Schluesseltech_39 (1)

Exercises

Note: indicates an increased difficulty. Solve the easier problems first.

E12.1 Electronic structure and Mott transition

a)In modeling the electronic structure of crystalline solids, what is the typical starting assumption to separate the electronic structure from the lattice dynamics, and why does it work?

b)In which of the three simplest models of electrons in a solid are the electronic correlations taken into account at least approximately?

c)Neglecting electronic correlations, would you predict NaCl to be an insulator or a metal? Why?

d)The competition of which two contributions to the total energy of the electrons is crucial for the Mott-transition? Which further contributions to the total energy are neglected in the simplest model?

e)Assume that a particular material is a Mott-insulator, but just barely so (i.e. the relevant energy contributions are almost equal). What would you predict to happen when sufficiently high pressure is applied, and why?

E12.2 Electronic ordering in correlated-electron materials

a)List and very briefly explain three “electronic degrees of freedom”, which can become ordered.

b)Discuss why electronic correlations favor ordering processes of these electronic degrees of freedom.

c)What, if any, connection is there between orbital order and orbital magnetic momentum?

d)To order of which of the electronic degrees of freedom is neutron scattering directly sensitive, and to which not?

e)For those electronic degrees of freedom, to which neutron is not directly sensitive, neutron scattering can still be used to deduce an ordered arrangement: How and why? Is there a more direct scattering method than neutron scattering?

E12.3 Crystal field

Fe has atomic number 26 and in oxides typically has valence states 2+ or 3+.

a)Determine the electronic configuration of free Fe2+ and Fe3+ ions (hint: as for Mn the outermost s-electrons are lost first upon ionization).

b)From Hund’s rules determine the values of the spin S, orbital angular momentum L, and total angular momentum J of Fe2+ and Fe3+ ions.

Calculate the expected effective moment in units of μB of Fe2+ and Fe3+ ions, i) assuming S, L, and J as determined in b) and ii) setting L = 0 (‘quenched orbital momentum’). Compare with the experimental values of 5.88 μB for Fe3+ and 5.25 − 5.53 μB for Fe2+.

d)The negatively charged oxygen ions surrounding the Fe ions in an oxide solid influence the energy of the different orbitals. Plot the expected energy level diagram for the case of an octahedral environment of nearest-neighbor O2−. How does the total spin moment of Fe2+ change between weak and strong crystal field splittings (relative to intra-atomic “Hund’s” exchange)?

e)(optional) In a tetrahedral environment the energy levels of the orbitals are reversed compared to an octahedral environment. Determine the spin moment of Fe2+ in a tetrahedral environment with strong crystal field splitting. Is an orbital angular momentum possible in this case? How about when a Jahn-Teller-distortion leads to a further splitting of the energy levels?

E12.4 Orbital and Magnetic order in LaMnO3 (Optional!)

The orbital and magnetic order in LaMnO3 is sketched in Fig. 12.9 (page 11 of the chapter) on the left. One crystallographic unit cell a × b × c is shown.

a)Why is there no charge order in LaMnO3?

b)What are the smallest unit cells (sketch in relation to the crystallographic cell) that can describe i) magnetic order, ii) orbital order (Hint: consider also centered cells), iii) both magnetic and orbital order.

c)Make a plot of reciprocal space in the a -c -plane indicating the positions, where you expect nuclear, orbital, and magnetic Bragg peaks to occur.

d)As c), but for the a -c -plane.

13 Flexible Polymers

Dieter Richter

Jülich Centre for Neutron Science & Institute for Neutron Scattering Research Center Jülich

Content

1 Introduction and Overview

Look around the house and you will see that you are surrounded by many kinds of polymers: plastic containers, surface coatings in the kitchen, toys and clothes in the living room and bedroom. Modern equipment and components in your car and work place are often made of or coated with polymer composite materials. Today traditional materials, such as metal, ceramics and wood have partly been replaced by synthetic polymers which may be stronger, lighter and cheaper and which through scientific research can be tailored to specific requirements.

It is hard to imagine life without modern synthetic plastics and rubbers. These polymers can be moulded into almost any shape, extruded into thin firms and fibres, applied as coatings and given bright colours or made transparent. New polymer composites are continually being developed including reinforced rubbers or construction materials even for aeroplanes.

Polymers can be categorized in three classes:

Thermoplastics. They can be repeatedly melted upon the application of heat and they are considered to be recyclable because of that.

