Lattice model dispersions

Prausnitz and coworkers [91,92] developed a model which accounts for nonideal entropic effects by deriving a partition function based on a lattice model with three categories of interaction sites hydrogen bond donors, hydrogen bond acceptors, and dispersion force contact sites. A different approach was taken by Marchetti et al. [93,94] and others [95-98], who developed a mean field theory... 

Fig. 2. Water proton spin-lattice relaxation dispersion in Cab-O-Sil M- samples with two different degrees of compression. The solid lines were computed using the model as in Ref. (45).

On the basis of a lattice model, upper and lower bounds have been established for the entropy of dispersion of spherical globules of radius r and volume fraction in the continuous phase (ref. 20). Here r and are the "actual" radius of the globules (including the adsorbed layer of surfactants) and the corresponding... 

A/ is calculated in what follows by neglecting the interactions among globules. Expressions for the entropy of dispersion of the globules in the continuous phase were derived by Ruckenstein and Chi [12] on the basis of a lattice model, assuming (as is usually done in this kind of model) that the volume of a site is equal to the volume of a molecule... 

The Direct Lattice Sum. Dispersion forces between two atoms can be described by a potential function expressed in terms containing inverse powers of the internuclear separations, s. The simplest function of this sort includes a potential energy of attraction proportional to the inverse sixth power of the separation and a repulsion that is zero at distances of separation greater than a particular value se and infinite at separations less than sc. This is the so-called hard sphere or van der Waals model. Such an approximate potential function can be improved in two respects. Investigations of the second virial coefficient have revealed that the potential energy of repulsion is best described as proportional to the inverse twelfth power of the separation and the term in sr9, which accounts for the greater part of the total attraction potential, due to the attraction of mutually induced dipoles, should have added to it the dipole-quadrupole and quadrupole-quadru-pole attractions, expressed as terms in sr8 and s-10, respectively. The complete potential function for the forces between two atoms is, therefore ... 

Lattice theories [37] enable one to consider nonspecific physical forces (e.g., molecular dipole moments, induction effects, and London dispersion forces) and have been applied successfully to model nonideality in a wide range of mixtmes. Guggenheim [43] was the first to develop a quasichemical theory using lattice models. Wilson [44], Renon and Prausnitz [45], Abrams and Prausnitz [46], and Vera et al. [47] modified it for nomandom mixtures. Panayiotou and Vera... 

The Flory theory discussed in the next section is another important theory on rigid liquid crystalline polymers. Because of its clear picture of the lattice model and the incorporation of the Onsager theory, it has become a basic method for the theoretical study of liquid crystalline polymers. As a result of the constant efforts of Flory and his co-workers, the theory has been applied to binary and poly-disperse systems and also includes the soft interactions. 

The occurrence of a secondary phase separation inside dispersed phase particles, associated with the low conversion level of the p-phase when compared to the overall conversion, explains the experimental observation that phase separation is still going on in the system even after gelation or vitrification of the a-phase [26-31]. A similar thermodynamic analysis was performed by Clarke et al. [105], who analyzed the phase behaviour of a linear monodisperse polymer with a branched polydisperse polymer, within the framework of the Flory-Huggins lattice model. The polydispersity of the branched polymer was treated with a power law statistics, cut off at some upper degree of polymerization dependent on conversion and functionality of the starting monomer. Cloud-point and coexistence curves were calculated numerically for various conversions. Spinodal curves were calculated analytically up to the gel point. It was shown that secondary phase separation was not only possible but highly probable, as previously discussed. 

Wigner Lattice Model.—Marcel j a et al. have considered a Wigner lattice model in which each particle moves about in a sphere containing an equal and opposite charge centred on the lattice site. The surface potential of each particle in the dispersion is determined by solving the Poisson-Boltzmann... 

Bohlin has developed a statistical theory, applicable to colloidal dispersions, in which macroscopic flow is the consequence of co-operative rearrangements of microscopic elements. The model is based on a lattice microstructure whose elements can exist in stressed or relaxed states. Stress relaxation at constant strain is co-operative owing to the presence of an energy of interaction between neighbouring elements in different states. In common with all lattice models, the predictions are sensitive to the (somewhat arbitrary) choice of co-ordination number. 

Using die same lattice model, the change in enthalpy AH ix, for replacing one species by the other in adjacent cells might be derived. The van der Waals type interactions involved may arise fixim permanent or induced dipole-dipole and dispersion mechanisms. 

Lattice models seek to describe microemulsion structural domains down to the near-molecular level. These include models of amphiphilicity, imposed by mean field attractive and repulsive interactions applied to simplified diatomic or oligomeric amphiphiles, oil and water. Recent versions of these models include bending energy contributions for surfactants meeting at angles to approximate realistic molecular features and accurately capture microemulsion phase diagram. In all cases, microemulsion phase behavior is most accurately captured when the models consider key physical attributes (either exphc-itly or implicitly), including the balance between entropic (which tend to disperse oil-water into ever finer domains) and interfacial (which drive phase separation and place limits on domain curvature) contributions. 

Figure 16 shows typical proton spin-lattice relaxation dispersion data for polyethylene melts as an illustration of the three-component behavior of polymer melts. For comparison with model theories the chain-mode regime represented by component B is suited best and will be discussed in detail. It will be shown that the NMR relaxometry frequency window of typically 10 Hz< V <10 Hz (for proton resonance) almost exclusively probes the influence of chain modes represented by component B (compare Fig. 5). That is, the correlation function experimentally relevant for spin-lattice relaxation dispersion may be identified with component B according to...

Fig. 27a-c. Proton spin-lattice relaxation dispersion under conditions where Rouse dynamics is expected to apply. The theoretical curves have been calculated with the aid of Eq. 64. The validity of this model is restricted to (0 Ts. The positions on the frequency axes where the condition q)Ts=1 apphes are indicated by arrows for the segment fluctuation time Ts fitted to the experimental data. The Tj values are in accord with those derived from the Ti minimum data (see Fig. 14) where applicable, a Polyisobutylene (Mm =4,700 < 15,000), melt at 357 K [49]. b Polydimethylsiloxane (M, =5,200... 

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