Polymers interaction between spheres

In addition to an array of experimental methods, we also consider a more diverse assortment of polymeric systems than has been true in other chapters. Besides synthetic polymer solutions, we also consider aqueous protein solutions. The former polymers are well represented by the random coil model the latter are approximated by rigid ellipsoids or spheres. For random coils changes in the goodness of the solvent affects coil dimensions. For aqueous proteins the solvent-solute interaction results in various degrees of hydration, which also changes the size of the molecules. Hence the methods we discuss are all potential sources of information about these interactions between polymers and their solvent environments. 

Several theoretical tentatives have been proposed to explain the empirical equations between [r ] and M. The effects of hydrodynamic interactions between the elements of a Gaussian chain were taken into account by Kirkwood and Riseman [46] in their theory of intrinsic viscosity describing the permeability of the polymer coil. Later, it was found that the Kirdwood - Riseman treatment contained errors which led to overestimate of hydrodynamic radii Rv Flory [47] has pointed out that most polymer chains with an appreciable molecular weight approximate the behavior of impermeable coils, and this leads to a great simplification in the interpretation of intrinsic viscosity. Substituting for the polymer coil a hydrodynamically equivalent sphere with a molar volume Ve, it was possible to obtain... 

Here /ie and are effective masses of electron and hole, respectively. Near to bottom of conductivity band and near to top of valent band where dependence E from k is close to parabolic, electron and hole move under action of a field as particles with effective masses fie — h2l(d2Ec(k)ldk1) and jUh = —h2l( E (k)ldk ) [6]. In particular, in above-considered onedimensional polymer semiconductor /ie — /ih — h2AEQj2PiP2d2 [6]. As a first approximation, it is possible to present nanocrystal as a sphere with radius R, which can be considered as a potential well with infinite walls [6], The value of AE in such nanocrystal is determined by the transition energy between quantum levels of electron and hole, with the account Coulomb interaction between these nanoparticles. 

The interactions between two parallel plates or spheres in a solution of nonadsorbing polymers assumed to be spherical have been extensively investigated experimentally,4-6 theoretically,7-18 and by simulations.14,15 The... 

One model which has been extensively used to model polymers in the continuum is the bead-spring model. In this model a polymer chain consists of Nbeads (mers) connected by a spring. The easiest way to include excluded volume interactions is to represent the beads as spheres centered at each connection point on the chain. The spheres can either be hard or soft. For soft spheres, a Lennard-Jones interaction is often used, where the interaction between monomers is... 

The hard sphere interaction energy is an accurate approximation for short-range interactions between particles. This occurs when we have steric stabilization [33,34] due to polymer adsorption and electrostatic stabilization with a thin double layer [35,36] (i.e., high ionic... 

Two Spheres. The interaction of spheres which are covered with adsorbed polymer layers is expected to follow that of macroscopic surfaces. If the surfaces of the spheres are saturated, the force between them will be repulsive, and if they are unsaturated there will be an attraction caused by bridging (2.3). However special effects are expected to arise due to the finite sizes of spheres and macromolecules. Firstly, when the spheres are small, their radii may be comparable with the range of the bridging attraction. Secondly, when the macromolecules are large, one of them may saturate two spheres then it may keep them bound to each other even though they are both saturated. 

The virial coefficients reflect interactions between polymer solute molecules because such a solute excludes other molecules from the space that it pervades. The excluded volume of a hypothetical rigid spherical solute is easily calculated, since the closest distance that the center of one sphere can approach the center of another is twice the radius of the sphere. Estimation of the excluded volume of llexible polymeric coils is a much more formidable task, but it has been shown that it is directly proportional to the second virial coefficient, at given solute molecular weight. 

The SEC partition coefficient [6] (.K sec) was measured on a Superose 6 column for three sets of well-characterized symmetrical solutes the compact, densely branched nonionic polysaccharide, Ficoll the flexible chain nonionic polysaccharide, pullulan and compact, anionic synthetic polymers, carboxylated starburst dendrimers. All three solutes display a congruent dependence of K ec on solute radius, R. In accord with a simple geometric model for SEC, all of these data conform to the same linear plot of i sEc versus R. This plot reveals the behavior of noninteracting spheres on this column. The mobile phase for the first two solutes was 0.2M NaH2P04-Na2HP04, pH 7.0. In order to ensure the suppression of electrostatic repulsive interactions between the dendrimer and the packing, the ionic strength was increased to 0.30M for that solute. 

Star polymers are known to interact through an ultrasoft pair potential that is very different from that of the other soft spheres described above [123]. The energy of interaction between two identical stars with effective diameter <7 is of the form ... 

PMMA sufficiently close to the mixed phase that spin diffusion from it is important. Radial spin diffusion was assumed, with a sphere of intimately mixed phase of radius A and a shell of close PMMA of thickness L-A (up to a point of abutment with a neighbouring sphere). Figure 18.16 shows experimental data for the 60 40 PMMA/PVDF blend, which is fitted to A = 6 A, L = 12 A and 30% isolated PMMA. The same experiment was then employed [81], together with F- C H, F CP, to study the influence of PMMA tacticity on PMMA/PVDF miscibility, yielding evidence for a specific interaction between segments of the two polymers. The mixed PMMA/PVDF phase was determined to have a mean radius of 6-8 A, with some dependence on PMMA tacticity. Large differences were found in the... 

An analysis of the previously defined function B(T) for dilute solutions of polymers in polar solvents may be helpful for the understanding of the interaction between polymer and these solvents.33 Applying Frohlich s theory, in which deformation polarization is treated macroscopically, we consider the solution as a cpntinuous medium containing polar units. The dielectric constant of the continuous medium is taken as equal to the square of the refractive index of the solution n0. This value is very close to that of the solvent. Each polar unit is represented as a sphere of dielectric constant 0a, haying a point dipole located at its center. It must be stressed that polar units may be either whole... 

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