSN: 2191-0421
Printed on acid-free paper
http://dnb.d-nb.de
V
Contents
ListofContributors XI Preface XV
1 ConvergenceofExperimentalandModelingStudies 1 VikasMittal1.1 Introduction 11.2 Review of Various Model Systems 1 References 10
2 Self-ConsistentFieldTheoryModelingofPolymerNanocomposites 11 ValeriyV.Ginzburg2.1 Introduction 112.2 Theoretical Methods 132.2.1 Incompressible SCFT 132.2.2 Compressible SCFT 172.3 Applications of SCFT Modeling: Predicting the Nanocomposite Phase
Behavior 182.3.1 Organically Modified Nanoclays in a Homopolymer Matrix 182.3.2 Organically Modified Nanoclays in a Binary Blend Containing End-
Functionalized Polymers 242.4 Summary and Outlook 32 Acknowledgments 33 References 33
3 ModernExperimentalandTheoreticalAnalysisMethodsofParticulate-FilledNanocompositesStructure 39
GeorgiiV.Kozlov,YuriiG.Yanovskii,andGennadiiE.Zaikov3.1 Introduction 393.2 Experimental 403.3 Results and Discussion 423.4 Conclusions 60 References 61
VI Contents
4 ReptationModelfortheDynamicsandRheologyofParticleReinforcedPolymerChains 63
KalonjiK.KabanemiandJean-FrançoisHétu4.1 Introduction 634.2 Terminal Relaxation Time 664.2.1 Linear Entangled Chains 664.2.2 Linear Entangled Chains with Rigid Spherical
Nanoparticles 664.3 Detachment/Reattachment Dynamics 724.4 Constitutive Equation 744.5 Numerical Results 754.5.1 Step Shear Strain 754.5.2 Steady Shear Flow 784.5.3 Start-up of Steady Shear Flow 844.5.4 Experimental Validation 854.6 Discussion and Generalization of the Model 884.6.1 Preliminaries 884.6.2 Diffusion of an Attached Chain 894.6.3 Multimode Constitutive Equation 914.7 Conclusions 92 References 93
5 MultiscaleModelingApproachforPolymericNanocomposites 95
PaolaPosocco,SabrinaPricl,andMaurizioFermeglia5.1 Multiscale Modeling of Polymer-Based Nanocomposite Materials:
Toward “Virtual Design” 955.2 Atomistic Scale: Basic Instincts 1015.2.1 Sodium Montmorillonite Silylation: Unexpected Effect of the
Aminosilane Chain Length 1015.2.2 Water-Based Montmorillonite/Poly(Ethylene Oxide)
Nanocomposites: A Molecular Viewpoint 1065.3 Mesoscale: Connecting Structure to Properties 1095.3.1 Water-Based Montmorillonite/Poly(Ethylene Oxide)
Nanocomposites at the Mesoscale 1095.3.2 Nanoparticles at the Right Place: Tuning Nanostructure
Morphology of Self-Assembled Nanoparticles in Diblock Copolymers 112
5.4 Macroscale: Where Is the Detail? The Matter at Continuum 119
5.4.1 Small Is Different. Size and Shape Effects of Nanoparticles on the Enhancement Efficiency in PCNs 119
5.5 Conclusions 123 References 125
Contents VII
6 ModelingofOxygenPermeationandMechanicalPropertiesofPolypropylene-LayeredSilicateNanocompositesUsingDoEDesigns 129
VikasMittal6.1 Introduction 1296.2 Materials and Methods 1316.2.1 Materials 1316.2.2 Filler Surface Modification and Composite Preparation 1316.2.3 Characterization and Modeling Techniques 1316.3 Results and Discussion 1326.4 Conclusions 141 Acknowledgment 141 References 141
7 MultiscaleStochasticFiniteElementsModelingofPolymerNanocomposites 143
AntoniosKontsosandJeffersonA.Cuadra7.1 Introduction 1437.2 Multiscale Stochastic Finite Elements Method 1447.2.1 Modeling State-of-the-Art and MSFEM Motivation 1447.2.2 Definition of a Representative Material Region (MR) 1457.2.3 Spatial Randomness Identification 1467.2.4 Multiscale Homogenization Model 1487.2.5 Monte Carlo Finite Element Model 1527.3 Applications and Results 1537.3.1 Estimation of Bulk Mechanical Properties 1537.3.2 Modeling of Nanoindentation Data 161 References 165
8 ModelingofThermalConductivityofPolymerNanocomposites 169 WeiLin8.1 Models for Thermal Conductivity of Polymer Composites – A Historical
Review on Effective Medium Approximations and Micromechanical Models 169
