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# Relation between Fractal Dimension of Holes and Porosity

Next, we will consider the classification including porous materials from a different point of view. When the shapes of pores change with porosity, fractal shapes might be found. Previously, the fractal dimension of the silica part in the gel was used for the classification of materials. A similar treatment is possible for the water part as well as the salt part, if necessary. If one removes the water part from the structures of colloidal solutions after gelation, a porous structure is obtained. Thus, the structure of the pores will be comparable to that of the water part. A similar argument holds for ion channels in ionically conducting glasses. For instance, as shown in Fig. 2.11, the shape covered by the trajectories of Li ions in the Li2Si03 glass is comparable to the ion channel formed by networks of Si04 units. Thus, it is quite natural that the structures of holes correspond to those of filled materials within the holes. This means the fractal dimension analysis for pores, cavity, and holes is applicable reasonably. Now the definition of the dimension is expanded to include fractal dimensions of assembly of structural units.

In general, a packing factor can be defined as the fraction of materials in a container with a total volume of Vtotai, that is Vm/VtoUX. For example, the atomic packing factor in the crystal structure is the fraction of volume in a unit cell. On the other hand, porosity is usually defined as a fraction of pores or voids in the material as already mentioned, that is, it can be defined as Vp0re/^totai- Thus, porosity is a counterpart of the packing factor. In this case, we can define two kinds of fractal dimensions even in a simple system. That is, one is for the structure of material and the other is the structure of holes (pores, channels).

Figure 2.11 Structure of ion channel visualized by accumulated positions of three Li ions in lithium metasilicate glass obtained by MD simulations during 1 ns run at 700 K. Each ion has a different color. The shapes of the paths of ions correspond to the channels formed by the network of Si04 units. Zig-zag shapes of ion channel are clearly found by the trajectories of ions. As in a similar manner, holes in the gel can be visualized as the water part filled in gels (see Chapter 9).

In Fig. 2.12, the situation of different packing factors (or fractions) is schematically compared with different porosities in the system. As found in Fig. 2.12a, if the content of material increases, the packing factor increases and fractal dimension of the material will also increase. One can also consider the situation with different contents of pores. An opposite but a quite similar situation can be seen for the increase of pores, cavities, channels (hereafter we use the word "holes" to include these different shapes) in the system as shown in Fig. 2.12b. With increase of the porosity of the system, the content of the holes increases. With this change, fractal dimension of holes will increase. Yu and Li [37] have examined the fractal character of the porous media, and it was shown that the fractal dimension Df is a function of porosity,

and the ratio of the lower limit to the upper limit of self-similar regions for Sierpinski carpets. That is,

Figure 2.12 Concept of negative dimension: Relation between the packing factor and porosity in 3D space, (a) In a real space, with an increase of packing factor of contents, particles tend to form larger units, clusters. With this change, the fractal dimension of materials also increases, (b) Opposite but a quite similar situation can be seen for the increase of holes (pores, cavities, channels) in the system. With increase of porosity, fractal dimension of the empty part increases. If one considered the filled cubic as the standard point, increases of fractal dimension of holes can be regarded as the increase of negative dimension. Another possible method for the classification of porous materials is thus by "fractal dimension of holes" or by using "negative dimension." Note that this concept is different from that used in Fig. 2.9.

where 2 < Df < 3. They argued that these relations hold for a disordered system having statistically fractal nature. Since the porosity is the counterpart of the packing factor, comparable equations should hold for the relation between fractal dimension of material and the packing fraction of it.

2.4.1: Concept of the Negative Dimension

If one uses "fractal dimension of holes” instead of "fractal dimension of the structure formed by substances,” the dimension might be regarded as the negative dimension [38]. In other words, if we considered the filled cubic by a matrix as the standard point in Fig. 2.12, "fractal dimension of holes” can be regarded as the negative dimension. Let's consider the situation of carbon nanotubes mentioned before. It can be classified by the dimension of the hole, if one would like to emphasize the porous part of the structures and it might be indexed as 3D(-md) or -mdf. Here "-m” means the fractal dimension of holes.

