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Predicting diffusion coefficients of chemicals in and through packaging materials

Abstract

Most of the physicochemical properties in polymers such as activity and partition coefficients, diffusion coefficients and their activation with temperature are accessible to direct calculations from first principles. Such predictions are particularly relevant for food packaging as they can be used (1) to demonstrate the compliance or safety of numerous polymer materials and of their constitutive substances (e.g. additives, residues…), when they are used: as containers, coatings, sealants, gaskets, printing inks, etc. (2) or to predict the indirect contamination of food by pollutants (e.g. from recycled polymers, storage ambiance…) (3) or to assess the plasticization of materials in contact by food constituents (e.g. fat matter, aroma…). This review article summarizes the classical and last mechanistic descriptions of diffusion in polymers and discusses the reliability of semi-empirical approaches used for compliance testing both in EU and US. It is concluded that simulation of diffusion in or through polymers is not limited to worst-case assumptions but could also be applied to real cases for risk assessment, designing packaging with low leaching risk or to synthesize plastic additives with low diffusion rates.
Keywords: diffusion, packaging, mathematical modeling, molecular modeling, migration

Introduction

The evolution of our urban lifestyles (takeout food, portioned packaging food, ready-to-eat food or microwaved food…) inevitably leads to a lot of concerns not only of sustainability impacts of packaging (Lewis et al., 2010) but also of packaging food safety involving an increase of the surface area of the materials in contact with food and consequently to a repeated exposure to chemical substances leached by these materials (Delmaar et al., 2005; van Leeuwen and Vermeire, 2007; Halden, 2010). Even if food contact materials are not the only source of exposure, such a chronic exposure starts from the first stages of life: during fetal life with the food ingested by the mother (Ranjit et al., 2010) and baby foods (Muncke, 2011). The exposure related to ubiquitous substances (i.e. highly frequent in food) depends on the considered substance or family, its frequency of occurrence, the time and temperature of contact between the food and its packaging (Vitrac and Hayert, 2005, 2007a; Vitrac and Leblanc, 2007; Poças et al., 2010) and in a less extent additional physicochemical factors such as pH and ozone content, which were found significant for the migration of bisphenol A (Mercea, 2009). Calculation methods for assessing consumer exposure of chemicals from packaging materials have been reviewed by Poças and Hogg (2007). They attract nowadays more and more attention due to the high concern for the contamination of packaged food product by endocrine disruptors (Vandenberg et al., 2009; Wagner and Oehlmann, 2009; Tacker, 2011; Batra, 2011; du Yeon et al., 2012) or cocktail of substances (Muncke, 2009; Zeliger, 2011). As a result, packaging materials are involved in strong scientific controversies propagated by evocative titles or editorials in both magazine and scientific literature such as: « How dangerous is Plastic » in Time Magazine of April 12, 2010 (Walsh, 2010); « …the drinking water left in a hot car can cause breast cancer » in Nature Reviews Endocrinology of May, 2010 (Heath, 2010). Two controversies have found large echoes in the scientific literature: the contamination of drinking water stored in polyethylene terephthalate bottles (Bach et al., 2012) and the role of packaging on the exposure to bisphenol A (Vandenberg et al., 2009; Goetz et al., 2010; Sharpe, 2010; Siva, 2012). Without necessarily similar audience, many surveys tend to incriminate