Tangent sphere model

Figure 4.10 AX4, AX3E, and AX2E2 molecules (a) tangent sphere models or domain models with spherical domains B is a bonding pair and E is a lone pair and (b) conventional bond line structures.

For AX molecules with no lone pairs in the valence shell of A, both the VSEPR model and the LCP model predict the same geometries, namely AX2 linear, AX3 equilateral triangular, AX4 tetrahedral, AX5 trigonal bipyramidal, and AX octahedral. Indeed Bent s tangent sphere model can be used equally as a model of the packing of spherical electron pair domains and as a model of the close packing of spherical ligands around the core of the central atom. 

Fig. 15. Tangent-sphere models of CF4, SiF4, hypothetical CPI , and SiFg-, based on conventional ionic and covalent radii (columns 1 and 2) and the electride ion model (column 3)...

Figure 8.1 The process of computing the incremental chemical potential involves adding one extra segment to an M - 1 segment chain moving in the solvent. The tangent hard sphere model of a (M — l)-mer (M = 5) is shown here. The dashed circles enclose the volume excluded to the centers of the solvent spheres.

Equation of State. A number of modeling studies of amorphous polymers have included PVT behavior, either as an end in itself, as a means to validate the potential ftinctions, or to investigate other aspects of polsrmer behavior (411). Polymer melts and glasses that have been studied in some detail include polyethylene (23,370,412), poly(dimethylsiloxane) (413), poly(ethylene oxide) (414), and poly(ethylene terephthalate) (78). Dendrimers having a variety of generations have been compared with linear chains in the respect of their equation of state by means of a tangent hard sphere model by Lue (415). 

Here 2. is the height of the equimolar dividing surface, and D is a measure of the thickness. Althoi a hyperbolic tangent arises naturally for the penetrable-sphere model (S S.S) and in the van der Waals theory of a system near its gas-liquid critical point (S 9.1) its use here is purely empirical indeed, we shall see in S 7.S that an exponential decay of p(z) at large values of 2 -2. is not correct for a Lennard-Jones potential that runs to r=9c. This equation is, however, a convenient one since it can be fitted to experimental points by inverting it to give... 

The focus of this chapter is on an intermediate class of models, a picture of which is shown in Fig. 1. The polymer molecule is a string of beads that interact via simple site-site interaction potentials. The simplest model is the freely jointed hard-sphere chain model where each molecule consists of a pearl necklace of tangent hard spheres of diameter a. There are no additional bending or torsional potentials. The next level of complexity is when a stiffness is introduced that is a function of the bond angle. In the semiflexible chain model, each molecule consists of a string of hard spheres with an additional bending potential, EB = kBTe( 1 + cos 0), where kB is Boltzmann s constant, T is... 

The wall-PRISM equation has been implemented for a number of hard-chain models including freely jointed [94] and semiflexible [96] tangent hard-sphere chains, freely rotating fused-hard-sphere chains [97], and united atom models of alkanes, isotactic polypropylene, polyisobutylene, and polydimethyl siloxane [95]. In all implementations to date, to my knowledge, the theory has been used exclusively for the stmcture of hard-sphere chains at smooth structureless hard walls. 

It might be said in extenuation of the hydridic model that, according to the ideas of Kimball (19), the H atom should enter a more or less spherical electron cloud representing an unpaired electron associated with a metal atom. Thus, LiH is represented in Kimball s theory as a pair of tangent electron cloud spheres, or spheroids, each comprising two electrons of opposite spin centered about a +3 and a +1 nucleus, respectively. This picture is equally applicable to the hydrogen in CH4, HC1, or a metallic hydride—i.e., in all cases hydrogen is surrounded by a pair of electrons. 

To address the hmitations of ancestral polymer solution theories, recent work has studied specific molecular models - the tangent hard-sphere chain model of a polymer molecule - in high detail, and has developed a generalized Rory theory (Dickman and Hall (1986) Yethiraj and Hall, 1991). The justification for this simplification is the van der Waals model of solution thermodynamics, see Section 4.1, p. 61 attractive interactions that stabilize the liquid at low pressure are considered to have weak structural effects, and are included finally at the level of first-order perturbation theory. The packing problems remaining are attacked on the basis of a hard-core model reference system. 

Thus, we first consider Eq. (8.10) for hard-core chain models, specifically tangent hard-sphere chain models (Dickman and Hall (1986) Yethiraj and Hall, 1991). Models and theories of the packing problems associated with hard-core molecules have been treated in Sections 4.3, 6.1, 7.5, and 7.6. We recall... 

Since the spring constant is complex due to dissipalion, the denominator never becomes zero. Equation 9 was first proposed by Dybwad [19]. In the limits of cal ca s > Eq. 9 reproduces the Sauerbrey equation (Eq. 2) and the simple-spring model (Eq. 4), respectively. Equation 9 can also be derived from Eq. 91 in Chap. 2 in this volume by expanding all tangents to first order. This amounts to a continuum model of the same experimental situation, where the contacts and the spheres correspond to a soft , first layer and a hard , second layer, respectively. 

As a first approximation " one can model the vinyl polymer as a freely jointed, tangent hard sphere chain as depicted on the second line of Figure 1. Thus each bond (of fixed length) is completely flexible with each site, including the side group site C, acting as a universal joint. Invoking the Flory ideality hypothesis, the intramolecular structure functions in Eq. (2.4) become "... 

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