Bandstra Joel Z, Brantley Susan L
Department of Math, Engineering, and Computer Science, Saint Francis University, P.O. Box 600, Loretto, Pennsylvania 15541, USA.
Earth and Environmental Systems Institute, and Department of Geosciences, Pennsylvania State University, 2217 EES Building, University Park, Pennsylvania 16802, USA.
Phys Rev E Stat Nonlin Soft Matter Phys. 2015 Dec;92(6):062114. doi: 10.1103/PhysRevE.92.062114. Epub 2015 Dec 8.
The topography of a reactive surface contains information about the reactions that form or modify the surface and, therefore, it should be possible to characterize reactivity using topography parameters such as surface area, roughness, or fractal dimension. As a test of this idea, we consider a two-dimensional (2D) lattice model for crystal dissolution and examine a suite of topography parameters to determine which may be useful for predicting rates and mechanisms of dissolution. The model is based on the assumption that the reactivity of a surface site decreases with the number of nearest neighbors. We show that the steady-state surface topography in our model system is a function of, at most, two variables: the ratio of the rate of loss of sites with two neighbors versus three neighbors (d(2)/d(3)) and the ratio of the rate of loss of sites with one neighbor versus three neighbors (d(1)/d(3)). This means that relative rates can be determined from two parameters characterizing the topography of a surface provided that the two parameters are independent of one another. It also means that absolute rates cannot be determined from measurements of surface topography alone. To identify independent sets of topography parameters, we simulated surfaces from a broad range of d(1)/d(3) and d(2)/d(3) and computed a suite of common topography parameters for each surface. Our results indicate that the fractal dimension D and the average spacing between steps, E[s], can serve to uniquely determine d(1)/d(3) and d(2)/d(3) provided that sufficiently strong correlations exist between the steps. Sufficiently strong correlations exist in our model system when D>1.5 (which corresponds to D>2.5 for real 3D reactive surfaces). When steps are uncorrelated, surface topography becomes independent of step retreat rate and D is equal to 1.5. Under these conditions, measures of surface topography are not independent and any single topography parameter contains all of the available mechanistic information about the surface. Our results also indicate that root-mean-square roughness cannot be used to reliably characterize the surface topography of fractal surfaces because it is an inherently noisy parameter for such surfaces with the scale of the noise being independent of length scale.
反应性表面的形貌包含有关形成或修饰该表面的反应的信息，因此，应该可以使用诸如表面积、粗糙度或分形维数等形貌参数来表征反应活性。作为对这一想法的检验，我们考虑一个用于晶体溶解的二维（2D）晶格模型，并研究一组形貌参数，以确定哪些参数可能有助于预测溶解速率和机制。该模型基于这样的假设：表面位点的反应活性随最近邻数量的增加而降低。我们表明，在我们的模型系统中，稳态表面形貌最多是两个变量的函数：具有两个邻居的位点与具有三个邻居的位点的损失速率之比（d(2)/d(3)）以及具有一个邻居的位点与具有三个邻居的位点的损失速率之比（d(1)/d(3)）。这意味着，只要这两个参数相互独立，就可以从表征表面形貌的两个参数中确定相对速率。这也意味着仅通过表面形貌测量无法确定绝对速率。为了确定独立的形貌参数集，我们模拟了一系列d(1)/d(3)和d(2)/d(3)的表面，并为每个表面计算了一组常见的形貌参数。我们的结果表明，只要台阶之间存在足够强的相关性，分形维数D和台阶之间的平均间距E[s]就可以唯一地确定d(1)/d(3)和d(2)/d(3)。当D>1.5时（对于实际的三维反应性表面，这对应于D>2.5），我们的模型系统中存在足够强的相关性。当台阶不相关时，表面形貌变得与台阶后退速率无关，且D等于1.5。在这些条件下，表面形貌的测量值不是独立的，任何单个形貌参数都包含有关表面的所有可用机制信息。我们的结果还表明，均方根粗糙度不能可靠地表征分形表面的表面形貌，因为对于此类表面，它本质上是一个噪声参数，噪声的尺度与长度尺度无关。
