Polymer brushes analytical approaches

Besides the analytical techniques, the theoretical description of polymer brushes allows a deeper understanding of the complex dynamic behavior of polymers on surfaces and is useful for future developments. Here, Roland Netz gives - also for the non-expert - a very helpful theoretical background on the theoretical approaches for the description of neutral and charged polymer brushes. 

This name covers all polymer chains (diblocks and others) attached by one end (or end-block) at ( external ) solid/liquid, liquid/air or ( internal ) liquid/liq-uid interfaces [226-228]. Usually this is achieved by the modified chain end, which adsorbs to the surface or is chemically bound to it. Double brushes may be also formed, e.g., by the copolymers A-N, when the joints of two blocks are located at a liquid/liquid interface and each of the blocks is immersed in different liquid. A number of theoretical models have dealt specifically with the case of brush layers immersed in polymer melts (and in solutions of homopolymers). These models include scaling approaches [229, 230], simple Flory-type mean field models [230-233], theories solving self-consistent mean field (SCMF) equations analytically [234,235] or numerically [236-238]. Also first computer simulations have recently been reported for brushes immersed in a melt [239]. 

Note that the sharpness of the transition in the change on going from the depletion zone to the parabolic zone is due to limitations in the analytical function and does not reflect real transition behaviour. The more gentle transitions indicated in the theoretical SCF profile are more realistic. In the self-consistent field calculation a lattice model is not presumed the volume fraction of the tethered chains is calculated from a diffusion equation that involves polymer propagators and a (z-dependent) potential function that includes enthalpic interactions between the two copolymer blocks and between each block and the solvent. Initially the potential function is set to zero, the pol)nner propagators are calculated and then the volume fraction variation of the tethered block. A new potential is calculated from this volume fraction profile and the process reiterated imtil the difference in volume fraction profiles calculated by sequential iterations is smaller than some defined tolerance. The approach bears similarities to the SCF approach of Shull (1991) but makes no allowance for the dry brush case, i.e. that in which the relative molecular mass of the solvent approaches that of the tethered polymer molecule. 

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