Elastomers. These rubbery materials can be stretched many times their original length. They do not melt upon the application of heat but they will degrade if heated to high enough temperatures.

Thermosets. These are generally rigid material that can withstand higher temperature than elastomers. They do not melt and will degrade if heated to high enough temperature.

As it is true for all classifications there are sometimes exceptions to the rules when describing the behaviour of polymers and intermixed behaviour is perfectly possible.

Polymers are one of the most important products of chemical industry. The development of this industry in Germany started in the second half of the 19th century. BASF was founded in 1866. In 1885 it had already 2335 employees. In 1900 BASF had grown to 6771 employees. Similarly Bayer was founded in 1881. In 1885 Bayer employed 24 chemists and 300 workers. Just 11 years later it had grown to 104 chemists and 2644 workers. The 2006 turnout of chemicals in Germany is given in Table 1. Among these products polymers are on the third rank with 25.88 billion Euros. Thus, polymers are indeed a very important commodity.

Table 1:

Turnover along products (Germany 2006)

Before coming to the present world of polymers, I like to briefly sketch the history of the development and understanding of polymer materials. Polymer production and use started with the search for a materials replacement of horn. This search started the modern plastics industry around 1800. The problem with horn was that it was not of uniform quality and that it was quite difficult to work with horn. Furthermore, though it had limited mold ability, it had unique properties and aesthetic appeal. The second natural material that came into focus was shellac. Shellac is a resin that is secreted by a particular beetle. It found use as a coating of furniture and was moulded into boxes, buttons, combs and electrical insulators and later the famous phonograph records were made out of it. Gutta-percha was the next natural material that came into use. Gutta-percha is attained from the palaquium tree and found use as an electrical insulator. For example, the first transatlantic cable was insulated with Gutta-percha and today still billiard balls are made from this material. Then it was realized that some natural polymers were becoming very useful, if they were modified chemically. The best example is rubber made from natural polyisoprene taken from the rubber tree. Charles Goodyear discovered the vulcanization of natural rubber using some sulphur compounds. With that rubber it turned out to be a very useful material and was one of the earliest and most important discoveries in the polymer industry. Celluloid was another early modified polymer. It was in addition the first polymer that was truly capable of imitating the aesthetics of horn and shell. Celluloid was a transparent material that could be coloured to imitate the patterns of horn and shell. One of its unique applications was the film industry where it was used as a substrate for the first motion picture film.

However it turned out that the physical properties of these natural polymers were still not sufficient for broad applications. In 1909 the first synthetic polymer Bakelite was developed. It is a composition of phenol and formaldehyde and opened the way for many applications outside natural polymers. Then, in the 30ties major breakthroughs in synthetic polymers occurred by luck. In 1933 Fawcett and Gibson discovered polyethylene in a botched experiment. They heated a mixture of ethylene and benzaldehyde to 170°C at a pressure of 1700 bar. Similarly Teflon was invented accidently by Roy Plunkett at Dupont in 1938 when he was experimenting with Freon coolants under pressure. Another most important discovery was the synthesis of synthetic rubbers – a mixture of butadiene and styrene monomers. The Buna called product was produced in an industrial scale first in 1937.

Science and technical developments always depends crucially on people, therefore, I like to finish this introductory paragraph in paying tribute to some of the most important pioneers in the field. The notion of polymers as macromolecules build from monomeric building blocks was only realized in the 20ties of the last century. At that time Staudinger in Freiburg (Nobel Prize 1953) developed his macromolecular hypothesis namely that polymers are molecules from covalently bonded units and not colloids as it was considered at that time. He could show the macromolecular character of the polymers in detailed studies of the viscosity and osmotic pressure that showed that polymers are indeed large molecules. In the 1930ies polymer chemistry got a very important boost by Carothers’ research on condensation reactions. He developed the synthesis for the first synthetic fibres of Nylon 6.6. Then in the 50ties the basis for coordination polymerization was found. Ziegler and Natta developed catalysers that allowed to synthesize stereo regular polymers, for example syndiotactic or isotactic polystyrene. For this very important development they received the Nobel Prize in chemistry in 1955.

