The Segment Potentials

Unless one is willing to become involved in many intricacies, a lattice model with united atoms (segments) features segments which are all of equal size. The price we have to pay for this is that there is no unique way to convert from lattice units to real space coordinates. We will discuss this point in the Result sections in more detail. 

Without going into too much detail, we can note the segment potentials are composed of four terms. For a unit of type A they are given by  

The first term is a contribution to the segment potential that does not depend on the segment type. The value of this term is chosen such that the predicted density of molecules in the system is consistent with the incompressibility 

The second term in equation (9) is the usual electrostatic term. Here vA is the valency of the unit and e is the elementary charge, and ip(z) is the electrostatic potential. This second term is the well-known contribution accounted for in the classical Poisson-Boltzmann (Gouy -Chapman) equation that describes the electric double layer. The electrostatic potential can be computed from the charge distribution, as explained below. 

The fourth term is a polarisation term. Here E(z) = di/z/dz is the electric field at position z. In previously published SCF results for charged bilayers, this last term is typically absent. It can be shown that the polarisation term is necessary to obtain accurate thermodynamic data. We note that all qualitative results of previous calculations remain valid and that, for example, properties such as the equilibrium membrane thickness are not affected significantly. The polarisation term represents relatively straightforward physics. If a (united) atom with a finite polarisability of erA is introduced from the bulk where the potential is zero to the coordinate z where a finite electric field exists, it will be polarised. The dipole that forms is proportional to the electric field and the relative dielectric permittivity of the (united) atom. The energy gain due to this is also proportional to the electric field, hence this term is proportional to the square of the electric field. The polarisation of the molecule also has an entropic consequence. It can be shown that the free energy effect for the polarisation, which should be included in the segment potential, is just half this value 

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The SCF solution, i.e. the condition that the segment densities and the segment potentials are consistent with each other, is found for a canonical ensemble. This means that the number of molecules of each molecule type is fixed. As explained above, membranes should be modelled in a (N, p, y, T), i.e. 

The right-hand side of (132) indicates that the segment volume fractions are uniquely computed from the segment potentials. As mentioned above, we implement a freely jointed chain model, which ensures the chain connectivity, but which does not prevent backfolding of the chain to previously occupied lattice sites. For this chain model, the volume fractions can be computed efficiently using a propagator formalism, which is intimately related to the Edwards equation [91]. 

To solve the SCF equations, we make use of the discretisation scheme of Scheutjens and Fleer [69], It is understood that here we cannot give full details on the SCF machinery. For this we refer to the literature [67,70-72]. However, pertinent issues and approximations will be mentioned in passing. The radial coordinate system is implemented using spherical lattice layers r = 1,..., tm, where layers r = 1,..., 5 are reserved for the solid particle. The number of sites per layer is a quadratic function of the layer number, L r) o= i. The mean-field approximation is applied within each layer, which means that we only collect the fraction of lattice sites occupied by segments. These dimensionless concentrations are referred to as volume fraction (p r). We assume that the system is fully incompressible, which means that in each layer the volume fraction of solvent = 1 — (r) — volume fractions are the segment potentials u r). The segment potentials can be computed from the volume fractions as briefly mentioned below. 

With this coordinate system in place, we can now focus on the computation of the volume fractions and the segment potentials. 

In the antisymmetric case the possible reaction path ranges from the MEP (when cui and coq are comparable) to the sudden path (coi <4 coq), when the system waits until the q vibration symmetrizes the potential (the segment of path with Q = Qo) and then instantaneously tunnels in the symmetric potential along the line q = 0. All of these types of paths are depicted in fig. 17. ... 

The explicit mathematical treatment for such stationary-state situations at certain ion-selective membranes was performed by Iljuschenko and Mirkin 106). As the publication is in Russian and in a not widely distributed journal, their work will be cited in the appendix. The authors obtain an equation (s. (34) on page 28) similar to the one developed by Eisenman et al. 6) for glass membranes using the three-segment potential approach. However, the mobilities used in the stationary-state treatment are those which describe the ion migration in an electric field through a diffusion layer at the phase boundary. A diffusion process through the entire membrane with constant ion mobilities does not have to be assumed. The non-Nernstian behavior of extremely thin layers (i.e., ISFET) can therefore also be described, as well as the role of an electron transfer at solid-state membranes. 

Fig. 2. Sketch of the interaction potential between segments m and n. The potential can be decomposed into a hard core repulsive potential Unm (hard) and a weak attractive potential Unn, (attr)...

In order to relax 1 mol of compacted polymeric segments, the material has to be subjected to an anodic potential (E) higher than the oxidation potential (E0) of the conducting polymer (the starting oxidation potential of the nonstoichiometric compound in the absence of any conformational control). Since the relaxation-nucleation processes (Fig. 37) are faster the higher the anodic limit of a potential step from the same cathodic potential limit, we assume that the energy involved in this relaxation is proportional to the anodic overpotential (rj)... 

As predicted, the experimental results follow a semilogarithmic dependence of /max on Ec (Fig. 41). When the experiments were repeated by potential steps to different anodic potentials, from different anodic potentials every time, parallel lines were obtained. The slopes are related to the charge consumption required during cathodic polarization to close and compact 1 mol of polymeric segments. A value of 4626 C mol-1, independent of the anodic potential, was obtained for z. ... 

This equation makes it possible to obtain the dependence between the degree of closure and the cathodic potential at which the polymer is reduced. The probability ( ) of a conformational change that will allow the reduction and compaction of a segment can be expressed as the inverse of the relaxation time. If all the other terms of Eq. (9) are included in P, then... 

The very first question that comes to mind when dealing with giant telescopes is the cost-effective feasibility of its optics. Assuming classical materials for the segments blanks, however, there is no need for a very substantial increase in production capacity from existing suppliers provided that the segment size remains below 2-m. Moderately lightweight Silicon Carbide is also considered as a serious and potentially cost-effective candidate, for its superior thermal performance and specific stiffness. 

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