# Different Kinds of Classifications

These classifications and definitions of units suggested in Sections 2.2 and 2.3 seem to be general but still are not definitive.

For example, in the case of magnetic properties, Leslie-Pelecky and Rieke [39] have proposed a classification shown in Fig. 2.13. Here the classification was designed to emphasize the physical mechanisms responsible for the magnetic behavior.

Properties of systems can also be treated as functions of other characteristics of the systems.

Core-shell particles seem to be an important class of NSM, which consists of two or more materials with the core part and surrounding part. In ref. [11], core-shell-type particles are classified by using different criteria. One method is based on the combinations of organic and inorganic materials for constituents. That is, four classes, (i) inorganic/inorganic, (ii) inorganic/organic, (iii) organic/inorganic, and (iv) organic/organic can be defined.

Another classification found in that reference of core/shell nanoparticles is as follows:

(a) spherical core/shell nanoparticles; (b) hexagonal core/ shell nanoparticles; (c) multiple small-core materials coated by single-shell material; (d) nanomatryoshka material; (e) movable core within hollow shell material.

Within the second classification, (a)-(c) can be indexed by our scheme, with the addition of subindexes for shapes and/or numbers of elements. Interesting structure found there is (d) nanomatryushka material. The hierarchy of structures can be represented by the hierarchy of parentheses in our indexing like 0D(0d(0d(0d))). The structure (e) seems to correspond to the right-hand side of Fig. 2.10.

Figure 2.13 Schematic representation of the different types of magnetic nanostructures. Type-А materials include the ideal ultrafine particle system, with interparticle spacing large enough to approximate the particles as noninteracting. Ferrofluids, in which magnetic particles are surrounded by a surfactant preventing interactions, are a subgroup of Type-А materials. Type-В materials are ultrafine particles with a core-shell morphology. Type-C nanocomposites are composed of small magnetic particles embedded in a chemically dissimilar matrix. The matrix may or may not be magnetoactive. Type-D materials consist of small crystallites dispersed in a nanocrystalline matrix. The nanostructure may be two-phase, in which nanocrystallites are a distinct phase from the matrix, or the ideal case, in which both the crystallites and the matrix are made of the same material. Reprinted with permission from Leslie-Pelecky, D. L., Rieke, R. D. (1996), Magnetic properties of nanostructured materials, Chem. Mater., 8, pp. 1770-1783. Copyright (1996) American Chemical Society.

# Common Possible Measures of Nanostructured Materials

## Effects of Nano-Sizing

Some fundamental effects of the nano-sizing considered in this chapter are as follows:

1. Size effects: There are several kinds of size effects so far argued.

In ref. [6], Gleiter argued that size effect is observed when the size of the system is comparable with the characteristic lengths of the phenomena. Here characteristic lengths may depend on the property to be discussed. Namely, it can be de Broglie wave (in the case of quantum dots), correlation length, length among sites, etc.

The size effects of nanoparticles tend to be argued with extremely large surface areas. Some of the size effects related to the surface areas/regions will be argued later.

Besides such effects, finite-size effects are known for electrostatic calculation as well as molecular dynamics simulations. Finite system size effects in molecular dynamics simulations are caused by the periodicity of MD cells, imposed by periodic boundary conditions.

One should not forget that if the system size is reduced, the length scale found in the system is cut or modified. That is, the structure itself is modified by the system size. Therefore, a comparison of system size with the characteristic length of the phenomena is not necessarily enough to discuss the size effects. Some methods to measure length scales from MD data will be introduced later.

