Model continuum

Theoretical models may be tested against more accurate theoretical treatments or against experiment. In this chapter, we concentrate on the latter. Because water is not only the solvent for which the most data appear to exist, 34-337 but is moreover the most important from a biological standpoint, it is the solvent on which we will focus. 

It is important to emphasize that only the solvent-accessible surface area (SASA), the generalized Born/surface area (GB/SA), and the full AMI—SM2 models purport to address local, nonelectrostatic effects. There is no a priori reason to expect the remaining purely electrostatic models to correlate closely with experiment nevertheless, it is worthwhile to examine the cross-correlations. We will highlight some of the most interesting trends. 

The SASA modeU i enjoys the second-best correlation to experiment for the uncharged solutes, with slope and intercept values quite near the ideal unity and zero, respectively. This is particularly impressive given its great simplicity and extremely rapid application. On the other hand, the range of molecules to which it has been applied is fairly simple—a small handful of functionalities on simple alkyl chains of varying length. It is noteworthy that as functionality becomes more complex, performance appears to degrade, as for acetamide. 

Free Energies, and Experimental Free Energies (kcal/mol) of Solvation for Neutral and (continued) 

The GB/SA model also correlates quite well with experiment for the neutral solutes. As expected, the regression slope and intercept are also nearly ideal. Conversely, based on the four ions for which results have been reported, there seems to be a tendency to overestimate ionic solvation free energies, but definite conclusions cannot be drawn from so small a sampling. Whereas the available data span a larger range of functionality than do those from the SASA model, there is still a paucity of results for complex and polyfunctional solutes. It would be very interesting to see how robust the model is in such instances. 

The modeling of solids as a continuum with a given shear strength, and the like is often used for predicting mechanical properties. These are modeled using hnite element or hnite difference techniques. This type of modeling is usually employed by engineers for structural analysis. It will not be discussed further here. 

One way to model a solid is to use software designed for gas-phase molecular computations. A large enough piece of the solid can be modeled so that the region in the center for practical purposes describes the region at the center of an infinite crystal. This is called a cluster calculation. 

As described in the chapter on band structures, these calculations reproduce the electronic structure of infinite solids. This is important for a number of types of studies, such as modeling compounds for use in solar cells, in which it is important to know whether the band gap is a direct or indirect gap. Band structure calculations are ideal for modeling an infinite regular crystal, but not for modeling surface chemistry or defect sites. 

Traditionally, modeling in chemical engineering has invoked continuum descriptions of momentum, mass, and energy conservation (Bird et al., 1960) 

When applied to partide-reinforced polymer composites, micromechanics models usually follow certain basic assumptions linear elasticity of fillers and polymer matrix the fillers are axisymmetric, identical in shape and size, and can be characterized by parameters such as aspect ratio well-bonded filler-polymer interface, so no interfacial slip is considered filler-matrix debonding and matrix microcraddng. Further details of some important preliminary concepts such as hnear elastidty, average stress and strain, composites average properties, and the strain concentration and stress concentration tensors can be found in preview literature [48-50]. 

The Halpin-Tsai equations and the Mori-Tanaka model are the most used to predict mechanical properties of composites. The Halpin-Tsai equations predict stiffness of the unidirectional composites as a function of aspect ratio. In this model, the longitudinal stiffness and transverse engineering moduli are expressed in the following general form  

The Mori-Tanaka model is derived based on the principles of Eshdby s inclusion model for predicting an elastic stress field in and around ellipsoidal filler in an infinite matrix. The complete analytical solutions for longitudinal ( u) and transverse ( 22) elastic moduli of an isotropic matrix filled with aligned spherical inclusions are [51] 

There are numerous advantages of using continuum models. They are widely used for system design and optimization. Continuum models tell us important information about the system, e g., discharge curves, state-of-health of the battery, cycle life behavior and subsequently capacity fade rate, etc. Battery models are also useful in predicting non-measurable internal variables such as solution phase concentration, solid phase concentration etc. This can be used to observe or measure buildup or loss of a certain chemical species within the domain of the battery and can be used efficient- 

