The ellipsoids axes determine the shape of the ellipsoid, while the sigmac parameter is the contact distance between two ellipsoids. For small sigmac values, the minimum of the Gay-Berne potential is close to the ellipsoid axes. For increasing large values of sigmac, the minimum of the potential shifts to higher values and the attractive curve becomes more shallow. Here is a demonstration with the following parameters:
eps0 = 5.5
sigmax = 7.6
sigmay = 6.2
sigmaz = 4.3
epsx = 0.2
epsy = 0.4
epsz = 1
upsilon=1
The choice of sigmac affects pretty much the stickiness of the particle, as you can see from the shape of the potential. In this paper, I have used a reference potential computed on atomistic models to fit the Gay-Berne parameters. In that case, sigmac is just another free parameter controlling the shape of the potential.