8.1.1 Parallel and Series Models 1708.1.2 Maxwell’s Model (Maxwell–Garnett Equation) 1728.1.3 Fricke’s Model 1728.1.4 Hamilton–Crosser Model 1748.1.5 Hashin’s Model 1758.1.6 Nielsen’s Micromechanics Model 1768.1.7 Equivalent Inclusion Method 1788.1.8 Benveniste–Miloh Model 1808.1.9 Davis’ Model 1828.1.10 Empirical Model by Agari and Uno 1828.1.11 Hasselman–Johnson Model 183
VIII Contents
8.1.12 Bruggeman Asymmetric Equation 1838.1.13 Felske’s Model 1858.2 A Generalized Effective Medium Theory 1868.2.1 ATA 1878.2.2 CPA 1888.2.3 Further Extension of ATA and CPA to Anisotropic Filler with
Orientation Distributions 1898.2.4 Incorporation of Size Distribution Functions into ATA and CPA 1908.2.5 Incorporation of Interfacial Thermal Resistance into ATA and CPA 1918.3 Challenges for Modeling Thermal Conductivity of Polymer
Nanocomposites 1918.3.1 Size Effect and Surface Effect 1918.3.2 Sensitivity of κf to a Specific Environment 1928.3.3 Interfacial Resistance Plays a Very Important Role 1938.3.4 Filler-Induced Change in κm 1958.3.5 Dispersion and Distribution 196 Acknowledgments 196 References 197
9 Numerical–AnalyticalModelforNanotube-ReinforcedNanocomposites 201
AntonioPantano9.1 Introduction 2019.2 Numerical–Analytical Model 2049.2.1 The Mori–Tanaka Method 2049.2.1.1 Calculation of the Correlation Matrix A1dil 2069.2.1.2 Calculation of the Stiffness Matrix of the Equivalent Inclusion Cl 2079.2.2 FEM Model Design 2079.2.2.1 RVE Geometry 2079.2.2.2 Matrix Constitutive Model 2089.2.2.3 Carbon Nanotube 2089.2.2.4 Contact Model 2089.2.2.5 Deformation Mode 2099.2.2.6 Calculation of the Equivalent Young’s Modulus of the MWCNT 2099.2.2.7 Calculation of the Eshelby Tensor 2099.3 Results 2109.4 Conclusions 212 Appendix 9.A 212 References 213
10 DissipativeParticlesDynamicsModelforPolymerNanocomposites 215
Shin-PonJu,Yao-ChunWang,andWen-JayLee10.1 Introduction 21510.2 Scheme for Multiscale Modeling 218
Contents IX
10.2.1 Dissipative Particle Dynamics Simulation Method 21910.2.2 Coarse-Grained Mapping 21910.2.3 Mixing Energy and Compressibility 22010.2.4 Dissipative Particle Dynamics Scales to Physical Scales 22210.3 Two Case Studies 22210.3.1 PE/PLLA Composite 22210.3.2 CNT/PE/PLLA Composite 22810.4 Future Work 234 References 234
11 Computer-AidedProductDesignofWheatStrawPolypropyleneComposites 237
RoisFatoni,AliAlmansoori,AliElkamel,andLeonardoSimon11.1 Natural Fiber Plastic Composites 23711.1.1 History and Current Market Situation 23711.1.2 Technical Issues and Current Research Progress 23811.2 Wheat Straw Polypropylene Composites 24011.3 Product Design and Computer-Aided Product Design 24211.4 Modeling Natural Fiber Polymer Composites 24511.5 Mixture Design of Experiments 247 References 252
12 ModelingoftheChemorheologicalBehaviorofThermosettingPolymerNanocomposites 255
LuigiTorre,DeboraPuglia,AntonioIannoni,andAndreaTerenzi12.1 Introduction 25512.2 The Cure Kinetics Model 25812.3 The Chemoviscosity Model 26312.4 Relationship between Tg and α 26812.5 Case Study 1: Carbon Nanofibers in Unsaturated Polyester 26812.5.1 Cure Kinetic Analysis 27112.5.2 Chemorheological Analysis 27512.6 Case Study 2: Montmorillonite in Epoxy Resin 27712.6.1 Cure Kinetic Analysis 27912.6.2 Relation between Tg and Degree of Cure 28112.6.3 Chemorheological Analysis 282 References 285
Index 289
XI
ListofContributors
Ali AlmansooriThe Petroleum InstituteDepartment of Chemical EngineeringP.O. Box 2533Abu DhabiUAE
Chetan ChanmalNational Chemical LaboratoryPolymer Science and Engineering DivisionDr. Homi Bhabha Road, PashanPuneMaharashtra 411008India
Ali ElkamelUniversity of WaterlooDepartment of Chemical Engineering200 University Avenue WestWaterloo, ONCanada N2L 3G1
Rois FatoniUniversity of WaterlooDepartment of Chemical Engineering200 University Avenue WestWaterloo, ONCanada N2L 3G1
Maurizio FermegliaUniversity of TriesteDepartment of Industrial Engineering and Information Technology (DI3)Via Valerio 1034127 TriesteItaly