So far the developments in polymers were dominated by chemical aspects. Polymer physics was largely developed by Flory in Stanford who layed ground for a statistical theory of polymer conformation and thermodynamics. For his outstanding achievements he received the Nobel prize in 1974. For the theme of this lecture the research of Flory is fundamental. He developed most of the theories that deal with polymer conformations that I discuss here. While the basics of polymer conformational in statistics were developed in the late 40ties and 50ties the understanding of polymer dynamics both in solution and in the melt took 25 years more. Pierre Gilles de Gennes who received the Nobel Prize in 1991 may be considered as a most eminent scientist in this field who laid ground for most of the basic concepts in polymer dynamics. In particularly famous is his reptation model where he discovered the basic motional mechanism of polymer chains in the melt. The last eminent scientist I like to mention is Alan Heeger who dealt with conductive polymers that are based on conjugated electrons in π-orbitals that may delocalize along the chain. Figure 1 presents photographs of these polymer pioneers.

(d) (e)

Fig. 1: Nobel Prize Winners in the field of polymer science. (a) Hermann Staudinger (Nobel Prize 1953), (b) Karl Ziegler and Giulio Natta (Nobel Prize 1955), (c) Paul J. Flory (Nobel Prize 1974), (d) Pierre-Gilles de Gennes (Nobel Prize 1991), (e) Alan Heeger (Nobel Prize 2000).

2 Basic definitions and properties

Polymers are chain molecules built from monomers . As an example Figure 2 presents the chemical structure of polystyrene together with the styrene monomer. During polymerization the double bond between the carbon atoms in styrene is opened and polymerization reaction connects the different carbon atoms to a long chain molecule. The polymerization mechanism

of such a molecule makes it clear that the molecular size in principle is ill defined and spans a very broad range from the monomer to the oligomer to the polymer with in general a large size distribution. As we shall see later that the size of the polymer plays an important role in its properties.

Styrene Polystyrene

Polymer backbones may be formed from a variety of atoms. Figure 3 shows a number of characteristic chain molecules that are often used in applications. The first and most common backbone is that of carbon atoms as shown before for polystyrene. Another important variety is that of a carbon oxygen backbone as presented by polyethyleneoxide where two carbons are separated by an oxygen atom along the chain. Within the molecules of life the polypeptides play an important role. There the backbone is built of carbon nitrogen chains and finally there exist also inorganic polymers like polydimethylsiloxane, a chain which is built from alternating silicon and oxygen atoms. In each case the monomer is always the smallest repeating unit along the chain. Thus, for polystyrene it contains two carbons along the backbone, for polyethyleneoxide two carbons and one oxygen and for polydimethylsiloxane one oxygen and one silicon atom along the backbone.

Fig. 3: Different polymer backbones. (a) carbon backbone, example: polystyrene, (b) carbon

oxygen backbone, polyethyleneoxide; (c) carbon nitrogene backbone: polypeptides; (d) inorganic polymers: polydimethylsiloxane.

2.1 Molecular weight and weight distributions

The molecular weight is of crucial importance for a large number of polymer properties. As the molecular weight increases so increases also the tensile strength that defines the elongation to break ,the impact strength that measures the reversible elasticity, the toughness which is related to the melting temperature, the creep resistance that follows from the melt viscosity and the stress crack resistance that is important for processing for example.

Fig. 4: Chain length dependence of the boiling and melting points for hydrocarbons.

Figure 4 presents the length dependence of the boiling point and the melting point for paraffins that become polyethylene if the chain lengths N grows substantially. The growth of the boiling temperature is limited by the disintegration of the material, while the melting temperature saturates around N ≈ 100 to a little bit above 100°C. Table 2 gives property dependences on chain length for hydrocarbons. Examples and uses are given for different classes of chain length. The table manifests the large variety of properties and uses that can be achieved by the same material just at different chain lengths.

Table 2:

Properties in dependence of chain length: hydrocarbons

Since apparently the molecular mass and also its distribution is very important for polymer properties, in the following we will discuss in more depth the relevant definitions and the

characterization of these quantities. Let us begin with the IUPAC (International Union of Pure and Applied Chemistry) definition of a polymer:

“A molecule of high relative molecular mass, the structure of which essentially comprises the multiple repetition of units derived actually or conceptually from molecules of low relative molecular mass.

There are two essential quantities that characterise these materials (i) the degree of polymerization or the number of units N of monomers and (ii) the distribution of the degree of polymerization N. The most common averages that are taken are the number average.

Thereby nN NA is the number of molecules with a degree of polymerization N, mn NA is the mass of the molecules with a degree of polymerization N and NA is the Avogadro number.

Further important definitions relate to the weight fraction.

with MN = nNM0, where M0 is the monomer mass. Finally, we consider the moments of the distributions.

With these definitions we can write the different averages in terms of the moments of the distribution. The number average becomes