Scientific resources related to the connection between 3D structure and material properties in different application areas are listed in scientific-resources/#material-properties. Some examples from this list are:
- anisotropic and heterogeneous connectivity which can negatively affect the performance, durability and safety of re-chargeable batteries (Ebner et al., 2014; Harris & Lu, 2013; Kehrwald et al., 2011; Lu et al., 2020; Müller et al., 2018)
- low connectivity which can significantly slow down mass transport (Barman et al., 2021; Siegel, 2012)
- anisotropic pore shapes, resulting from compression, which can decrease the wettability of battery electrodes (Lee et al., 2014)
- connectivity of porous networks in rock which affects oil recovery and CO2 storage (Alhosani et al., 2020; Saif et al., 2017)
- the porosity profile and pore connectivity which influences the strength and durability of materials like cement (Bossa et al., 2015)
General geometrical characteristics that are considered important for connecting 3D structure with material properties are discussed in Tutorial, Part I, Anisotropy, heterogeneity and connectivity. Below is a summary of how the characterization methods in Mist can be used to connect 3D structure to specific material properties. Details about the specific characterization methods are found in the tutorial of each method, see Tutorial, Part II, Characterization methods and Tutorial, Advanced use.
A powerful structure/property formula
Examples of studies where statistical learning of structure-property relationships, with models fitted to large datasets, can be found in the references listed in scientific-resources/#structure-property-models. A simple but powerful formula from this list of references, which is used in Barman et al. (2019), Prifling et al. (2021) and Stenzel et al. (2017), is
![\[a\frac{\epsilon^b \beta^c}{\tau^d}\]](https://mist.math.chalmers.se/wp-content/ql-cache/quicklatex.com-40097a9227d50cf9b50117c4bfe977a9_l3.png "Rendered by QuickLaTeX.com")
where ε is the porosity (or porosity of connected pores), β is a bottleneck factor and τ is a measure of tortuosity. a, b, c and d are model parameters that can be fitted to data.
Predicts material properties with high accuracy
This formula has been shown to be able to predict diffusion and electric conductivity with high accuracy. The formula was found to explain around 90-99% of the variation in diffusion/electrical conductivity in the two studies Barman et al. (2019) and Stenzel et al. (2017), and performed almost as well as a general machine learning model (Stenzel et al., 2017).
Similar results were found in a recent large-scale simulation study, see Prifling et al. (2021). This study compares several structure-property formulas, for diffusion/electrical & thermal conductivity and for permeability.
Porosity, connectivity and bottleneck effects
It is well known that the porosity is an important for material properties such as mass transport or conductivity.
It is also well known that the connectivity, tortuosity τ, plays a key role. However, there are many definitions of τ. The geodesic tortuosity used here, and in Barman et al. (2019), Prifling et al. (2021) and Stenzel et al. (2017), is purely a geometric feature. In contrast to definitions such as the diffusive tortuosity, a geometric tortuosity allows us to separate effects otherwise lumped together in the tortuosity factor. The geodesic tortuosity used in Mist has been shown to have strong relationships with the materials properties. The mean geodesic tortuosity was included in all structure-property formulas in Prifling et al. (2021). The geodesic tortuosity on its own usually explains a large part of variation in material properties. More details are given below.
It has been known a long time that bottlenecks influence material properties. However, methods for measuring bottlenecks in 3D structures have only been introduced relatively recently (Barman et al., 2021, Holzer et al., 2013). These methods are all implemented in Mist, see below.
How to use Mist to compute the formula
- The porosity, alt. porosity of connected pores, ε, and mean geodesic tortuosity, τ, are returned as summary statistics from the geodesic tortuosity method.
- The bottleneck factor β can be measured in Mist using: standard deviation of geodesic tortuosity; geodesic channel strength; and constrictivity and accessible pore size.
Note that the constrictivity and geodesic tortuosity/geodesic channel strength are measuring two different types of bottleneck effects. The constrictivity and accessible pore size are related to bottlenecks caused by variations in pore size (see Holzer et al., 2013, Prifling et al., 2021, Stenzel et al., 2017). The standard deviation of geodesic tortuosity and the geodesic channel strength are related to bottleneck effects caused by poorly connected pores (path-bottlneecks, see Barman et al., 2019, Barman et al., 2021, Prifling et al., 2021 [note, Barman et al. (2019) and Prifling et al. (2021) do not talk explicitly about path-bottlenecks, but use the standard deviation of geodesic tortuosity in the best performing structure-property formulas]). See Tutorial, Part I, Specific connectivity relationships for a visual illustration of these two bottleneck types.
Other structure/property relationships
Prifling et al. (2021) include several structure/property formulas, fitted to a large set of 3D structures. Other resources include Nishiyama & Yokoyama (2017), which details how bottlenecks caused by variations in pore size, measured in Mist by the constrictivity and accessible pore size, correlate with permeability.
