MANY CHAIN SYSTEMS

Off-lattice models consider chains composed of interacting units in the free space. Single chains or simulation boxes containing many-chain systems can be investigated. Usually the solvent is only considered according to its quality effects in thermal systems. Therefore it is assumed to fill the remaining space act-... 

Fig. 2. The tube model replaces the many-chain system left) with an effective constraint on each single chain right). The tube permits diffusion of chains along their own contours only...

Therefore, the principal difficulty connected with the application of Eq. (12) is due to the incompleteness of the Gauss invariant. So, the use of the Gauss invariant for adequate classification of topologically different states in many-chain systems is very problematic. Nevertheless, that approach was used repeatedly for consideration of such physically important question as the high-elasticity of polymer networks with topological constraints [15]. Unfortunately,... 

Although most theorists have focused on the origin of the counterion-mediated attraction, the main focus of our work has been to explore the consequences of the interaction in many-chain systems. The charge fluctuation approach is particularly well suited to many-chain systems because it allows an analytical approach [30]. This is particularly important because the counterion-mediated interaction is not pairwise additive. We have found that, in equilibrium, the system should phase separate on a macroscopic scale into a dilute phase in coexistence with a concentrated phase of parallel chains. 

Summarizing, we may conclude that reported sedimentation and diffusion data on semi-dilute solutions do not always lend support to the predictions of the Brochard-de Gennes blob theory. Certainly, this theory oversimplifies the complex hydrodynamic interactions involved in many-chain systems. Nonetheless we should not underestimate its merit that has sparked off many sedimentation and diffusion measurements on concentrated polymer solutions in recent years. 

The reptation idea premises that the topological constraint exists and is sufficiently strong. In principle, we should be able to determine whether it is right or not by solving the equations of motion governing the long-timescale dynamics of many-chain systems. Recently, some attempts to this very difficult problem have appeared. Here we mention two of them. 

Having discussed static properties, we now consider dynamical problems in many chain systems. Here we have to consider another very important type of interaction which arises from the very nature of the polymer polymers are one-dimensionally connected objects and cannot cross each other. A good way of looking at this is to imagine that the polymers have no thickness, and no attractive force, like a mathematical curve in space. Qearly the excluded volume of such a chain is zero. However, even such chains can interact strongly due to the topological constraints that chains cannot cross each other. 

Although not a simulation method, the numerical solution of the SOFT equations is a powerful technique applicable to a variety of dense polymer systems. There are many iterative strategies that have been applied to solve the SOFT equations.Beginning with homogeneously disordered fields, these equations are iterated until a minimum free energy is obtained. When the mean field to is known, in principle, any property of the many-chain system can be accessed (e.g., the free energy is given by F s H [< ]). 

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