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## The method for determining nano-contact angle

Cui Shu-Wen, Zhu Ru-Zeng, Wei Jiu-An, Wang Xiao-Song, Yang Hong-Xiu, Xu Sheng-Hua, Sun Zhi-Wei
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• #### 摘要

对纳观接触角的确定曾有过许多研究工作, 本文对各种理论进行分析评论, 指出其各自的优缺点甚至错误, 认为最为简单实用的理论是朱如曾于1995年在《大学物理》((Vol. 14(2))) 的文章中对前人的宏观接触角的错误理论采用澄清接触角概念的方法所得到的纳观接触角的近似理论及近似公式α = (1-2EPS/EPL)π (其中EPL和EPS分别表示液体内部一个液体分子的势能和固体表面一个液态分子与固体的相互作用势能, 并可用分子动力学(MD) 模拟得到), 此理论属于纳观接触角的分子动力学理论的近似简化形式, 值得进一步发展. 为此, 本文根据物理分析假设Gibbs张力表面上位于非三相接触区的一个液体分子的势能为EPL/2x, 三相接触线上一个液体分子与其余液体的相互作用势能为(1+kEPS/EPL)α EPL/2xπ, 其中x和k 为优化参数. 根据Gibbs分界面上处处势能相等条件, 得到改进的纳观接触角的近似公式α = π({1-2xEPS/EPL)/(1+kEPS/EPL)．对固体表面的氩纳米液柱, 在温度90K下对液体分子之间采用林纳德－琼斯(L-J) 势, 液体分子与固体原子间采用带有可变强度参数a的 L-J 势, 对0.650a a值进行了MD模拟．得到了相应的Gibbs 张力面．将其纳观底角视为近似纳观接触角, 结合物理条件(当EPS/EPL=0时, α = π)用最小二乘法得到优化参数值x=0.7141, k=1.6051和相关系数0.9997. 这一充分接近于1的相关系数表明, 对于不同相互作用强度的纳米液固接触系统, 优化参数x和k确实可近似视为常数, 由此确认我们提出的利用MD模拟来确定纳观接触角近似公式中优化参数的可行性和该近似公式的一般适用性.

#### Abstract

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.

#### 作者及机构信息

###### 1. 沧州师范学院物理与电子信息系, 沧州 061001; 2. 中国科学院力学研究所非线性力学国家重点实验室, 北京 100190; 3. Silfex, a Division of Lam Research, 950 South Franklin Street, Eaton, Ohio, 45320, America; 4. 河南理工大学机械与动力学院, 焦作 454003; 5. 沧州师范学院图书馆, 沧州 061001; 6. 中国科学院力学研究所微重力国家实验室, 北京 100190
• 基金项目: 国家自然科学基金(批准号:11072242)、河南省教育厅科学技术研究重点项目(批准号:15A130001)和河南理工大学博士基金(批准号:72515-466)资助的课题.

#### Authors and contacts

###### 1. Department of Physics and Electronic Information, Normal University, Cangzhou 061001, China; 2. State Key Laboratory of Nonlinear Mechanics, Institute of Mechanics, Chinese Academy of Sciences, Beijing 100190, China; 3. Silfex, a Division of Lam Research, 950 South Franklin Street, Eaton, Ohio, 45320, America; 4. Institute of Mechanical and Power Engineering Henan Polytechnic Univeraity, Jiaozuo 454003, China; 5. CangZhou Normal University Library, Cangzhou Normal University, Cangzhou 061001, China; 6. Key Laboratory of Microgravity, Institute of Mechanics, Chinese Academy of Science, Beijing 100190, China
• Funds: Project supported by the National Natural Science Foundation of China (Grant No. 11072242), the Key Project of Science of the Education Bureau of Henan Province (Grant No. 15A130001), and the Doctor Research Foundation of Henan Polytechnic University (Grant No. 72515-466).

#### 参考文献

 [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)

#### 施引文献

•  [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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##### 出版历程
• 收稿日期:  2014-10-09
• 修回日期:  2015-03-03
• 刊出日期:  2015-06-05

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