# Changes in the Network Structures and the Formation of Gels

- Reconstruction of Si-O Bonds during Formation of Gels and Distribution of Q„ Structures
- Pair Correlation Functions and Running Coordination Numbers
- Barrier for Coagulation
- Concept of the Potential of Mean Force
- Structure Surrounding Colloidal Silica–Running Coordination Numbers
- Comparison with Description by the DLVO Model

In both I-A and I-B, the colloidal units form one large aggregate. In the I-B system, which has a larger NaCl content, the connections among the units are more developed.

The instantaneous structures of the systems I-A and I-B obtained from the MD simulations after equilibration are depicted in Ref. [1]. Figures 8.2 and 8.3 show the structures of system II-A and system I-C after gelation, respectively. The former system is a reference without salt. A three-dimensional view of II-A reveals that some small clusters are separated from the largest cluster. The size of these clusters is several times larger than that of the original Si0_{2} colloidal units in our systems.

As shown in this figure, spherical shapes of colloidal units are not completely lost in clusters and some Si0_{4} units are connected by sharing edges (or corners, faces) to form a larger cluster. Naturally, in systems II-A and B, which contain a larger amount of water (by a factor of ~two) than systems I-A and I-B, the network is less developed. In the system I-C, following a short run (~several hundred ps at 400 K) after the long run at 300 K, a gradual shrinkage (approximately 19% in volume) of the system in *NPT* conditions is found. As shown later, this process occurred in parallel with the slowing down of the system including the water as well as the formation of three-dimensional (infinitive) networks of silica. The networks are formed of -Si-O-Si- bonds and the water molecules were trapped in the network. We regard this formation of a 3D percolated structure as gelation. In Fig. 8.3, the structure of the gel thus spontaneously formed is represented.

Figure 8.2 Structure of system II-A at 300 К within a basic cell of MD. It contains 19 clusters in the basic cell of the simulation Blue: Si and Red: 0 are connected by bonds. Pale blue polyhedra: Ow (oxygen in water molecules) surrounded by Hw (Hydrogen in water molecules) atoms. Silica nanocolloids are separated by networks of water in this system.

Figure 8.3 Structure of the system I-C (after gelation). Green: Na, Yellow: Cl. The system forms infinitive networks of both silica and water networks with periodic boundary conditions.

The system seems to be more ordered than in the case of typical aggregates, suggesting that the long-range structure of the system may be affected by a periodic boundary condition. This finding suggests the existence of long-range cooperative motions during the formation of gels. This process may also be affected by the initial arrangements of colloidal units. The dynamics of each constituent of the system, including the solvent, changed considerably with the formation of the gel, as demonstrated later.

## Reconstruction of Si-O Bonds during Formation of Gels and Distribution of Q„ Structures

How does the number of connections in the network change with the formation of gels? The coordination number for Si around Si represents the possible connections in the network through Si-O-Si. The number of Si-0 bonds in each system was counted using a cut-off length of 2.1 A. The mean number of bridging oxygens, N_{b}._{0} (for Si-(0)-Si connections), in each system was determined. The number was found to be 3.7-3.9 for these systems. From these results, one can conclude that the formation of gels occurred with the combination of bond-break and bond-formation with keeping the total number of bonds. Using the same definition of Si-0 bonds, the distribution of *Q _{n }*structures (where

*n*is the number of bridging oxygen atoms in the Si0

_{4}units) was determined to characterize each network system (see Fig. 8.4 for typical

*Q„*structures). The distributions of

*Q„*structures obtained by MD simulations are shown in Fig. 8.5. All systems show the peaks at

*n*= 4 of

*Q*structures. In several systems, a negligibly small number of

_{n}*Q*structures was found. In the series of I in Fig. 8.5a, there are small differences in the distributions, while for the series of II in Fig. 8.5b, the distributions are almost coincident with each other. The cluster structure is thus formed by the reconstruction of networks, that is, both bonds breaking within each colloidal unit and the formation of new bonds among units occurred.

_{s}Further discussion of changes in *Q _{n}* distribution will be given in Chapter 9.

Figure 8.4 Schematic description of *Q _{n}* structures with

*n*= 1-4, where

*n*means a number of bridging oxygen atoms. Si atoms connected to the bridging oxygen atoms are also shown. Reproduced with some modification from Habasaki, J., Leon, C., Ngai, K. L. (20017), Chapter 9, Molecular dynamics simulations of silicate glasses, in

*Dynamics of Glassy, Crystalline and Liquid Ionic Conductors,*2017, Springer, with permission.

Figure 8.5 Distribution of *Q _{n}* values (number of

*n*is taken as axis number), (a) For colloidal silica unit surrounded by water and systems in series I with smaller water contents, (b) For series II with larger water contents. Three kinds of distributions overlap in this figure.

