Modelling and simulation of the permanganic etching of banded spherulitic polyethylene: correlation with AFM observations

Atomic force microscopy (AFM) has been used to examine the 3D profile of some permanganically etched metallocene catalysed high-density polyethylenes. Banded spherulites were observed with an asymmetric band radial profile, having a characteristic shape with specific ‘inner’ and ‘outer’ band slopes, the inner slope being much lower than the outer slope. Permanganic etching is known to produce surface etching that is more rapid for the amorphous part of the polymer than for the crystalline part. A mathematical model of the etching process was developed to demonstrate that band asymmetry can stem from differential etching and to fully simulate the entire resultant relief. Etching was modelled analytically in a 3D geometry, based on the scalar product of two etch vectors, the first of which, c, is locally perpendicular to the lamella and secondly p, locally perpendicular to the specimen surface. By modelling a banded spherulite with vector c constantly perpendicular to the spherulite radius and having its extremity following a helical trajectory, it is possible to calculate the scalar product c·p at any point on the surface. This product gives a formula capable of calculating etch profile development as a function of time and for any initial surface profile, with the following parameters: etching rates of groups of flat-on lamellae and groups of edge-on lamellae, interband spacing, distance of the spherulite centre from the initial surface. Simulations of the etching of banded spherulites were performed and the parameters adjusted to give a very food fit to many AFM observations. The model confirms that the observed AFM profile of banded spherulites in polyethylene is probably of geometric origin alone. It is shown that the observed slope asymmetry is not material dependant, i.e. molecular weight or other structural parameters. The model not only explains the slope asymmetry but also gives a good simulation of band height and its radial variation, etching depth and apparent band period. It could also be used to estimate the relative speed of the permanganic etching of different crystal facets and the amorphous phase. Further refinement of the modelling work on other semicrystalline polymers amenable to permanganic etching is also a possibility.