Average cluster size

Repeat the studies, selecting intermediate temperature values using the relationships between Pq and J values for water shown in Table 3.2. For example, use Pb(WW) = 0.50 and J(WW) = 0.71 in Example 3.2. Compare the fx values with the Studies in Examples 3. land 3.2. A plot of each value from Studies in 3.1 and 3.2 will reveal the influence on these attributes with and without a density consideration. Also compare the average cluster sizes between these two groups of studies. How much difference is found in the fx values when water density is accounted for ... 

Simulated water temperature (°C) /o(ST Average cluster size Average munber of solute bonds... 

Repeat this example using 2060 water cells and 40 solute cells in the Example 4.2 Parameter Setup. This is approximately a 2% solution. Repeat the dynamics again with a higher concentration such as 2020 water cells and 80 solute cells, using Example 4.2 Parameter Setup. Compare the structures of water as characterized by their fx profiles and average cluster sizes. Some measures of the structure change in water as a fimction of the concentration are shown in Table 4.2. 

Record the average fx values for W and the average cluster size for W at the end of each run. 

Increase the nonpolar character of the solute by using rules Pb(WS) = 0.5 and J(WS) = 0.7. Keep all the other rules constant. Run each experiment 10 times and collect and average the fi values. Repeat the study using a more nonpolar parameter set for the solute, for example Pb(WS) = 0.8 and J(WS) = 0.25. Other parameters are retained as in Example 4.3. Record the fx values and the average cluster size for water at the end of each run. 

The reader is invited to examine this phenomenon by running the models described above, by varying these two sets of parameters. The solute is modeled as a 10 X 10 block of 100 cells in the center of a 55 x 55 cell grid. The water content of the grid is 69% of the spaces around the solute block, randomly placed at the beginning of each run. The water temperature (WW), solute-solute afiinity (SS), and hydropathic character of the solute (WS) are presented in the parameter setup for Example 4.4. The extent of dissolution as a function of the rules and time (5000 iterations) is recorded as the fo and the average cluster size of the solute (S). 

The reader is encouraged to run this model and collect the average cluster size of amphiphile cells. Observing the run reveals a view of the emergent property known as micelle formation. Periodic halting of the run when these micelles are prominent will be of interest. Try a screen grab of several good examples. 

Run this example using the following parameter setup. Record the average cluster size and the percent of percolations over a number of runs. The number of iterations will be zero and only the initial grid configuration will be used. A suggested number of runs is 1000. 

Illustration Short-time behavior in well mixed systems. Consider the initial evolution of the size distribution of an aggregation process for small deviations from monodisperse initial conditions. Assume, as well, that the system is well-mixed so that spatial inhomogeneities may be ignored. Of particular interest is the growth rate of the average cluster size and how the polydispersity scales with the average cluster size. 

Here s0 denotes the initial average cluster size. 

It should be noted that the predictions for the number average cluster size and polydispersity agree with analytical results for K(x, y) = 1, x + y, and xy. Furthermore, the short-time form of number average size in Eq. (81) matches the form of s(t) predicted by the scaling ansatz. Computational simulations (Hansen and Ottino, 1996b) also verify these predictions (Fig. 38). 

Fig. 38. Variation of polydispersity with average cluster size at short times in a journal bearing flow. The symbols are from simulations and the lines are fits from Eq. (82). The regular flow is the journal bearing flow with only the inner cylinder rotating (Hansen and Ottino, 1996b).

As previously discussed, we expect the scaling to hold if the polydisper-sity, P, remains constant with respect to time. For the well-mixed system the polydispersity reaches about 2 when the average cluster size is approximately 10 particles, and statistically fluctuates about 2 until the mean field approximation and the scaling break down, when the number of clusters remaining in the system is about 100 or so. The polydispersity of the size distribution in the poorly mixed system never reaches a steady value. The ratio which is constant if the scaling holds and mass is conserved,... 

Pig. 40. Growth of average cluster size for area conserving clusters and fractal clusters in the journal bearing flow (Hansen and Ottino, 1996b). 

Note that only one system, the one corresponding to constant capture radius clusters in chaotic flows, behaves as expected via mean field predictions. In general, the average cluster size grows fastest in the well-mixed system. However, in some cases the average cluster size in the regular flow grows faster than in the poorly mixed system. 

Model computations indicate that the average cluster size ... 

Employing experimental supersaturated solution diffusion coefficient data and the cluster di sion theory of Cussler (22), Myerson and Lo (27 attempted to estimate the average cluster size in supersaturated glycine solutions. They estimated an average cluster size on the order of two molecules. Their calculations indicated that while the average cluster size was small, large clusters of hundreds of molecules existed, only there were very few of them. Most of the molecular association was in the form of dimers and trimers. 

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