List of Additional Expressions Defining Effeetive lonie Condnetivity, Effeetive Eleetronie Condnetivity, Effeetive Diffnsivity Term Correeted for Porosity and Open Cirenit Potentials for a LiCoOi-LiCe Li-Ion Battery 

Continmrm models are also an important design tool. They are used to optimize system design data (e.g., length and thickness of 

To overcome problems arising fi-om the finite system size used in MC or MD simulation, boundary conditions are imposed using periodic-stochastic approximations or continuum models. In particular, in stochastic boundary conditions the finite system is not duplicated but a boundary force is applied to interact with atoms of the system. This force is set as to reproduce the solvent regions that have been neglected. Anyway, in general any of the methods used to impose boundary conditions in MC or MD can be used in the QM/MM approach. 

Originally, continuum models of solvent were formulated as dielectric models for electrostatic effects. In a dielectric model the solvent is modeled as a continuous medium, usually assumed homogeneous and isotropic, characterized by a scalar, static dielectric constant e. This model of the solvent, that can be referred to the original work by Bom, Onsager and Kirkwood 60-80 years ago, assumes linear response of the solvent to a perturbing electric field due to the presence of the molecular solute. 

This simple definition then has been largely extended to treat more complex phenomena, including not only electrostatic effects and nowadays continuum solvation models represent very articulate methodologies able to describe different systems of increasing complexity. 

The history of, and the theory behind, continuum solvation models have been described exhaustively in many reviews and articles in the past, so we prefer not to repeat them here. In addition, so large and continuously increasing is the amount of examples of theoretical developments on one hand, and of numerical applications on the other, that we shall limit our attention to a brief review of the basic characteristics of these models which have gained wide acceptance and are in use by various research groups. 

Jacopo Tomasi, Benedetta Mennucci, Chiara Cappelli 

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Within the framework of the same dielectric continuum model for the solvent, the Gibbs free energy of solvation of an ion of radius and charge may be estimated by calculating the electrostatic work done when hypothetically charging a sphere at constant radius from q = 0 q = This yields the Bom equation [13]... 

Mineva T, Russo N and Sicilia E 1998 Solvation effects on reaction profiles by the polarizable continuum model coupled with Gaussian density functional method J. Oomp. Ohem. 19 290-9... 

Van der Zwan G and Hynes J T 1982 Dynamical polar solvent effects on solution reactions A simple continuum model J. Chem. Phys. 76 2993-3001... 

Continuum models go one step frirtlier and drop the notion of particles altogether. Two classes of models shall be discussed field theoretical models that describe the equilibrium properties in temis of spatially varying fields of mesoscopic quantities (e.g., density or composition of a mixture) and effective interface models that describe the state of the system only in temis of the position of mterfaces. Sometimes these models can be derived from a mesoscopic model (e.g., the Edwards Hamiltonian for polymeric systems) but often the Hamiltonians are based on general symmetry considerations (e.g., Landau-Ginzburg models). These models are well suited to examine the generic universal features of mesoscopic behaviour. 

An even coarser description is attempted in Ginzburg-Landau-type models. These continuum models describe the system configuration in temis of one or several, continuous order parameter fields. These fields are thought to describe the spatial variation of the composition. Similar to spin models, the amphiphilic properties are incorporated into the Flamiltonian by construction. The Flamiltonians are motivated by fiindamental synnnetry and stability criteria and offer a unified view on the general features of self-assembly. The universal, generic behaviour—tlie possible morphologies and effects of fluctuations, for instance—rather than the description of a specific material is the subject of these models. 

