What is a 'corona phase'?
One of the authors has recently established a research group at my university, so I spoke to a member of their lab:

The corona phase is not exactly a phase in the thermodynamic sense, but rather the portion of the polymer in contact with the nanotube. In the below image (taken from the Nature Nanotechnology paper in question), the blue portion and the parts of the red portion in contact with the tube would be the corona phase.