Simulation of Wessling Model for Network Formation in Dispersed Polymer Phases

C. Roselieb

Institute for Theoretical Physics, Cologne University, D-50923 Köln, Germany

For a comment please look at
B.Wessling, Comments on C. Roselieb's "Simulation of Wessling Model for Network Formation in Dispersions"

Abstract: The model, developed by Wessling to simulate the formation of network structures formed in dispersed polymer phases, was investigated with new programs and systems of up to 10⁹ sites. The results of Wessling are confirmed. A new model is presented by the author, which is forming stable networks.

Introduction

In our daily life we are using a lot of different polymer products. They are often composed of two ore more phases. Sometimes one of the phases is no polymer and in this cases we call these compositions (polymer) "compounds".

These compounds can have very useful properties, like conductivity or impact resistance. To get for example conductive compounds, one disperses a conductive phase, like carbon black, in a matrix polymer, like polyethylen or polystyrol. The conductivity becomes possible by the formation of networks by particles of the dispersed phase through flocculation, if the concentration of this phase is above some critical value [9]. To describe this phenomenon Wessling developed a new thermodynamic model [6,7,8,9] based on his experimental results [5]. This model and the experimental results lead Wessling to the assumption that the formation of networks in compounds is not well described by the simulations based on percolation theory [1]. So he developed a new model based on a "cellular automata" program [4].

The Wessling Model

Cellular automata deal with usually two- dimensional (also higher dimensions are possible) square latices, in which every site can assume exactly two states. These states in biological processes are usually called "dead" and "alive", in a computer "0" and "1", in physics often "spin down" and "spin up". During each iteration step the state of each site is new defined in dependence of the states of its next neighbour sites. This process is regulated by different rules for every possible combination of "dead" and "alive" neighbour sites. For example in the famous "Game of Life" a "dead" site becomes "alive", if and only if exactly three neighbours are "alive". A "alive" site stays in it`s state, if two or three neighbours are "alive", in all other cases the site will become "dead" in the next generation. An easy way to denote these rules is to write down the number of "alive" neighbour sites, needed for one site to become "alive", with an minus sign and the number of "alive" site needed to stay "alive" with a plus. The "Game of Life" would be consequently noticed as [- 3, +23].

In the Wessling model a "dead" site represents a dispersed particle which is not integrated in the network. An "alive" site represents a flocculated particle, which can be a part of a network, if it also has flocculated neighbours. In graphical presentations the "dispersed" sites are shown as white fields, the "flocculated" sites as black ones. All sites are representing particles of the dispersed phase, not of the matrix polymer. That is similar to real systems, where the particles of the dispersed phase are close together in flat mono- layers. They are not all in contact, only flocculated particles are able to contact each other and form the network [5].

Wessling used a program named "cellular automata", which is available as an interactive demo program version [2], for his experiments. So the lattice size was restricted to 250 x 250 sites. But it was able to change the rules freely and to choose between four (just the nearest ones) and 8 neighbours (also the diagonal ones). Wesslings inquiries lead him to the conclusion that [- 234,+12] is the best rule for simulating dispersed polymer phases, considering 8 neighbours [4].

Investigation of the Wessling Model

The new investigations [3] about the Wessling model were made with much bigger systems, up to 37800 x 37800 sites ¹ and with faster programs (about 20 times faster then the demo program), up to 37.4 10⁶ updates/sec on a SUN workstation². Using new programs, it was possible to make special investigations about the structure of the formed networks.

The aim of this simulation is to get network formations of flocculated particles simular to the networks pictured with a SEM [5,10]. So it was necessary to develop criteria to evaluate the network structures. A useful criterion was the next neighbour correlation of flocculated sites.

I summarised the correlations into three groups. A site said to belong to a chain of flocculated sites has two neighbours which are also flocculated. If a chain is branching, there is a site with 3 flocculated neighbours at the branch. Both types together are called "Chains". Sites with only one or without flocculated neighbours not really belong to the network, so they are collected as "Dead Ends". Sites with more then 3 flocculated neighbours are not a part of chains, so they are summed up as "Clusters". This classification is not always objective, because for example a crossing is a flocculated site with 4 also flocculated neighbours and a cluster of 3 flocculated sites looks like a part of a chain. But it is very helpful together with the pictures, especially to show the changes during the iterations.

