The statistical mechanical theory of rubber elasticity

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The Polyurethane (PU) Networks

In this thesis we prepared three polyurethane model networks (elastomers) starting from diols and a triisocyanate. In this section we present the general aspects of polyurethane synthesis, including the main and secondary reactions.

Brief History

In 1849, Würtz was the first using a reaction of a glycol with an isocyanate, but it was Otto Bayer in 1937 who discovered how to transform the product into a useful plastic. The pioneering work on polyurethane polymers was conducted by Otto Bayer and his coworkers at the laboratories of I.G. Farben in Leverkusen, Germany [Farben 1937]. They recognized that using the polyaddition principle to produce polyurethanes from liquid diisocyanates and liquid polyether or polyester diols seemed to point to special opportunities. Their objective was to obtain synthetic fibers and elastomers to substitute natural rubber. The new monomer combination also circumvented existing patents obtained by Wallace Carothers on polyesters. Initially, work focused on the production of fibres and flexible foams. With development constrained by World War II (when PU’s were applied on a limited scale as aircraft coating), it was not until 1952 that polyisocyanates became commercially available. Commercial production of flexible polyurethane foam began in 1954, based on toluene diisocyanate (TDI) and polyester polyols. The invention of these foams (initially called imitation swiss cheese by the inventors) was thanks to water accidentally introduced in the reaction mix. These materials were also used to produce rigid foams, gum rubber, and elastomers. The first commercially available polyether polyol, poly(tetramethylene ether) glycol), was introduced by DuPont in 1956 by polymerizing tetrahydrofuran. Between 1965 and 1980 were developed foams based on MDI (diphenylmethane diisocyanate). More recently, building on existing polyurethane spray coating technology and polyetheramine chemistry, extensive development of two-component polyurea spray elastomers took place in the 1990s. Their fast reactivity and relative insensitivity to moisture make them useful coatings for large surface area projects, such as secondary containment, manhole and tunnel coatings, and tank liners.

Polyurethane components

Polyols

A polyol is a molecule with two or more hydroxyl functional groups R’-(OH)n ≥ 2. These hydroxyl functional groups are available for organic reactions. A molecule with two hydroxyl groups is a diol (for linear chains), one with three is a triol (for network chains), one with four is a tetrol (for special foams) and so on. The main use of polymeric polyols is as reactants to make other polymers. They can be reacted with isocyanates to make polyurethanes, and this use consumes most polyether polyols.
In practice, polyols are distinguished from short chain or low-molecular weight glycol chain extenders and crosslinkers such as ethylene glycol (EG), 1,4-butanediol (BDO), diethylene glycol (DEG), glycerine, and trimethylol propane (TMP). Polyols are polymers in their own. They are formed by base-catalyzed addition of propylene oxide (PO), ethylene oxide (EO) onto a hydroxyl or amine containing initiator, or by polyesterification of a di-acid, such as adipic acid, with glycols, such as ethylene glycol or dipropylene glycol (DPG). Polyols extended with PO or EO are polyether polyols (see Figure 1.1). Polyols formed by polyesterification are polyester polyols. The choice of initiator, extender, and molecular weight of the polyol greatly affect its physical state, and the physical properties of the polyurethane polymer. Important characteristics of polyols are their molecular backbone, initiator, molecular weight, percentage of primary hydroxyl groups, functionality, and viscosity.

Polyisocyanates

A polyisocyanate is a molecule with two or more isocyanate functional groups R-(N=C=O)n ≥ 2. The isocyanates are very reactive components characterized by the presence of the group –N=C=O. The rate of the reaction depends on the family of the isocyanate, the most important parameters being the amount of reactive groups and the functionality. The polyisocyanates are produced by the phosgenation of an amine. Molecules that contain two isocyanate groups are called diisocyanates, and molecules containing three isocyanate groups are called triisocyanates. There exist two main families of aromatic polyisocyanates: TDI (toluene diisocyanate) and MDI (diphenylmethane diisocyanate) (see Figure 1.2). There exist also aliphatic polyisocyantaes, such as hexamethylene diisocyanate (HDI) or isophorone diisocyanate (IPDI).
The isocyanates present a double bond N=C which is quite polar and react with all the components having a mobile hydrogen. Depending on the kind of isocyanate, the charges can be presented in two mesomeric forms [Abder-Rahim 2001] (see Figure 1.3).

