Reviews in Computational Chemistry

224 Cellular Automata Models of Aqueous Solution Systems

relationship between water and solute. These patterns ultimately translate into simulations of solubility, the hydrophobic effect, and diffusion preferences.

STUDIES OF WATER AND SOLUTION PHENOMENA

A Cellular Automata Model of Water

In a study of water, Kier and Cheng36 found relationships between the average cluster size and viscosity, and between the fraction of unbound cells, f0, and the vapor pressure. The number of water-designated cells was chosen to be 69% of the cells in the grid, this value giving a concentration of unbound water molecules in agreement with earlier predictions and experiments.30,39 More recent studies have examined all the fx attributes from the rules PB and J, whereby the PB(W) value was systematically varied. A profile of water cells, described by the fraction of each bonding type, f0–f4, for each ‘‘temperature’’, is shown in Figure 14. The values of f0 correspond closely to the values used by Walrafen39 and Haggis et al.40 to predict the dielectric constant and heat capacity of water. This finding led to the conjecture that the rules chosen give rise to configurations mirroring physical reality. A test of this possibility is reported in Table 1 where a set of equations relating several physical properties of water to attributes from the cellular automata simulations are shown. In addition to the examples in the table, the free hydrogen fraction, fH, is close to the value of about 15% reported from infrared (IR) evidence of the free OH content.41 These correlations between simulated water attributes and various physical properties suggest that the water model created with cellular automata has validity.36

The Hydrophobic Effect

The hydrophobic effect is a term describing the influence of relatively nonpolar (lipophilic) substances on the collective behavior of water molecules in their vicinity. The common expression is that water is ‘‘more structured’’ or

organized when in contact with a lipophilic solute. This behavior was observed in a cellular automata model of a solute in water,42,43 which led to a study in

more detail.44 The hydrophobic effect was modeled by systematically increasing the breaking probability, PB(WS), value, encoding an increasing probability of a solute molecule, S, not to associate with water.

It was observed that low PB(WS) values, modeling a polar molecule, produced configurations in which the solute molecules were extensively surrounded by water molecules, a pattern simulating hydration or electrostriction. Conversely, with high values of PB(WS) most of the solute molecules were found outside of the water clusters and within the cavities. This configuration leaves the water clusters relatively free of solute; hence they are more

226 Cellular Automata Models of Aqueous Solution Systems

Table 1 Water Properties Related to Cellular Automata Attributes

relative lipophilicity of a solute molecule is a useful rule for further studies on solution phenomena.

Solute Dissolution

Cellular automata simulations of the dissolution process have been described.42 The solute molecules, S, started in a solid block of cells at the center of the grid. They are endowed with rules PB(S), J(S), PB(WS), and J(WS). The attributes recorded from the dynamics were the f0(S), fraction of solutes unbound to other solute molecules, plus the average number of solute–solute joined faces, T(S), and the average distance that solute molecules have traveled from the center of the block at some specific iteration, D(S). The f0(S) values were interpreted to represent the extent of dissolution of the solute. The decrease in the T(S) values characterize the extent of disruption of the solute block, whereas the D(S) values quantify the extent of diffusion of solutes into the surrounding water.

The effects of the four joining and breaking parameters, PB(WS), J(WS), PB(S), and J(S), were studied using a high and a low numerical value for each. From the results several conclusions were drawn about the influence of the rules and about some aspects of the process. The extent and rate of the solute cluster disruption, which is reflected by T(S), is primarily a function of the PB(S) rule with secondary influence from the PB(WS) rule. A high value of PB(S) is a rule governing a high probability of solute molecules separating from each other; hence the rule may be viewed as the effective melting process relating to crystal disruption. The dissolved state was considered to exist when an isolated solute cell is present. The f0 fraction is a measure of the extent of the modeled solubility. The extent of dissolution depends on both the PB(S) and the PB(WS) rules. High values of PB(S) and low values of PB(WS) promote an extensive degree of dissolution. A low value of PB(WS) characterizes a solute that is relatively polar and hydrophilic. The studies also showed that solutes with high PB(WS) values diffuse more rapidly than those with low values. The interpretation is that the more lipophilic the solute molecule, the greater its diffusion rate. It was also observed that simulations of higher temperatures led to faster disruption of the block of solute, more f0 molecular

Studies of Water and Solution Phenomena 227

Figure 15 A model of the hydrophobic effect. The dark cells represent water, the white cells are cavities, and the gray cells are hydrophobic solute molecules.

configurations simulating greater dissolution, and more extensive diffusion of these molecules through the bulk water. This profile is the expected pattern of effects characteristic of increased temperature.

