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Expansion of the Classification Using Fractal Dimension for Assembly of Structural UnitsTable of Contents:
Many NSMs might have more complex structures and not necessarily in the crystalline state. In Chapters 59, the following nanostructured materials are treated.
Porous systems treated in Chapters 6 and 7 in case 1 mentioned above are starting from glassy systems and have both disorder and porosity, while systems such as gel and aggregate (Chapters 8 and 9) in case 2 show fractal shapes. Now a target of debates is disordered systems and not polycrystalline nor partially crystalized systems, although methods and concepts discussed in the book apply to both. In these systems, changes among related systems are rather gradual and may have complicated heterogeneous structures. In such cases, the dimensionality of assembly of structural units will be discussed by the fractal dimension [4, 5] rather than dimension of space. In the following sections, classifications will be expanded to fractal (and multifractal) and porous systems. Classification by Using Fractal DimensionsStructures of spontaneously formed gel obtained by full atomistic MD simulationsIn Fig. 2.7, an example of the structure of the gel obtained from the full atomistic MD simulations of silicananocolloidwaterNaCl systems [4] is shown for a slice. As shown in this figure, several components and networks formed complicated structures. Some characteristics of structures are picked up here. First, the silica part forms infinitive networks (with periodic boundary conditions of MD). Na and Cl partially form networks but are surrounded by networks of water. The water part and the silica part also interact. That is, the silica network has a complicated shape and the water part is surrounded by it. In Chapters 8 and 9, the structure will be characterized by several methods. Figure 2.7 An example of a nanostructure obtained by Molecular Dynamics Simulations. A slice (with a thickness of 10 A) of the structure of spontaneously formed gel from silicananocolloidwaterNaCl systems [4]. Details for this simulation and obtained structures will be shown in Chapters 8 and 9. Here the Hw and Ow mean H and О atoms in water. At first glance, the water part seems to be separated by the networks formed by Si0_{4} units in a gel, when it is shown by the slice; however, the water part is connected to the next slices, if we examine the structure in detail. As shown in this example, 3D connections of structures may be missing in the representations, by a slice or by projection to a plane. Therefore, as it will be discussed in Chapter 9, caution is required to treat fractal structures by 2D images (such as transmission electron microscopy (ТЕМ) images). How can the structure of the gel be classified? One of the possible choices for expansion of the classification will be achieved by introducing fractal dimension [9, 10], including the concept of multifractal [1113] for assembly of structural units. Concepts of fractal and/or multifractal and methods of fractal dimension analyses will be shown in Chapter 4. Monofractal and multifractal structures and classificationsThe units to form a fractal structure can be any of the ones in Fig. 2.3. The systems with matrixes in Fig. 2.6 can be also a target of such treatment. If the structure is monofractal, it is simply characterized as one fractal dimension, d_{{}. If the 1d structure formed the fractal structure with the fractal dimension 2.5, then the index such as 2.5d_{f}(ld) can be used. In the case of gel in [4], fractal dimension of the silica part for a short length scale was found to be 2.25, while that for the longer length scale was found to be 3.01, when we used the running coordination numbers of SiSi pair to calculate them (see Chapters 8 and 9). Existence of more than one exponent means the multifractality of the system (details of the explanation of this concept will be given in Chapter 4). In Fig. 2.8, the values are plotted against values for networks of the silica part of systems obtained by atomistic simulation of silicananocolloidwaterNaCl systems with several conditions. Systems that include clusters, aggregates, and gels are found in the silicananocolloidwaterNaCl systems of series I and II. Here, contents of salt change in the order of in each series. Series II systems are more waterrich than series I. II A is a reference system without salt. In IIA, several clusters are found, while aggregates are developed with the concentration of salt as shown in IIB and IIC. A large aggregate developed in IA. When the system IB was annealed at 400 K, gel was formed in IC. When value increases and exceeds a certain value, the gel will be formed. The value seems to be ~2.5, which is comparable to the threshold value for the 3D percolation (2.53 [29]), or a related one. In this example, systems can be classified for colloids, clusters, aggregates, and gels. That is, the value of d_{f}._{long }changes in the order of clusterof longranged structures is well represented by d_{f}._{long}, while the values for c/_{f}._{short} seem to show fluctuation related to the regularity of the porous parts of the pattern (lacunarity) [30]. Multifractality can be regarded as a kind of the presentation of the heterogeneity [1], and hence, the plot using both length scales can classify heterogeneity of systems. Generally, some characteristic regions can be defined on the plot, if there are enough number of data points. By using plots, one can classify each system by its region where it belongs to (see Chapter 9). Thus, mapping of related and other systems will be useful for classifications of complex systems as well as for clarifications of the mechanism of coagulation. Another possible way of classification can be based on multifractal analyses [1416]. Parameters of singularity spectra, /(a), can be used for such purpose. Examples of analyses as well as applications to other systems will be discussed in Chapter 4 for the case of porous systems. Characteristics of Porous MaterialsIn ref. [7], inverse nanostructures with cavity building units are not regarded as a separate one. It seems to be a reasonable choice to simplify the