Introduction

With their ease of processing and high performance, polymeric materials have become a common and important part of modern life. However, because almost all polymers are composed predominately of hydrocarbons, these materials are flammable and thus greatly increase fire hazard to human life and property. As estimated for the United States, there are approximately 400,000 residential fires each year, 20% involving electrical distribution and appliances, and 10% concerning upholstered furniture and mattresses. These fires kill about 4,000 people, injure 20,000 people, and result in property losses amounting to about US$4.5 billion. Flame retardants are additives that can make flammable materials more difficult to ignite and significantly reduce the spread of fire. Use of flame retardants plays a major role in fire safety, saving lives, and preventing injuries and property damage. For example, in 1974, the number of recorded television set fires in the United Kingdom was more than 2,300, whereas this number had decreased to 470 in 1989, despite the number of television sets in use increasing many times. This is because effective flame retardants were developed for television sets.

Introduction

It has been reported that approximately 12 persons are killed and 120 are severely injured because of fire every day in Europe. Fire has considerable impact on the environment in terms of destruction of substructures and production of toxic and/or corrosive compounds such as CO, dioxins, HCN, and polycyclic aromatic compounds. Consequently, it is necessary to limit this kind of risk by designing new materials with improved flammability properties. Nowadays, many companies (building and civil engineering, transportation, cable-making and electrotechnical material, etc.) are directly concerned with this topic.

Buildings contain increasing calorific value in the form of highly combustible polymeric materials replacing more traditional materials (wood, alloys, metals, etc.) with the aim of improving the comfort of occupants (pieces of furniture, carpets, toys, household and leisure electric components, and data processing equipment, etc.). Potential sources of fire tend to growwith the multiplication of electric and electronic devices. The increasing sophistication and miniaturization of electronics (with increasingly powerful and fast microprocessors) have as a consequence a stronger concentration of energy, leading to an increased risk of localized overheating and thus of fire.

Layered silicates

The idea of flame retardant materials dates back to about 450 BC, when the Egyptians used alum to reduce the flammability of wood. The Romans (in about 200 BC) used a mixture of alum and vinegar to reduce the combustibility of wood. Today, there are more than 175 chemicals classified as flame retardants. The major groups are inorganic, halogenated, organic, organophosphorus, and nitrogen-based flame retardants, which account for 50%, 25%, 20%, and >5% of the annual production, respectively.

In many cases, existing flame retardant systems show considerable disadvantages. The application of aluminum trihydrate and magnesium hydroxide requires a very high portion of the filler to be deployed within the polymer matrix; filling levels of more than 60 wt% are necessary to achieve suitable flame retardancy, for example, in cables and wires. Clear disadvantages of these filling levels are the high density and the lack of flexibility of end products, the poor mechanical properties, and the problematic compounding and extrusion steps.

Background

Organic polymers are rapidly and increasingly taking the place of traditional inorganic and metallic materials in various fields owing to their excellent properties, such as low density, resistance to erosion, and ease of processing. However, organic polymers are inherently flammable; their use can cause the occurrence of large fires and, consequently, loss of lives and properties. Thus, enhancing the flame retardancy of these organic polymers is becoming more and more imperative with their wider application, especially in fields such as electronics where high flame retardancy is required.

For traditional flame retardants, on one hand, a very high loading is usually needed to meet flame retardancy demands, which can lead to the deterioration of mechanical properties; on the other hand, utilization of flame retardants can cause environmental problems.

Introduction

Thermodynamically, the introduction of a solid particle into a polymer matrix either decreases or increases the interfacial energy, depending on the degree of interaction between polymer chains and solid surfaces. If strong absorption of the polymer chains on the surfaces takes place, the system can be approached through minimization of the interfacial energy, reducing the energy factors. Furthermore, the minimization of interfacial energy can be optimized by increasing the interfacial area of solid particles. Therefore, in order to maximize reduction of the interfacial energy, the solid particles need a large aspect ratio, making both layered silicates and carbon nanotubes (CNTs) good candidates. In particular, layered silicates cation-exchanged with organophilic surfactants can be delaminated into a single silicate sheet in a polymer matrix and remain as nanosheets with aspect ratio 100–1000. Because of this unique delamination of organophilic silicates, polymer–organoclay nanocomposites are of great interest in industry and academia. Numerous research groups have characterized and predicted the microstructures of polymer/organoclay nanocomposites using advanced techniques.

In Appendix A, Section A.2, we considered the oscillations of particles using Newton's laws of classical mechanics. The distinctive feature of Newton's mechanics, when considering interactions between particles (for example, the gravitational interaction), is the instantaneous transmission of the interaction between particles, i.e., transmission occurs with infinite speed. The interaction between charged particles is realized through an electromagnetic field, which possesses energy and momentum and is carried through space with finite speed. Electromagnetic waves can exist without any charges in a space in which there is no substance, i.e., in vacuum. This is substantiated by the fact that the equations of classical electrodynamics allow solutions in the form of electromagnetic waves for such conditions.

