Solomentsev Y, White LR
Department of Chemical Engineering, Carnegie-Mellon University, Pittsburgh, Pennsylvania, 15213-3890
J Colloid Interface Sci. 1999 Oct 1;218(1):122-136. doi: 10.1006/jcis.1999.6389.
The line tension associated with the three-phase contact line of a liquid drop on a rigid substrate is calculated as the excess free energy per unit length of the contact line relative to the free energy of the drop/substrate system when the interaction between the free liquid interface and the substrate is turned off. To perform this calculation, the interaction energy is treated in the Derjaguin approximation and the microscopic drop profile determined accordingly. An explicit expression for the line tension as a function of the macroscopic contact angle for the case of an infinite drop is obtained as a quadrature over the interaction energy per unit area between planar half-spaces. The magnitude of the line tension is shown to be the product of the free liquid surface tension and the length scale associated with the interaction energy. For attractive interaction energies the line tension is negative but when the interaction has both an attractive component (which must dominate at small separations) and a repulsive component which dominates at larger separations, the line tension can be positive. Some simple physical models of the interaction are considered to illustrate these points. For the case of a finite drop, we show that the concept of line tension as a macroscopic quantity which may be added to the free energy balance, giving rise to deviation from Young's equation for the contact angle, is a valid one in the sense that a full treatment of the microscopic drop shape yields the same macroscopic picture of the drop near contact. The line tension for a finite drop is demonstrated to be the value calculated for the infinite drop to within terms of the order of the ratio of interaction energy range to drop radius. Hence, the treatment of line tension as a macroscopic concept is valid down to drop sizes of the order of the range of the interaction energy. Finally, by applying the Derjaguin approximation in a different (but equally valid) way and deriving an alternate expression for the line tension, we are able to elucidate the range of validity of the approximation itself. Line tensions calculated by these methods would appear to be accurate for contact angles up to approximately 20 degrees. Copyright 1999 Academic Press.
与刚性基底上液滴的三相接触线相关的线张力，被计算为接触线每单位长度的过剩自由能，相对于自由液体界面与基底之间相互作用关闭时液滴/基底系统的自由能。为了进行此计算，相互作用能在德亚金近似中处理，并据此确定微观液滴轮廓。对于无限大液滴的情况，得到了线张力作为宏观接触角函数的显式表达式，它是平面半空间之间每单位面积相互作用能的积分。线张力的大小被证明是自由液体表面张力与与相互作用能相关的长度尺度的乘积。对于吸引性相互作用能，线张力为负，但当相互作用既有吸引分量（在小间距时必须占主导）又有排斥分量（在大间距时占主导）时，线张力可以为正。考虑了一些简单的相互作用物理模型来说明这些要点。对于有限液滴的情况，我们表明线张力作为一个宏观量的概念，可以添加到自由能平衡中，从而导致接触角偏离杨氏方程，从微观液滴形状的完整处理产生与接触附近液滴相同的宏观图像的意义上来说，这是一个有效的概念。有限液滴的线张力被证明在相互作用能范围与液滴半径之比的量级项内与无限大液滴计算的值相同。因此，将线张力作为宏观概念的处理对于与相互作用能范围量级相当的液滴尺寸是有效的。最后，通过以不同（但同样有效）的方式应用德亚金近似并推导出线张力的另一种表达式，我们能够阐明该近似本身的有效性范围。通过这些方法计算的线张力对于高达约20度的接触角似乎是准确的。版权所有1999年学术出版社。
