Schluesseltech_39 (1)

Similarly also higher moments averages may be defined such as the Z average which relates to the ratio of m3/m2.

These moments are accessible by different experimental techniques. E.g. measurements of the osmotic pressure reveal the number average. The forward scattering from a polymer ensemble gives the weight average and last but not least scattering experiments on the size of a polymer reveal the Z average.

Finally in order to give an example and to provide some impression about the different averages we will take a simple example. Consider 1000 spheres of steal with masses of 10g, 50g and 250g. Their numbers are 900 and 50 for the higher weights respectively. Following the prescriptions of Eq.[3] to [6] for the moments we find

From that the different averages are easily found.

These numbers are very instructive. Very small fractions of high weight lead to drastic differences in the averages and also may lead to inequate perceptions. If you look e.g. on the Z average you would not think that most of these spheres are small.

2.2 Polymer structure

In a covalently bonded system the microstructure i.e. the organization of atoms along the chain is fixed by the polymerization process. There exist several isomerism’s that influence the local structure. First let us consider structural isomerism. Figure 5 displays two examples for this phenomenon. The first regards the formation of polypropylene. There the CH3 side groups may be either placed adjacent to each other – this is the head to head configuration or they may be placed on every second carbon. This leads us to a so called head to tail configuration. Similarly polybutadiene may exist in three structural isomers: The cis-form, where the chain is symmetrically arranged with respect to the double bond, the trans form is an asymmetric configuration and finally the vinyl form, where the double bond sticks out of the main chain and the main chain itself is formed by a single bonded carbons.

Fig. 5: Polymer microstructure. (a) The structural isomerism in the case of polypropylene.

(b) Structural isomers of polybutadiene.

The stereoisomerism is another form of structural isomerism on the local scale. Figure 6 depicts the three forms of stereo isomerism for polystyrene. Isotactic polystyrene is a form where the phenyl ring is always on the same side of the chain. A syndiotactic form is reached when the direction of the phenyl ring is alternating along the chain and finally the atactic form that is produced by non –stereo specific polymerization mechanisms has a random sequence of phenyl rings with respect to the chain backbone. While the first two may crystallize, because of its irregular arrangement of the phenyl rings atactic polystyrene stays amorphous. Thus, small changes on the local chain structure may have large consequences for the polymer material as a whole.

Polymers may also have structural differences on the large scale. The simplest case is the homopolymer which is built from just one monomer. Copolymers are polymers that are built from two or more different monomers and may exist at least in three different forms. The two monomers may be arranged in a random sequence – these copolymers are called random copolymers, the monomers may be arranged in blocks there would be an A block and a B block. Such a polymer is called diblock copolymer and finally the monomers could for example be arranged in an alternating way. Then we will have an alternating copolymer. But certain many other possibilities could be considered. Finally polymers can differ by their architecture.

Figure 7 displays a number of common polymer architectures. The simplest case is a linear chain, ring polymers are very interesting subjects scientifically, star polymers are entities where different linear polymers are connected at one central point. All the other polymers are different examples for branched materials. The simplest nontrivial branched molecule is a so called H polymer, where a central cross bar is connecting arms on both sides. Comb polymers are featuring side arms along a common backbone. Ladder polymers are combs with another backbone on the other side. Dendrimers are branched polymers originating from a central star that at each end of a branch bifurcate again, (h) is showing a general branched polymer. The rheological properties of these materials differ greatly and modern polymer design considers a deliberate selection of architecture, in order to promote certain properties.

Fig. 6: Sterio isomerism at the example of polystyrene. (a) Isotactic. (b) Syndiotactic. (c) Atactic.

Fig. 7: Examples for a polymer architecture. (a) Linear. (b) Ring. (c) Star. (d) H-polymer.

(e) Comb polymer. (f) Ladder polymer. (g) Dendrimer. (h) Randomly branched polymer.

Finally, I want to mention rubbers, where elastomers are crosslinked by the vulcanization process. In this process, at least in the original one sulfur compounds join different chains. They form sulfur bridges of several sulfur atoms between backbones of adjacent chains. Depending on the degree of vulcanization the stress strain behaviour of such a rubber can be significantly altered. Figure 8 displays the stress strain curves for long chain polyisoprenes in the unvulcanized and vulcanised state. The mechanical strength of the vulcanised rubber is strongly enhanced compared to the homopolymer.

