Water, theories Lattice model

In order to describe the collapse of a long-chain polymer in a poor solvent, Flory developed a nice and simple theory in terms of entropy and enthalpy of a solution of the polymer in water [14]. In order to obtain these two competing thermodynamic functions, he employed a lattice model which can be justified by the much larger size of the polymer than the solvent molecules. The polymer chains are represented as random walks on a lattice, each site being occupied either by one chain monomer or by a solvent molecule, as shown in Figure 15.8. The fraction of sites occupied by monomers of the polymer can be denoted as 0, which is related to the concentration c, i.e., the number of monomers per cm by 0 = ca, where is the volume of the unit cell in the cubic lattice. Though the lattice model is rather abstract, the essential features of the problem are largely preserved here. This theory provides a convenient framework to describe solutions of all concentrations. 

The difficulty with the lattice models of hydrophobicity is that they do not do full justice to the HB network of water. However, a molecular-level theory is also rather difficult because the hydrophobic effect is a complex collective phenomenon involving many water molecules. The hydrophobic solute perturbs the HB network of water, resulting in a change of both entropy and enthalpy of the solute-solvent... 

As to the first question, lattice models do exhibit oil/water inlerfacial tensions that are reduced to various degrees from the value in the absence of amphiphile. For example, in the three-component model solved within mean-field theory, a reduction on the order of 30 was found in the oil/water interfacial tension at three-phase coexistence with the microemulsion [101]. When simulated so that fluctuations were included [102], the reduction increased to about a factor of 100, which is characteristic of a weak amphiphile. Other lattice models [103] have obtained reductions as large as a factor of 800, larger than that provided by even the strong amphiphile C6E3 [104]. 

Lattice models for liquids are rarely used nowadays. The same is true of lattice models for water. Nevertheless, the model presented in this section is of interest for three reasons First, it presents a prototype of an interstitial model having features in common with many models proposed for water and used successfully to explain some of the outstanding properties of water and aqueous solutions. Second, this model demonstrates some general aspects of the mixture model approach to the theory of water, for which explicit expressions for all the thermodynamic quantities in terms of molecular properties may be obtained. Finally, the detailed study of this model has a didactic virtue, being an example of a simple and solvable model. 

Various equations of state have been developed to treat association ia supercritical fluids. Two of the most often used are the statistical association fluid theory (SAET) (60,61) and the lattice fluid hydrogen bonding model (LEHB) (62). These models iaclude parameters that describe the enthalpy and entropy of association. The most detailed description of association ia supercritical water has been obtained usiag molecular dynamics and Monte Carlo computer simulations (63), but this requires much larger amounts of computer time (64—66). 

Taking into account the modes in which the water can be sorbed in the resin, different models should be considered to describe the overall process. First, the ordinary dissolution of a substance in the polymer may be described by the Flory-Huggins theory which treats the random mixing of an unoriented polymer and a solvent by using the liquid lattice approach. If as is the penetrant external activity, vp the polymer volume fraction and the solvent-polymer interaction parameter, the relationship relating these variables in the case of polymer of infinite molecular weight is as follows ... 

Before proceeding, it is important to recall the significant feature which appears to distinguish the cluster model from the two other prominent mixture models—i.e., the broken-down ice lattice and the clathrate hydrate cage structures. The latter two theories allow for the existence of discrete sites in water, owing to the cavities present either in the ice... 

Finally, one word about the lattice theories of microemulsions [30 36]. In these models the space is divided into cells in which either water or oil can be found. This reduces the problepi to a kind of lattice gas, for which there is a rich literature in statistical mechanics that could be extended to microemulsions. A predictive treatment of both droplet and bicontinuous microemulsions was developed recently by Nagarajan and Ruckenstein [37], which, in contrast to the previous theoretical approaches, takes into account the molecular structures of the surfactant, cosurfactant, and hydrocarbon molecules. The treatment is similar to that employed by Nagarajan and Ruckenstein for solubilization [38]. 

Thus, in the new theory by CBL, a model was formulated in which the librational properties of water played an important part. Even that was not the whole story. The calculation of the specific rate of water reorientation is a complex task. One cannot consider the reorienting water molecule as an isolated entity. If that were so, then one could work on the only basis of the rate of rotation of a gas molecule and calculate the rate. However, the water molecule is hydrogen bonded to other water molecules within the 3D lattice and therefore the reorientation involves the torsional stretching and breaking of the hydrogen bonds—an attempt that seldom succeeds. 

The molecular models adopt a statistical mechanical treatment of the adsorbed layer. In most cases a lattice structure is assumed and the differences of the various models lie in the effects on which the emphasis is put. There are two main molecular approaches one has been developed by Guidelli and his colleagues and the other is based on the LBS theory. Guidelli s approach emphasizes local order and hydrogen bonding among adsorbed water (solvent) molecules, whereas the models based on the LBS theory disregard local order and focus their attention on the polarizability of the adsorbed molecules. 

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