Atomic force microscopy of polymer brushes: insights into controversies

Atomic force microscopes (AFM) and nanoindenters have been used for decades to evaluate mechanical properties of thin films at the nanoscale. It is argued that the elastic solutions to the indentation problem, which are most often associated with the names of Galin or Sneddon, may be used for extracting elastic contact modulus of bulk samples and continual films, while their application to contact between an AFM probe and a polymer brush is a priori questionable. This is because the character of compression of a polymer brush is drastically different from the response of an elastic half-space to indentation. In the present paper, a number of controversial issues related to the interpretation of the AFM data obtained for polymer brushes tested with a rigid probe are studied. In particular, a correct relation has been established between the constitutive equation for a single polymer brush in compression with a bare rigid surface and the constitutive equation for two identical polymer brushes in compression under the assumption of lack of interpenetration of compressed brushes. It is shown that the so-called apparent elastic modulus of a polymer brush introduced based on the Hertzian force-displacement relation depends on the indenter radius and, thus, may not serve as a characteristic of polymer brush. Also, the Derjaguin’s approximation-based method of identifying the point of initial contact is given in opposition to controversial methods, which are broadly based on the Hertzian contact mechanics