Hence, by iteratively searching over the boundary between the 5 and 3 roots domains, we can find an optimal combination, which is obtained for r C = 1.6217, r ∆pKa = 0.2862, ∆pK c a = 2.572. It corresponds to a situation where branches start to touch in the root space, as in the 3 weak acids case of section 4.1. Hence, for 4 weak acids mixture the optimal combination found here for an interval centered around pH = 7 is pK a1 = 5.714, pK a2 = 6.632, pK a3 = 7.368, pK a4 = 8.286 and r C = 1.6217, as depicted in figure .a (root space) and 8.b (corresponding buffer index curves at ∆pK a = ∆pK c a ).