Site incompressibility

The structure of a simple mixture is dominated by the repulsive forces between the molecules [15]. Any model of a liquid mixture and, a fortiori of a polymer solution, should therefore take proper account of the configurational entropy of the mixture [16-18]. In the standard lattice model of a polymer solution, it is assumed that polymers live on a regular lattice of n sites with coordination number q. If there are n2 polymer chains, each occupying r consecutive sites, then the remaining m single sites are occupied by the solvent. The total volume of the incompressible solution is n = m + m2. In the case r = 1, the combinatorial contribution of two kinds of molecules to the partition function is... 

One usually distinguishes two types of lattice models. The first type may be called lattice-gas models. In this case, the number of molecules in the system is less than the number of available sites. In other words, there are vacant sites. The second type of lattice models may be called lattice fluids. In this case, all lattice sites are filled exactly by the molecular components in the system the system is considered to be incompressible. It is easily shown that a two-component incompressible lattice fluid model can be mapped on a one-component lattice gas one. In other words, it is possible to interpret vacant sites to be occupied by a ghost ... 

For historical reasons, the incompressible lattice-fluid system description is used, even if the distribution of one of the components is coupled to the distribution of vacant sites. Constant pressure SCF calculations are the same as constant chemical potential calculations for the vacant sites. These conditions are used below. 

The site entropy is thus a sensible candidate for describing fluid relaxation outside the immediate vicinity of the glass transition. In a more precise language, is actually an entropy density, and the maximum in Sc T) derives from an interplay between changes in the entropy and fluid density as the temperature is varied. Explicit calculations demonstrate that the maximum in Sc T) disappears in the limit of an incompressible fluid, which is physically achieved in the limit of infinite pressure. The pressure dependence of Sc T) is described in Section X, where it is found that the maximum in Sc T) becomes progressively shallower and 7a becomes larger with increasing pressure. 

The solution is assumed ideal. It is incompressible all lattice sites are filled with some species of molecule. All species of molecules at the lattice sites are of equal (or nearly equal) size. For physical reasons, only one molecule can occupy each lattice site. Since there is a distribution of adsorptive energies within the zeolite, corresponding to the locally varying electrostatic field, the adsorption problem is approached from the standpoint of a superposition of several solutions - all the sites in each being identical. The number of solutions that must be considered equals the number of different adsorptive energy sites that are found within the zeolite. 

We consider a simple cubic lattice with a coordination number 2 = 6. For an incompressible polymer solution, each lattice site is occupied by a solvent molecule or by a segment of polymer chain. The attraction interactions between the nearest-neighbor sites are characterized by a reduced exchange energy = fi( pp + ss 2eps) between a segment p and a solvent s, where is the attractive energy of an i—j pair. 

The structure of the interface formed by coexisting phases is well described by the Cahn-Hilliard approach [53] (developed in a slightly different context by Landau and Lifshitz [54]) extended to incompressible binary polymer mixtures by several authors [4,49,55,56]. The central point of this approach is the free energy functional definition that describes two semi-infinite polymer phases <]), and 2 separated by a planar interface (at depth z=0) and the composition ( )(z) across this interface. The relevant functional Fb for the free energy of mixing per site volume Q (taken as equal to the average segmental volume V of both blend components) and the area A of the interface is expressed by... 

Recently some of our results [74] presented in Fig. 7 were analyzed by Dudo-wicz et al. [47]. In their lattice cluster model each segment can occupy several lattice sites in order to express the segment molecular structure and local correlations. Incompressibility is lifted and unoccupied lattice sites are introduced. The related theory [128] of interfacial properties independently describes the composition profiles of both blend components. Computations [47] performed by Dudowicz well evaluate qualitatively the coexistence curve, the interfacial width as well as the corresponding ( -dependent effective SANS interaction parameter [73] by very similar sets of three contact (van der Waals) energies eHH, eDD> and hd-... 

We now consider the problem of minimization of the free energy of a system with a fixed total volume and neglect compressions of the molecules. In the two-phase region of the phase diagram, the system consists of two coexisting phases (i = 1,2) with a volume fraction 0/ of B particles in each of the phases. In addition to the volume fractions, we have to specify the number of sites (i.e., volume in an incompressible system) occupied by each macroscopic phase and we denote as Ni the number of sites in each phase. Since the total number of sites, N, in the incompressible system is fixed, N and N2 are constrained ... 

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. 

If we now assume that the mixture is incompressible, every site must be occupied either by an A monomer or by a B monomer, so 0a + 0b = 1- We can express this as... 