Fig. 8: Stress strain relationships for vulcanized and non-vulcanized polyisoprene.

2.3 Polymer crystallization

Figure 9 displays a spherulite structure as it is formed in polymer crystallization. This structure contains an assembly of chain folded lamellar crystallites that built up the spherulite. Each of the lamellae is in the order of 10nm thick and stapled as seen on the right hand part of the figure. They grow from the centre and form the beautiful spherulites. Because of kinetic hindrance - polymers are entangled - a large single crystal never form. Instead, lamellar structures appear and push the entangled amorphous part into the intermediate regime in between the lamellae. Thus, crystallizable polymers are always semicrystalline. As mentioned already above atactic, configurations on a local scale prevent crystallization all together.

Fig. 9: Spherolitic structure of semicrystalline natural rubber. The chain folded lamellar crystallites are represented by the white lines.

In general the mechanical properties of polymers are strongly enhanced by crystallization. The tensile strength and the E modulus increase significantly with crystallinity. Semicrystalline polymers constitute the largest group of commercially useful polymers. These polymers exist as a viscous liquid at temperatures above the melting points of the crystals. Upon cooling crystals nucleate and strengthen the material significantly. Semicrystalline polymers are used for example as packaging materials, as plastic bags and all kinds of construction materials.

2.4 Mechanical behaviour

The mechanical behaviour of polymers is very rich. Figure 10 displays schematically the temperature dependence of the shear elasticity modulus E. We see a number of different regimes. At very low temperature the material is brittle. Then at a first transition temperature Tβ secondary relaxations become important that make the material more ductile. At a glass transition temperature Tg the modulus drops significantly but at the same time the elasticity of the material is enhanced. At Tm the crystallites within the semicrystalline polymer melt and the system becomes a highly viscoelastic liquid. Such a viscoelastic liquid displays a very interesting frequency or time dependent modulus. This is shown in Figure 11.

Here the real and imaginary part of the dynamic shear modulus is plotted for a high molecular weight polymer. The real part represent the elastic modulus and we see that within a certain frequency range the polymer melt basically reacts elastically. In this regime Hooke’s law is approximately valid. At low frequencies the material flows and there Newton’s law for viscous flow applies. The imaginary part, the loss modulus, shows a peak at low frequencies. This peak is characteristic for the longest relaxation frequency in the melt and also signifies the cross over from elastic to viscous behaviour. In the lecture on macromolecular dynamics this will be emphasized in much more detail.

Fig. 10:Schematic representation of the shear modulus for a semicrystalline polymer as a function of temperature. (1) Brittle solid, Tβ onset of β-relaxation. (2) Ductile solid, Tg glass transition. (3) Elastomeric character, Tm melting point. (4) Viscoelastic regime

Fig. 11: Dynamic modulus of a long chain polymer. G’ and G’’ real an imaginary part of the modulus.

2.5 Intermediate summary

Summarizing this part of the lecture polymers are a huge class of substances with very broad and different properties. They are an important industrial commodity and are extremely important for many aspects of our daily life. Polymers encompass synthetic, organic polymers, biopolymers like proteins, polypeptides, polynucleotides, polysaccharides, natural rubbers, semisynthetic polymers like chemically modified biopolymers and inorganic polymers like the siloxanes, the silanes and the phosphascences.

3 Polymer chain conformations – ideal chains

3.1 Ideal chains

The conformation of a polymer describes the spatial arrangements of its monomers that relates to its flexibility, to the interactions between the monomers and the interactions with the solvent. In order to illustrate the relevant magnitudes we consider a chain of N = 1010 bonds which could be realised by a DNA molecule and magnify it to the centimetre scale. Thus, we consider magnitudes under the condition that the bond length amounts to 1cm. If we would now collapse the coil, then the volume it would pervade is given by V = N A3 = 1010cm3. This is about the size of a lecture hall.

Now let the molecule perform a random walk. Under this circumstances we have R ≈ N1/2 A = 1km. This is the size of the Forschungszentrum. Now let’s consider an excluded

volume chain that we will discuss later. Under these circumstances R ≈ AN3/5 = 10km. And

finally if there would be a long range electrostatic interaction, the size of the chain would be the contour length R ≈ N1 = 105km or in the order of a third of the distance to the moon. Thus, polymer chains may pervade very different spatial regions depending on their conformation.

