Modeling and Prediction of Polymer Nanocomposite Properties E-Book

• Herausgeber: Wiley-VCH
• Kategorie: Wissenschaft und neue Technologien
• Serie: Polymer Nano-, Micro- and Macrocomposites
• Sprache: Englisch

Seitenzahl: 477

Veröffentlichungsjahr: 2012

Beschreibung

The book series 'Polymer Nano-, Micro- and Macrocomposites' provides complete and comprehensive information on all important
aspects of polymer composite research and development, including, but not limited to synthesis, filler modification, modeling,
characterization as well as application and commercialization issues. Each book focuses on a particular topic and gives a balanced in-depth overview of the respective subfi eld of polymer composite science and its relation to industrial applications. With the books the readers obtain dedicated resources with information relevant to their research, thereby helping to save time and money.

This book lays the theoretical foundations and emphasizes the close connection between theory and experiment to optimize models
and real-life procedures for the various stages of polymer composite development. As such, it covers quantum-mechanical approaches to
understand the chemical processes on an atomistic level, molecular mechanics simulations to predict the filler surface dynamics, finite
element methods to investigate the macro-mechanical behavior, and thermodynamic models to assess the temperature stability. The whole is
rounded off by a look at multiscale models that can simulate properties at various length and time scales in one go - and with predictive
accuracy.

Leseprobe

Table of Contents

Cover

Series page

Title page

Copyright page

List of Contributors

Preface

1 Convergence of Experimental and Modeling Studies

1.1 Introduction

1.2 Review of Various Model Systems

2 Self-Consistent Field Theory Modeling of Polymer Nanocomposites

2.1 Introduction

2.2 Theoretical Methods

2.3 Applications of SCFT Modeling: Predicting the Nanocomposite Phase Behavior

2.4 Summary and Outlook

Acknowledgments

3 Modern Experimental and Theoretical Analysis Methods of Particulate-Filled Nanocomposites Structure

3.1 Introduction

3.2 Experimental

3.3 Results and Discussion

3.4 Conclusions

4 Reptation Model for the Dynamics and Rheology of Particle Reinforced Polymer Chains

4.1 Introduction

4.2 Terminal Relaxation Time

4.3 Detachment/Reattachment Dynamics

4.4 Constitutive Equation

4.5 Numerical Results

4.6 Discussion and Generalization of the Model

4.7 Conclusions

5 Multiscale Modeling Approach for Polymeric Nanocomposites

5.1 Multiscale Modeling of Polymer-Based Nanocomposite Materials: Toward “Virtual Design”

5.2 Atomistic Scale: Basic Instincts

5.3 Mesoscale: Connecting Structure to Properties

5.4 Macroscale: Where Is the Detail? The Matter at Continuum

5.5 Conclusions

6 Modeling of Oxygen Permeation and Mechanical Properties of Polypropylene-Layered Silicate Nanocomposites Using DoE Designs

6.1 Introduction

6.2 Materials and Methods

6.3 Results and Discussion

6.4 Conclusions

Acknowledgment

7 Multiscale Stochastic Finite Elements Modeling of Polymer Nanocomposites

7.1 Introduction

7.2 Multiscale Stochastic Finite Elements Method

7.3 Applications and Results

8 Modeling of Thermal Conductivity of Polymer Nanocomposites

8.1 Models for Thermal Conductivity of Polymer Composites – A Historical Review on Effective Medium Approximations and Micromechanical Models

8.2 A Generalized Effective Medium Theory

8.3 Challenges for Modeling Thermal Conductivity of Polymer Nanocomposites

Acknowledgments

9 Numerical–Analytical Model for Nanotube-Reinforced Nanocomposites

9.1 Introduction

9.2 Numerical–Analytical Model

9.3 Results

9.4 Conclusions

Appendix 9.A

10 Dissipative Particles Dynamics Model for Polymer Nanocomposites

10.1 Introduction

10.2 Scheme for Multiscale Modeling

10.3 Two Case Studies

10.4 Future Work

11 Computer-Aided Product Design of Wheat Straw Polypropylene Composites

11.1 Natural Fiber Plastic Composites

11.2 Wheat Straw Polypropylene Composites

11.3 Product Design and Computer-Aided Product Design

11.4 Modeling Natural Fiber Polymer Composites

11.5 Mixture Design of Experiments

12 Modeling of the Chemorheological Behavior of Thermosetting Polymer Nanocomposites

12.1 Introduction

12.2 The Cure Kinetics Model

12.3 The Chemoviscosity Model

12.4 Relationship between Tg and α

12.5 Case Study 1: Carbon Nanofibers in Unsaturated Polyester

12.6 Case Study 2: Montmorillonite in Epoxy Resin

Index

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.)