See Ebner et al. (2014) and Kehrwald et al. (2011) for examples of how anisotropy and heterogeneity in lithium-ion battery electrodes can negatively affect their performance, durability and safety. In Mist you can measure anisotropy in pore size using the pore size methods or compute connectivity characteristics such as the geodesic tortuosity in different directions. The negative effect of heterogeneity in connectivity–where you have some areas that are well-connected and other areas that are poorly connected–is in part that it leads to increased stress on the well-connected areas since there is a higher load of transport in these areas (Müller et al., 2018). These well-connected areas can be seen as path-bottlenecks, which can be captured in Mist using the geodesic methods as discussed above.
Structure/property relationships using Mist
Step 1, visual exploration
It usually helps to start by exploring the 3D structure interactively using the Mapping data-feature in Mist, as illustrated in the videos in Tutorial, Part I. This allows you to understand how the differents aspects of the geometry of the 3D structure are related, e.g., if heterogeneity in porosity causes bottleneck effects.
Using Mapping data, you can choose to only show voxels that have values within a certain range. With the pore size-methods, you can show only voxels with a large pore size, to better understand inhomogeneities in pore size. With the geodesic tortuosity, you can show only voxels that are well connected, or voxels that are poorly connected, to explore how the connectivity varies throughout the 3D structure. The videos in Pore size methods and Geodesic tortuosity show examples of how this can be done using one of the demo structures.
As these videos illustrate, heterogeneity and anisotropy in porosity and pore size is relatively easy to get a feel for by simply looking at the 3D structure. The connectivity of the pore network, on the other hand, is more difficult to observe by visual inspection of the 3D structure alone. Visual exploration using the connectivity methods in Mist (connected components, geodesic methods and intrusion porosimetry), is therefore especially helpful for understanding the connectivity properties of a material’s 3D structure.
Step 2, quantifying geometrical properties
With the knowledge given by the visual exploration, you can quantify the most relevant geometrical properties of the 3D structure using the available summary statistics of the methods in Mist. See the reference manual, or the tutorials for explanations of the summary statistics.
You can load several 3D structures into Mist, and run the characterization methods with the same settings on all structures. The resulting summary statistics are saved to files as shown in Tutorial, Part III, Gettings started with the GUI. You can also automate the process for multiple structures by running Mist in batch-mode, separate from the user interface, as described here.
Step 3, relating the geometrical characterization to material properties
Use the geometry characterization from Mist to understand your material’s properties, e.g. using the formula above, as is done in Barman et al. (2019), Prifling et al. (2021) and Stenzel et al. (2017). To compute material properties numerically, you can use, e.g., some of the software listed here.
Methods in Mist — tailored for finding structure/property relationships
Porosity, connectivity and bottleneck effects are arguably the most important aspects influencing material properties such as diffusion and electrical conductivity. Connectivity and bottleneck effects are themselves often influenced by material inhomogeneities. Also, the material properties can vary in different directions due to anisotropy. Therefore, the characterization methods in Mist are focused on both visual exploration and numerical quantification of these aspects.
Here are some notes on the specific implementation of some of the characterization methods in Mist, and why those were chosen over other existing methods:
- The geodesic tortuosity that is used in Mist captures poorly connected areas well since it computes the tortuosity for all connected pore voxels. This captures effects not only of the length of paths through the pore system, but also poorly connected pores such as dead-ends (see illustrations in Tutorial, Part I, Specific connectivity relationships). Using the full volume tortuosity leads to better predictions of material properties compared to the standard inlet, or geometric tortuosity, see Barman et al. (2019) and the discussion here.
- Bottleneck effects caused by poor connectivity, path-bottlenecks, are not captured by existing bottleneck methods. The standard deviation of the full volume tortuosity (returned by the geodesic tortuosity-method), and the new channel strength-method in Mist can be used to quantify this type of effect.
- The pore size-methods implemented in Mist are well suited both for characterizing anisotropy and for capturing the size and shape of irregular pores in an intuitive way. Alternatives to pore size are:
The common method of contact distributions, e.g., is closely related to pore size. Advantages with using contact distributions are that they require about half the amount of effort to compute, and it is easier to establish theoretical results for contact distributions for simple statistical pore models. However, the contact distribution puts more weight on small sizes compared to pore size distribution. Therefore, it is not as well suited for visual exploration, and the interpretation of the contact distribution is not as intuitive as the pore size distribution.
The chord length distribution is another common characteristic of pore size and shape. It is closely related to line pore sizes as implemented in Mist. The chord length distribution however does not contain any information on directionality. Therefore, it is not suited for quantifying anisotropy, in contrast to the line pore size.