The basic building block for a polymer chain are the directional covalent bonds of their constituents. For carbon which is the most important polymer builder the local configurations are shown in Figure 12.

Fig. 12:Directionality of the covalent bonds of carbon. (a) Methane (tetrahedral arrangement). (b) Ethylene, easy rotation around C-C bond. (c) Coplanar butane.

Figure 12a displays the simplest carbon based molecule, the methane. Here the tetrahedral arrangement of the bonding orbitals are clearly seen. This spatial arrangement is very important for the conformational properties of a carbon based polymer. Figure 12b displays the ethane molecule where two carbons form the backbone. In this molecule an easy rotation around the C-C axis is possible. The potential minimum is reached if adjacent hydrogen atoms are displaced by 60°C with respect to each other. Finally Figure 12c shows the butane molecule which comprises 4 carbons along the backbone. Here the typical conformational states also of a long polymer chain are visible. There are three conformational possibilities: the molecule may be coplanar - this is called a trans conformation or we may witness a rotation by plus or minus 120°C around the centre carbon bond. These states are called gauche+ and gauche-.

Figure 13 displays the corresponding conformational potential for rotations. We see the three minima where the trans is the lowest state. For a hydrocarbon the barrier height from the trans to the gauche conformation is given by Eb = 1700K.

Fig. 13: Rotational potential for transand gauche states.

polymer exhibits high conformational mobility. With that we now may define a conformation of a polymer as all spatial arrangements that originate by rotations around the bonds.

We now consider a long chain and discuss its conformational characteristics. Figure 14 displays a polymer chain where the different monomers are described by little beats. ri are the bonds vectors and the ensemble of all ri describes the conformation. In order to measure the

statistical average then all positive and negative values of RE are equally possible and the average of RE is zero. Thus, we need to consider the mean square end to end distance.

In order to go further we have to evaluate the scalar products between the different bond vectors ri and rj, ri r j = A20 cos(θij ). With that we can write the mean square end to end distance.

The next step requires knowledge about the ensemble averages of the projection angles θij. Here different models for chain conformations come in. The simplest case is the freely jointed chain where each segment can rotate freely around its connection to the next. Under these

circumstances cos(θij ) = 0 for i ≠ j and the end to end vector becomes

Even if fixed bond angles are considered, in an ideal chain there are no long range correlations. Thus, we have

where CN is called characteristic ratio. For infinitely long chains CN = C∞ which is a characteristic quantity for a polymer chain.

Fig. 14: Schematic representation of a polymer chain. Ai are the monomers, Rn ≡ RE is the end to end vector connecting the first with the last monomer, θij is the angle between the bonds ri and rj.

In order to map a given real chain on an ideal chain the notion of the Kuhn length b has been introduced. With that one can define an equivalently freely jointed chain which has the same end to end distance based on a larger segment and smaller number of monomers. The requirement of the same contour length leads to n b = Rmax, where n is the number of Kuhn segments in the equivalent chain. Furthermore, we have to require that the end to end distance

of the real and the equivalent chain are the same. Thus, nb2 = RE2 = bRmax = C∞NA20 from these two requirements we can calculate the number of Kuhn segments

and the size of the Kuhn length itself

As an example we consider the case of polyethylene where C∞ = 7.4, A0 = 1.54 and θ = 68°. Thus, Rmax = NA0 cos(θ/2). With Eq.[12b] we arrive at b = 14Å. We may also define the mass of a Kuhn segment M0Kuhn = Nn mbond .

Table 3 displays characteristic conformational parameters for a number of polymers, including the characteristic ratio, the Kuhn length, the density and the molecular weight per Kuhn segment.

Table 3:

Characteristic ratios, Kuhn lengths and molar masses of Kuhn monomers for common polymers at 413K.

As a disclaimer I like to note that in the following we do not make an explicit difference between Kuhn chains and chains built from real monomers. We will use always Eq.[11b] without any further specification. If we address a Kuhn chain then this will be explicitly mentioned.

3.2 Fractality

The result for the mean square end to end distance as a function of chain length reminds on the mean square displacement for random walks. There