# Pair Correlation Functions and Running Coordination Numbers

A pair correlation function, g(r) and a running coordination number, *N{r),* are useful to characterize the networks and the formation of gels. The pair correlation function, *g[r),* for the Si-Si pairs (connected by bridging oxygen) was determined as shown in Fig. 8.6. In Fig. 8.6a, the pair correlation functions for the Si-Si pairs

**Figure 8.6 **(a) Pair correlation functions of Si-Si pairs in the systems I-A, I-C, II-A, and II-C on a log-log scale. A power-law region and a region corresponding to *g(r)* ~ 1 with some deviations are observed. The *g(r) *function displays a clear minimum near r = 30 (A) in the system without salt, in which the clusters are isolated. Minimum of the function corresponds to the maximum of the potential of the mean force *Wy.* That is, it corresponds to the barrier of the coagulation, (b) Pair correlation function of Si-Si pairs in double logarithm plots. The same pairs as in (a) are contained. The minimum in each curve is clearer. Figures are reproduced from Habasaki, J., Ishikawa, M. (2014), Molecular dynamics study of coagulation in silica-nanocolloid-water-NaCl systems based on the atomistic model, *Phys. Chetn. Chem. Phys.,* 16, 24000-24017, with permission from the PCCP Owner Societies.

of the systems I-A, I-C, II-A and II-C are plotted on a semi-log scale, whereas in Fig. 8.6b, these correlation functions are plotted on a double logarithmic scale. The series II systems exhibit larger peaks than the series I systems, which indicate more isolated clusters. Remarkable features observed are systematic changes in peak heights and positions of minima in the functions. I- A and I-C exhibit smaller peak heights than II-A and II-C. For a higher concentration of NaCl, the Si-Si peak heights for the first and several subsequent shells (for dense regions of the networks) decrease, and these dense regions become smaller in both series of systems. A small minimum (deviation from *g{r) =* 1) is observed in the long-length-scale regions (r > 12~35 A) of the dilute systems before g(r) = 1 is attained. This trend can be more clearly observed when a logarithmic scale is used for the *g[r)* value, as shown in Fig. 8.6b. The minimum is located at approximately half of the mean distance between clusters or aggregates when these structures are separated by the solvent.

The existence of this minimum is consistent with the results of off-lattice DCLA calculations performed by Hasmy et al. [18]. In the pair correlation function, the r = 0 position of the function is common to all Si atoms, including those located on the peripheries of clusters. The size of these clusters is several times larger than that of the original Si0_{2} colloidal units in our systems. The largest deviation from 1 is observed in the system without salt (II-A) at approximately *r =* 30 A. A decrease in the peak heights of g(r) in the short-length-scale region can be observed with increasing NaCl concentration for I-B (not shown); hence, changes occur with coagulation, while, higher temperature of system I-C has no significant effect on the function. Thus, in system I-C, the power-law region below r ~ 12 A represents the spreading of the network of clusters, and the longer-length-scale region represents the development of connections among these structures to form a gel.

# Barrier for Coagulation

## Concept of the Potential of Mean Force

The structure found in g(r) of Si-Si pairs shown in Fig. 8.6 is also useful to understand the barrier of coagulation process, which is related to the maximum observed in the pair correlation function, g(r) of Si-Si pairs.

As shown in the previous section, the minimum of £f(r) is located at approximately half of the mean distance between clusters or aggregates when these structures are separated by solvent. The existence of the minimum seems to be important to understand complex structures in gels and its mechanism of the formation.

Here we explain the physical meaning of the barrier based on the concept of the potential of mean force (PMF), which was introduced by Kirkwood [19]. Generally, the concept plays an important role in statistical mechanical theories of liquids. For equilibrated or nearly equilibrated systems, the PMF, И^(г), is related to the pair correlation function *g^r)* as follows:

From this functional form, the minimum of g(r) observed in Fig. 8.6 corresponds to the barrier for effective repulsive interaction among colloidal clusters. That is, in *g[r),* it corresponds to the boundary region separating clusters. As expected, the effective repulsive force is the largest for the system without salt, for which the deepest minimum is observed. In the gel, the length of the power-law region is the shortest among all investigated systems, and the minimum is not so clear. In this case, the structure of the original colloidal units rapidly decayed and spread to form the gel. The strength of the effective repulsive interaction among aggregates is controlled by the salt concentration and the packing density of Si0_{2} (Si0_{2}: water ratio) as systematic changes of g(r) with concentrations.

In the case of the DLVO model and related ones, one assumes the potential consists of the repulsive and attractive parts. In the case of MD simulations, the origin of each force is clear from the potential functions used and the £f(r) is determined by the sum of forces. Analytically, the force acting for an /-th particle is represented by [20-22]

where the *U _{N}* is for internal energy for

*n*body potentials of

*N*particles. In the case of the pair potential,

*n*= 2.

