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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° and sure being 180° 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 α = (1-2EPS/EPL)π,where EPL is the potential of a liquid molecule in the internal liquid and EPS is the interact potential between a liquid molecule and the solid on which it locats. Both EPS and EPL 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° and 180° 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 EPL/2x and that of a liquid molecule on the three-phase contact line is (1+kEPS/EPL)α EPL/2xπ, 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 α = π(1-2xEPS/EPL)/(1+kEPS/EPL) 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 EPS/EPL=0, α = π), 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.
[1] Young T 1805 Phil. Trans. 95 84
[2] Jameson G J, del Cerro M C G 1976 J. Chem. Soc. Furaduy I. 72 883
[3] White L R 1977 J. Chem. Soc. Faraday Trans 1. 73 390
[4] Zhu R Z 2001 Advances In Mechanics 31 489 (in Chinese) [朱如曾 2001 力学进展 31 489]
[5] Zhu R Z 2004 Advances In Applied Mechanics (Beijing:Science Press) p223 (in Chinese) [朱如曾 2004 应用力学进展 (北京:科学出版社) 第223页]
[6] Solomentsev Y, White L R 1999 J. Colloid Interface Sci. 218 122
[7] de Gennes P G, Brochard-Wyart F, Quere D 2004 Capillarity and wetting phenomena:drops, bubbles, pearls waves. Springer-Verlag, New York
[8] Berim G O, Ruckenstein E 2004 J. Phys. Chem. B 108 19330
[9] Berim G O, Ruckenstein E 2004 J. Phys. Chem. B 108 19339
[10] Ruckenstein E, Berim G O 2010 Adv. Colloid Interface Sci. 157 1
[11] Berim G O, Ruckenstein E 2009 J. Chem. Phys. 130 044709
[12] Saville G 1977 J. Chem. Soc. Faraday Trans. 73 1122
[13] Sikkenk J H, Indekeu J O, Menu G 1988 J. Stat. Phys. 52 23
[14] Nijmeijer M J P, Bruin C, Bakker A F 1990 Phys. Rev. A 42 6052
[15] Matsumoto S, Maruyama S, Saruwatari H 1995 ASME/JSME Therm. Eng. Conf. 2 557
[16] Kimura T, Maruyama S 2002 Microscale Therm. Eng. 6 3
[17] Maruyama S, Matsumoto S, Ogita A1994 Therm. Sci. Eng. 2 77
[18] Maruyama S 2000 Adv in Numerical Heat Transfer (Vol.2) (New York:Taylor & Francis) pp189-226
[19] Maruyama S, Kimura T, Lu M C 2002 Thermal Science & Engineering 6 23
[20] Sinha S 2004 Ph.D. Dissertation (University of California, Los Angeles)
[21] Shi B 2006 Ph.D. Dissertation (University of California, Los Angeles)
[22] Maruyama S, Kurshige T, Matsumoto S, Yamaguchi Y, Kimura T 1998 Microscale Thermophysi. Eng. 2 49
[23] Zhu R Z 1995 College Physic 14 1 (in Chinese) [朱如曾 1995 大学物理 14 1]
[24] Li P 1987 Thermology (Beijing:Beijing Normal University Press) p340 (in Chinese) [李平 1987 热学(北京:北京师范大学出版社科学出版社) 第340页]
[25] Gibbs J W 1928 Collected Works (New York:Longmans Green and Company Press) p219
[26] Cui S W, Wei J A, Wang X S, Xu S H, Sun Z W, Zhu R Z 2015 J. Comput. Theor. Nanosci. 12 189
[27] Adamson A W 1984 Physical Chemistry of Surfaces (Beijing:Science Press)
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[1] Young T 1805 Phil. Trans. 95 84
[2] Jameson G J, del Cerro M C G 1976 J. Chem. Soc. Furaduy I. 72 883
[3] White L R 1977 J. Chem. Soc. Faraday Trans 1. 73 390
[4] Zhu R Z 2001 Advances In Mechanics 31 489 (in Chinese) [朱如曾 2001 力学进展 31 489]
[5] Zhu R Z 2004 Advances In Applied Mechanics (Beijing:Science Press) p223 (in Chinese) [朱如曾 2004 应用力学进展 (北京:科学出版社) 第223页]
[6] Solomentsev Y, White L R 1999 J. Colloid Interface Sci. 218 122
[7] de Gennes P G, Brochard-Wyart F, Quere D 2004 Capillarity and wetting phenomena:drops, bubbles, pearls waves. Springer-Verlag, New York
[8] Berim G O, Ruckenstein E 2004 J. Phys. Chem. B 108 19330
[9] Berim G O, Ruckenstein E 2004 J. Phys. Chem. B 108 19339
[10] Ruckenstein E, Berim G O 2010 Adv. Colloid Interface Sci. 157 1
[11] Berim G O, Ruckenstein E 2009 J. Chem. Phys. 130 044709
[12] Saville G 1977 J. Chem. Soc. Faraday Trans. 73 1122
[13] Sikkenk J H, Indekeu J O, Menu G 1988 J. Stat. Phys. 52 23
[14] Nijmeijer M J P, Bruin C, Bakker A F 1990 Phys. Rev. A 42 6052
[15] Matsumoto S, Maruyama S, Saruwatari H 1995 ASME/JSME Therm. Eng. Conf. 2 557
[16] Kimura T, Maruyama S 2002 Microscale Therm. Eng. 6 3
[17] Maruyama S, Matsumoto S, Ogita A1994 Therm. Sci. Eng. 2 77
[18] Maruyama S 2000 Adv in Numerical Heat Transfer (Vol.2) (New York:Taylor & Francis) pp189-226
[19] Maruyama S, Kimura T, Lu M C 2002 Thermal Science & Engineering 6 23
[20] Sinha S 2004 Ph.D. Dissertation (University of California, Los Angeles)
[21] Shi B 2006 Ph.D. Dissertation (University of California, Los Angeles)
[22] Maruyama S, Kurshige T, Matsumoto S, Yamaguchi Y, Kimura T 1998 Microscale Thermophysi. Eng. 2 49
[23] Zhu R Z 1995 College Physic 14 1 (in Chinese) [朱如曾 1995 大学物理 14 1]
[24] Li P 1987 Thermology (Beijing:Beijing Normal University Press) p340 (in Chinese) [李平 1987 热学(北京:北京师范大学出版社科学出版社) 第340页]
[25] Gibbs J W 1928 Collected Works (New York:Longmans Green and Company Press) p219
[26] Cui S W, Wei J A, Wang X S, Xu S H, Sun Z W, Zhu R Z 2015 J. Comput. Theor. Nanosci. 12 189
[27] Adamson A W 1984 Physical Chemistry of Surfaces (Beijing:Science Press)
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