Soft spherical

In the case of finite star chains with very high functionality, the units are concentrated near and in the star core. Therefore, their theoretical behavior can approximately be described by a rigid sphere [2]. The form factor of a sphere presents a series of oscillations. The experimental data of stars with 128 arms [67] show a smooth function covering the first two oscillations of the sphere, followed by a peak coincident with the third oscillation and the asymptotic behavior for high q previously described for stars of lower functionalities. It seems that the chain resembles a soft spherical core with a peripheral region of considerably smaller density. 

Like many polymers, the fluidity or flow (rheological) properties of denrimers are of interest (Box 14.1). Dendrimers behave like soft, spherical particles, surrounded by a hard surface shell and hence are often described as core-shell type bodies. Hence the interior of dendrimers can be deformed according to the type of linkage and building blocks and my contain voids suitable for encapsulating guest species.2... 

Figure 8. Dependence of yield stress on silicone content of BPF carbonate-silicone block polymers. Line is calculated from Halpin-Tsai equations for moduli of composite of rigid matrix containing soft spherical inclusions.

Figure 2.5. Molecular D)oiainics simulation of self-diffusion in a dense fluid of "soft" spherical particles near a "hard" solid wail. The "wall" exerts no force on the particles but reverses the z-component of the velocity if a molecule attempts to cross it a is the length parameter in the repulsive part of the Lennard-Jones interaction. (Redrawm from J.N. Cape. J. Chem. Soc., Faraday Trans. II 78 (1982) 317.)...

Fig-6 Periodic box of concentrated dispersion of soft spherical particles. Each pair of particles at contact forms a facet, as shown in Fig. 4, that deforms according to Hertz s theory or similar law... 

Soft Spherical Confinement for Hydrogen Atom Revisited 148... 

The confinement model has been extensively used to analyze the hydrogen atom enclosed by hard and soft spherical boxes [1-98], with confining boxes of diverse geometrical shapes [31,88-98], and it has also been applied to studies of the helium atom [99-122], many-electron atoms [50, 52,55,123-131], molecules [132-142] and the harmonic oscillator [143-171], among others. 

SOFT SPHERICAL CONFINEMENT FOR HYDROGEN ATOM REVISITED... 

In this contribution, the adequacy of the TFDA.W method for the study of many-electron atom confinement within different confinement conditions -by closed and open boundaries - is explored. We begin by making a brief review of the main strategy followed in the TFDA.W method to account for the study of many-electron atoms confined by hard and soft spherical boxes. Here, important quantitative corrections to our previous studies are introduced, leading to better agreement with other reference calculations,... 

According to Ref. [65], an appropriate guiding criterion to define a physically plausible exterior orbital density consists in using the functional form for the wave function derived from exact solutions to the Schrodinger equation for the hydrogen atom confined by a soft spherical box [34], i.e. ... 

The stiffness of such a composite is close to the Hashin-Shtrikman lower bound for isotropic composites. Even if the spherical particles are perfectly rigid compared with the matrix, their stiffening effect at low concentrations is modest. Conversely, when the inclusions are more compliant than the matrix, spherical ones reduce the stiffness the least and platelet ones reduce it the most. Indeed, soft platelets are suggestive of crack-like defects. Soft platelets, therefore result not only in a compliant composite, but also a weak one. Soft spherical inclusions are used intentionally as crack stoppers to enhance the toughness of polymers such as polystyrene (high impact polystyrene), with a small sacrifice in stiffness. 

L6pez-Garcia, J.J., Grosse, C., and Homo, J., Numerical study of colloidal suspensions of soft spherical particles using the network method. 1. DC electrophoretic mohrUty, J. Colloid Interface Sci., 265, 327, 2003. 

The porous medium (sorbent, membrane, gel particle) can be made of soft spherical particles that swell quite a bit when immersed in a solvent they are called gek. If they swell in water, they are called hydrogels however, the polymers are crosslinked so that they retain their overall structure. Of course, the polymers are soft and compress under pressure a typical example would be gels (which are crosslinked) based on agarose (Figure 3.3.5B). Such gel particles are widely used in a variety of separation techniques for separating macromolecules/proteins, etc. 

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