Valeriy V. GinzburgThe Dow Chemical CompanyBuilding 1702Midland, MI 48674USA
Jean-François HétuNational Research Council of Canada (NRC)Industrial Materials Institute (IMI)75 de MortagneBoucherville, QCCanada J4B 6Y4
Antonio IannoniUniversity of PerugiaMaterial Engineering CenterDepartment of Civil and Environmental EngineeringStrada di Pentima 405100 TerniItaly
XII ListofContributors
Jyoti JogNational Chemical LaboratoryPolymer Science and Engineering DivisionDr. Homi Bhabha Road, PashanPuneMaharashtra 411008India
Shin-Pon JuNational Sun Yat-sen UniversityDepartment of Mechanical and Electro-Mechanical EngineeringCenter for Nano Science and Nano TechnologyKaohsiung 80424, Taiwan
Kalonji K. KabanemiNational Research Council of Canada (NRC)Industrial Materials Institute (IMI)75 de MortagneBoucherville, QCCanada J4B 6Y4
Antonios KontsosDrexel UniversityDepartment of Mechanical Engineering & Mechanics3141 Chestnut St., AEL 172 APhiladelphia, PA, 19104USA
Georgii V. KozlovInstitute of Applied Mechanics of Russian Academy of SciencesLeninskii pr., 32 aMoscow 119991Russian Federation
Wen-Jay LeeNational Center for High-Performance ComputingTainan 74147, Taiwan
Wei LinSchool of Materials Science and EngineeringGeorgia Institute of Technology771 Ferst Drive NWAtlanta, GA 30332USA
Vikas MittalThe Petroleum InstituteChemical Engineering DepartmentRoom 2204, Bu Hasa BuildingAbu Dhabi 2533United Arab Emirates
Antonio PantanoUniversità degli Studi di PalermoDipartimento di Ingegneria Chimica, Gestionale, Informatica e MeccanicaEdificio 8 – viale delle Scienze90128 PalermoItaly
Paola PosoccoUniversity of TriesteDepartment of Industrial Engineering and Information Technology (DI3)Via Valerio 1034127 TriesteItaly
Sabrina PriclUniversity of TriesteDepartment of Industrial Engineering and Information Technology (DI3)Via Valerio 1034127 TriesteItaly
Debora PugliaUniversity of PerugiaMaterial Engineering CenterDepartment of Civil and Environmental EngineeringStrada di Pentima 405100 TerniItaly
ListofContributors XIII
Leonardo SimonUniversity of WaterlooDepartment of Chemical Engineering200 University Avenue WestWaterloo, ONCanada N2L 3G1
Andrea TerenziUniversity of PerugiaMaterial Engineering CenterDepartment of Civil and Environmental EngineeringStrada di Pentima 405100 TerniItaly
Luigi TorreUniversity of PerugiaMaterial Engineering CenterDepartment of Civil and Environmental EngineeringStrada di Pentima 405100 TerniItaly
Yao-Chun WangNational Sun Yat-sen UniversityDepartment of Mechanical and Electro-Mechanical EngineeringCenter for Nano Science and Nano TechnologyKaohsiung 80424, Taiwan
Yurii G. YanovskiiInstitute of Applied Mechanics of Russian Academy of SciencesLeninskii pr., 32 aMoscow 119991Russian Federation
Gennadii E. ZaikovN.M. Emanuel Institute of Biochemical Physics of Russian Academy of SciencesKosygin st., 4Moscow 119334Russian Federation
XV
Preface
Modeling and prediction of the nanocomposite properties is generally achieved using different finite element, statistical and micromechanical models. These models help in predicting the properties of the nanomaterials, thus eliminating the need for synthesizing each and every composite first to ascertain its properties. A number of precautions are, however, necessary in order to avoid discrepancies in the model outcome, for example, the model used should not have unrealistic assumptions and the experimental results should be in plenty in order to have an accurate model. The validation of the model should also be achieved by a compari-son of the predicted values with the experimental values. The chapters contained in the book present examples of modeling and prediction of polymer clay nano-composite properties using various types of theoretical methods.
