The Site Model Theory

The site model theory is based on transition state theory, and although first developed to explain the dielectric behaviour of crystalline solids [11,12] has also been applied to mechanical relaxations in polymers [13]. 

Assume that the applied stress a causes a small linear shift in the free energies of the sites such that 

Combining these equations and making suitable approximations gives a rate equation 

In this equation e is the instantaneous or unrelaxed elastic deformation and it is considered that each change in site population produces a proportionate change in strain by an amount e. 

Since RT is usually small compared with the equilibrium free energy difference we may approximate to 

In the simplest form of this theory there are two sites , each representing a particular local conformational state of the molecule, separated by an energy barrier, as shown in fig. 5.20. The AGs are the Gibbs free-energy differences per mole (constant pressure conditions are assumed). 

Assume that there are i molecules in the conformational state 1 represented by site 1 at a particular time and that the probability that any molecule initially in state 1 will make the transition to state 2 in any small interval of time dt is dt. The total number of molecules moving from site 1 to site 2 in unit time is then iV]2. A similar argument applies for transitions in the reverse direction, with V21 replacing and 2 replacing i. Thus, if n° and are the equilibrium values of i and 2, respectively, and v°2 and V21 are the corresponding equilibrium values, then 

Assume that an applied stress a disturbs the equilibrium by causing small changes in the free energies of the sites (and hence in the transition probabilities) and that, at any subsequent time, there are n more molecules in site 1 than the original equilibrium number. Then 

Now assume that the strain e is equal to an, where a is a constant, i.e. that each time a molecule changes site it causes the same increment of strain. It then follows from equation (5.19) that 

The probabilities V12 dt and V21 dt must be proportional to the probabilities that a molecule gains enough energy to surmount the respective barriers between the sites. Provided that the effect of stress on AGi and AG2 is small compared with their equilibrium values, it follows that 

---

Yield involves an irreversible deformation and takes place by a shearing mechanism in which molecules slide past one another. If molecules are to slide past each other, energy barriers have to be overcome. Raising the temperature of the polymer will make it easier for these barriers to be overcome, as diseussed already in relation to the site-model theory of mechanical relaxation in section 5.7.3 and as discussed in section 8.2.5 of the present ehapter in relation to yielding. 

We now take the intuitive step (to be justified below by the site model theory) that the viscoelastic behaviour can be directly related to a controlling molecular rate process with a constant activation energy. 

We have so far discussed two types of theories, those based on the site model, and thosebased on the WLF equation and its ramifications, which deal with time-temperature equivalence. The site model theories predict constant activation energies and are more applicable to relaxation transitions originating from localised chain motions, whereas the WLF equation theories deal with the glass transition behaviour in amorphous polymers. 

The discussion so far considers the theoretical shear stress of a crystal in the absence of thermal fluctuations. Frank [81] considered that there are always local thermal fluctuations, which must be taken into account. At any temperature T, there is a significant chance of thermal fluctuations in the timescale of the experiments supplying an energy up to 50 kT. Furthermore, this discussion only relates the yield stress to the elastic energy whereas U is strictly the activation enthalpy. Analogous to the site model theory (see Section 7.3), we should discuss the Gibbs free energy AG, where AG — TAS and the shear strain rate is... 

The site model appears to provide an adequate description of basic features of the dynamics of molecular motions in secondary transitions. On the other hand, however, attempts to calculate and interpret the extent of energy losses, i.e. the relaxation strength, in terms of the barrier model have so far been less successful. Theories employing a model with the same depth of potential energy minima1,76,77 do not seem suitable... 

The PRISM (Polymer-Reference-Interaction-Site model) theory is an extension of the Ornstein-Zernike equation to molecular systems [20-22]. It connects the total correlation function h(r)=g(r) 1, where g(r) is the pair correlation function, with the direct correlation function c(r) and intramolecular correlation functions (co r)). For a primitive model of a polyelectrolyte solution with polymer chains and counterions only, there are three different relevant correlation functions the monomer-monomer, the counterion-counterion, and the monomer-counterion correlation function [23, 24]. Neglecting chain end effects and considering all monomers as equivalent, we obtain the following three PRISM equations for a homogeneous and isotropic system in Fourier space ... 

Statistical thermodynamics is also changing in recent years. Newly developed RISM (Reference Interactions Site Model) theory has no restriction on the shape of solute species, in contrast to old theories in which spherical species are usually assumed. Ab initio calculations are being combined with molecular dynamic simulations. This combination becomes possible because of the improvement of high-speed computers. The polarization effect and multibody problem will be... 

