Dai Bing, Leal L Gary, Redondo Antonio
Department of Chemical Engineering, University of California, Santa Barbara, Santa Barbara, California 93106, USA.
Phys Rev E Stat Nonlin Soft Matter Phys. 2008 Dec;78(6 Pt 1):061602. doi: 10.1103/PhysRevE.78.061602. Epub 2008 Dec 8.
The effect of the attractive forces originating from van der Waals interactions on the dynamics of thin films (<or= approximately 100 nm) is often approximated in fluid dynamics as the disjoining pressure between two unbounded parallel interfaces. However, it is known that this concept of the disjoining pressure, as a force per unit area between parallel interfaces, cannot generally be extended to films of nonuniform thickness. In this paper, we derive a formula for the disjoining pressure for a film of nonuniform thickness by minimizing the total Helmholtz free energy for a thin film residing on a solid substrate. Comparing to the augmented Young-Laplace equation, the disjoining pressure for a thin film of small slope on a flat substrate is shown to take the form Pi=-A_{123}(4-3h_{x};{2}+3hh_{xx})/24pih;{3} , where A123 is the Hamaker constant for phases 1 and 3 interacting through phase 2; h , h_{x} , and h_{xx} are the local film thickness, slope and second order derivative, respectively. For the limiting case of parallel interfaces (e.g., h_{x}=h_{xx} identical with 0 ), the disjoining pressure reduces to Pi=-A_{123}/6pih;{3} in agreement with the classical Lifshitz expression for the van der Waals force. The derivation can be readily extended to more general nonuniform films by constructing tangential planes at both interfaces of the films. Because of steric effects that prevent molecules from overlapping each other, the molecular size cannot be neglected when applying the mesoscopic concept of the disjoining pressure to films of thickness comparable to molecular scales.
源自范德华相互作用的吸引力对薄膜（厚度≤约100纳米）动力学的影响，在流体动力学中通常近似为两个无界平行界面之间的分离压力。然而，众所周知，这种作为平行界面之间单位面积力的分离压力概念，一般不能扩展到厚度不均匀的薄膜。在本文中，我们通过使位于固体基质上的薄膜的总亥姆霍兹自由能最小化，推导出了厚度不均匀薄膜的分离压力公式。与扩展的杨-拉普拉斯方程相比，在平坦基质上具有小斜率的薄膜的分离压力形式为Pi = -A₁₂₃(4 - 3hₓ² + 3hhₓₓ)/24πh³，其中A₁₂₃是通过相2相互作用的相1和相3的哈马克常数；h、hₓ和hₓₓ分别是局部薄膜厚度、斜率和二阶导数。对于平行界面的极限情况（例如，hₓ = hₓₓ = 0），分离压力简化为Pi = -A₁₂₃/6πh³，这与范德华力的经典 Lifshitz 表达式一致。通过在薄膜的两个界面处构建切平面，该推导可以很容易地扩展到更一般的非均匀薄膜。由于空间位阻效应阻止分子相互重叠，在将分离压力的介观概念应用于厚度与分子尺度相当的薄膜时，分子大小不能被忽略。