Chapter 1 comments on the convergence of the experimental and theoretical studies and reviews briefly the various kinds of melds used for the prediction of nanocomposite properties. Chapter 2 reviews the application of Self-Consistent Field Theory (SCFT) to prediction of polymer-clay nanocomposite morphology. Over the past decade, SCFT has been shown to qualitatively describe the factors influencing the polymer ability to intercalate or exfoliate the clay platelets. In Chapter 3, the experimental analysis of particulate-filled nanocomposites butadiene-styrene rubber/fullerene-containing mineral (nanoshungite) is ana-lyzed with the aid of force-atomic microscopy, nanoindentation methods, and computer treatment. The theoretical analysis is carried out within the frameworks of fractal analysis. Chapter 4 presents a reptation-based model that incorporates polymer-particle interactions and confinement to describe the dynamics and rheo-logical behaviors of linear entangled polymers filled with isotropic nanoscale par-ticles. In Chapter 5, a hierarchical procedure for bridging the gap between atomistic and macroscopic modeling via mesoscopic simulations is presented. The concept of multiscale modeling is outlined, and relevant examples of applications of single scale and multiscale procedures for nanostructured systems of industrial interest are illustrated. The behavior of polymer-layered silicate nanocomposites is modeled in Chapter 6 through various factorial and mixtures design methodologies in order to optimize the composite performance and to accurately predict the properties especially for the non-polar polymer systems. Chapter 7 introduces a hierarchical multiscale and stochastic Finite Element Method (MSFEM) to model the spatial
XVI Preface
randomness induced in polymers by the non-uniform distribution of nanophases including primarily single walled carbon nanotubes (SWCNT). In Chapter 8, a general effective medium model derived from “grain averaging theory”—in analogy to quantum scattering theory—is reviewed in which anisotropicity of the second phase (filler from hereafter) can be included. Chapter 9 presents a new technique that takes into account the curvature that the nanotubes show when immersed in the polymer, and is based on a numerical-analytical approach that has significant advances over micromechanical modeling and can be applied to several kinds of nanostructured composites. In Chapter 10, details of the coarse grain scheme from molecular dynamics (MD) to dissipative particle dynamics (DPD) modeling are discussed. Two polymer nanocomposite case studies – PE/PLLA (polyethylene/poly lactic acid) and PE/PLLA/CNT – are provided to demon-strate how multiscale simulation can describe the effects of volume fraction and mixing method on the structure. Chapter 11 presents a product design approach and strategy to design wheat straw polypropylene composites (WSPPC). In this approach, a product design problem is connected to and simultaneously solved with process-product problem to create new products that satisfy the market needs. In Chapter 12, a kinetic model is used to predict the reaction rate and the degree of cure as a function of time and temperature; whereas a rheological model describes viscosity as a function of time and temperature. Since viscosity is also dependent on the degree of cure, the rheological model combined with the kinetic model forms a chemorheological model.
I am indebted to Wiley-VCH for publication of the book. I am thankful to my family, especially to my wife Preeti for her continuous support during the prepara-tion of the manuscript.
Vikas MITTALAbu Dhabi
1
ConvergenceofExperimentalandModelingStudiesVikasMittal
1.1Introduction
Experimental results on composite properties are generally modeled using differ-ent finite element and micromechanical models to gain further insights into the experimental findings. Such models are also useful in predicting the properties of same or similar materials, thus eliminating the need for synthesizing each and every composite first to ascertain its properties. A number of precautions are, however, necessary to avoid discrepancies in the model outcome, for example, the model used should not have unrealistic assumptions, and the experimental results should be in plenty to have an accurate model. The following sections present some examples of modeling and prediction of polymer clay nanocomposite proper-ties using micromechanical, finite element, and factorial design methods.
