This project is part of the SPP 1273 of the DFG.

Model for Aggregate Restructuring

In colloidal suspensions, large aggregates are often formed due to strong cohesive forces between the primary particles. The structure of such aggregates is well understood in resting fluids. However, under process conditions, shear forces act on the aggregate. These forces affect the aggregate structure significantly. In the first funding period, different simulation techniques as rigorous computational fluid dynamics (CFD) and discrete element methods (DEM) was used in order to investigate the influence of shear flow on aggregate structure. In order to avoid unrealistic results in the DEM simulations, it was necessary to develop a novel model for the tangential interactions between bonded colloidal particles. In the next funding period, the employed models and methods will be further improved and validated. Also the gained insight will be used to predict restructuring rates for macroscopic population balance modeling. Eventually, such mechanistic modeling will allow to effectively control the structure and final size of colloidal aggregates by exposing them to a certain hydrodynamic environment.

Finite Element Simulations

Figure 1. Finite element mesh.
The surface of one particle is represented by 10.000 triangles.

We consider a colloidal aggregate in a shear flow. The aggregate consists of a small number (5 to 20) of spherical primary particles of some 10^2 nm size. Starting point for the simulation of the restructuring behavior is an aggregate generated by (a) diffusion limited cluster aggregation or (b) reaction limited cluster aggregation.

Our goal is to determine the hydrodynamic forces acting on the constituent particles of the aggregate. This information will later be used to refine the model that is being developed at TUM (link) in the joint project (B1) in the SPP 1273. The drag forces on the constituents of the aggregate depend on the shear rate and the relative position to the flow. Each configuration of neighboring particles will lead to different drag forces. Constituent primary particles which are located in the middle of the aggregate will experience smaller drag forces than particles at larger radii and large drag forces may lead to aggregate break up.

Aggregate rotations

Once the aggregate is subject to a shear flow, it will start rotating and restructuring, depending on the shear rate and overall fluid velocity. In order to allow for a three dimensional arbitrary rotation, we introduce an update layer. This is supposed to minimize the costly computations for grid remeshing; now only a minor fraction of the volume mesh is deformed by rotations and has to be reconnected. The reconnection algorithm is currently in development stage. The biggest task is the handling of the huge amount of mesh data with at least 64 processing units.

Figure 2. Update Layer:
The update layer surrounds the aggregate
in order to provide fast reconnection
after rotations of the aggregate.

The next movie shows a testcase for the reconnection algorithm. The software version combines XNS and Netgen via a python script. Each time step is started by python - file I/O and parameter file reading included. Another python script produces Ensight-readable command scripts and Ensight produces pictures of time steps. The tool enve, shipped with Ensight, is then applied to produce movies.

The movie shows a small (rigid) aggregate consisting of five primary particles which is subject to a shear flow (flow along x-axis, linear shear along z-axis.) and the update layer of (for the testcase rather large) elements which are reconnected if necessary. One can see that reconnection does not happen at each time step but only if the deformation of the update layer is too large. This reduces processing time for example at times when the motion of the aggregate is slower because no reconnection has to be calculated.

Free Draining Approximation

In the current version of the model, the hydrodynamic forces on the aggregate are approximated by Stokesian drag forces. This ansatz is called "free-draining approximation" in the literature. Each particle is considered separately and the fluid force is determined via the Stokes drag formula for uniform flow conditions. We expect different behavior especially for particles in inner regions of the aggregate. The outer particles will shield the inner particles from the flow. Furthermore, the shape of the fluid flow is expected to be of major influence; we consider simple shear flows which will induce rotations and reorganization of the aggregates.

The simulation tool which will be used for the rigorous modeling is called XNS (link). The colloidal aggregate and the surrounding fluid are discretized by a three dimensional finite element mesh. This mesh is generated from input data, containing positions, sizes, (and masses) of the primary particles, with the mesh generator Netgen (link). This finite element mesh is then used as input for the simulation tool XNS. XNS solves the Navier Stokes equation (in this case the Stokes equation) for the given geometry. Fluid velocity and pressure, as well as the drag forces on the constituent particles can be calculated, the latter as an integral over the constituent particles' surfaces.

Figure 3. Drag forces:
The figure shows the drag forces
acting on the constituent particles
of an aggregate with 20 particles.
The flow has a shear rate of 50/s in
z direction.

Our results have been summarized in our paper "Restructuring of colloidal aggregates in shear flows and limitations of the free-draining approximation". (Please contact Eva Schlauch.)

Links

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