The structure of grafted adsorbing polymers on surfaces is described as a statistical ensemble of loops generated by an one-dimensional random walk perpendicular to the surface. The configuration of each chain is considered as a succession of closed loops ended by an open loop (a tail). The probability of formation of each individual loop is the product between the probability of first return to the surface and a Boltzmann factor containing the free energy of the Flory-Huggins kind, which is approximated by the minimum free energy of all possible configurations of that loop. At high grafting densities, the attractive interactions between monomers and surface control the fraction of polymer belonging to either closed loops or tails, hence the formation of a stretched grafted brush. At low grafting densities, the increase of that interaction above a critical value generates an abrupt collapse of the brush on the surface. Whereas for long polymers (with more than about 100 Kuhn segments), the structure of the brush can be determined, in general, only via Monte-Carlo sampling, it is argued that the two structural transitions indicated above can be well predicted by simple approximations.
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