1.2ReviewofVariousModelSystems
A number of micromechanical models have been developed over the years to predict the mechanical behavior of particulate composites [1–4]. The Halpin–Tsai model has received special attention owing to better prediction of the properties for a variety of reinforcement geometries. The relative tensile modulus is expressed as
E E/ 1 / 1m f f= + −( ) ( )ζηϕ ηϕ
where E and Em correspond to the elastic moduli of composite and matrix, respec-tively, ζ represents the shape factor, which is dependent on filler geometry and loading direction and φf is the inorganic volume fraction. η is given by the expression
η ζ= − +( ) ( )E E E Ef m f m/ 1 / /
where Ef is the modulus of the filler. The η values need to be correctly defined in order to have better prediction of the properties. For the oriented discontinuous
1
Modeling and Prediction of Polymer Nanocomposite Properties, First Edition. Edited by Vikas Mittal.© 2013 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2013 by Wiley-VCH Verlag GmbH & Co. KGaA.
2 1 ConvergenceofExperimentalandModelingStudies
ribbon or lamellae, it is estimated to be twice the aspect ratio. It has been reported to overpredict the stiffness in this case; therefore, its value was reported be 2/3 times the aspect ratio [5]. Nevertheless, several assumptions prevent the theory to correctly predict the stiffness of the layered silicate nanocomposites. Assumptions like firm bonding of filler and matrix, perfect alignment of the platelets in the matrix, and uniform shape and size of the filler particles in the matrix make it very difficult to correctly predict the nanocomposites properties. Incomplete exfoliation of the nanocomposites, thus, the presence of a distribution of tactoid thicknesses, is another concern. The model has recently been modified to accom-modate the effect of incomplete exfoliation and misorientation of the filler, but the effect of imperfect adhesion at the surface still needs to be incorporated [6, 7].
As a case study, tensile properties of polypropylene (PP) nanocomposites con-taining dioctadecyldimethylammonium-modified montmorillonite (2C18•M880) using different filler inorganic volume fraction were modeled using these micro-mechanical approaches [8]. The modulus of the composites linearly increased with volume fraction with an increase of 45% at 4 vol% as compared to the pure PP. As shown in Figure 1.1a, the data were fitted to the conventional Halpin–Tsai equation with η = 1, which gives a value of 10.1 for ζ, indicating that possibly in these nanocomposites it cannot be simply taken as twice the aspect ratio as gener-ally used [9]. To account for the incomplete filler exfoliation and the presence of tactoid stacks in the composites, thickness of the particle was explained by the following equation:
t d n tparticle 1 platelet1= − +00 ( )
where d001 is the basal plane spacing of 001 plane, n is the number of the platelets in the stack, and tplatelet is the thickness of one platelet in the pristine montmorillo-nite. Thus, in this approach, filler particles were replaced by the stacks of filler platelets [6]. Applying this treatment to the Halpin–Tsai equation, different curves have been generated based on the number of platelets present in the stack as shown in Figure 1.1b. As is evident from the figure, the experimental value of relative tensile modulus for 1 vol% OMMT composites lies near to theoretical curve with 50 platelets in the stack, but the composites with higher volume fractions of the filler could not follow the predicted rise in the modulus. The observed behavior underlines another important limitation of the theoretical models for their inability to take into account the possible decrease in d-spacing with increasing volume frac-tion. Besides, the effect of misoriented platelets on the modulus also needs to be incorporated in the model. As can be seen in the SEM micrographs in Figure 1.2, for the 3 vol% 2C18•M880 OMMT-PP nanocomposites, the filler platelets can be safely treated as random and misaligned in the matrix. Figure 1.3a shows the result-ing comparison when the effects of incomplete exfoliation combined with the platelets misalignment considerations were incorporated in the Halpin–Tsai model for random 3D platelets [5]. As can be seen the number of platelets in the stack for 1 vol% composites now lie between 30 and 50 (∼40). Brune and Bicerano have also refined the predictions for the behavior of nanocomposites based on the combina-tion of incomplete exfoliation and misorientation [7]. Comparing the suggested
1.2ReviewofVariousModelSystems 3
treatment with the experimental data, Figure 1.3b showed that the number of plate-lets in the stacks in 1 vol% composite was observed to be between 20 and 25, which gives an aspect ratio of about 15 for these composites. However, one major limita-tion of the mechanical models is the assumption of perfect adhesion at the inter-face, whereas the polyolefin composites studied in fact lack this adhesion, as only weak van der Waals forces can exist in the studied polymer organic monolayer systems. The theoretical results predicted above were therefore only able to match the experimental results of polar polymers due to the same reason [10, 11].