Electropolymerization

Concepts, Materials and Applications

2010

ISBN: 978-3-527-32414-9

Mittal, V. (ed.)

Polymer Nanotube Nanocomposites

Synthesis, 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

The Editor

Dr. Vikas Mittal

The Petroleum Institute

Chemical Engineering Department

Bu Hasa Building, Room 2204

Abu Dhabi

UAE

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© 2013 Wiley-VCH Verlag & Co. KGaA, Boschstr. 12, 69469 Weinheim, Germany

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Print ISBN: 978-3-527-33150-5

ePDF ISBN: 978-3-527-64436-0

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mobi ISBN: 978-3-527-64437-7

oBook ISBN: 978-3-527-64434-6

ISSN: 2191-0421

List of Contributors

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

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

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

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 comparison of the predicted values with the experimental values. The chapters contained in the book present examples of modeling and prediction of polymer clay nanocomposite 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 analyzed 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 rheological behaviors of linear entangled polymers filled with isotropic nanoscale particles. 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 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 demonstrate 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 preparation of the manuscript.

Vikas MITTALAbu Dhabi

1

Convergence of Experimental and Modeling Studies

Vikas Mittal

1.1 Introduction

Experimental results on composite properties are generally modeled using different 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 properties using micromechanical, finite element, and factorial design methods.

1.2 Review of Various Model Systems

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

where E and Em correspond to the elastic moduli of composite and matrix, respectively, 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

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 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 accommodate 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 containing dioctadecyldimethylammonium-modified montmorillonite (2C18•M880) using different filler inorganic volume fraction were modeled using these micromechanical 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 generally 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:

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 montmorillonite. 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 fraction. 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 resulting 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 combination of incomplete exfoliation and misorientation [7]. Comparing the suggested treatment with the experimental data, Figure 1.3b showed that the number of platelets 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 limitation of the mechanical models is the assumption of perfect adhesion at the interface, 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].

Figure 1.1 (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).

Figure 1.2 SEM micrographs of 3 vol% PP nanocomposites.

Reproduced from Ref. [8] with permission (Sage Publishers).

Figure 1.3 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).

Nicolais and Nicodemo [12] suggested a simple model to predict the tensile strength of the filled polymers described by the equation

where P1 is stress concentration-related constant with a value of 1.21 for the spherical particles having no adhesion with the matrix and P2 is geometry-related constant 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 com­posites 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, 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 compatibilizers, an increase in the yield strengths were reported probably due to better adhesion, 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 predicted by the simple equation as

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 suggested above and the strain hardening of the confined polymer. It also indicates that the brittleness increased on increasing the filler volume fraction.

Figure 1.4 (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).

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 permeability 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 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 effectiveness. Similarly, Figure 1.7 plots the comparison of measured permeability through polypropylene nanocomposites with numerical predictions for composites of parallel 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.

Figure 1.5 (a) A computer model comprising 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).

Figure 1.6 Dependence of the gas permeation 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).

Figure 1.7 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).

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 montmorillonite and compatibilizer have been shown. Like conventional models, which depend on oversimplified assumptions, these models do not suffer from these limitations and can still predict the composite properties using a set of simple equations.

Figure 1.8 Mixture plot for the prediction of oxygen permeation of polyethylene nanocomposites with different amounts of components: polymer (P), organically modified montmorillonite (OM) and compatibilizer (Compat).

Figure 1.9 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).

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.

2

Self-Consistent Field Theory Modeling of Polymer Nanocomposites

Valeriy V. Ginzburg

2.1 Introduction

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, polyurethane, 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 mechanical and barrier properties of a nanocomposite depend on the dispersion (macroscopic) 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 formulated 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 ().