• 2. Change of the dimensionality of the system: The change of dimension in a nanosystem is concerned with the dimension of the structural units as well as the formation of larger structures such as clusters, aggregates, and gels. Therefore, this concept includes fractals and multifractals [14-16]. That is, the complexity of the structures containing more than one exponent is considered. Examples for gel and related systems will be shown. Several methods for measuring fractal dimensions for both the structure and dynamics of systems based on the MD data will be explained in Chapter 4. The dimensionality of the system has been already used as the basis of the classification mentioned in the previous sections because the properties of nanostructured materials are affected by their dimensionality.
• 3. Atomic structure near the surface: If a nanostructure is introduced, local structures tend to be modified. The coordination number of atoms near the surface as well as shapes of coordination polyhedra must be changed. If the medium- and long-range interactions are cut, the forces acting on the surface atoms are also different from the bulk. These changes are caused by nano-sizing.

Some of these effects will be discussed in the following subsections. Other factors considered in nanoionics will be discussed in Chapters 3, 6, and 7.

## Effects Related to the Surface Areas or Surface Regions and Possible Measures of Characteristics of Nano-Sizing

Let's consider the situation of one particle with different sizes. The ratio of the surface area and volume may be used as a measure to characterize nanoparticles. If we consider the case of cubic particles, the ratio is always 6 P/I3 = 6/1, where / is the side length. That is, the ratio is simply inversely proportional to the size of each cubic particle. An important problem to be considered here seems to be the thickness of the surface region. If it does not change with the particle size, the ratio of the surface region becomes larger and larger with nano-sizing. That is, in nano-sized particles, almost parts of the system have the characteristics as surfaces. If the thickness is 1 nm, it can be negligibly small compared with a large particle of >100 nm size, while in the particle of 10 nm size, the width cannot be negligible. Therefore, one needs to examine how large this thickness is and how it changes with size. Thus, one of the possible measures can be the length scale of the surface structure.

On the other hand, with nano-sizing, total surface areas and regions change drastically when density and total volume of the system are kept unchanged. For example, if one considers the cubic systems with side length /L = 1 nm and 12= 100 nm, with the same total volume l = 1,000,000 (nm)3, the total surface areas are 6,000,000 and 600 nm2, respectively. That is, it increases with the number of partitions (with the power law with the exponent of 3). Therefore, if the phenomena depended on the surface areas only, such power law dependence or its function of the particle size might be expected. This large dependence is coming from the difference in dimensionality of the area and volume, that is the difference in the complexity of dimensions of space. If one considered a more complicated situation with fractal nature, a measure of the complexity is a fractal dimension. Thus, another possible measure of the nanostructured materials can be "fractal dimensions.” Such dependence should also be modified if one considered the width of the surface regions and/or active regions characteristic to the phenomena.

If the properties of materials are related to the surface area, they will be affected considerably by the size of particles. One can easily imagine how large the difference in the solubility of powder sugar and sugar candy is. Melting is also affected by size. One can imagine the situation of snow and a large block of ice if one neglects the difference in phases.

In the field of pharmaceuticals, the dissolution process is known to be a function of the surface area [40] and such a process is important to control the dissolution of pharmaceuticals [41].

Adsorption is closely related to surface areas or regions and might be used to estimate the specific surface area. Experimentally, adsorption in porous media tends to be treated by Brunauer, Emmet, and Teller's (BET) methods [42] and used to measure the surface area. However, caution is required for the applicability of methods such as the BET equation in the case of nanostructured materials [43]. The limitation and the applicability of the method are also argued by using the binary Lennard-Jones model [44].

### Dissolution and nucleation of nanoparticles: Nernst–Noyes–nanoparticles: Nernst–Noyes–Whitney equation

If the dissolution of a solid is governed by the diffusion obeying Fick's law, the process can be described by the following Nernst- Noyes-Whitney equation [40]:

where S is the surface area of the particle, D is the diffusion coefficient, V is the volume of solvent, d is the thickness of the diffusive layer. Cs is the saturated solubility of the particle in solution in the diffusion layer and C the concentration of the particle. Here D can be correlated with viscosity by the Stokes-Einstein equation, D = RT/6mjrN for the case of sphere particles, with a radius r. If D/[V6) is regarded as a constant k, the Noyes-Whitney equation is obtained as follows:

As found in this relation, when the particle size is reduced, the total effective surface area of a particle increases, and it results in the increase of the dissolution rate.