almost all available materials in the market including: plastics (Wittassek and Angerer, 2008; Felix et al., 2008; Guart et al., 2011; Bach et al., 2012; Kappenstein et al., 2012), can coatings (Poole et al., 2004), paper and board (D’Hollander et al., 2010; Vollmer et al., 2011). These experimental studies are macroscopic and usually neglect the physicochemical details and the conditions, where the amounts transferred to the food are significant. Such phenomena have been reviewed by Lau and Wong (2000), Piringer and Baner (2000, 2008), Helmroth et al. (2002a), Arvanitoyannis and Bosnea (2004), Poças et al. (2008). They all conclude on the key role of diffusion and its activation on migration of organic substances (e.g. additives, polymer residues) and mineral substances (e.g. catalyst residues) (Fordham et al., 1995; Kawamura et al., 2009; Welle and Franz, 2010; Haldimann et al., 2012). Diffusion mechanisms in solid polymers have been discussed in several reference textbooks (Mehrer, 2010; Neogi, 1996; Stastna and De Kee, 1995; Vieth, 1991) and reviews. They tend however to focus either on the diffusion of gas molecules in polymers (Alexander Stern, 1994; Klopffer and Flaconneche, 2001) or large molecules in gels (Masaro and Zhu, 1999). Hence, there is a general opinion according to: diffusion coefficients of additive-like molecules could not be predicted accurately (page 156 of Cussler (2009)) or related to the chemical structures of the diffusants (page 135 of Piringer and Baner (2008)). The practical consequence is that migration modeling concepts used to check the compliance of food contact materials or for evaluating consumer exposure to packaging substances rely on models (Helmroth et al., 2002a; Begley et al., 2005) disconnected from the progress gained in the field of Polymer Science or more broadly in Chemical Engineering over the last decade. The question is all the more relevant than it could be expected that the same science might be used to design low migration materials assemblies and to assess the safety of materials (Vitrac and Hayert, 2007a; Nguyen et al., 2013). For complementary properties, such as partition coefficients (Tehrany and Desobry, 2004) and their activation with temperature, it has been demonstrated that both molecular dynamics (Hess et al., 2008; Hess and van der Vegt, 2008; Özal et al., 2008; Boulougouris, 2010, 2011; De Angelis et al., 2010) and advanced molecular simulation techniques (Gillet et al., 2009a, 2010; Vitrac and Gillet, 2010) enable tailored and accurate estimations of partition coefficients of additive and polymer residues in rubber and glassy polymers (Lipscomb, 1990) without requiring any fitting procedure or experimental data. Similar trends have been obtained for diffusion coefficients, by simulating hundreds of configurations with coarse-grained molecular dynamics (Durand et al., 2010) and by bridging free-volume theories for small and rigid solutes with the theory of flexible solutes in solid polymers (Fang et al., 2013).
This review aims at filling the gap between disciplines to encourage a more critical use of physical models of diffusion rather than empirical approaches to extend the applications where migration modeling can be used for decision making (Brandsch et al., 2002; Arvanitoyannis and Bosnea, 2004; Vergnaud and Rosca, 2006; Vitrac and Hayert, 2007a ; Gillet et al., 2009b). Such contributions could be also thought to be used to assess consumer exposure to arbitrary substances whatever the availability of contamination data and to develop safe materials. In particular, the concepts of homologies which enable to extrapolate the diffusion coefficient of from one molecule to a close one or from a polymer to another one are detailed in depth beyond early attempts by Reynier et al. (2001a), Reynier et al. (2001b) and Vitrac et al. (2006).