Polycondensation of isocyanate-polyol to produce Polyurethane

The polycondensation reaction is exothermic and consists of the chemical reaction of two molecules with different functional groups. For the polyurethane formation (polycondensate) it consists on the reaction of a hydroxyl group (-OH), the presence of a free “active” hydrogen and an isocyanate group (-NCO) (see Figure 1.4). The base of the chemistry of polyurethanes is the high reactivity of the isocyanates. Since the isocyanate group (-NCO) can react with alcohols, amines, carboxylic acids and water (see Figure 1.5), it can form bond of urethane, urea and amides. Reaction of an isocyanate with an alcohol yields a urethane, reaction of an isocyanate with an amine yields a urea, and reaction of an isocyanate with water results in intermediates which decompose to yield carbon dioxide and an amine, which further reacts to again form an urea. The reaction with water is used for the production of foams.
Polyurethanes are in the class of compounds called reaction polymer [Oertel 1985, Ulrich 1996, Woods 1990] and are formed by reacting a monomer containing at least two isocyanate functional groups with another monomer containing at least two alcohol groups, in the presence (or not) of a catalyst. To form a network a reaction either of a triol with a diisocyanate or of a triisocyanate with a diol is needed. The rate of the reaction varies as a function of the kind of alcohol used in the following order: primary alcohol > secondary alcohol > tertiary alcohol > phenol.
Though the properties of the polyurethane are determined mainly by the choice of the polyol, the isocyanate exerts some influence. The cure rate is influenced by the functional group reactivity and the number of functional isocyanate groups per reactive molecule. Polyurethanes are based upon a well-defined stoichiometry. The choice of isocyanate also affects the stability of the polyurethane upon exposure to light. Polyurethanes made with aromatic isocyanates yellow with exposure to light, then the use of stabilizers may be included in the formulation; whereas those made with aliphatic isocyanates are stable [Randall 2002]. There are several kinds of polyurethanes: cellular, compacts, elastomers, coatings and adhesives. In this thesis we prepared polyurethane elastomers from polyether polyols and a trifunctional isocyanate.

Secondary reactions

The urethane and urea groups already formed as presented in Figure 1.6, have other reactive hydrogen atoms that can react with another isocyanate giving as secondary products allophanates and biurets as presented in Figure 1.7.
Figure 1.6: Secondary reactions of the isocyanate with urethane and urea groups (taken from “techniques de l’ingenieur”).
In the presence of certain “activators”, the isocyanates can react with each other and form by oligomerization, urediones (dimers), isocyanurates (trimers) or carbodiimides, as shown in Figure 1.7.
Figure 1.7: Secondary reactions of oligomerization of the isocyanates (taken from “techniques de l’ingenieur”).

Additives for polyurethanes’ preparation

The main additives used to produce polyurethanes are catalysts, chains extenders, surfactants, dyes, blowing agents, fillers, etc. The catalysts are accelerators of the reactions and for the polyurethane production they are used to reduce the curing times. There are two main catalysts used, the amines (i.e. Trietilendiamine or TEDA) and metallic salts (i.e. Dibutyldilaurate DBTL). In our case the catalyst was not used due to too fast reactions.
Polyurethanes made with aromatic isocyanates yellow with exposure to light, and the use of stabilizers may be included in the formulation (e.g. Irganox); whereas those made with aliphatic isocyanates are stable [Randall 2002].

