Theory of Lubrication due to Poly-Electrolyte Polymer Brushes
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Theory of Lubrication due to Poly-Electrolyte Polymer Brushes Jeffrey B. Sokoloff Physics, Northeastern University, 110 Forsyth Street, Boston, MA, 02115 ABSTRACT It is shown using a method based on the mean field theory of Miklavic Marcelja [1] that it should be possible for osmotic pressure due to the counterions associated with the two polyelectrolyte polymer brush coated surfaces to support a reasonable load (i.e., about 105 Pa) with the brushes held sufficiently far apart to prevent entanglement of polymers belonging to the two brushes, thus avoiding what is likely to be the dominant mechanisms for static and dry friction. INTRODUCTION Polymer brush coatings on solid surfaces, illustrated in Fig. 1, provide very effective lubrication, in the sense that they are able to support significant load, but have exceedingly low friction coefficients [1]. Human and animal joints are known to exhibit very low friction and wear. The outer surfaces of the cartilage coating these joints have polymeric molecules protruding from them [3–15], which suggests the strong possibility that their very effective lubrication is a result of polymer brush lubrication.
FIG. 1: The geometry of two polyelectrolyte polymer brush coated surfaces with the load pushing the surfaces together supported by osmotic pressure due to counterions in the interface regions separating the tops of the brushes is illustrated schematically. The dots located among the polymer chains and in the interface region between the two brushes represent the counterions. As illustrated here, D denotes the spacing of the surfaces and h denotes the polymer brush height.
Raviv, et. al. [16], have found that polyelectrolyte brushes exhibit remarkably low friction coefficients (10−3 or less) compared to the friction coefficients typically found for neutral
polymer brushes [2, 17, 18], which already exhibit low friction. In an effort to determine a possible mechanism for this, a solution for the Poisson-Boltzmann equation beyond the DebyeHuckel approximation will be used to determine the concentration of counterions in a region located midway between the two polyelectrolyte brushes. This result is used to show that for polyelectrolyte brushes, osmotic pressure due to the counterions is capable of supporting about 105Pa of load, even with the tops of the mean field theoretic monomer profiles of the two brushes about 100Ao apart. In fact, it will be argued that this result is valid even for salt concentrations up to 0.1 M, the salt concentration in living matter. As a consequence, there will be little entanglement of the polymers belonging to the two brushes. It was argued in Ref. [19] that such entanglement of polymer brushes leads to static and nonzero slow sliding speed limit kinetic friction, and is likely responsible for wear as well. THEORY Ref. [19] shows that the methods of Ref. [1] may be applied to polyelectrolyte brushes to obtain the probability that monomers stick out a distance z0 − h into the interface region between the two brushes, Posm ∝ exp[
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