Figure1.1 (a) Relative tensile modulus of PP nanocomposites plotted as a function of inorganic volume fraction. The solid line represents the fitting using the unmodified Halpin–Tsai equation. (b) Relative tensile
modulus of the above-mentioned composites (:- experimental) compared with the values considering different number of platelets in the stack. Reproduced from reference 8 with permission (Sage Publishers).
(a)
(b)
RR
I
I
4 1 ConvergenceofExperimentalandModelingStudies
Figure1.2 SEM micrographs of 3 vol% PP nanocomposites. Reproduced from Ref. [8] with permission (Sage Publishers).
(a)
(b)
1.2ReviewofVariousModelSystems 5
Nicolais and Nicodemo [12] suggested a simple model to predict the tensile strength of the filled polymers described by the equation
σ σ ϕ/ 11 1 P2= − P
where P1 is stress concentration-related constant with a value of 1.21 for the spheri-cal particles having no adhesion with the matrix and P2 is geometry-related con-stant with a value of 0.67 when the sample fails by random failure. The yield strength, yield strain, and stress at break for the 2C18•M880 OMMT-PP com-posites as a function of inorganic filler volume fraction have been plotted in Figure 1.4. The yield strength decayed with augmenting the filler volume fraction,
Figure1.3 Relative tensile modulus of PP nanocomposites at different inorganic volume fraction (:- experimental) compared with the values considering different number
of platelets in the stack applying the platelet misorientation corrections. Reproduced from Ref. [8] with permission (Sage Publishers).
RR
I
I
(a)
(b)
6 1 ConvergenceofExperimentalandModelingStudies
indicating the lack of adhesion at the interface and brittleness as shown in Figure 1.4a. As described earlier, with the addition of low-molecular-weight compatibiliz-ers, an increase in the yield strengths were reported probably due to better adhe-sion, higher extents of delamination, and plasticization effects. The platelets in the present case, which may have been only kinetically trapped, also lead to straining of the confined polymer chains. Fitting the values of yield strength in the Nicolais and Nicodemo model yielded P1 as 2.30 and P2 as 0.63, thus deviating from the values marked for the spherical particles [9]. The stress at break also decreased nonlinearly with filler volume fraction owing to similar reasons and the presence of tactoids. The fitting of stress at break values (Figure 1.4b) in the model yielded P1 and P2 as 6.13 and 1.03, respectively, which shows higher deviation from the spherical particle predictions. Nielsen [13] suggested that the strain can be pre-dicted by the simple equation as
ε ε ϕc m f1 3/ 1= − /
Figure1.4 (a) Relative yield strength, (b) relative stress at break, and (c) relative yield strain of PP nanocomposites plotted as a function of inorganic volume fraction. The
solid lines represent the fitting using the theoretical equations, whereas the dotted line serves simply as a guide. Reproduced from Ref. [8] with permission (Sage Publishers).
(b) (a)
Rel
ativ
e yi
eld
stre
ngth
Rel
ativ
e yi
eld
stra
in
Rel
ativ
e st
ress
at b
reak
Inorganic volume fraction
Inorganic volume fraction
Inorganic volume fraction
(c)
1.2ReviewofVariousModelSystems 7
where εc and εm are the yield strains of the composite and matrix, respectively, and φf is the filler volume fraction. It was assumed that the polymer breaks at the same elongation in the filled composite as the bulk unfilled polymer does. The much lower experimental values (Figure 1.4c) agree with the lack of adhesion as sug-gested above and the strain hardening of the confined polymer. It also indicates that the brittleness increased on increasing the filler volume fraction.
Figure 1.5a shows a typical finite element model of round platelets with an aspect ratio of 50 at 3 vol% loading, while Figure 1.5b presents a 2D-cut through the center of the model [14]. The solid lines in Figure 1.6a represent the numerical predictions for the relative permeability of composites as a function of increasing volume fraction of misaligned platelets with an aspect ratio of 50 or 100. As noted, there is excellent agreement between the experimentally measured oxygen perme-ability and the numerical predictions up to ca. 3 vol%. Above this concentration, it seems that the number of exfoliated layers decreases, leading to a lower average aspect ratio in both epoxy (EP) and polyurethane (PU) composites. An average
Figure1.5 (a) A computer model compris-ing 50 randomly distributed and oriented round platelets with an aspect ratio of 50 at 3 vol% loading, periodic boundary conditions
applied; (b) cross section through the center of the model. Reproduced from Ref. [14] with permission (Wiley).