Here, ( ) = Pm Vm/tj is the volume fraction of sites of type M, Vm is the volume of a site of type M, Vq is a reference volume , and Sm is the total neutron scattering length of a site of species M. Note that this chi-parameter will generally diverge as < )m - 0 due to the unrealistic incompressibility assumption. The SANS chi is, by construction, equivalent to the incompressible Flory value at the spinodal... 

The incompressible chi-parameter defined in Eq. (6.16) has also been extensively studied. Many of the numerical results for site volume and/or statistical segment length asymmetric athermal Gaussian drain blends are adequately reproduced at a qualitative level by the analytic thread model discussed in Sect. 2. For an athermal stiffness blend of very long Gaussian threads the k = 0 direct correlation functions are [23,62] ... 

The Flory-Huggins theory in its simplest form considers all lattice sites to be occupied by one component or the other, in other words, incompressible solutions. It can be applied only to solutions sufficiently concentrated to have uniform segment density. It also assumes that there are no energetically preferred arrangements of polymer segments and solvent molecules in the lattice. Newer theories have extended the treatment to lattices with empty sites. 

A serious deficiency in the classical theory, as exemplified by equation (4.5), is the assumption of incompressibility. This deficiency can easily be remedied by the addition of free volume in the form of holes to the system. These holes will be about the size of a mer and occupy one lattice site. In materials science and engineering, holes are frequently called vacancies. Imagine that a multicomponent mixture is mixed with No holes of volume fraction vo. Then the entropy of mixing is... 

The design of a PEM-type electrolyzer is relatively simple. As Figure 2.10 illustrates, it comprises a stack of elementary cells connected in a series by bipolar plates. Each cell comprises two electrodes separated by a Proton Exchange Membrane (PEM). Each electrode is made up of a thin catalytic layer which is the site of the oxidation (anode) or reduction reaction (cathode), and layers of porous materials which act as current distributors/receivers depending on the electrode in question. In general, this porous material will be incompressible (titanium) at the anode and compressible (carbon) at the cathode in order to add mechanical flexibility when the whole ensemble is compressed. 

Solids are nearly incompressible and are rigid, not flnid. Kinetic-molecular theory explains this by saying that the particles making up a solid (which may be atoms, molecules, or ions) exist in close contact and (unlike those in a gas or liquid) do not move about but oscillate or vibrate about fixed sites. This explains the rigidity of a solid. And, of course, the compact structure explains its incompressibility. Figure 11.2 compares the gaseous, liquid, and solid states in kinetic-molecular terms. 

A solid is a nearly incompressible state of matter with a well-defined shape, because the units (atoms, molecules, or ions) making up the solid are in close contact and in fixed positions or sites. In the next section, we will look at the kinds of forces holding the units together in different types of solids. In later sections, we will look at crystalline solids and their structure. 

Gases are composed of molecules in constant random motion throughout mostly empty space. This explains why gases are compressible fluids. Liquids are also composed of molecules in constant random motion, but the molecules are more tightly packed. Thus, liquids are incompressible fluids. Solids are composed of atoms, molecules, or ions that are in close contact and oscillate about fixed sites. Thus, solids are incompressible and rigid rather than fluid. 

Ceo is ti = 1.034 nm = 10.34 A). The lattice constant of the FCC crystal composed ofCeo units is 14.16 A,just3% smaller than the value 14.62A, which is what we would expect for incompressible spheres in contact with their nearest neighbors. The free space between such spheres is enough to accommodate other atoms. In fact, structures in which this space is occupied by alkali atoms (Li, Na, K, Rb and Cs) have proven quite remarkable they exhibit superconductivity at relatively high transition temperatures (see Table 13.1). The positions occupied by the alkali atoms are the eight tetrahedral and four octahedral sites of the cubic cell, shown in Fig. 13.7. When all these sites are occupied by alkali atoms the structure has the composition M3C60. Solids with one, two, four and six alkali atoms per Ceo have also been observed but they correspond to structures with lower symmetry or to close-packed structures of a different type (like the BCC crystal). All these crystals are collectively referred to as doped fullerides . 

Avogadro s number Wa. (bj is the coherent scattering length of atom j and the volume of both a lattice site and a monomer). The definition of Eq. 4 implies an incompressible melt as no free volume has been considered. The structure factor S(Q) represents an average over all chain conformations as indicated by the brackets in Eq. 5 it is defined in units of a molar volume [cm /mol] and is identical to the partial structure factor Saa(Q) determined by intra- and intermolecular correlation between the monomers of polymer A. A separation of Saa(Q) into intra- E(Q) and intermolecular W(Q) interference gives... 

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