 critical assumption in their model is that the ion and electron densities can be written as a function of the local electrostatic potential alone. Although this assumption sounds reasonable for instance in the collisionless limit where OML theory should apply, its validity must be tested carefully. On the other hand, our first principles approach free from such an assumption also demonstrates a repulsive nature for the electrostatic interaction. Furthermore, the fact that the electric field around the grain is consistent with the OML theory indicates the assumption made by Markes and Williams (2000) is indeed reasonable.

One might argue that the fact that $\alpha\neq1$ explains the discrepancy between the simulation results and ODS theory, but this is not the case. Assuming that linear superposition of the potential is also possible for $\alpha\neq1$, we can easily calculate the ODS attractive force for this case as well. The resulting attractive potential force may be written as $$\label{ml-j}
    q{\phi_{{\rm ODS}}}\left(d\right)=\alpha\frac{q^{2}}{\lambda_{{\rm D}}}\left(\frac{\lambda_{{\rm D}}}{d}-\frac{\alpha}{2}\right)\exp\left(-\frac{d}{\lambda_{{\rm D}}} \right),$$ which is shown in Fig.\[mod\] for $\alpha=1, 1.2, 1.4$. It can be easily understood that the potential minimum moves inward and the depth increases as $\alpha$ increases. In fact, an easy analytical calculation confirms this tendency. Clearly, $\alpha\neq1$ does not help to explain the discrepancy.

![Modified ODS attractive potential given by Eq. (\[ml-j\]). Red, green, and blue lines represent $\alpha = 1, 1.2, 1.4$, respectively.[]{data-label="mod"}](modod.eps){width="90mm"}

Although it is not easy to analytically determine the value of $\alpha$ in general, we can estimate the upper and lower bounds as follows. We define $r_{{\rm c}}$ as a solution to the equation $$\label{exp-equ}
        \frac{\alpha}{4\pi\Lambda}\frac{q}{e}\frac{\lambda_{{\rm D}}}{r_{{\rm c}}}\exp{\left(-\frac{r_{{\rm c}}}{\lambda_{{\rm D}}}\right)}=1,$$ where the left