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Hyperscaling relation between the interfacial tension of liquids and their correlation length near the critical point

Interfaces involving coexisting phases in condensed matter are essential in various examples of soft matter phenomena such as wetting, nucleation, morphology, phase separation kinetics, membranes, phase coexistence in nanomaterials, etc. Most analytical theories available use concepts derived from mean field theory which does not describe adequately these systems. Satisfactory numerical simulations for interfaces at atomistic to mesoscopic scales remains a challenge. In the present work, the interfacial tension between mixtures of organic solvents and water is obtained from mesoscopic computer simulations. The temperature dependence of the interfacial tension is found to obey a scaling law with an average critical exponent mu = 1.23. Additionally, we calculate the evolution of the correlation length, defined as the thickness of the interface between the immiscible fluids, as a function of temperature and find that it obeys also a scaling law with the average critical exponent being nu = 0.67. Lastly, we show that the comparison of mu and nu for these binary mixtures constitutes the first test of Widom hyperscaling relation between these exponents in 3d, expressed as mu = nu (d - 1). Based on these values and those for the 3d Ising model it is argued that both systems belong to the same universality class, which opens up the way for the calculation of new scaling exponents.