The concepts of “generally recognized diffusion models” in legal US and EU systems

US and EU manage the risk of contamination of food by packaging substances by two closely related concepts but with different application modalities: “food contact substance notifications” under the US law and “inert food contact materials” principles in EU regulations.
According to US law, only the regulatory status of the components of a food contact material is tested and not the whole material itself. Under section 409(h)(2)(C) of the Federal Food, Drug, and Cosmetic Act (CFR, 2011) , a “food contact substance” is defined as a special (i.e., indirect (Till et al., 1987)) food additive “intended for use as a component of materials used in manufacturing, packing, packaging, transporting, or holding food if such use is not intended to have any technical effect in such food”. Coatings, plastics, paper, adhesives, as well as colorants, antimicrobials and antioxidants found in packaging fall into this category (Shanklin and Sánchez, 2005). Any substance, which was not generally considered as safe (GRAS) in food (CFR, 2012a) or in food packaging (CFR, 2012b) and not subjected to any Threshold of Regulation Exemption (CFR, 2012c; Munro et al., 2002), must be listed in the inventory of effective food contact substance (FDA, 2012b), which includes 952 substances at the beginning of 2013 or listed in the CFR parts 174-181. If the substance does not fulfill any of the previous criteria because the substance is not listed and must be used for a different intended use, a notification must be submitted. The notification stepwise process (FDA, 2012a) authorizes diffusion modeling as a substitute of experimental migration testing or to extrapolate the data at a different temperature as soon as a “predictable migration-time behavior (e.g. Fickian diffusion)” has been established. An example cited: “migration for two hours of retorting at 121°C can be estimated and added to migration after 238 hours at 40°C”. Without citing it, the reasoning assumes several properties or approximations. Firstly, that the transferred amount is invariant with the product Dt or Dt (see Eq. 4.18 of Crank’s book (Crank, 1979) and section 2.1.4.1.4), where D is the diffusion coefficient and t is contact time. Secondly, it assumes an Arrhenius behavior over the whole range of temperature between 40°C and 121°C with activation energy of ca. 60 kJ⋅mol-1 (≈ 8.31 10 3 ln 238 2 1 273 40 1273 121 ). Is it true for every polymer? Even if the glass transition temperature is crossed? Even if the material is closer to its melting point/flow threshold than its glass transition temperature? For any migrant regardless its size, geometry and flexibility? Similar extrapolations are used in US system to assess the safety of recycled materials. In this case, surrogates are used to simulate the misuse of materials before recycling (FDA, 2006). They should include: a volatile polar organic substance, a volatile non-polar organic substance, a non-volatile polar organic substance, a non-volatile non-polar organic substance, and a heavy-metal salt. As reported in Appendix 1 of (FDA, 2006), most of the D values used in numeric challenge tests originate also from mathematical models with a goal of extrapolating D values from one molecule to a next one. The safety of packaging for irradiated foods (Paquette Kristina, 2004) and of so-called “functional barriers” (see section 2.1.4.1.4) for both food (FDA, 2006, 2007) and drug (MAPP 5015.5, 2011) applications are supported by very similar arguments.
The EU regulation system uses in a very similar fashion the concept of migration rate or diffusion rate. Article 3 of the EU framework regulation 1935/2004/EC (EC, 2004) defines a so-called “inert packaging” as “manufactured in compliance with good manufacturing practice so that, under normal or foreseeable conditions of use, they do not transfer their constituents to food in quantities which could endanger human health…”. Migration modeling is a legal concept introduced in EU, initially via the article 14 of the Directive 72/2002/EC (EC, 2002a): “For certain types of plastics the availability of generally recognized diffusion models based on experimental data allows the estimation of the migration level of a substance under certain conditions, therefore avoiding complex, costly and time-consuming testing”. The concept has been reformulated in the Regulation 10/2011/EC: “generally recognized diffusion models based on scientific evidence that are constructed such as to overestimate real migration.” In other words, the estimation of diffusion parameters (e.g. diffusion coefficient, activation energies) and partitioning is assumed to be conservative and not the real values. But, How conservative are they? Is it safe to extrapolate the behavior from a small molecule to a large one? From low temperatures to higher temperatures? In EU, the results of the project SMT-CT98-7513(EC, 2002b) and published as a collective work by Begley et al. (2005) is usually chosen as reference (Poças et al., 2008), whereas the Food and Drug administration (FDA, 2006) recommends earlier or alternative versions of these models (Baner et al., 1996; Limm and Hollifield, 1996).
The sources of diffusion coefficients and activation energies are scarce and underline the needs of reliable models in absence of a generic database of diffusion coefficients. A bibliometric analysis ([ISI Web of Knowledge v5.9, Thomson Reuters, USA – on Feb 3rd, 2013]) shows that the number of specific studies of diffusion coefficients in polymers is in particularly low regarding the importance of the task (e.g. number of substances and polymers): it is usually thought that between 5000 and 8000 different substances would enter into the composition of food contact materials with 885 substances for the sole positive list of additives and monomers for plastics in EU (EC, 2011, 2012a, b). Since 1979, 86 articles have been published in Macromolecules – the first journal in Polymer Science by its number citations –with “Diffusion Coefficient” in the title (over a total of 774 with “Diffusion” in the title). The collected effort tended to be reported to less specialized journals such as Food Additives and Contaminants, which has reported diffusion coefficient values in 45 articles, since 1990. The concept of “generally recognized diffusion model” is even more difficult to establish. The 90th edition of CRC Handbook of Chemistry and Physics (Lide, 2009) reports diffusion coefficients in gases, liquids and semi-conductors but none for polymers. The Physical Properties of Polymers Handbook (Mark, 2007) and The Polymer Data Handbook (Mark, 1999) report only diffusion coefficients of gases. Only the 4th edition of the Polymer Handbook (Brandrup et al., 1999) includes some diffusion coefficients for organic compounds but without inferring any generic rules to extrapolate to other substances and polymers.

