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For the first time, the classical density functional theory (DFT) is numerically solved in three- and two-dimensional spaces for a two sphere model of electrostatic interactions between two spherical nanoscale colloids immersed in a primitive model electrolyte solution. Two scientific anomalies are found that (i) contrary to what is often asserted that presence of multivalent counter ion is necessary to induce a like-charge attraction (LCA), univalent counter ion also induces the LCA only if bulk electrolyte concentration and colloid surface charge are high enough, and (ii) although the LCA in general becomes stronger with the bulk electrolyte concentration, adverse effects unexpectedly occur if the colloid surface charge quantity rises sufficiently. In addition, effects of counter ion and co-ion diameters in eliciting the LCA are first investigated and several novel phenomena such as monotonic and non-monotonic dependence of the LCA well depth on the counter ion diameter in different colloid surface charge zones are confirmed. Based these findings, a hydrogen bonding style mechanism is suggested and surprisingly, by appealing to fairly common-sense concepts such as bond energy, bond length, number of hydrogen bonds formed, and counter ion single-layer saturation adsorption capacity, self-consistently explains origin of the LCA between two spherical nanoscale particles, and all phenomena previously reported and observed in this study.


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