This classification leads for the Wessling model to the following (only watching the flocculated sites): Dead Ends 0.9 %, Chains 34.1 %, Cluster 65.0 %.

If a statistical distribution at the beginning is used, the values become after 30 iteration steps nearly constant (+/- 0.1 % in a 37800 x 37800 lattice), after some oscillations (fig. 1). The biggest part of Chains we get after three iterations (57 %) independent of the statistical distribution at the beginning and the size of the lattice. After that the oscillation becomes smaller and smaller. The fraction of flocculated sites among all sites is 40 %.

But that the fraction of the Dead Ends, Chains and Clusters become stable does not mean that the network is not changing anymore. The opposite is the case. Observing the number of iteration steps a site has the same neighbour configurations shows us that the network is broken down and rebuilt all the time. A flocculated site with three or more flocculated neighbour sites can't survive at all, because of the rules [- 234,+12]. So it is only interesting to watch the sites with one or two neighbours. From the 0.8 % sites with one neighbour less then 0,1 % are detected in the following iteration step and none in the second following one. 16.3 % of the sites have two flocculated neighbours, just 0.3 % stay in this state for one iteration, 0.02 % for two and none for three iterations.

Also the sites with one or two flocculated neighbours are unable to stay in this configuration for more then three generations.

The New Model

The new model is using cellular automata on a triangular lattice. This makes sense, because the particles seem to be round (compare SEM- Picture [4, 10]) and if they try to be as close to each other as possible this arrangement is the best. Consequently every lattice site has 6 nearest neighbours which all have the same distance to it (fig. 2).

The rule that seems to be best is [- 2,+123]. Using the same classification, the new model has a much bigger fraction of Chains 90.0 % and Dead Ends 9.2 %. The fraction of Clusters is 0.8 %. All together 52 % of the lattice are flocculated sites. This is 12 % more than in the Wessling Model.

Another difference between the two models is, that the new one is forming networks which are stable. That means the biggest part of the sites which are forming the network is a long time (may be infinite) in the same situation of neighbourhood.

Less than 4 % of the sites have a changing part of flocculated neighbours or change themselves to be dispersed. This state is reached after many iteration steps, depending on the fraction of flocculated sites at the beginning. For an initial fraction 50 % of sites it just takes about 15 iterations until a stable structure is built. For smaller fractions, for example 10 %, it takes a longer time, about 25 iterations. In this case, the process is looks like growth (fig. 3). After a few iterations there are some islands of networks and big parts of the lattice are empty (just dispersed sites). But the islands are becoming bigger and bigger. After a number of iterations the networks are all over the lattice (fig. 4, 5, 6).

Conclusion

The new investigations confirm the results of Wessling for bigger systems. They also show that the changes in the formated networks become very small after 30 iterations, if the lattice size is big enough. Initially the fraction of Chains and Clusters are oscillating. Especially in the third iteration step the part of Chains become very high and the part of Clusters very low. After more than 30 iterations the part of Chains and Clusters is stable.

In the new model the part of Chains is growing during the first 25 iterations and reaches 90 %. This seems to be more similar to the the real systems, if we compare the new model with the SEM- pictures. Data about the fraction of Chains, Cluster and Dead Ends in real systems would be helpful.

The biggest contrast between the models is the stability of the networks. In the Wessling model the networks are instable, they are always broken down and rebuilt, but the networks are always all over the lattice. In the new model only one network is formed. This network needs a number of iteration steps to become stable, after that it does not change very much any more.

Both models produce networks, which are simular to real systems. To get a exact comparison between the models and the real systems we would need precise values about the structure and dynamics of the formed networks in real systems.

Acknowledgements

I would like to thank especially D. Stauffer, who made several valuable proposals for the investigations and helped me to write this article. I also want to thank B. Wessling, who discussed his theory and investigations with me and helped to understand these things better.