Adsorption isotherm and mechanism of $\mathrm{Ca^{2+}}$ binding to polyelectrolyte

Polyelectrolytes, such as polyacrylic acid (PAA), can effectively mitigate $\mathrm{CaCO_3}$ scale formation. Despite their success as antiscalants, the underlying mechanism of $\mathrm{Ca^{2+}}$ binding to polyelectrolyte chains remains unresolved. Through all-atom molecular dynamics simulations, we construct an adsorption isotherm of $\mathrm{Ca^{2+}}$ binding to sodium polyacrylate ($\mathrm{NaPAA}$) and investigate the associated binding mechanism. We find that the number of calcium ions adsorbed $[\mathrm{Ca^{2+}_{ads}}]$ to the polymer saturates at moderately high concentrations of free calcium ions $[\mathrm{Ca^{2+}_{aq}}]$ in the solution. This saturation value is intricately connected with the binding modes accessible to $\mathrm{Ca^{2+}}$ ions when they bind to the polyelectrolyte chain. We identify two dominant binding modes: the first involves binding to at most two carboxylate oxygens on a polyacrylate chain, and the second, termed the high binding mode, involves binding to four or more carboxylate oxygens. As the concentration of free calcium ions $[\mathrm{Ca^{2+}_{aq}}]$ increases from low to moderate levels, the polyelectrolyte chain undergoes a conformational transition from an extended coil to a hairpin-like structure, enhancing the accessibility to the high binding mode. At moderate concentrations of $[\mathrm{Ca^{2+}_{aq}}]$, the high binding mode accounts for at least a third of all binding events. The chain's conformational change and its consequent access to the high binding mode is found to increase the overall $\mathrm{Ca^{2+}}$ ion binding capacity of the polyelectrolyte chain.