Kinoshita, M. Okamoto, Y. Hirata, F. (1997) Solvation structure and stability of peptides in aqueous solutions analyzed by the reference interaction site model theory. Journal of Chemical Physics 107, 1586-1599... 

An alternative theoretical approach to determining gas solubility utilizes the reference interaction site model theory. This theory has been applied for the noble gases and is found to give reasonable agreement to experimental data... 

Kinoshita M, Hirata F (1997) Analysis of salt effects on solubility of noble gases in water using the reference interaction site model theory. J Chem Phys 106 5202-5215... 

T. Imai, A. Kovalenko and F. Hirata. Partial molar volume of proteins studied by the three-dimensional reference interaction site model theory. J. Phys. Ghem. B 109, 2005, 6658-6665. 

M.o. theory and the transition state treatment In 1942 Wheland proposed a simple model for the transition state of electrophilic substitution in which a pair of electrons is localised at the site of substitution, and the carbon atom at that site has changed from the sp to the sp state of hybridisation. Such a structure, originally proposed as a model for the transition state is now known to describe the (T-complexes which are intermediates in electrophilic substitutions... 

The logic that leads us to this last result also limits the applicability of the ensuing derivation. Applying the fraction of total lattice sites vacant to the immediate vicinity of the first segment makes the model descriptive of a relatively concentrated solution. This is somewhat novel in itself, since theories of solutions more commonly assume dilute conditions. More to the point, the model is unrealistic for dilute solutions where the site occupancy within the domain of a dissolved polymer coil is greater than that for the solution as a whole. We shall return to a model more appropriate for dilute solutions below. For now we continue with the case of the more concentrated solution, realizing... 

The lattice model that served as the basis for calculating ASj in the last section continues to characterize the Flory-Huggins theory in the development of an expression for AHj . Specifically, we are concerned with the change in enthalpy which occurs when one species is replaced by another in adjacent lattice sites. The situation can be represented in the notation of a chemical reaction ... 

One important class of integral equation theories is based on the reference interaction site model (RISM) proposed by Chandler [77]. These RISM theories have been used to smdy the confonnation of small peptides in liquid water [78-80]. However, the approach is not appropriate for large molecular solutes such as proteins and nucleic acids. Because RISM is based on a reduction to site-site, solute-solvent radially symmetrical distribution functions, there is a loss of infonnation about the tliree-dimensional spatial organization of the solvent density around a macromolecular solute of irregular shape. To circumvent this limitation, extensions of RISM-like theories for tliree-dimensional space (3d-RISM) have been proposed [81,82],... 

The integral equation method is free of the disadvantages of the continuum model and simulation techniques mentioned in the foregoing, and it gives a microscopic picture of the solvent effect within a reasonable computational time. Since details of the RISM-SCF/ MCSCF method are discussed in the following section we here briefly sketch the reference interaction site model (RISM) theory. 

We recently proposed a new method referred to as RISM-SCF/MCSCF based on the ab initio electronic structure theory and the integral equation theory of molecular liquids (RISM). Ten-no et al. [12,13] proposed the original RISM-SCF method in 1993. The basic idea of the method is to replace the reaction field in the continuum models with a microscopic expression in terms of the site-site radial distribution functions between solute and solvent, which can be calculated from the RISM theory. Exploiting the microscopic reaction field, the Fock operator of a molecule in solution can be expressed by... 

II. The change of the sign between these two models may be also important for clustering and the surface growth mechanism. In the selfconsistent theory, the silver atoms tend to establish bonds with palladium atoms, while in the non-selfconsistent theory, the Ag-Ag bonds are favored. This explains the differences between the concentration profiles calculated within the models I and II and the fact that they do not simply follow the variations of the on-site terms. 

A model called histone code theory includes more aspects of chromatin regulation which have been identified. The histone code theory predicts that histone acetylation and other posttranslational histone modifications serve as binding sites for regulatory proteins which mediate processes like gene transcription upon recruitment (see Fig. 2b) [3]. In this context histone modifications can be understood as... 

An alternative stream came from the valence bond (VB) theory. Ovchinnikov judged the ground-state spin for the alternant diradicals by half the difference between the number of starred and unstarred ir-sites, i.e., S = (n -n)l2 [72]. It is the simplest way to predict the spin preference of ground states just on the basis of the molecular graph theory, and in many cases its results are parallel to those obtained from the NBMO analysis and from the sophisticated MO or DFT (density functional theory) calculations. However, this simple VB rule cannot be applied to the non-alternate diradicals. The exact solutions of semi-empirical VB, Hubbard, and PPP models shed light on the nature of spin correlation [37, 73-77]. 

---