The method for determining nano-contact angle

Theoretical analyses are given to the known approaches of nano-contact angle and arrive at the conclusions: 1) All the approaches based on the assumptions of Qusi-uniform liquid film, or uniform liquid molecular density, or uniform liquid molecular densities respectively inside and outside the interface layer cannot give the correct nano-contact angle, and it is difficult to improve them. Among these approaches, both the conclusions of nano-contact angle sure being 0 degrees and sure being 180 degrees are false. 2) Density functional theory (DFT) approach and Molecular Dynamics (MD) approach are capable to treat of nano-contact angle, however, the work is very heavy for using the DFT approach. 3) In 1995, Ruzeng Zhu (College Physic [Vol. 14 (2), p1-4 (in Chinese)], corrected the concept of contact angle in a earlier false theory for macro contact angle and obtained the most simple and convenient approximate formula of nano-contact angle alpha = (1 - 2 E-PS/E-PL) pi, where E-PL is the potential of a liquid molecule in the internal liquid and E-PS is the interact potential between a liquid molecule and the solid on which it locats. Both E-PS and E-PL can be obtained by MD, therefore this theory as a approximate simplified form belongs to Molecular Dynamics approach of nano-contact angle. The results of 0 degrees and 180 degrees for complete wetting and complete non-wetting given by this formula are correct under the assumption of incompressible fluid, therefore, this theory is worthy of further development. For this end, based on the physical analysis, we assume that the potential energy of a liquid molecule on the Gibss surface of tension outside the three-phase contact area is E-PL/2x and that of a liquid molecule on the three-phase contact line is (1 + kE(PS)/E-PL) alpha E-PL/2x pi, where x and k are optimal parameters. According to the condition that the potential energy is the same everywhere on the Gibss surface of tension, an improved approximate formula for nano-contact angle alpha = pi(1 - 2xE(PS)/E-PL)/(1 + kE(PS)/E-PL) is obtained. To obtain the value of x and k, MD simulations are carried on argon liquid cylinders placed on the solid surface under the temperature 90 K, by using the lennard - Jones (LJ) potentials for the interaction between liquid molecules and for that between a liquid molecule and a solid molecule with the variable coefficient of strength a. Eight values of a between 0.650 and 0.825 are used. The Gibss surfaces of tension are obtained by simulations and their bottom angles are treated as the approximate nano-contact angles. Combining these data with the physical conditions (when E-PS/E-PL = 0, alpha = pi), the optimized parameter values x = 0.7141, k = 1.6051 with the correlation coefficient 0.9997 are obtained by least square method. This correlation coefficient close enough to 1 indicates that for nano liquid solid contact system with different interaction strength, the parameter of optimization x and k really can be viewed as constants, so that our using MD simulation to determine of the optimized parameters is feasible and our approximate formula is of general applicability.