Davidson School of Chemical Engineering, Purdue University , 480 Stadium Mall Drive, West Lafayette, Indiana 47907-2100, United States.
Langmuir. 2017 Jan 24;33(3):714-725. doi: 10.1021/acs.langmuir.6b04063. Epub 2017 Jan 12.
In order to minimize the effects of surface roughness and deformation, a new method for estimating the Hamaker constant, A, of solids using the approach-to-contact regime of an atomic force microscope (AFM) is presented. First, a previous "jump-into-contact" quasi-static method for determining A from AFM measurements is analyzed and then extended to include various AFM tip-surface force models of interest. Then, to test the efficacy of the "jump-into-contact" method, a dynamic model of the AFM tip motion is developed. For finite AFM cantilever-surface approach speeds, a true "jump" point, or limit of stability, is found not to appear, and the quasi-static model fails to represent the dynamic tip behavior at close tip-surface separations. Hence, a new "quasi-dynamic" method for estimating A is proposed that uses the dynamically well-defined deflection at which the tip and surface first come into contact, d, instead of the dynamically ill-defined "jump" point. With the new method, an apparent Hamaker constant, A, is calculated from d and a corresponding quasi-static-based equation. Since A depends on the cantilever's approach speed, v, and the AFM's sampling resolution, δ, a double extrapolation procedure is used to determine A in the quasi-static (v → 0) and continuous sampling (δ → 0) limits, thereby recovering the "true" value of A. The accuracy of the new method is validated using simulated AFM data. To enable the experimental implementation of this method, a new dimensionless parameter τ is introduced to guide cantilever selection and the AFM operating conditions. The value of τ quantifies how close a given cantilever is to its quasi-static limit for a chosen cantilever-surface approach speed. For sufficiently small values of τ (i.e., a cantilever that effectively behaves "quasi-statically"), simulated data indicate that A will be within ∼3% or less of the inputted value of the Hamaker constant. This implies that Hamaker constants can be reliably estimated using a single measurement taken with an appropriately chosen cantilever and a slow, yet practical, approach speed (with no extrapolation required). This result is confirmed by the very good agreement found between the experimental AFM results obtained using this new method and previously reported predictions of A for amorphous silica, polystyrene, and α-AlO substrates obtained using the Lifshitz method.
为了最小化表面粗糙度和变形的影响，提出了一种使用原子力显微镜（AFM）接触接近阶段来估计固体 Hamaker 常数 A 的新方法。首先，分析了以前用于从 AFM 测量中确定 A 的“跳入接触”准静态方法，然后将其扩展到包括各种感兴趣的 AFM 针尖-表面力模型。然后，为了测试“跳入接触”方法的有效性，开发了 AFM 针尖运动的动力学模型。对于有限的 AFM 悬臂-表面接近速度，发现不存在真正的“跳跃”点或稳定性极限，准静态模型无法在接近针尖-表面分离时代表动态针尖行为。因此，提出了一种新的“准动态”方法来估计 A，该方法使用针尖和表面首次接触时动态定义良好的挠度 d，而不是动态定义不明确的“跳跃”点。使用新方法，从 d 和相应的准静态方程计算出表观 Hamaker 常数 A。由于 A 取决于悬臂的接近速度 v 和 AFM 的采样分辨率 δ，因此使用双外推程序来确定准静态（v→0）和连续采样（δ→0）极限中的 A，从而恢复 A 的“真实”值。使用模拟 AFM 数据验证了新方法的准确性。为了使该方法在实验中得以实施，引入了一个新的无量纲参数 τ 来指导悬臂的选择和 AFM 的操作条件。τ 的值量化了给定悬臂在选择的悬臂-表面接近速度下接近其准静态极限的程度。对于足够小的 τ 值（即，有效地表现出“准静态”的悬臂），模拟数据表明 A 将在输入的 Hamaker 常数值的 3%或更小范围内。这意味着可以使用适当选择的悬臂和缓慢但实际的接近速度（无需外推）进行单次测量来可靠地估计 Hamaker 常数。这一结果得到了很好的验证，使用这种新方法获得的实验 AFM 结果与使用 Lifshitz 方法获得的无定形二氧化硅、聚苯乙烯和α-AlO 衬底的 A 的先前报告预测非常吻合。
