Star polymers coefficient

The branching coefficient, of star polymers is defined as the ratio of the radius of gyration of branched and linear polymers having the same molecular weight [67],i.e.,... 

It is not clear why this transition should occur at such a higher level of arm entanglement for polystyrene stars than for other star polymers. This observation is in direct conflict with the standard assumption that through a proper scaling of plateau modulus (Go) and monomeric friction coefficient (0 that rheological behavior should be dependent only on molecular topology and be independent of molecular chemical structure. This standard assumption was demonstrated to hold fairly well for the linear viscoelastic response of well-entangled monodisperse linear polyisoprene, polybutadiene, and polystyrene melts by McLeish and Milner [24]. 

Entangled star polymers relax by arm retractions with relaxation times and viscosities exponentially large in the number of entanglements per arm NJNg [Eqs (9.58) and (9.61)]. This leads to exponentially small diffusion coefficients [Eq. (9.62)] for entangled star polymers. 

It should also be noted that a similar treatment is possible for the translational hydrodynamic radius, Rhj, obtained from measurements of translational diffusion coefficients or sedimentation coefficients of branched polymers. One may define a parameter gn = Rh,fb/Rhji - the ratio of the hydrodynamic radius of the branched polymer relative to that of a linear polymer of the same molecular weight. Again, it is expected that gH < 1. For star polymers with uniform subchain lengths having... 

The osmotic coefficient 0 has been measured both in solutions of strongly dissociating poly[2-(methacryloyloxy)ethyl]-trimethylammonium iodide (PMETAI) [48], and weakly dissociating PAA [43] star polymers. 

In Fig. 8b, we show the osmotic coefficients for PAA stars that differ with respect to the numbers of arms at the given degree of neutralization. In accordance with theoretical predictions, the osmotic coefficient (j> decreases (i.e., the degree of localization of counterions increases) upon an increase in the number of arms, p, in the star. Note that in the case of star polymers with relatively small number of arms, the osmotic coefficient is significantly larger (by two orders of magnitude) than that measured previously in the solutions of colloidal PE brushes [44,45]. 

An increase in concentration Oion of added salt ions, leads to the penetration of salt ions into the star interior and a decrease in the differential osmotic pressure. When the concentration of added ions sufficiently exceeds the average concentration of counterions in the osmotic star, the polyion is found in the so-called salt-dominated regime. Here, the differential osmotic pressure of ions is equivalent to that created by binary monomer-monomer interactions with an effective second viral coefficient Ueff = a /24>ion. As a result, one recovers the same scaling dependence for the size of a PE star as that found for neutral star polymer under good solvent conditions, (4), with replacement u —> Uetr ... 

Figure 37 Relative zero-shear viscosity (normalized to the solvent tis) as a function of the effective volume fraction

The results of eqns (7.263) and (7.264) are in qualitative agreement with experimental results the viscosity increases steeply because of the exponential factor, and the steady state compliance is pri rtional to M. However, the quantitative agreement is not satisfactory. The observed viscosity is smaller than the calculated one, and the best fit with experiments is obtained only when the numerical coefficient in the exponential of eqn (7.263) is replaced by a smaller number (about 1/2) instead of lS/8. This suggests that relaxation mechanisms other than the contour length fluctuations are important for star polymers. Indeed it has been pointed out that in the case of star polymers the constraint release, and perhaps other tube reorganization processes, are as important as the contour length fluctuation. 

Equation [55] can be used to evaluate the decrease in size of a star polymer as compared to that of a linear macromolecule with the same degree of polymerization/ V, which is measured by the coefficient of the star contraaion ... 

Dynamic light scattering from dilute solutions provides the value of the diffusion coefficient, which can be converted to hydrodynamic radius J h,star of the star polymer. The ratio Rh/(Rg) characterizes the compactness of the macromolecule for the uniform hard sphere impenettable for the flow, it is Rb/Rg= (5/3) 1.29, whereas for the Gaussian coil, Rh/(Rg) = 3a- / /8 0.66. For ideal stars (without excluded-volume interactions), the ratio Rh/(Rg) can be derived within the Kirkwood-Riseman approximation, which gives the value of Rh/(Rg) 0.93. Reported experimental values of the Rh/(Rg) ratio for star polymers and starlike block copolymer micelles are usually found close... 

When a is close to unity this equation will predict strongly enhanced values of the temperature coefficient of a for brandied polymers because of the presence of the term in poor agreement with perturbation theory which predicts quite small effects. For a six-branched star polymer for example Eq. (15) predicts a 300% increase while perturbation theory gives only 13%. It seems that i values obtained in this manner will be seriously underestimated. 

Second virial coefficient of polybutadiene (star polymers)... 

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