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Colloidal gels are intermediates in the production of highly porous particle systems. In the production process, the gels are fragmented after their creation. These gel fragments consolidate to particles whose application-technological properties are determined by their size and porosity. A model of the consolidation process is proposed: The consolidation process of a gel fragment is simulated with the Molecular Dynamics (MD) method with the assumption of van der Waals forces in interplay with the thermal motion as driving forces for the consolidation. The simulation results are compared with experimental data and with a Monte Carlo (MC) simulation.

Precipitated silica is used as a filling material in the manufacture of shoes, tires and general rubber goods. It is used in tires in order to reduce the rolling resistance, while the abrasion resistance and wet grip are increased [^{2}/g [

In the industrial inorganic precipitation of silica, sodium silicate solution and sulfuric acid are fed into a water tank while the solution is stirred. The solution gelates after about 30 min. This gel is fragmented by the stirrer. The gel fragments consolidate to aggregates which are characterized by their size and porosity.

The simulation of aggregates makes the investigation of the development of the aggregate structure as a function of time possible. These insights into the aggregate structure can be used to tailor the aggregate structure by designing new processes in the future [

So far, the consolidation of aggregates has been modeled

The Stokesian dynamics method, as described in [

Here, _{H}_{P}_{B}

First attempts of the modeling of the structure of colloidal aggregates can be found in [_{R}

Furthermore, the consolidation of gel fragments is simulated by statistical methods, such as the Monte Carlo method. The model described in Schlomach (2007) [

MD simulation is used here in consideration of the argument above. Further arguments for this decision are given in Section 2.1.2.

The paper is organized as follows: In Section 2.1, a model of the consolidation of a gel fragment is introduced taking van der Waals interactions and thermal motion into account. In Section 2.2, simulation results of the consolidation behavior with the help of MD simulation are presented. The consolidation of a gel fragment is computed and compared with experimental data and a former MC simulation of Schlomach and Kind (2007). The temporal development of the structural factor that is measured in Schlomach (2004) [

It is assumed that the driving forces for the consolidation are van der Waals forces [_{LJ}

The properties (in physics: position and momentum) of the systems of many particles (e.g., atoms, molecules, particles) are analyzed in statistical physics. Statistical ensembles are thus an important tool in statistical physics. An

The Ergodic hypothesis is the fundament of statistical physics. In the Ergodic hypothesis, it is assumed that the temporal average of a measured variable (e.g., of the pressure) is equal to the ensemble average:

Monte-Carlo (MC) simulations are especially appropriate for the calculation of statistical average values of a certain quantity. The Monte-Carlo method is—in the form it was developed by Metropolis

The consolidation of gel fragments is simulated by the MD tool, LAMMPS (Large-scale Atomic/Molecular Massively Parallel Simulator) [

A fractal aggregate according to Schlomach and Kind (2007) [

The solvent environment in the simulation is achieved by a high density lattice of solvent molecules. The Lennard-Jones (12-6) potential

_{R}_{A}

A colloidal particle has a size of 2_{1} = _{2} = _{cc}

The colloid-solvent interaction energy is given by
_{cs}_{El}_{DLVO}

Here, _{0}

In

A cluster consisting of 1,000 particles in the environment of ∼400,000 solvent molecules is densified in LAMMPS as an isothermal isobaric (NPT) ensemble under the assumption of the interaction energies between the particles according to Equations (

In the following, these simulation results are compared with the experiment of Schlomach and Kind [_{f}_{agg}_{p}

Furthermore, the electrostatic repulsive force according to

In order to simulate the shrinkage of an expanded gel sample, the highly porous cluster (left side of ^{6} particles is generated. The kind of arrangement is shown in

The simulation leads to the results shown in

The sharp bend of the consolidation ratio computed is because of the low resolution of the time step in the initial phase of this square-root-of-time plot.

The consolidation of gel fragments and of a particle network consisting of gel fragments have been investigated on the assumption of the interplay of van der Waals energies and thermal motion as driving forces for the shrinkage. Two molecular dynamics simulations have been performed with the tool LAMMPS: The fractal dimension of the MD simulation of the shrinkage of the gel fragment was compared to the results of a former MC simulation and also to experimental data. After time scale adjustment (factor 100) the simulation data could be aligned in agreement with the experimental data. Thus, the assumptions of the simulation have apparently been correct. Van der Waals forces in interplay with the thermal motion are considered to be driving forces for consolidation.

Furthermore, an MD simulation of the consolidation of a gel volume element was performed. The consolidation ratio of the consolidating particle network was compared to experimental data from oedometer tests. Good agreement of experiment and simulated data was also found here after time scale adjustment (factor 10). However, in order to describe the temporal course of the consolidation in more thoroughly, the fluid flow through the particle network needs to be taken into consideration. This has been neglected in the simulation so far. This coupling will, however, be investigated in the future.

Lennard-Jones (12-6) potential.

Schematic illustration of the particle motion.

Structure of the model gel fragment (cluster of 1,000 particles) [

Lennard-Jones (12-6) and DLVO potential.

Structure of the model gel fragment (cluster of 1,000 particles) before and after shrinkage.

Experimental data of the fractal dimension

Arrangement of the clusters (2 × 2 clusters).

Computed consolidation ratio (MD-simulation) of the particulate network and experimental data from [

The equations of motion used are those of Shinoda _{i} and _{i} are the position and momentum of atom _{i} are the forces on atom _{g}_{k}_{ξk}^{th} thermostat of the Nosé Hoover chain (with length _{i}_{g}_{k}^{th} thermostat, respectively. The latter two are used to adjust the frequency by which these variables fluctuate [_{f} is the number of degrees of freedom, _{ext}_{ext}_{int}_{int}

The matrix Σ is defined as

Here, ^{t}h

Thus, the partition function becomes [

If

The time integration schemes closely follow the time-reversible measure-preserving Verlet [

Here,

Where _{i}_{i}_{i}_{i}_{g}_{g}_{g}_{ξk}_{ξk}_{k}

One of us (H.S.) wants to express that all praise is due to God, the Lord of the Worlds.

This research project is financially supported by the German Research Foundation, DFG, within the priority program “SPP 1273 Kolloidverfahrenstechnik” (Colloid Process Engineering).

(finite) distance at which inter-particle potential is zero

distance between the particles

interaction energy

_{cc}

Hamaker constant for colloid-colloid interactions

_{cs}

Hamaker constant for colloid-solvent interactions

radius of the colloidal particles

_{c}

cutoff distance

_{f}

fractal dimension

number of primary particles in the aggregate

_{agg}

radius of the aggregate

_{p}

radius of the primary particles