Polymer radius of gyration

Fig. XI-6. Polymer segment volume fraction profiles for N = 10, = 0-5, and Xi = 1, on a semilogarithinic plot against distance from the surface scaled on the polymer radius of gyration showing contributions from loops and tails. The inset shows the overall profile on a linear scale, from Ref. 65.

In the following sections, synthesis of the anionic polymers, copolymer molecular weight, limiting viscosity number, electrolyte effects, solution shear thinning, screen factor, polymer radius of gyration, and solution aging will be discussed and data on the copolymers presented. 

Rg is the polymer radius of gyration, Xs is the value of the x parameter (see Section 2.3.1) at the spinodal point, and D is the mutual diffusion coefficient of the two polymer components. Bates and Wiltzius (1989) have confirmed the predictions of Eqs. (9-4) and (9-5) for early-time SD of binary blends of perdeuterated and protonated 1,4-polybutadiene. Neutron-scattering studies of SD on a similar system by Jiimai et al. (1993a, 1993b) also confirm the Cahn theory at early times, but the spinodal growth rates deviate somewhat from Eq. (9-5). 

Linear polymers in the presence of spherical colloids with radius comparable to or smaller than the polymer radius of gyration have also been studied by... 

An alternative method commonly used to determine the polymer radius of gyration is the Zimm s method [49], It follows from Eq. 8 upon assumption of weak interparticle correlations [p(6) = P(0) = P(q)] and on the assumption that particles (polymers) interact with each other through only one segment of each particle at a time, which is the better fulfilled the lower the concentration. It holds then that... 

Here, q j is the solvent viscosity and rg is the polymer radius of gyration. Using an exact expression for tg [61], Dp was rewritten as... 

There have been several attempts to extend the above Flory theory to correct for its failure for the i/g estimate (in d = 2) and for estimating i/, and vg on fractals. We give here a simple and elegant one (cf. [7]) which starts with a functional estimate of the radius of gyration distribution P(R) exp[—F(iJ)], instead of that for the free energy F(R). The form of the distribution function P R) of the polymer radius of gyration is given by... 

Various quantities, such as the fraction of repeating xmits (monomers) captured at the interface (which serves as an order parameter of the localization phase transition) and the components of the polymer radius of gyration parallel (I g ) and perpendicular (/ gj ) to the phase boundary between the immiscible hquids, can be then studied in order to verify the predictions of the pertinent scaling analysis by comparison with results from Monte Carlo simulations [36,45-47]. As an example, we show the changing degree of copolymer localization (Fig. 8a) and the ensuing... 

Figure 1. 0(eq. 5) against probe diameter for various polymer concentrations. Ri, and Rg are the polymer radius of gyration and hydrodynamic radius. is the estimated correlation length at 0.5, 3, and 7 g/L. 

Figure 19-11. Reduced viscosities as a function of the reduced shear stress of colloidal silica suspensions (diameter of 100 nm) in the presence of addedpolymer (polystyrene). The solvent used is decalin which is a near theta solvent for polystyrene. The size ratio of the polymer radius of gyration to the colloid radius (Rg/R) is 0.02S. The colloid volume fraction ((f>) is kept fixed at 0.4. In the absence of added polymer (Cp/c = 0), the particles behave as hard spheres and as more polymer is added to the system, the particles begin to feel an attraction. The colloid-polymer suspensions at (p of 0.4 shear thin between a zero rate viscosity of r o and a high shear rate plateau viscosity r]x,. The shear thinning behavior (in the absence and presence of polymer) is well captured by equation (19-10) with n = 1.4. Note rjo, rjao and cTc are functions of volume fraction and strengths of attraction but weakly dependent on range of attraction (Shah, 2003c Rueb, 1997).

Interpolymer Interactions Angular Dependent Scattering In the case where the polymer size is not constrained to be much smaller than X and dn/dc of one or more comonomers is close to 0, the angular dependence of the Zimm plot KclR vs. (f) becomes distorted and the obtained is now an apparent value with a complicated relationship to the copolymeric contributions to both polymer radius of gyration and refractivity [44,45]. 

Figure 7.12 shows examples of experimental I(q) as measured at different temperature and fits using the analytical Leibler function. The experimental data are indeed fitted very well by the structure factor of the RPA theory. The solid curves shown in the figure represents best fits convoluted by the experimental resolution function. In the insert is shown the effect of instrumental smearing. In typical data analysis, both the polymer radius of gyration Rg and the Flory-Huggins interaction parameter x are used as adjustable parameters. 

The mean field critical amplitudes of the correlation length of both blends are the same within experimental uncertainty and increase by about 10% for a 80% polymer content. The Ising amplitudes are smaller than their mean field numbers, and, instead, they decrease with polymer content by about 20%. In addition, the amplitudes are different in both blends PB(1,4)/PS shows slightly more than 10% larger values than PB(1,2 1,4)/PS. The mean field amplitude is proportional to the polymer radius of gyration and the square root of (1 + pr /rc) (Eq. 29) the polymer parameters are very similar, (Table 2) so the result of equal amplitudes is not surprising. On the other hand, the Ising critical amplitudes seem to depend more sensitively... 

Their radius must be much higher than the polymer radius of gyration. 

Their radius, which is equivalent to the size of the nanoparticles, should be less or equal to the polymer radius of gyration... 

The enthalpic considerations on which Eq. (7.4) is based are probably suEhdent to describe relatively large nanoparticles with amphiphilic surface chemistry (e.g., particles covered with random AB-copolymers). However, for a particle with smaller radii (comparable to the polymer radius of gyration) and/or more preferential surface treatment, interfacial segregation is not very strong. In modeling their dynamic and thermodynamic behavior in polymers, we need to consider carefully both enthalpic and entropic factors, as will be discussed in the following sections. 

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