By virtue of their simple stnicture, some properties of continuum models can be solved analytically in a mean field approxunation. The phase behaviour interfacial properties and the wetting properties have been explored. The effect of fluctuations is hrvestigated in Monte Carlo simulations as well as non-equilibrium phenomena (e.g., phase separation kinetics). Extensions of this one-order-parameter model are described in the review by Gompper and Schick [76]. A very interesting feature of tiiese models is that effective quantities of the interface—like the interfacial tension and the bending moduli—can be expressed as a fiinctional of the order parameter profiles across an interface [78]. These quantities can then be used as input for an even more coarse-grained description. 

Ire boundary element method of Kashin is similar in spirit to the polarisable continuum model, lut the surface of the cavity is taken to be the molecular surface of the solute [Kashin and lamboodiri 1987 Kashin 1990]. This cavity surface is divided into small boimdary elements, he solute is modelled as a set of atoms with point polarisabilities. The electric field induces 1 dipole proportional to its polarisability. The electric field at an atom has contributions from lipoles on other atoms in the molecule, from polarisation charges on the boundary, and where appropriate) from the charges of electrolytes in the solution. The charge density is issumed to be constant within each boundary element but is not reduced to a single )oint as in the PCM model. A set of linear equations can be set up to describe the electrostatic nteractions within the system. The solutions to these equations give the boundary element harge distribution and the induced dipoles, from which thermodynamic quantities can be letermined. 

Claverie P, J P Daudey, J Lmglet, B Pullman, D Piazzola and M J Huron 1978. Studies of Solvent Effects. I. Discrete, Continuum and Discrete-Continuum Models and Their Comparison for Some Simple Cases NH, CH3OH and substituted NH4. Journal of Physical Chemistry 82 405-418. 

Constanciel R and R Contreras 1984. Self-Consistent Field Theory of Solvent Effects Representation by Continuum Models - Introduction of Desolvation Contribution. Theoretica Chimica Acta 65 1-11. 

Qiu D, P S Shenl F P HoUinger and W C Still 1997. The GB/SA Continuum Model for Solvation. / Fast Anal5dical Method for the Calculation of Approximate Bom Radii. Journal of Physical Chcniistr 101 3005-3014. 

The most accurate calculations are those that use a layer of explicit solvent molecules surrounded, in turn, by a continuum model. This adds the additional... 

The present discussion of continuum modeling of dynamic fracture is not an exhaustive review. Rather, it points out the variety of approaches which have been, and are still being, pursued to provide methods for calculating dynamic fracture phenomena. Such work is still quite active and considerable effort... 

Although the continuum model of the ion could be analyzed by Gauss law together with spherical symmetry, in order to treat more general continuum models of electrostatics such as solvated proteins we need to consider media that have a position-specific permittivity e(r). For these a more general variant of Poisson s equation holds ... 

The continuum model, in which solvent is regarded as a continuum dielectric, has been used to study solvent effects for a long time [2,3]. Because the electrostatic interaction in a polar system dominates over other forces such as van der Waals interactions, solvation energies can be approximated by a reaction field due to polarization of the dielectric continuum as solvent. Other contributions such as dispersion interactions, which must be explicitly considered for nonpolar solvent systems, have usually been treated with empirical quantity such as macroscopic surface tension of solvent. 

A variety of methodologies have been implemented for the reaction field. The basic equation for the dielectric continuum model is the Poisson-Laplace equation, by which the electrostatic field in a cavity with an arbitrary shape and size is calculated, although some methods do not satisfy the equation. Because the solute s electronic strucmre and the reaction field depend on each other, a nonlinear equation (modified Schrddinger equation) has to be solved in an iterative manner. In practice this is achieved by modifying the electronic Hamiltonian or Fock operator, which is defined through the shape and size of the cavity and the description of the solute s electronic distribution. If one takes a dipole moment approximation for the solute s electronic distribution and a spherical cavity (Onsager s reaction field), the interaction can be derived rather easily and an analytical expression of theFock operator is obtained. However, such an expression is not feasible for an arbitrary electronic distribution in an arbitrary cavity fitted to the molecular shape. In this case the Fock operator is very complicated and has to be prepared by a numerical procedure. 

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... 

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