Model Networks

The rubbery networks prepared by random crosslinking of the precursor chains have inhomogeneous structures with a broad length distribution of the network strands; in addition, the characterization of the strand length distribution in elastomers is not possible by current analytical techniques. However end-linking end-reactive precursor chains of known molecular weight using multifunctional crosslinkers afford a tailor-made model network with a well characterized structure. In the case of complete reaction, the molecular mass of the network strands between neighboring crosslinks (Mc) is identical to that of the precursor chains, and the junction functionality ( fe) is the same as the functionality of the crosslinker. To consider the effect of the incomplete reaction on Mc and fe, the degree of the end-linking reaction (p) is estimated from the amount of soluble species extracted after the reaction [Andrady et al. 1991, Urayama 2008].
An elastomer model network should have precursors with well known molecular weight and multifunctional crosslinker to be very close of the stoichimetric conditions, controlled lengths of the network strands, reduced amount of trapped entanglements and inexistent or very few dangling chains [Urayama et al. 2009].

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Bimodal Networks

Experimentally, a bimodal network combines long- and short-polymer chains, presenting variable crosslinking density. Typically, bimodal networks result from blending difunctional long and short chains and crosslinking them together. They exhibit values of the modulus which increase very substantially at high elongations, thus giving unusually large values of the ultimate strength. This improvement in mechanical properties has been attributed to the limited extensibility of the short chains [Mark 1985, Mark and Erman 1988].
The theoretical analysis of bimodal networks, was first performed by Higgs and Ball [Higgs and Ball 1988]. These networks are composed of two types of chains, conveniently referred to as short and long chains, differing either in their molecular weight or in their chemical structure, thus obeying two distinct probability distribution functions for their end-to-end separations. The original theoretical approach, based on Gaussian phantom network chains for both components, was essentially developed for random bimodal networks with a random number of short or long chains connected at a given junction. Later, Kloczkowski et al. [Kloczkowski et al. 1991] considered the statistical mechanics of regular bimodal networks, which, by definition, have a fixed number Φs and ΦL of short and long chains, respectively, at every junction and hence lend themselves to analytical solutions.

Self-Assembled Monolayers (SAMs)

During several stages of sample preparation we prepared self-assembled monolayers (SAMs). The main goal of the SAMs preparation was to modify glass surfaces (glass lenses and glass plates) to bond the polyurethane to the glass in a covalent way (as presented in chapter 4). We also modified the surface of metallic molds with SAMs to make an easy release fluoro-terminated coating (also presented in chapter 4).
A self assembled monolayer (SAM) is an organized layer of amphiphilic molecules in which one end of the molecule, the “head group” shows a special affinity for a substrate (glass in our case); SAMs also consist of a tail with a functional group at the terminal end as seen in Figure 1.8 (b) and (c). The SAMs are created by the chemisorption of hydrophilic “head groups” onto a substrate from either the vapor or liquid phase [Schwartz 2001] followed by a slow two-dimensional organization of the “tail groups” (see Figure 1.8(a)) [Wnek and Bowlin 2004]. Initially, adsorbate molecules form either a disordered mass of molecules or form a “lying down phase” [Schwartz 2001], and over a period of hours, begin to form crystalline or semicrystalline structures on the substrate surface [Love et al. 2005, Vos et al. 2003]. The hydrophilic “head groups” assemble together on the substrate, while the tail groups (that can be hydrophilic or hydrophobic) assemble far from the substrate. Areas of close-packed molecules nucleate and grow until the surface of the substrate is covered in a single monolayer. Adsorbate molecules adsorb readily because they lower the surface energy of the substrate [Love et al. 2005] and are stable due to the strong chemisorption of the “head groups.” These bonds create monolayers that are more stable than the physisorbed bonds of Langmuir-Blodgett films. The monolayer packs tightly due to van der Waals interactions, thereby reducing its own free energy [Love et al. 2005, Sullivan 2003, Oclin 2004]. Self-assembled monolayers (SAMs) offer a unique way to confine molecules in two dimensions. Figure 1.1 presents a schematic of self-assembled monolayers structures.
Figure 1.8: Schematic of SAM structure: (a) from solution to SAM; (b) schematic of head and functional group on the substrate, and (c) n-alkyl silane on glass.