The main equations of classical electrodynamics are Maxwell's equations, which were formulated after analyzing numerous experimental data. In this sense they are analogous to Newton's equations of classical mechanics. Maxwell's equations are the basis for electrical and radio engineering, television and radiolocation, integrated and fiber optics, and numerous phenomena and processes that take place in materials placed in an electromagnetic field. Together with Newton's equations, Maxwell's equations are the fundamental equations of classical physics. Just like Newton's equations, Maxwell's equations have their limits of applicability. For example, they do not sufficiently well describe the state of an electromagnetic field in a medium at frequencies higher than 1014–1015 Hz.

The solution of most problems associated with electron quantum states in physical systems and structures (atoms, molecules, quantum nanostructure objects, and crystals) is hard to find because of the mathematical difficulties of getting exact solutions of the Schrödinger equation. Therefore, approximate methods of solving such problems are of special interest. We will consider some of these methods, such as the adiabatic approximation now and later the effective-mass method, using real physical systems as examples. In this chapter we will consider several widely used approximation methods for finding the wavefunctions and energies of quantum states as well as the probabilities of transitions between quantum states. First of all, we will consider stationary and non-stationary perturbation theories. What is common to these two theories is that it is assumed that the perturbation is weak and that it changes negligibly the state of the unperturbed system. Stationary perturbation theory is used for the approximate description of a system's behavior if the Hamiltonian of the quantum system being considered does not directly depend on time. In the opposite case, non-stationary theory is used. Then, we will briefly consider the quasiclassical approximation, which is used for the problems of quantum mechanics which are close to analogous problems of classical mechanics.

Stationary perturbation theory for a system with non-degenerate states

This theory is used for the approximate calculation of the energy levels and the wavefunctions of stationary states of systems that are subjected to the influence of small perturbations.

Three-dimensional quantum-dot superlattices can be considered as nanocrystals. Spherical nanoparticles consisting of a big enough number (from 10 to 1000) of atoms or ions, which are connected with each other and are ordered in a certain fashion, can be considered as the structural units of such nanocrystals. Examples of nanocrystals that are of natural origin are the crystalline modifications of boron and carbon which have as their structural units the molecules B12 and C60. The boron molecule B12 consists of 12 boron atoms, and the carbon molecule C60, which is called fullerene, consists of 60 carbon atoms. The fullerene molecule resembles a soccer ball, i.e., it consists of 12 pentagons and 20 hexagons, with carbon atoms at their corners. These nanoparticles form face-centered superlattices with a period of about 1−10 nm. At these distances between molecules of C60 weak molecular forces, which provide the crystalline state of fullerene, act.

In addition to nanocrystals of natural origin, numerous artificial three-dimensional superlattices consisting of various types of nanoparticles have been fabricated. The variety of nanocrystalline structures as well as of conventional crystals is defined by the differences in the distribution of electrons over the quantum states of atoms. The most significant role in the formation of individual nanoparticles as well as of crystals is played by the electrons in the outer shells of atoms.

In the previous chapter we have analyzed the peculiarities of quantized electron motion in layered structures with one-dimensional potential wells. From the mathematical point of view, the solution of the Schrödinger equation for one-dimensional potential profiles is much simpler. However, many quantum objects, such as atoms, molecules, and quantum dots, are three-dimensional objects. Thus, in order to analyze electron motion in such objects we need to find solutions of the Schrödinger equation for three-dimensional potential profiles. The electron motion in spaces with dimensionality higher than one, especially for rectangular potential profiles with infinite potential barriers, is not so difficult to analyze. At the same time we have to keep in mind that such potential profiles frequently represent some approximation of the more complex, real potential profiles. Depending on the type of structure and on the form of the potential profile, the electron motion may be limited in two directions (two-dimensional quantization) or in three directions (three-dimensional quantization). In this chapter we will show that the existence of discrete energy levels in the electron spectrum is an intrinsic feature of electron motion in potential wells of any form and dimensionality.

An electron in a rectangular potential well (quantum box)

In the previous chapter we studied the electron motion in one-dimensional potential wells. An electron's motion along one direction was confined by the potential profile and the momentum in this direction was quantized.

In Chapters 3 and 4 we have discussed electron behavior in potential wells of various profiles and dimensionalities. We have established that localization of electrons in such potential wells, regardless of their form, leads to the discretization of the electron energy spectrum whereby the distance between energy levels substantially depends on the geometrical size of the potential wells. If this size is macroscopic then the distance between the energy levels is so small that we can consider the energy spectrum to be practically continuous (or quasicontinuous). Electrons in metallic samples of macroscopic sizes have this kind of energy spectrum. Another limiting case is that of small clusters consisting of just a few atoms, where the distance between energy levels is of the order of electron-volts. Gradual decrease of one or several geometrical dimensions of the potential well from macroscopic to about 1 μm practically does not change the form of the electron energy spectrum. Very often macroscopic materials (or macroscopic crystals) are referred to as bulk materials or bulk crystals. Changes happen only when the size of structures is of the order of or less than 100 nm. Such structures are called nanostructures. The change of the electron spectrum from quasicontinuous to discrete implies changes in most of the physical properties of nanostructures compared with those in bulk crystals. In this chapter we will consider the main peculiarities of the electron energy spectrum in nanostructures of various dimensionalities.