Table C1.

Possible formulations of electrostatic potential energy functions assuming point charges (coulomb and Debye-Hückel) or charges homogeneously distributed on surfaces (sphere or cylinder with radius r_act)

Formulation	Coulomb	Debye-Hückel/Nakajima	Sphere (radius ract)	Cylinder (radius ract, length ${\ell }_c$)
Potential $\text{Ψ}$ [J]	${\text{Ψ}}_C\left(d\right)=-\frac{A}{d}$	${\text{Ψ}}_{DH}\left(d\right)=-A\cdot \frac{\text{exp}\left(-d/\lambda \right)}{d}$	${\text{Ψ}}_{DH\text{sph}}\left(d\right)=-A\cdot \frac{\text{exp}\left(-\left(d-r_{\text{act}}\right)/\lambda \right)}{\left(1+r_{\text{act}}/\lambda \right)\cdot d}$	${\text{Ψ}}_{DH\text{cyl}}\left(d\right)=-A\cdot \frac{2\cdot \lambda \cdot K_0\left(d/\lambda \right)}{r_{\text{act}}\cdot {\ell }_c\cdot K_1\left(r_{\text{act}}/\lambda \right)}$
Force F [N]	$F_C\left(d\right)=\frac{A}{d^2}$	$F_{DH}\left(d\right)=A\cdot \frac{\text{exp}\left(-d/\lambda \right)\cdot \left(\lambda +d\right)}{\lambda \cdot d^2}$	$F_{DH\text{sph}}\left(d\right)=A\cdot \frac{\text{exp}\left(-\left(d-r_{\text{act}}\right)/\lambda \right)\cdot \left(\lambda +d\right)}{\left(\lambda +r_{\text{act}}\right)\cdot d^2}$	$F_{DH\text{cyl}}\left(d\right)=A\cdot \frac{2\cdot K_1\left(d/\lambda \right)}{r_{\text{act}}\cdot {\ell }_c\cdot K_1\left(r_{\text{act}}/\lambda \right)}$

Regarding the forces, we note that, for equal ϵ_r and I values, $F_C\left(r_{\text{act}}\right)=F_{DH\text{sph}}\left(r_{\text{act}}\right)=F_{DH\text{cyl}}\left(r_{\text{act}}\right)=A\cdot r_{\text{act}}^{-2}$, where the latter equation requires ${\ell }_c=2\cdot r_{\text{act}}$ (as chosen here).