Rubber elasticity

In this section we present some fundamental concepts on the elasticity observed in elastomers also called rubber elasticity. These fundamentals are needed to better understand the physical and mechanical characterization of the polyurethane model networks presented in the next chapters. Historically, the term rubber was used to refer to natural rubber only. The more modern term elastomer is sometimes employed in relation to synthetic materials having rubber-like properties, regardless of their chemical composition. The most important physical characteristic of the rubber-like state is a high degree of deformability exhibited under the action of relatively small stresses [Treloar 2005]. High extensibility and low Young’s modulus are the most remarkable properties of the rubber-like material; additionally, elastomers present particular thermal or thermoelastic properties. The thermoelastic effect dates from Gough [Gough 1805] that made some experimental observations: 1) the elastomer in the stretched state, under a constant load, contracts (reversibly) on heating, and 2) the elastomer gives out heat (reversibly) when stretched. These observations were confirmed fifty years later by Joule [Joule 1859] who worked with perfectly reversible vulcanized rubber.
The typical high elasticity of rubber arises from its molecular structure. Because the linear molecules are long and flexible, they take up random configurations under Brownian Thermal motions, like agitated snakes. When they are straightened out by an applied force, and released, they spring back to random shapes as fast as their thermal motion allows. In practice, the molecules are tied together by a few permanent chemical bonds, by a process known as “crosslinking”, to give the material a permanent shape. Thus, after crosslinking, rubber becomes a soft, highly elastic solid. Although rubber has a characteristic ability to undergo large elastic deformations, in practice many rubber springs are subjected only to relatively small strains, rarely exceeding 25% in extension or compression. A good approximation for the corresponding stresses is then given by conventional elastic analysis, assuming simple linear stress-strain relationships, because, like all solids, rubber behaves as a linearly elastic material at small strains. But some features of the behaviour of rubber can be understood only in terms of its response to large deformations. To treat large elastic deformations, we must consider how to characterize the elastic properties of highly extensible, nonlinearly elastic materials when a simple modulus of elasticity is not longer enough.

Table of contents :

Introduction
1.1.- The Networks
1.1.1.- The Polyurethane (PU) Networks
1.1.1.1.- Brief History
1.1.1.2.- Polyurethane components
1.1.1.2.1.- Polyols
1.1.1.2.2.- Polyisocyanates
1.1.1.3.- Polycondensation of isocyanate-polyol to produce Polyurethane
1.1.1.3.1.- Secondary reactions
1.1.1.4.- Additives for polyurethanes’ preparation
1.1.2.- Model Networks
1.1.3.- Bimodal Networks
1.2.- Self-Assembled Monolayers (SAMs)
1.3.- Rubber elasticity
1.3.1.- Continuum Mechanics, Small and Large Strain Elasticity
1.3.2.- Elastic properties at small strains
1.3.3.- The Strain Energy Function W
1.3.4.- Strain Invariants
1.3.5.- The statistical mechanical theory of rubber elasticity
1.3.5.1.- Statistical theory of rubber elasticity
1.3.5.1.1- Affine network model
1.3.5.1.2- Phantom network model
1.3.6.- Thermodynamics of elastomers deformation
1.3.7.- Deviations from rubber elasticity theory
1.3.7.1.- Chain entanglements
1.3.7.2.- Finite extensibility
1.4.- Compression or Tension under confined conditions
1.4.1.- Cavitation
1.4.1.1- Inflation of a Spherical Shell (Balloon)
1.5.- Viscoelasticity: Effects of Temperature and Frequency
1.6.- Fracture behaviour
1.6.1.- The energy balance approach
1.6.2.- The Threshold Energy: Molecular model
1.7.- Molecular rate processes with a constant activation energy: relevance for fracture processes
Bibliography

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