Onset of percolation

FIG. 2 Low-frequency conductivity at 45°C as a function of composition, a (weight fraction decane relative to decane and brine) for brine, decane, and AOT microemulsions exhibiting the phase behavior illustrated in Fig. 1. The breakpoint at a = 0.85 corresponds to the onset of percolation. This conductivity increases by two orders as a decreases from 0.85 to 0.7. (Reproduced by permission of the American Institute of Physics from Ref. 37.)... 

FIG. 5 Order parameter for disperse pseudophase water (percolating clusters versus isolated swollen micelles and nonpercolating clusters) derived from self-diffusion data for brine, decane, and AOT microemulsion system of single-phase region illustrated in Fig. 1. The a and arrow denote the onset of percolation in low-frequency conductivity and a breakpoint in water self-diffusion increase. The other arrow (b) indicates where AOT self-diffusion begins to increase. 

As described in the introduction, certain cosurfactants appear able to drive percolation transitions. Variations in the cosurfactant chemical potential, RT n (where is cosurfactant concentration or activity), holding other compositional features constant, provide the driving force for these percolation transitions. A water, toluene, and AOT microemulsion system using acrylamide as cosurfactant exhibited percolation type behavior for a variety of redox electron-transfer processes. The corresponding low-frequency electrical conductivity data for such a system is illustrated in Fig. 8, where the water, toluene, and AOT mole ratio (11.2 19.2 1.00) is held approximately constant, and the acrylamide concentration, is varied from 0 to 6% (w/w). At about = 1.2%, the arrow labeled in Fig. 8 indicates the onset of percolation in electrical conductivity. 

FIG. 8 Low-frequency conductivity (a) of water, toluene, and AOT reverse microemulsions at 25°C as a function of acrylamide (cosurfactant) concentration, (wt%). The Op and arrow at f = 1.2% shows the approximate onset of percolation in low-frequency conductivity. 

FIG. 9 Measured self-diffusion coefficients at 25°C for toluene (A), water ( ), acrylamide ( , and AOT ( ) in water, toluene, and AOT reverse microemulsions as a function of cosurfactant (acrylamide) concentration, f (wt%). The breakpoint at about 1.2% acrylamide approximately denotes, the onset of percolation in electrical conductivity. 

Several unifying conclusions may be based upon the order parameter results illustrated here for microstructural transitions driven by three different field variables, (1) disperse phase volume fraction, (2) temperature, and (3) chemical potential. It appears that the onset of percolating cluster formation may be experimentally and quantitatively distinguished from the onset of irregular bicontinuous structure formation. It also appears that... 

Balberg I, Anderson C, Alexander S, Wagner N. Excluded volume and its relation to the onset of percolation. Physical Review B. 1984 30(7) 3933-43. 

The onset of percolation and the conditions that produce this phase-transition phenomenon have been of considerable interest to many disciplines of science. The classical studies have focused on immobile ingredients in a system that increase in concentration by randomly adding to the collection of particles. It is possible to estimate the conditions leading to the onset of percolation under these circumstances. When the ingredients are in motion, this estimation is far more difficult. It is an obvious challenge that was tackled using cellular automata. 

A series of runs were made starting with a concentration of 100 occupied cells in a grid of 3025 cells. Successive increases in the occupied cell concentration were introduced in increments of 100. The concentrations of the five configurations, f0 through f4, were recorded for each occupied cell concentration. The size of the largest cluster was also recorded at each occupied cell concentration. Finally, the occupied cell concentrations at which the onset of percolation takes place and at which there is a 50% probability of percolation were recorded. 

Particle settling is another factor that may influence the observed onset of percolation. Depending on how the electrical properties of the sample are measured, the observed value for Pc may be either higher or lower than the value of Pc in a truly random system. 

Figure 16.5. Snapshots of flie simulation of different binary mixtures - water-DMSO in the top panel, water-eflianol in die middle, and water-TBA in the bottom panel. Water moleeules are shown in silver. Co-solvents (DMSO, edianol, and TBA) are represented in blue. The snapshot is shown at two different coneentrations - one before the onset of percolation to show the microheterogeneity in the system, and one after the onset of percolation to show the spanning cluster of the cosolvent. Figure adapted with permission from J. Phys. Chem. B, 115 (2011), 685. Copyright (2011) American Chemical Society.

As a final comment, let us briefly discuss the permittivity of the studied microemulsions. Figure 22 shows the temperature dependencies of the experimental permittivity and the results of the calculations on the basis of the developed model performed by using Eqs (67) and (71). The difference between the values of e obtained from these formulas can only be observed at high s. The calculated values of s agree well with the experimental data in the region far below the onset of percolation (T < Tqjj), where the assumptions of the model are fulfilled. At temperatures close... 

There are several exact results available in this model to serve as checks on approximate calculations of and L(E). In the limit of small x or 6, all averaged quantities may be expanded in powers of a small parameter. For arbitrary x and 5, a number of moments of the density of states can be calculated exactly (Velicky, 1968). When 5 > 2, the density of states is split into two separated sub-bands, centered about and e , each of width B. Thus in the limit 5 -> a site containing a B atom is forbidden to an electron with energy near and percolation theory (Frisch, 1963) may be used to determine the probability that such an electron is trapped or free to move across the crystal. When x is greater than x, the criticd value for the onset of percolation, there will be extended states in the A sub-band. Since x, is less than 1/2 for all three-dimensional lattices, we observe that at least one of the strongly split sub-bands will always contain extended states, in contrast to the complete localization observed for r>Fg in the Lorentzian model of the preceding section. 

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