the density of counterions and ϕ(z) the electrostatic potential. The first integral describes the energy of the bottom brush, the second integral describes the energy of the top brush, the third integral describes the energy of the counterions, the fourth integral describes the energy in the interpenetration zone and the fifth line contains three integral describing the electrostatic interactions in the system. In the first and second integrals, the first terms accounts for the entropic elasticity of the brush chains, the second terms shows the excluded volume interactions between monomers and the third terms are constraints for fixing the number of monomers. In the third integral, the first term accounts for the entropy of the counterions and the second term is the constraint to fix the number of counterions. Eq.