Of course, this equation is important for controlling the solubility of drugs. Furthermore, the nucleation theory is closely related to this process [45]. Namely, particle growth can be calculated using the Noyes-Whitney equation with the negative concentration gradient.

Again, caution is required for the applicability of such relations at the nanoscale. As pointed out in ref. [45], it may not work for the nanoscale. A similar situation is found in the self-healing process discussed in Chapter 7. When we consider the properties of nanoscale, they are often related to the events of both short and long time scales. For example, it was found that the self-healing process in porous silicates found in a short time scale region is not controlled by the motion in diffusive regimes. As shown in these examples, the applicability of such an equation in the nanoscale should be examined carefully.

## Determination of Size and Strain of Nanoparticles by X-ray Diffraction Patterns

In nanostructured materials, nanoparticles and/or nano-sized domains are contained in many cases. They can be dispersed, mixed in layers, pellets, and films. In this situation, the determination of the particle size would be an important task in experimental approaches. There are several methods such as direct observations by SEM, determination of the (secondary) particle size, and distribution by dynamic light scattering (DLS). Since each method has its characteristics and errors, careful arguments are required.

In this subsection, the experimental method to determine the size of nanoparticles by X-ray diffraction will be explained briefly. The line breadths in the powder X-ray diffraction pattern are affected by both small particle size and strain broadening. In 1918, Scherrer proposed the method to determine the size of crystals from the width of the diffraction peak [46]. If other effects such as a strain of the system are negligibly small, mean crystallite sizes, L, can be determined from the broadening of the peak, /3, by the following relation:

where К is a constant and / is a wavelength, 9 is a Bragg angle.

Here the constant К depends on the definition of the size of the particle and that of the broadening. A careful choice of value is recommended [47]. On the other hand, if the contribution of the strain is considered, the following relation by Williamson and Hall is a good representation in many cases [48]:

This equation was derived from Eq. (2.5) for the size broadening and the following Eq. (2.7) concerned with the strain broadening:

From the sum of (2.6) and (2.7),

If both sides of the Eq. (2.8) are multiplied by cos в, observed for Eq. (2.6) can be obtained.

In Fig. 2.14, the Williamson-Hall plot observed for BaF2 nanoparticles [49] is shown. In this case, results are found to depend on hydrothermal treatment times.

Figure 2.14 Representative Williamson-Hall plots of the BaF2 nanoparticles prepared using different hydrothermal treatment times. Reprinted with permission from Andrade, A. B., Ferreira, N. S., and Valerio, M. E. G. (2017), Particle size effects on structural and optical properties of BaF2 nanoparticles, RCS Adv., 43, pp. 26839-26848. The Royal Society of Chemistry, 2017.

As shown in this example, if /3totaiC0S\$ is plotted against sin в, lines are obtained. From the slope, Ce can be determined, while from the intersection, Kl/L can be obtained. As shown in these relations, both strain and size are considered as the function of the lattice parameters. It means that nano-sizing of particles affects the lattice constant as well as the mechanical properties of materials.

New methods are still developing [50]. Comparisons of methods are found in several references [45, 51].

## Effect of Changes of the Atomic Structure: Coordination Number and Formation of Different Phases

When the particle size is reduced, structures of materials are modified near the surface. For example, coordination number of the particle on the surface must be different from the bulk one. This situation can be found in some catalysts. Because the activity of the catalyst is affected by the coordination number [52], such changes should be considered.

The reduction of the size also causes the formation of different phases [53, 54]. For example, high activity of catalyst face-centered cubic (fee) Ru nanoparticles has been reported by Kumara and coworkers [54]. Recently, it was reported that the Ru nanoparticle with the fee structure confined in a porous cage (as-synthesized Ru NPs@PCC-2 composite) exhibited record-high catalytic activity in methanolysis of ammonia borane, which is important in chemical hydrogen storage [55].

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