The next accurate approximation is Eq. ( ) for the microion contribution, with a relative error that is almost one order of magnitude larger than Eq. ( ), reaching values of up to 0.03 for Φ ≈ 0.5 Finally, Eq. ( ) for the basic-site contribution yields the largest error, with a relative error typically two orders of magnitude larger than for the acid case and values around 0.2 for Φ ≈ 0.5. We have also derived an alternative expression of c BH + (r) in terms of the OH -concentration using water autodissociation (see §2 of the SI). We have seen that the corresponding nonlinear expression and the linear approximation produce the same results as Eqs. ( ) and (36). From the analysis of the approximations, we can anticipate that the linearized equation will be less accurate inside the microgel as compared to the outside, due to the larger accumulated error stemming from the charge-regulating charge distributions as