Fixed charge approximation

One of the deficiencies of the force fields in use today is that they commonly use the fixed charge approximation, whereas in reality the atomic charges vary in response to changes of both the molecular conformation and the environment. Obviously, a scheme with geometry-dependent charges would extend capabilities of the force field methods. 

The dipole moment components as a function of time are usually computed with a fixed charge approximation. Values of atomic charges are often taken from quantum chemical calculations of representative models, although absolute charge values play no role (other than influencing the MD trajectory) unless absolute infrared intensities are to be computed. FFowever, the relative values of charges are of importance for reproducing relative band intensities. 

With reference to an actual polymer molecule we should, of course, speak of the potential at a point within the molecule, since the potential will decrease radially from its center in the manner dictated by the spatial distribution of the fixed charges (which like the segment density, may often be approximated by a Gaussian distribution) and that of the counter-ions. For the purpose of the present qualitative discussion, however, we refer merely to the potential finside the molecule. 

Fig. 23. Typical variation of the ratio n+ / na of the concentrations of H+ and H°, respectively, across ap-n junction, assuming rapid charge-change processes. The dotted, dashed and full curves were calculated assuming no bias, 2.02 V, and 9.88 V reverse bias, respectively, with a distribution of fixed charge in the junction approximately the same as that of the sample used for Fig. 21 before passivation and with the additional (arbitrary) choice of parameters eT+ = em, ed= -0.25 eV, t o/t0+ =. 001, and T = 200°C. 

One of the simplest equations is obtained using the Debye-Hiickel approximation (for low potentials) and the superposition principle. The latter assumes that the unperturbed potential near a charged surface can be simply added to that potential due to the other (unperturbed) surface. Thus, for the example shown in the Figure 6.12, it follows that /m = 2 /d/2- This is precisely valid for Coulomb-type interactions, where the potential at any point can be calculated from the potentials produced by each fixed charge, individually. However, the Poisson-Boltzmann equation is non-linear (this has to do with the fact that in the diffuse double-layer the ions are not fixed but move because of their kinetic energy) and so this is formally not correct although it still offers a useful approximation. 

A terminological remark is due. An equilibrium between two media with different fixed charge density (e.g., an ion-exchanger in contact with an electrolyte solution) is occasionally termed the Donnan equilibrium. The corresponding potential drop between the bulks of the respective media is then termed the Donnan potential. By the same token, we speak of the local Donnan equilibrium and the local Donnan potential, referring, respectively, to the local equilibrium and the interface potential jump at the surface of discontinuity of the fixed charge density, considered in the framework of the LEN approximation. 

One s first reaction is to reject the adequacy of casting the problem as a linear one, but, as Grossmann and Santibanez show, the use of discrete (zero/one) decisions allow one to include to a very good approximation many of the nonlinearities. For example, a zero/one variable can be associated with the existence or non-existence of a unit. In the cost function that discrete variable can cause one to add in a fixed charge for the unit only if it exists. Also one can define a continuous "flow11 variable for the unit which can be forced to zero if the unit does not exist by the linear constraint ... 

In this chapter, mathematical procedures for the estimation of the electrical interactions between particles covered by an ion-penetrable membrane immersed in a general electrolyte solution is introduced. The treatment is similar to that for rigid particles, except that fixed charges are distributed over a finite volume in space, rather than over a rigid surface. This introduces some complexities. Several approximate methods for the resolution of the Poisson-Boltzmann equation are discussed. The basic thermodynamic properties of an electrical double layer, including Helmholtz free energy, amount of ion adsorption, and entropy are then estimated on the basis of the results obtained, followed by the evaluation of the critical coagulation concentration of counterions and the stability ratio of the system under consideration. 

We consider spherical particles, and, without loss of generality, the fixed charge in the membrane is assumed to be negative. For simplicity, we assume that the distribution of fixed charge is uniform. The variation of the electrical potential is governed by Eq. (1) with m = 2, and the boundary conditions described by Eqs. (2)-(5). We use the approximate perturbation solution expressed by Eqs. (37) and (38)-(40). Suppose that the membrane is thick, and i Don and [/d are related by Eq. (48). 

Certain expenses are always present in an industrial plant whether or not the manufacturing process is in operation. Costs that are invariant with the amount of production are designated as fixed costs or fixed charges. These include costs for depreciation, local property taxes, insurance, and rent. Expenses of this type are a direct function of the capital investment. As a rough approximation, these charges amount to about 10 to 20 percent of the total product cost. 

As indicated in Fig. 11-7, the optimum reflux ratio occurs at the point where the sum of fixed charges and operating costs is a minimum. As a rough approximation, the optimum reflux mho usually falls in the range of 1.1 to 1.3 times the minimum reflux ratio. The following example illustrates the general method for determining the optimum reflux ratio in distillation operations. 

We give below an approximate expression for sed for fh case where the electrolyte is symmetrical with valence z and bulk concentration (number density) n°°, and the fixed charge density is constant, Pflx(r) = p = constant. We consider the case where... 

FIGURE 28.3 Membrane potential E, contributions from the diffusion potential and the Donnan potential difference AiJ/y)on — / don / don functions of the density of memhrane-fixed charges N. The values of the parameters used in the calculation are the same as those in Fig. 28.2. The dashed line is the approximate result for E (Eq. (28.21)). As A—> CX3, ni tends to the Nemst potential for cations (59 mV in the present case). From Ref. [7]. 

Nevertheless, these methods are mostly applied with fixed charges (even if these are chosen in a sophisticated way) and with pairwise additivity approximation as well as with the neglect of nuclear quantum effects. Suggestions for polarizable models appeared in literature mainly for water [23], The quality of potential parameterization varies from system to system and from quantity to quantity, raising the question of transferability. Spontaneous events like reactions cannot appear in simulations unless the event is included in the parameterization. Despite these problems, it is possible to reproduce important quantities as structural, thermodynamic and transport properties with traditional MD (MC) mainly due to the condition of the nanosecond time scale and the large system size in which the simulation takes place [24],... 

---