classification. However, porous materials are an important class of materials in many kinds of applications, and properties of such systems are different from the packed ones. Therefore, we consider it separately and then relations among these cases are discussed later. Structures of pores are closely related to the mechanical properties, electrical properties, solubility, and characteristics for storages and can be characterized by their shapes, connectivity, porosity, and lacunarity [30] and/or fractal dimensions. Porous materials can be subdivided into three categories by sizes of pores, set out by IUPAC [31]. That is, the pores with widths exceeding about 50 nm are called macropores; the pores of widths between 2 and 50 nm are called mesopores; and the pores with widths not exceeding about 2 nm are called micropores. Also, in the recommendation by IUPAC [32], authors can classify the pores according to their availability to an external fluid (closed pore, open pores, blind pores) and/or shapes (cylindrical, inkbottle shaped). In the report, the possibility to use fractal dimension analysis for the characterization of porous systems is also referred to. That is, characteristics of surface properties such as pore volume are considered to be proportional to (resolution of analysis)^{0}, where D is the fractal dimension of the surface for which the property is relevant. Porous materials gain the attention of researchers in many fields. For example, transport in the porous media is discussed by Adler and Thovert [33] and the importance of the fractal characters is discussed. The character seems to be necessary to clarify the effects of nanosizing on several properties. O'Neill and coworkers [34] have examined the porous materials as a candidate for lowdielectricconstant materials, which is useful for IC manufacturing at 45 nm features and beyond. In that work, MD simulations (in NPT conditions) were used to examine the relation between pore size and mechanical properties. In their simulations, the modulus decreases with porosity as ~(1  <^>)^{5}, where ip is the porosity of the system. This relation is more rapid in most oxides with macro porosity and faster than the upper bound of the HashinShtrikman theory [35, 36] as discussed by them. Hence, the classification of porous materials will be useful for systematic studies of the properties of them. Furthermore, the interconnectivity of pores is an important property in many applications of porous materials. In the percolation theory [29], the percolation threshold for twophase systems is known to be ~0.16, if we use the packing fraction (in volume) as an index. Thus, the connectivity is closely related to the porosity. Classification of Porous Materials Based on the Apparent Dimension of the StructuresIn each class of structure, systems with different porosity may exist or can be prepared. In this section, the classification of porous systems will be tried similarly as in Fig. 2.3. Figure 2.9 Examples of substructures of porous materials: Classification is by apparent dimension of the structures plotted against space dimension. Possible measures to characterize structures are porosity in each class, fractal dimensions and so on. The index can be like 3D(ld), for the example of the stacked pipes (The third case in the right column.). As shown in the next figure, dimension of the porous materials can be measured for both material and holes. Characteristics of apparent structures are analogous to the structure in Fig. 2.3 except for pores and therefore, the explanations in Fig. 2.3 for each position is applicable. These structures are comparable to the situations in Fig. 2.3. In a similar manner, dispersions of holes can be represented as in Fig. 2.6. In Fig. 2.9, examples of typical structures of porous materials are shown. In this figure, the classification is done now by apparent dimension (formed by both rigid and porous parts) of the structures plotted against the space dimension. The index can be determined like 3D(ld). Here d is the apparent dimension of the structure containing the porous part. As shown in Fig. 2.10, the merit of this indexing is the simplicity of the expression of the combination of dense and porous materials with distinguishing characteristics of each component. An example on the lefthand side of the figure is for stacked pipes with and without holes, which can be indexed as 3D(ld,ld). On the righthand side of the figure, the structure corresponds to a kind of coreshell particle [11] with pore and rigid parts and can be indexed as 0D(0d,0d). One can change the index order by size or by other properties. Figure 2.10 Examples (images) of possible combinations of porous and nonporous substructures; left: coexistence of rods and pipes, where only diameters are in nanosize; right: a nanoparticle in a nanopore. Using the classification of porous systems as in Figs. 2.3 and 2.9, both cases can be treated, qualitatively. The possible indexes for left and right cases are 3D(ld,ld) and 0D(0d,0d), respectively. Other possible methods for indexing of porous materials are discussed in the text. These situations are not the same as classification in ref. [7]. Both skeletons of fibers and nanotubes are classified into 3D1 in that reference based on the dimensionality of the material part surrounding the voids, while in our classification, the skeleton of fibers is indexed as 3D(3d) while nanotubes are indexed as 3D(ld), respectively. This distinction between the two cases seems to be useful to consider mechanical and related properties. Of course, one of the possible measures to characterize porous structures is density and/or porosity in each class as used in many works. For more quantitative treatment, the index such as 3D(ld( = a)) can represent stacked pipes, where is the porosity. In Chapters 6 and 7, porous silicate systems are treated simply as a function of density, since the density was found to be a function of porosity. In the case with porosity, the change of the shape of the system with introducing pores is not directly included. Naturally, the classification can include combinations of materials with different porosity. 
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