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Some generalities about diffusion

Mass transfer from and to food packaging materials

In most of the cases, the reality of leaching of substances by materials in contact into food cannot be observed by naked eyes. Migrating substances are indeed usually colorless, odorless and tasteless. Even with analytical methods, identifying an unknown chemical among all the food constituents is a cumbersome task (Himmelsbach et al., 2009; Silva et al., 2006; Simal-Gandara et al., 2002). By contrast, the reverse process is easier to highlight. In the everyday life, we shall have already noticed that transparent plastic tableware and containers can be easily tainted by food pigments, or their surface properties can be affected by oily contact. For both transfer from or to food packaging materials, diffusional transport is involved.
2.1.4.1.1 Sorption of food constituent into food packaging: first observation of the reality of diffusion
A change of refraction index associated to the sorption of decane, simulating an oily contact, in polystyrene is presented in Figure 2-1 based on the observations of Morrissey and Vesely (2000) but also described by Feigenbaum et al., (1991). A moving migration front separates an outer region with polystyrene swollen by decane and a dry region, where the polymer remains at glassy state (T<Tg with a glass transition temperature Tg of ca. 95 °C). From a physicochemical point of view, the depicted sorption involves a complex transport mechanism combining the solute concentration gradient and the gradient of elastic stresses as detailed in (Del Nobile et al., 1994; Lipscomb, 1990; Mensitieri et al., 1991; Miller-Chou and Koenig, 2003).
Figure 2-1 Microscopic observations in visible light and its molecular interpretation of the sorption of decane in atactic polystyrene at 70 °C (after Morrissey and Vesely (2000)).

Cross mass transfer in multicomponent food packaging systems

Main migrants from packaging materials reported in the literature fall into two categories (Brimer and Skaanild, 2011; Crompton, 2007; Deshpande, 2002; Rahman, 2007):
1. Intentionally added substances known as additives to aid processing or end-service (life-time, mechanical properties…), including antioxidants, antiblocking agents, antifungal agents, bactericidal agents, brighteners and whiteners, colorants, expanding agents, impact improvers, ultraviolet protective agents and ultraviolet degradation inhibitors, printing ink adhesives, gas barrier packaging oxygen scavengers, antisplit agents, antistatic agents, heat and light stabilizers, melt strength improvers, plasticizers, lubricants and slip agents, pigments, fillers, mold release agents, and fungicides.
2. Polymerization residues, including monomers, oligomers (with a molecular weight of up to 200), catalysts (mainly metallic salts and organic peroxides), solvents,
emulsifiers and wetting agents, raw material impurities, plant contaminants, inhibitors, decomposition and side reaction products.
Apart of plasticizers, additives are usually hindered and bulky substances with relatively low diffusion coefficients, well-known primary distribution in packaging assemblies and initial concentrations typically lower than 0.005 kg⋅kg-1. Liquid plasticizers (Patrick and Limited, 2005) are by contrast small and low branched molecules used in high concentration (above 0.1 kg⋅kg-1) with a much higher migration power. They tend to be ubiquitous not only in cling films but also in printing ink, adhesives, sealing closures, etc. Residues exhibit much broader chemical structures and migration behaviors: polymer degradation products, incomplete cross-linking reaction products, polymerization catalysts and initiators in curing reactions, processing aids such as solvents and surface agents… The occurrence of such substances and their migration routes are far less documented. According to (Deshpande, 2002), the more volatile gaseous monomers, e.g. ethylene, propylene, and vinyl chloride, usually decrease in concentration with time, but very low levels may persist in the finished product almost indefinitely. Styrene and acrylonitrile residues are generally the most difficult to remove.
Typical migrants with molecular weight ranging from 100 to 2300 g⋅mol-1 are listed in Table 2-1. As detailed in Nguyen et al. (2013), recent crises such as those involving printing ink residues arose from an insufficient description and understanding of diffusional transport along the packaging and recycling supply chain. Ink components can be redistributed during the storage of films before contact (Nguyen et al., 2013) or be present in recycled paper and board fibers and permeate across the primary packaging and contact layers (Biedermann et al., 2011). In simple words, the list of possible contaminants is neither limited to the components of the layer in contact nor to the primary packaging.
Possible migrants which are not in contact with food need to diffuse before contaminating the food. More generally, diffusion is the limiting mechanism as soon as the concentration profile in the any layer (in direct contact or not with food) is not uniform. The identified sources and routes of all transfers are reported in Table 2-2; possible couplings due to cross-transfer are sketched in Figure 2-2 in agreement with the more general discussion found in Vergnaud and Rosca, (2006). Activation of desorption of packaging constituents into food due to plasticization of the contact layer by food constituents (see Figure 2-1) is poorly described the literature and very often referred as “oil extraction” (Helmroth et al., 2002b; Riquet et al., 1998). It is, however, underlined that not only triacylglycerols but many hydrophobic food constituents such as the aroma can also be absorbed in layers in contact (Ducruet et al., 2007).