For obtaining the PMF (or related free energy) from simulation runs along with reaction coordinates, techniques such as umbrella sampling [23, 24] are required for obtaining good statistics. For example, Allen et al. [25] have examined the potential of mean force to understand the behavior of particles in narrow channels. However, this is not the case for the discussion of the PMF from the pair correlation function, *g(r),* using Eq. (8.1), because the pair correlation function can be obtained with good statistics in MD using many numbers of initial configurations and many particles. That is, the trend in the PMF can be determined based on the behavior of g(r). From Fig. 8.6, we have judged that the system sizes are large enough to observe and argue the systematic changes in the minimum position of the *g[r)*

related to the separation of colloids or clusters. If the calculation resources and times were available and if actual calculation could be done within an acceptable real time, larger systems may be favored for obtaining precise characteristics of the barrier. Although the explanation was given for the Si-Si pair, it is needless to say that one must include interactions of all pairs, to consider the structure of the system.

## Structure Surrounding Colloidal Silica–Running Coordination Numbers

In Fig. 8.7, the structure of a cluster taken from the gel, defined by a sphere with a radius of 12 A around a certain Si atom, is shown. One can observe how structures are mixing when it is observed from a central Si ion. As shown in this figure, the silica part and water part are mixing as well as Naions and Cl ions. In Fig. 8.8, pair correlation function, *g[r),* of pairs of species in I-C and II-A are compared for the power law region. All pairs including Si atoms are shown here. From these curves, one can understand the mean behavior of how structures are mixing when we observe it from all central Si ions.

**Figure 8.7 **A structure of a cluster (radius = 12 A) taken from a gel. Blue: Si, Red: 0. Green: Cl. Yellow: Na. Blue: 0W. Pale blue: HW.

Figure 8.8 Comparison of pair correlation functions, *g[r),* in the power- law region for systems 1-C (solid curves) and II-A (dashed curves). Black: Si-Si, Red: Si-0, Blue: Si-OW, Pink: Si-HW, Green: Si-Na*, Brown: Si—Cl^{-}. In II-A, the contribution of the Si0_{2} component decreases with increasing r, while that of the water component gradually increases. In I-C, the slopes of these components flatten, indicating the mixing of these structures. This figure is reproduced from Habasaki, J., Ishikawa, M. (2014), Molecular dynamics study of coagulation in silica-nanocolloid- water-NaCl systems based on the atomistic model, *Phys. Chem. Chem. Phys.,* 16, 24000-24017, with permission from PCCP Owner Societies.

In II-A, upper curves (silica part) for Si-Si and Si-0 pairs are fairly separated from those for Si-OW and Si-HW pairs, while the curves are not separated in I-C (gel). Here 0W is for the oxygen and HW is for the hydrogen of water molecules.

It is useful to compare the running coordination numbers *N{r*) for every species around Si atoms to understand how *g(r) *of Si-Si pairs is determined in the system.

The running coordination number *N[r*) is defined as follows:

where is the number density of the species *j* in the system.

In Fig. 8.9, corresponding running coordination numbers are shown for I-C and II-A systems. As seen in these figures, the numbers corresponding to HW and OW around Si atoms differ notably in these systems. The running coordination numbers of the water components rapidly increase with increasing r. That is, larger changes in the *N(r)* slopes are observed for Si-HW and Si-OW than for Si-Si or Si-0 in both II-A and I-C. In system I-C, a larger amount of solvent exists near Si than in system II-A. This means that the structure of water is a counterpart of the silica network.

**Figure 8.9 **(a) Running coordination numbers for System II-A. (b) Running coordination numbers for System I-C. The colors in (a) and (b) are the same as in Fig. 8.7. The functions for the Si-Si and Si-0 pairs are the counterparts of the functions for the water components (Si-OW and Si-HW), which are affected by the salt concentration. This figure is reproduced from Habasaki, J., Ishikawa, M. (2014), Molecular dynamics study of coagulation in silica-nanocolloid-water-NaCl systems based on the atomistic model, *Phys. Chem. Chem. Phys.,* 16, 24000-24017, with permission from PCCP Owner Societies.

Of course, the contribution of the salt in I-C increases as a function of r; however, the coordination number of Na^{+} and/or Ch in System I-C (gel) is 20~30 times smaller than that of water in the shorter-r region (at around 3 A) and ~10 times smaller in the longer-r region. This result means that the concentration of NaCl is not homogeneous when we consider the nanoscopic structures of colloids solution and/or the formation of gels. From these results, we can conclude that the system is not a simple mixture of silica part and solvent but each component of silica, water, and NaCl plays a role to form complicated nanostructures. Further details of the structures of the silica part such as fractal dimension and connections of Si0_{4} units represented by a dendrogram will be shown in Chapter 9.

## Comparison with Description by the DLVO Model

The DLVO model [2, 3] is often used to explain the behavior of the colloidal systems. This model accounts for two types of forces that are present in a stable colloid: the van der Waals force and electrostatic repulsion. The total interaction potential can be calculated as a function of distance, and the colloid is stable when the two forces balance each other.

It can successfully explain the stability of colloidal systems in several cases but not necessarily adequate for other cases.

One can find that the explanation of barrier for coagulation shown in the previous section is not necessarily the same as that based on the DLVO model, concerning with the origin of the force determining the barrier of the coagulations and each contribution of terms in the potential functions can be separated in MD simulations if necessary. Therefore, the information from the atomistic simulations will be useful for the origins of forces in related systems without assuming the DLVO model a priori.