Bullard Jeffrey W, Garboczi Edward J
Materials and Construction Research Division, National Institute of Standards and Technology, Gaithersburg, Maryland 20899-8615, USA.
Phys Rev E Stat Nonlin Soft Matter Phys. 2009 Jan;79(1 Pt 1):011604. doi: 10.1103/PhysRevE.79.011604. Epub 2009 Jan 14.
Minimization of free energy is used to calculate the equilibrium vertical rise and meniscus shape of a liquid column between two closely spaced, parallel planar surfaces that are inert and immobile. States of minimum free energy are found using standard variational principles, which lead not only to an Euler-Lagrange differential equation for the meniscus shape and elevation, but also to the boundary conditions at the three-phase junction where the liquid meniscus intersects the solid walls. The analysis shows that the classical Young-Dupré equation for the thermodynamic contact angle is valid at the three-phase junction, as already shown for sessile drops with or without the influence of a gravitational field. Integration of the Euler-Lagrange equation shows that a generalized Laplace-Young (LY) equation first proposed by O'Brien, Craig, and Peyton [J. Colloid Interface Sci. 26, 500 (1968)] gives an exact prediction of the mean elevation of the meniscus at any wall separation, whereas the classical LY equation for the elevation of the midpoint of the meniscus is accurate only when the separation approaches zero or infinity. When both walls are identical, the meniscus is symmetric about the midpoint, and the midpoint elevation is a more traditional and convenient measure of capillary rise than the mean elevation. Therefore, for this symmetric system a different equation is fitted to numerical predictions of the midpoint elevation and is shown to give excellent agreement for contact angles between 15 degrees and 160 degrees and wall separations up to 30mm . When the walls have dissimilar surface properties, the meniscus generally assumes an asymmetric shape, and significant elevation of the liquid column can occur even when one of the walls has a contact angle significantly greater than 90 degrees . The height of the capillary rise depends on the spacing between the walls and also on the difference in contact angles at the two surfaces. When the contact angle at one wall is greater than 90 degrees but the contact angle at the other wall is less than 90 degrees , the meniscus can have an inflection point separating a region of positive curvature from a region of negative curvature, the inflection point being pinned at zero height. However, this condition arises only when the spacing between the walls exceeds a threshold value that depends on the difference in contact angles.
自由能最小化用于计算在两个紧密间隔、平行的惰性且固定不动的平面表面之间的液柱的平衡垂直上升高度和弯月面形状。利用标准变分原理找到自由能最小的状态，这不仅会导出一个关于弯月面形状和高度的欧拉 - 拉格朗日微分方程，还会导出液弯月面与固体壁相交处三相交界处的边界条件。分析表明，经典的用于热力学接触角的杨 - 杜普雷方程在三相交界处是有效的，这在有或没有重力场影响的 sessile 液滴的情况下已经得到证明。欧拉 - 拉格朗日方程的积分表明，由奥布赖恩、克雷格和佩顿 [《胶体与界面科学杂志》26, 500 (1968)] 首次提出的广义拉普拉斯 - 杨（LY）方程能够精确预测在任何壁间距下弯月面的平均高度，而经典的关于弯月面中点高度的 LY 方程仅在间距接近零或无穷大时才准确。当两壁相同时，弯月面对称于中点，并且中点高度比平均高度是更传统且方便的毛细上升量度。因此，对于这个对称系统，一个不同的方程被拟合到中点高度的数值预测上，并显示在接触角介于 15 度和 160 度之间且壁间距高达 30 毫米时能给出极好的一致性。当壁具有不同的表面性质时，弯月面通常呈现不对称形状，并且即使其中一个壁的接触角显著大于 90 度，液柱也可能出现显著升高。毛细上升的高度取决于壁之间的间距，也取决于两个表面上接触角的差异。当一个壁上的接触角大于 90 度而另一个壁上的接触角小于 90 度时，弯月面可能有一个拐点，将正曲率区域与负曲率区域分开，该拐点固定在零高度处。然而，这种情况仅在壁之间的间距超过一个取决于接触角差异的阈值时才会出现。