Table of contents :

Chapter 1. Introduction
Chapter 2. Literature review
Diffusion in food packaging materials
2.1 Abstract
2.1.1 Introduction
2.1.2 The concepts of “generally recognized diffusion models” in legal US and EU
2.1.3systems
Some generalities about diffusion
2.1.42.1.4.1 Mass transfer from and to food packaging materials
2.1.4.1.1 Sorption of food constituent into food packaging: first observation of the reality of diffusion
2.1.4.1.2 Cross mass transfer in multicomponent food packaging systems
2.1.4.1.3 Mass transfer controlled by diffusion in the packaging materials
2.1.4.1.4 The concept of functional barrier
2.1.4.2 Molecular diffusion
2.1.4.2.1 A macroscopic definition
2.1.4.2.2 A microscopic definition
2.1.4.2.3 Trace diffusion and random walks
2.1.4.2.4 Mutual diffusion
2.1.4.3 Diffusion in thermoplastic and elastomers
Scaling laws and friction models
2.1.52.1.5.1 Overview
2.1.5.1.1 Hydrodynamic theory of diffusion
2.1.5.1.2 Theoretical composition laws for diffusants consisting in N repeated patterns
2.1.5.2 Experimental data of diffusion coefficients in solid polymers
2.1.5.2.1 Linear substances
2.1.5.2.2 Additive type substances
2.1.5.2.3 Combined effect of T, Tg and geometry
2.1.6.1 Common assumptions
2.1.6.2 Vrentas and Duda theory for rigid solutes
2.1.6.3 Extension to flexible solutes
Activation models and data
2.1.72.1.7.1 Apparent effects of temperature and pressure
2.1.7.2 Activation energies
2.1.7.2.1 Effect of the molecular mass
2.1.7.2.2 Polymer effects
2.1.7.2.3 Combined solute and polymer effects
Alternative models to predict D or to overestimate D
2.1.82.1.8.1 The justification of alternative models
2.1.8.2 Models overestimating D
2.1.8.3 Prediction of D based on decision trees and molecular descriptors
Conclusions
2.1.9 References
2.1.10 Barrier materials
2.2 Some definitions and choices
2.2.1 Non-reactive barrier systems
2.2.22.2.2.1 Overview
2.2.2.2 Common transfer models for passive/active systems
2.2.2.2.1 Out-of-equilibrium approaches
2.2.2.2.2 Approaches at equilibrium
2.2.2.3 Reported experimental performances of passive barrier systems
2.2.2.4 New concepts of active systems
Reactive barriers
2.2.32.2.3.1 The concept of sacrificial reagent to increase barrier properties
2.2.3.2 Performances of reactive barriers
Conclusions
Chapter 3. Objectives and approaches
General objectives
3.1 Particular objectives
III Approaches followed in this thesis
3.3Chapter 4. Materials and methods
CHapter 4 Materials
4.1 Solutes
4.1.1 Polymer
4.1.2 Nano-adsorbents
4.1.3 Polymer/nanocomposite film processing and formulation
4.1.44.1.4.1 Virgin films
4.1.4.2 Source films
4.1.4.3 Nanocomposite film processing
4.2 Differential scanning calorimetry (DSC)
4.2.1 Polarized optical microscopy
4.2.2 Measurement of diffusion coefficients
4.2.34.2.3.1 Single film imaging by DUV/fluorescence microspectroscopy
4.2.3.1.1 Sample preparation
4.2.3.1.2 Theoretical concentration profiles
4.2.3.1.3 Data analyzing
4.2.3.2 14 films contact method
4.2.3.2.1 Sample preparation
4.2.3.2.2 Theoretical concentration profiles
4.2.3.2.3 Data analyzing
Chapter 5. Results and discussion
Diffusion in bulk polymers
5.1 Beyond tortuosity: how negative correlations decrease D values with Non-obstacle related effects on D
5.1.1considered time scale
5.1.2 Study and model development
5.1.35.1.3.1 Abstract
5.1.3.3 Theory
5.1.3.3.1 Scaling of D with the number of jumping units and temperature for linear solutes
5.1.3.3.2 Conventional free-volume theories
5.1.3.3.3 Extended free-volume models for aromatic solutes in aliphatic polymers
5.1.3.3.4 Modeling of activation terms for oligophenyl solutes
