A thermodynamic gauge for mobile counter-ions from colloids and nanoparticles

A thermodynamic equilibrium sensor is proposed that measures the ratio of the number of elementary charges z to the mass m of charged solutes such as charged colloids and nanoparticles. The sensor comprises a small, membrane-encapsulated salt solution volume that absorbs neutral salt molecules in response to the release of mobile counter-ions by charge carriers in the surrounding suspension. The sensor state emerges as a limiting case of the equilibrium salt imbalance, and the ensuing osmotic pressure difference, between arbitrary salt and suspension volumes. A weight concentration of charge carriers c is predicted to significantly increase the sensor's salt number density from its initial value ρ_s,0 to ρ^R_s, according to the relation (ρ^R_s/ρ_s,0)² - 1 = zc/mρ_s,0, under the assumption that the mobile ions involved in the thermodynamic sensor-suspension equilibrium are ideal and homogeneously distributed.