(a)
(b)
8 1 ConvergenceofExperimentalandModelingStudies
aspect ratio of the montmorillonite platelets in nanocomposites can be estimated from the relative permeability at 3 vol% loading. The effect of misalignment on the barrier performance of platelets with different aspect ratios at 3 vol% loading, as predicted by computer models, is shown in Figure 1.6b.With increasing aspect ratio, it becomes necessary to align the platelets in order not to lose their effective-ness. Similarly, Figure 1.7 plots the comparison of measured permeability through polypropylene nanocomposites with numerical predictions for composites of par-allel oriented and misaligned disk-shaped impermeable inclusions with aspect ratios (diameter/thickness) 30 and 100, respectively [15]. From this comparison, a macroscopic average of the aspect ratio for the inclusions is estimated to be between 30 and 100. However, a more precise estimation can only be made when the degree of orientation is experimentally determined and an orientation-dependent term is included in the numerical calculation.
Figure1.6 Dependence of the gas permea-tion through nanocomposites on the inorganic volume fraction, aspect ratio, and orientation of the platelets: (a) comparison between the measured relative oxygen permeability in EP- and PU-nanocomposites
and numerical predictions; (b) influence of misalignment on the performance of platelets as permeation barrier at 3 vol% loading as predicted numerically. Reproduced from Ref. [14] with permission (Wiley).
(a)
(b)
Inorganic volume fraction (f)
Aspect ratio (a)
misaligned
aligned
0 50 100 150 200
0.00
1.0
0.8
0.6
0.4
0.2
0.0
0.02 0.04 0.06
EPPU
a = 50
Rel
ativ
e tr
ansm
issi
on r
ate
(TC/T
P)
Rel
ativ
e tr
ansm
issi
on r
ate
(TC/T
P)
a = 100
1.0
0.8
0.6
1.2ReviewofVariousModelSystems 9
Figures 1.8 and 1.9 also demonstrate the possibility of modeling and prediction of polyethylene clay nanocomposite properties using mixture design methods. Examples of both oxygen permeation as well as tensile modulus as a function of different amounts of different components polymer, organically modified montmo-rillonite and compatibilizer have been shown. Like conventional models, which depend on oversimplified assumptions, these models do not suffer from these
Figure1.7 Relative permeability of the 2C18 – M880-PP nanocomposites as a function of the inorganic volume fraction. The lines represent numerical predictions for composites of parallel oriented and
misaligned disk-shaped impermeable inclusions with aspect ratio (diameter/thickness) of 30 and 100, respectively. Reproduced from Ref. [15] with permission (Wiley).
a = 100 misaligned
Inorganic volume fraction0.00
Rel
ativ
e pe
rmea
bilit
y1.0
0.9
0.8
0.7
0.60.01 0.02 0.03 0.04
a = 30 aligned
Figure1.8 Mixture plot for the prediction of oxygen permeation of polyethylene nanocompos-ites with different amounts of components: polymer (P), organically modified montmorillonite (OM) and compatibilizer (Compat).
10 1 ConvergenceofExperimentalandModelingStudies
Figure1.9 Mixture plot for the prediction of tensile modulus of polyethylene nanocomposites with different amounts of components: polymer (P), organically modified montmorillonite (OM), and compatibilizer (Compat).
limitations and can still predict the composite properties using a set of simple equations.
References
1 Kerner, E.H. (1956) Proc. Phys. Soc., B69, 808.
2 Hashin, Z., and Shtrikman, S. (1963) J. Mech. Phys. Solids, 11, 127.
3 Halpin, J.C. (1969) J. Compos. Mater., 3, 732.
4 Halpin, J.C. (1992) Primer on Composite Materials Analysis, Technomic, Lancaster.
5 van Es, M., Xiqiao, F., van Turnhout, J., and van der Giessen, E. (2001) Specialty Polymer Additives: Principles and Application (eds S. Al-Malaika, A.W. Golovoy, and C.A. Wilkie), Blackwell Science, CA Melden, MA, pp. 391–414.