An interesting and unexpected observation arises from the graphical display of the disintegration of the solute block. The early stages of the disruption occurred as a series of intrusions of cavities rather than water molecules into the block. The cavities roamed throughout the block, behaving as ‘‘particles’’. The entrance of water into the block structure appeared much later, after there had occurred significant disruption and loss of solute molecules from the block. An image of this behavior is depicted in Figure 16. This observation was found for all parameter sets used in this study. The cellular automata model of the dissolution process is in substantial agreement with experimental evidence. This study produced some results suggesting ideas concerning a familiar process.

228 Cellular Automata Models of Aqueous Solution Systems

Figure 16 A model of the disruption of a crystal in water. The dark cells represent water, the white cells are cavities, and the gray cells are solute molecules.

Aqueous Diffusion

A series of studies have been reported modeling the diffusion process in water.49 Using the rules previously defined, we examined several characteristics of a system to determine their influence on diffusion. The first study revealed that solutes of high lipophilicity (low polarity) diffuse faster than those of low lipophilicity (high polarity). This result is not commonly considered or reported. Diffusion studies are numerous in the literature, but compar-

isons with solute polarity are very scarce. Two such studies, however, support the cellular automata model of this phenomenon.50,51

A model of relative diffusion rates as a function of water temperature produced the expected result of greater diffusion with higher temperature. Additional studies49 were conducted in an attempt to model the influence of solution characteristics on solute diffusion. A series of dilute solutions were

Studies of Water and Solution Phenomena 229

modeled whereby the relative lipophilicity of a solute S1 was varied. A second solute, S2, was introduced at the center of the grid. The dynamics showed that the solute S2 diffused faster when there is no cosolute and when the polarity of S2 is low. The presence of the cosolute revealed that the diffusion of S2 was fastest when the cosolvent S1 was lipophilic.

In another study,49 the grid was divided into two halves, the upper half, containing a solution of a relatively nonpolar solute, while the lower half contained a solution of a relatively polar solute. Between these two halves, a thin layer of cells simulated a solution made up of a solute of intermediate polarity. The dynamics created a model in which the solute in the middle layer preferentially diffused into the upper layer containing the nonpolar solute. A second study simulated an aqueous solution of intermediate polarity in the middle of the grid. Surrounding this solution were regions with highly polar, intermediately polar, and nonpolar solutes in solution. A fourth region around the center was pure water. The solute in the central section of the grid was allowed to diffuse freely. It diffused more rapidly into the pure water; however, it secondarily preferred the more nonpolar solution quadrant. This finding is consistent with the results from the studies described above. The diffusion of a solute is modeled to be faster when it is lipophilic and when cosolutes are lipophilic.

Immiscible Liquids and Partitioning

Cheng and Kier modeled the separation of immiscible liquids and the partitioning of a solute between them.52,53 For this study, the cellular automata grid had to be modified to create a model of the differential effect on the ingredients due to ‘‘gravity’’. To accomplish this, boundary conditions were imposed at the upper and lower edges of the grid to contain the liquids in a closed system. The differential effect of gravity was modeled by introducing the rules described earlier.

The water cells were ruled to have high self-affinities and low affinities with the other solvent. A PB(WS1) rule was chosen with a high value to endow the two liquids with markedly different polarities. The emerging configuration can thus be described as being an immiscible, two-phase system. Each experiment began with a random distribution of equal numbers of water and S1 cells, the sum totaling 69% of the grid space. The breaking and joining parameters were chosen to create significant self-affinity between each type of ingredient and a weak affinity between different types. The gravity terms were chosen to favor the water cell moving to the lower position relative to the other solvent.

