Microscopy-image analysis provides precious information on size and structure of colloidal aggregates and agglomerates. The structure of colloids is often characterized using the mass fractal dimension 𝑑𝑓$d_f$$d_f$, which is different from the two-dimensional (2D) fractal dimension 𝑑𝑝$d_p$$d_p$ that can be computed from microscopy images. In this work, we propose to use a recent morphological aggregation model to find a relationship between 2D image fractal dimension and 3D mass fractal dimension of aggregates and agglomerates. Our case study is represented by scanning transmission electron microscopy images of boehmite colloidal suspensions. The behaviour of the computed 𝑑𝑓$d_f$$d_f$ at different acid and base concentration shows a fair agreement with the results of small angle x-ray scattering and with the literature, enabling to use the 𝑑𝑓$d_f$$d_f$ versus 𝑑𝑝$d_p$$d_p$ relationship to study the impact of the composition of the colloidal suspension on the density of colloidal aggregates and agglomerates.

中文翻译：

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通过具有可变紧密度的聚合模型从二维显微镜图像中获取质量分形维数

显微图像分析提供了有关胶体聚集体和附聚物的大小和结构的宝贵信息。胶体的结构通常使用质量分形维数来表征𝑑𝑓$d_F$$d_f$，这与二维 (2D) 分形维数不同𝑑𝑝$d_p$$d_p$可以从显微镜图像中计算出来。在这项工作中，我们建议使用最近的形态聚合模型来寻找 2D 图像分形维数和 3D 质量分形维数之间的关系。我们的案例研究以勃姆石胶体悬浮液的扫描透射电子显微镜图像为代表。计算的行为𝑑𝑓$d_F$$d_f$在不同的酸碱浓度下，与小角 X 射线散射的结果和文献具有较好的一致性，能够使用𝑑𝑓$d_F$$d_f$相对𝑑𝑝$d_p$$d_p$研究胶体悬浮液的组成对胶体聚集体和附聚物密度的影响的关系。