Durán wrote down an explicit formula for such map in ["Pointed Wiedersehen Metrics on Exotic Spheres and Diffeomorphisms of $S^6$"][1].  That is he wrote an explicit formula for an exotic diffeomorphism from $S^6$ to $S^6$ which is homotopic but not isotopic to the identity. This the produces an explicit homeomorphism from $S^7$ to an exotic sphere by glueing as described by Ryan in his [answer][2].

Geometric properties of that particular map were later studied by various people.
For example, it's written down explicitly on page 1 in ["Bootstrapping $Ad$-Equivariant Maps, Diffeomorphisms and Involutions"][3]  by Durán and Rigas and Sperança (this link is freely accessible unlike the first one).


  [1]: https://doi.org/10.1023/A:1013163427655 "Geometriae Dedicata 88, 199–210 (2001). zbMATH review at https://zbmath.org/1002.53026"
  [2]: https://mathoverflow.net/a/93488
  [3]: https://mc.sbm.org.br/wp-content/uploads/sites/9/sites/9/2021/12/35-2.pdf "Mat. Contemp. 35, 27-39 (2008), doi:10.21711/231766362008/rmc352. zbMATH review at https://zbmath.org/1204.57031"