Expanding the Schulze-Hardy Rule and the Hofmeister Series to Nanometer-Scaled Hydrophilic Macroions

The Schulze-Hardy rule is a well-established observation in colloid science (can be derived from the DLVO theory) that demonstrates the relationship between the critical coagulation concentration (CCC) of colloids and the valence of extra counterionic electrolytes (z), with a simple mathematical relationship of CCC approximate to z(-6). Here the Schulze-Hardy Rule is expanded to much smaller, nano-scaled soluble macroions in aqueous solution, by examining the stability of the macroions in the presence of additional electrolytes. The CCC values of the macroions follow the general trend of CCC approximate to z(-n) but the n value is significantly dependent on the surface charge density of the macroions, ranging from n = 2 at very low surface charge density to n = 6 at a high surface charge density. In addition, different cations with the same valence showed clear different impacts on the CCC values, with an interesting trend being connected to the Hofmeister series originally discovered in protein solutions.