6 Fornes, T.D., and Paul, D.R. (2003) Polymer, 44, 4993.
7 Brune, D.A., and Bicerano, J. (2002) Polymer, 43, 369.
8 Mittal, V. (2007) J. Thermoplastic Compos. Mater., 20, 575.
9 Osman, M.A., Rupp, J.E.P., and Suter, U.W. (2005) Polymer, 46, 1653.
10 Luo, J.J., and Daniel, I.M. (2003) Compos. Sci. Technol., 63, 1607.
11 Wu, Y.P., Jia, Q.X., Yu, D.S., and Zhang, L.Q. (2004) Polym. Test., 23, 903.
12 Nicolais, L., and Nicodemo, L. (1973) Polym. Eng. Sci., 13, 469.
13 Nielsen, L.E. (1966) J. Appl. Polym. Sci., 10, 97.
14 Osman, M.A., Mittal, V., and Lusti, H.R. (2004) Macromol. Rapid Commun., 25, 1145.
15 Osman, M.A., Mittal, V., and Suter, U.W. (2007) Macromol. Chem. Phys., 208, 68.
11
Self-ConsistentFieldTheoryModelingofPolymerNanocompositesValeriyV.Ginzburg
2.1Introduction
Polymer–clay nanocomposites have been studied extensively over the past three decades because of their potential utility in various applications [1–15]. The Toyota research group has shown that mixing 1–3 wt% clay into nylon-6 polymer can result in 2×–4× increase in the stiffness (or Young’s modulus) compared to the pure polymer [16–18]. Polymer–clay nanocomposites were also developed for many other polymers (natural rubber, polyethylene, polypropylene, epoxy, poly-urethane, etc.) [19–26]. However, in many cases, reproducing the initial success turned out to be a challenge. Specifically, it was shown very early on that mechani-cal and barrier properties of a nanocomposite depend on the dispersion (macro-scopic) and exfoliation (microscopic) of clay platelets in the polymer matrix. If the platelets are dispersed uniformly and are not aggregated, the “interphase” area (where polymer chains interact with the clays) is very large; if, on the other hand, platelets are aggregated into tactoids, the nanocomposite behaves essentially as a conventional composite with micron-sized fillers. We refer the readers to papers by Paul et al. [3, 27] and Bicerano et al. [13, 28, 29] for more details.
Very early on, Vaia and Giannelis [30, 31] realized that successful exfoliation of clay platelets in the melt is related to the polarity of the polymer and effective energy of the interaction between the polymer and the clay platelets. They formu-lated a simple thermodynamic model aimed at estimating the free energy of the matrix polymer going into the space between two nearby clay platelets. This free energy depends on the enthalpy of the polymer–clay interaction and the entropy of the matrix chain when it is confined in the gallery between the platelets. Thus, the exfoliated, intercalated, and immiscible morphologies (Figure 2.1) could be predicted based on the shape of the free energy profile as a function of the clay–clay separation (Figure 2.2).
Balazs and co-workers [32–40] extended this approach and utilized the lattice self-consistent field theory (SCFT) formalism of Scheutjens and Fleer [41–44].
2
Modeling and Prediction of Polymer Nanocomposite Properties, First Edition. Edited by Vikas Mittal.© 2013 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2013 by Wiley-VCH Verlag GmbH & Co. KGaA.
12 2 Self-ConsistentFieldTheoryModelingofPolymerNanocomposites
These studies provided important guidance into the role played by the organic modifiers (surfactants) in promoting exfoliation of organoclays in polymers. Some predictions from the models were nontrivial (e.g., it was shown that it was better to graft longer chains at lower grafting density instead of shorter chains at higher grafting density) but were ultimately borne out by experiment [19, 45–47]. Balazs and co-workers also emphasized the thermodynamic nature of their models and pointed out that the actual morphology often depends on the processing history (e.g., whether the composite was made by melt-compounding, in situ polymeriza-tion, or solvent route). Other groups have also utilized SCFT for nanocomposite modeling in recent years [48, 49]. In addition, molecular-level [50–58] and coarse-grained [2, 59–67] particle-based simulations (molecular dynamics and Monte Carlo) have been widely used to understand the dynamics of exfoliation and inter-calation and provide important insights into the process. Still, SCFT remains probably the quickest and a conceptually simplest way to qualitatively screen the possibility that a given nanocomposite formulation could result in a thermody-namically stable exfoliated or intercalated morphology.
Figure2.1 Schematic representation of main morphologies of polymer–clay nanocomposites. Black lines represent clay platelets. Gray lines in the middle panel show the polymer chains intercalating the galleries between the platelets. Matrix polymers are not shown.
Figure2.2 Thermodynamic framework for predicting nanocomposite morphologies. As the distance between two adjacent platelets H (a) is changed, the free energy per unit
area changes as well (b). Morphology is then deduced based on the shape of the curve. For more details, see text.
(a) (b)
Intercalated
Exfoliated
Plate separation, H
Immiscible
Free
ene
rgy
per
unit
area
