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    Polymers

Text "Polymers"	Kurt Kremer, Max Planck Institute for Polymer Research, Mainz
Coagulation of Chain Polymers	Helge Frauenkron, Peter Grassberger, NIC Research Group for Many-Particle Physics
A Long Chain Polymer on the Surface of a Cylinder	Helge Frauenkron, Maria Serena Causo, Peter Grassberger, NIC Research Group for Many-Particle Physics
Nematically Ordered Polymer Melts	Henning Weber, Wolfgang Paul and Kurt Binder, Institute of Physics, University of Mainz
Simulations Concerning Polyelectrolytes	Kurt Kremer, Max Planck Institute for Polymer Research, Mainz

Text "Polymers"

In our daily life, we are inevitably constantly confronted with the properties of polymer materials. On the one hand, we ourselves largely consist of polymers (biopolymers such as DNA) in addition to water and, on the other hand, our daily life is no longer conceivable without polymer additives and polymer materials. They range from simple commodity polymer materials (plastics), such as polyethylene for plastic bags, through technical polymers, such as polycarbonate for CDs or coatings for printed circuit boards for the semiconductor industry, to water-soluble polymers in the environment and hygiene sector (water treatment, superabsorbers) up to food additives enabling the fabrication of fat-reduced products in the dietetic sector. They therefore have an extremely great significance in our life.

A characteristic feature of polymers is their structure. In the simplest case, a polymer consists of a chain molecule of repeated identical units. Much more complex structures play an important role both in nature and in chemistry. In addition to mixtures of a large number of different repeating units (so-called copolymers) the simple form of a chain is also often varied so that branched star-like objects (star polymers), comb-like structures, cross-linked structures (rubber, elastomers) up to so-called polymeric "bottle brushes" are observed, i.e. objects in which a large number of side-threads emanate from one strand so that the large molecule (macromolecule) looks like a molecular bottle brush. The properties of macromolecular substances are mainly determined by two factors: on the one hand, the length of the chain (we confine ourselves to simple chain polymers) plays an important role. For example, the viscosity of a polymer melt will approximately increase tenfold if the chain length is doubled, and this is independent of the chemical structure of the repeating unit. These properties, which are the same for all systems, are termed "universal"; they are covered by so-called "scale laws" (e.g. viscosity ~ N^3.4, where N is the number of repeating units. On the other hand, the chemical structure of the repeating units is of decisive significance for the magnitude e.g. of viscosity or expansion of the entire chain molecule in the solution or melt. For the example of viscosity this means that:

= A x N^3.4,

where A is a function of temperature or chemical structure of the repeating units. The viscosity can be varied by several orders of magnitude by varying A and by varying the chain length.

In order to understand the macroscopic behaviour of such materials, both parameters must be precisely understood. The simulations carried out at NIC primarily serve to investigate generic and thus universal properties. Within the framework of projects dealt with at NIC, however, there are also intensive attempts at considering both aspects and connect them with each other. Since the generic aspects are independent of the detailed structure of monomers, it is appropriate for reasons of computer time savings alone to develop models as simple as possible which have the decisive properties. The connectivity of the chain as well as the volume of the individual repeating unit, which is excluded for the other particles, must be appropriately taken into account in the model. Such models are considered both in continuum and as paths on lattices. All the examples of polymer simulations mentioned here consider variants of this model under different physical conditions.

These are only some of the large number of simulations carried out at NIC in the "soft-matter" range. In spite of the fact that generic rather than material-specific properties are primarily examined, direct starting points have already been evolved for industrial research. This is thus a field in which basic research continues to have a direct relation to problems in application.

(Kurt Kremer, Max Planck Institute for Polymer Research, Mainz)

Coagulation of Chain Polymers

The picture shows eight polymer chains with 128 monomers each dissolved in a poor solvent. The molecules of the solvent and the other polymer chains are not shown.

In a good solvent, polymers repel each other and are more stretched than random walks due to the repulsive monomer-monomer interaction in one and the same chain. In a poor solvent, it is more difficult for the constituents (monomers) of a polymer to come into contact with the molecules of the solvent, so that they must form more contacts among each other.

This has two consequences: on the one hand, the chains become increasingly coiled and, on the other hand, they are clumped together. This finally leads to a separation into a very concentrated, "clumped" phase and a very dilute phase. The transition between these phases should follow the same laws as the transition between gaseous and liquid phases (e.g. the evaporation of water). There should be a "critical point" at which the system is described by the well-known Ising model. In addition, however, there should also be universal dependencies on the chain length.

The latter can only be measured experimentally with great difficulty and are also difficult to simulate with conventional methods. The picture was obtained with a new mathematical method permitting the simulation of longer chains than before and leading to a new understanding of the experiments.

(Helge Frauenkron, Peter Grassberger, NIC Research Group for Many-Particle Physics)

A Long Chain Polymer on the Surface of a Cylinder

The behaviour of chain polymers in good solvents is described by the model of "self-avoiding random walks" on large length scales. This model leads to anomalous scale laws with "universal" exponents. Thus, for example, the diameter of such a random walk grows with the number N of steps (monomers) like R ~ N , where ˜ 0.588 in three dimensions and = 3/4 in two dimensions. does not depend on microscopic details. The fact that can be exactly calculated in two dimensions is related to the so-called "conformal symmetry" which plays a much greater role in two than in three dimensions. It is thus possible to conformally map a plane from which a small circle has been cut onto the surface of a cylinder. The investigation of self-avoiding random walks on a cylinder can therefore provide important information about such random walks in the plane. Simulations such as that in the picture have refuted a suggestion made about five years ago and have drawn attention to a subtle point in the application of conformal mappings previously overlooked.

(Helge Frauenkron, Maria Serena Causo, Peter Grassberger, NIC Research Group for Many-Particle Physics)

Nematically Ordered Polymer Melts

Nematic order means that the rod-like macromolecules which only interact with their neighbours by hard wall repulsion are oriented spontaneously along a preferred direction. For the chain centres, all the positions in this direction remain equally probable in this state, i.e. they remain disordered. The orientation of the macromolecules can be controlled by applying an electric field, an effect utilized for liquid crystals.

(Henning Weber, Wolfgang Paul and Kurt Binder, Institute of Physics, University of Mainz)

Simulations Concerning Polyelectrolytes

Polyelectrolytes are polymers which can split off ions in water so that they are electrically charged. Nearly all biological polymers are polyelectrolytes. The monomers thus strongly repel each other so that the chains are extensively stretched. There are two classes of polyelectrolytes: those with a hydrophilic chain backbone (they are rather seldom) and those with a hydrophobic backbone. In the latter case, an interesting competitive situation arises between electrostatic repulsion of the charged monomers and effective attraction caused by the water. The simulations compare the two cases as a function of concentration. All the ions are explicitly simulated and the full Ewald sum is taken into account for the interaction. The two upper pictures show the hydrophilic and hydrophobic cases at very low concentration, the bottom pictures the analogous comparison in the semidilute system. The case of a collapsed phase, which is nevertheless stable as a colloidal suspension (bottom right), is not possible with neutral chains.

(Kurt Kremer, Max Planck Institute for Polymer Research, Mainz)

Introduction	Super- computing	Astro- physics	Elementary Particles	Many Particles	Polymers	Chemistry	Environment	Other Fields