At a very early iteration time, there formed small, vertically oriented stacks of water cells in the upper half of the grid and similar stacks of S1 in the lower half of the grid. These stacks steadily enlarged and moved toward the central part of the grid. Aggregation of each common type of cell stack occurred as they moved toward the central section of the grid. An interface formed with a greater concentration of water in the lower half while the

Studies of Water and Solution Phenomena 231

observed that the solute was associating with the patches of solvent to which it had the closest parameter-governed affinity. As a relatively stable configuration developed, the ratio of the solute molecules among the two phases became relatively constant. This ratio is the partition coefficient of the solute between the two phases. The dominant rules influencing the partition coefficient were PB(WS2) and PB(S1S2), where the latter parameter reflects the affinity of the solute for the two liquids W and S1, where S2 is the ‘‘solute’’ molecule.

Observing the course of the dynamics, we see a constantly changing pattern from the random configuration at the outset to the eventual formation of a disturbed interface and separated compartments of the two solvents. The solute molecules moved rapidly to the patches in which the rules have ordained an affinity. The solute molecules partitioned themselves among the patches long before the two phases and the interface have formed.

Micelle Formation

The organization in a system of two immiscible liquids results from a general pattern of behavior found among two molecules with little attraction to each other. Another example of this can be found in the behavior of a molecule that has two substructures with significantly different polarities. A term applied to these molecules is amphiphile, denoting a molecule with a dual polarity character. Under certain conditions, these molecules may interact with one another to form large, long-lasting, functional structures called micelles.60,61 These are organized and capable of entrapping solute molecules and forming membranes in biological systems. A micelle is a structure formed from the close interaction of the lipophilic fragments of amphiphiles plus the electrostatic encounters of the polar end of the amphiphile with the surrounding water. Typically, the micelles assume a spherical structure with the nonpolar fragments in the interior and the polar fragments on the periphery, interacting with the aqueous solution.

The formation of micellular structures is a dynamic process that has been modeled using cellular automata.62 The model of an amphiphile was created by treating each face of a square automaton cell as an independent structure, as described in Figure 4. Each face of this variegated cell can have its own set of PB(X) and J(X) values. Three of the faces were considered as equivalent and were endowed with rules modeling a lipophilic or nonpolar part of the amphiphile. The other face was treated as a polar fragment of the molecule and assigned characteristic rules. The outcome of the dynamics was the creation of structures in which the nonpolar fragments were in the interior of an aggregation of cells while the polar fragment lay on the periphery as seen in Figure 18. The interpretation of these organized clusters is that they model a micelle. The dominant influence on the formation of these structures is the extent of nonpolar character of the designated three sides of the cell. Of secondary influence is the polarity of the remaining face of the cell. If this is too

232 Cellular Automata Models of Aqueous Solution Systems

Figure 18 Two examples of micelles formed from the dynamics of amphiphile interactions. The dark faces represent the polar exterior of the molecules, and the light faces represent the lipophilic fragments of the amphiphiles.

polar, the micelle formation is retarded. Both of these influences produce models mimicking experiment.63,64

Another study on these variegated cells depicting an amphiphile revealed a temperature effect on the critical micelle concentration (cmc) that was minimal at about PB(W) ¼ 0.25. Experimentally, the minimal cmc value occurs at about 25 C.64 The onset of the cmc was also modeled and shown to be dependent on a modestly polar fragment of the amphiphile.

Membrane Permeability

An extension of the micelle and diffusion models was a simulation of the diffusion of a solute through a layer of lipophilic cells simulating a membrane separating two water compartments.65 A layer five cells wide was positioned on a grid between two compartments, each filled with water. The membrane cells were endowed with a PB(WS) rule making them lipophilic. The membrane cells could move about within the layer according to their rule response, but they could not escape from the layer. The two water cell compartments on either side were assigned identical rules but were ‘‘colored’’ differently in order to monitor their origins after some movement into and through the membrane layer. This model included a section of a membrane with aqueous compartments on either side. In biological structures, this might be a living cell membrane or tissue in an aqueous environment. Of interest was how effectively this cellular automata model reflected experimental evidence about diffusion through a membrane. It is recognized that water and some small molecules pass through membranes by passive diffusion.66 The cellular automata dynamics revealed that water molecules from both compartments pass into and through the membrane as expected.

To model the behavior of a solute in a membrane environment, 15 cells simulating solute molecules at a concentration of 0.01 were positioned randomly near the lower edge of the membrane surface. These cells were endowed with rules making them hydrophilic. As the dynamics proceeded, it was