5.1.3.4 Experimental section
5.1.3.4.1 Materials
5.1.3.4.2 Film processing and formulation
5.1.3.4.3 Methods
5.1.3.5 Results and discussion
5.1.3.5.1 Comparison of the scaling of D between linear aliphatic solutes and aromatic solutes
5.1.3.5.2 Scaling diffusion coefficients according to Eqs. (5-11), (5-30)-(5-31)
5.1.3.5.3 Polymer effects as probed with diphenyl alkanes
5.1.3.5.4 Solute activation parameters of oligophenyls
5.1.3.5.5 Mechanisms of translation of oligophenyls in aliphatic polymers
5.1.3.6 Conclusions
5.1.3.7 Author information
5.1.3.8 Acknowledgements
5.1.3.9 References
5.1.4 Characterization and thermodynamic properties of nano-clays
5.2 Introduction
5.2.1 Simulation and experimental study
5.2.25.2.2.1 Abstract
5.2.2.2 Introduction
5.2.2.3 Material and method
5.2.2.3.1 Materials
5.2.2.3.1.1 Nanoclay
5.2.2.3.1.2 Probe solutes
5.2.2.3.2 Material characterization
5.2.2.3.2.1 X-ray diffraction analysis (XRD)
5.2.2.3.2.2 Thermogravimetric analysis (TGA)
5.2.2.3.2.3 Specific surface area
5.2.2.3.3 Sorption properties at infinite dilution
5.2.2.3.4 Sorption isotherms of anisole
5.2.2.3.5 Molecular modeling strategies
5.2.2.3.5.1 Preparation of neat clay crystal structure
5.2.2.3.5.2 Preparation of surface modified clays
5.2.2.3.5.3 Sorption calculations at diluted state
5.2.2.3.5.4 Sorption calculations at concentrated state
5.2.2.3.5.5 Sorption properties of organic solutes in surfactants
5.2.2.4 Results and Discussion
5.2.2.4.1 Experimental characterization of clays
5.2.2.4.1.1 Organic composition
5.2.2.4.1.2 Gallery structure
5.2.2.4.2 Clay atomistic model and its characterization
5.2.2.4.3 Sorption properties at infinite dilution
5.2.2.4.3.1 Simulated Henry coefficient and isosteric heat of sorption
5.2.2.4.3.2 Comparison with experimental adsorption enthalpy
5.2.2.4.4 Sorption behavior at concentrated state
5.2.2.4.4.1 Simulated isotherms
5.2.2.4.4.2 Experimental sorption isotherms
5.2.2.4.4.3 Isosteric heats of sorption
5.2.2.5 Conclusions
5.2.2.6 References
Applications for developing polymer nanocomposite systems
5.2.3 Proof of the concepts of barrier materials including nano-
5.3adsorbents
5.3.1 Experimental study and interpretation
5.3.25.3.2.1 Abstract
5.3.2.2 Introduction
5.3.2.3 Theory of polymer barrier materials
5.3.2.3.1 Beyond tortuosity concepts.
VI 5.3.2.3.2 Reduction of D due to asymmetric barriers to translation: a two states toy model.
5.3.2.3.3 Inverse gas chromatrography (IGC) as a one-dimensional physical model.
5.3.2.3.4 D reduction due to entropic trapping: a first approach
5.3.2.4 Materials and methods
5.3.2.4.1 Polymer and nano-adsorbents.
5.3.2.4.2 Solute probes
5.3.2.4.3 Estimation of γi,MMT(T)
5.3.2.4.4 Estimation γi,P(T) by a generalized off-lattice Flory-Huggins method
5.3.2.4.5 Diffusion coefficient (D) determinations.
5.3.2.5 Results and discussion
5.3.2.5.1 Effect of crystallite morphology on diffusion coefficients
5.3.2.5.2 Partitioning between nanoclays and amorphous regions of PCL and PVA. 254
5.3.2.5.3 Reduction of diffusion coefficients in PCL nanocomposite.
5.3.2.5.4 Reduction of diffusion coefficients in PVA nanocomposite.
5.3.2.6 Conclusions
5.3.2.7 References
Does the nano-adsorbent concept work?
5.3.3 Extended results and discussion
5.4 The concept of increasing dwelling times to lessen D
5.4.1 Do we expect a change in the molecular mechanism of diffusion in presence 5
Chapter 6. Conclusions and perspectives
Overview
6.1 Diffusion of tracers in bulk polymers
6.2 A new concept of barrier material: chaotic materials
6.4Chapter
7. References

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