Polymer radius

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. 

The polymer radius has to be larger than 80% of the particle radius to avoid adsorption limitation under orthokinetic conditions. As a rule of thumb a particle diameter of about 1 pm marks the transition between perikinetic and orthokinetic coagulation (and flocculation). The effective size of a polymeric flocculant must clearly be very large to avoid adsorption limitation. However, if the polymer is sufficiently small, the Brownian diffusion rate may be fast enough to prevent adsorption limitation. For example, if the particle radius is 0.535 pm and the shear rate is 1800 s-, then tAp due to Brownian motion will be shorter than t 0 for r < 0.001, i.e., for a polymer with a... 

In Fig. 3 we show the density profiles (normalized to unity) for four different values of obtained with the full mean-field theory [52]. hi (a) the distance from the grafting surface is rescaled by the scahng prediction for the brush height, ho and in (b) it is rescaled by the imperturbed polymer radius Ro. 

In a better solution than that provided by a theta solvent the polymer coil will be more expanded. The radius of gyration will exceed the which is characteristic of the bulk amorphous state or a theta solution. If the polymer radius in a good solvent is times its unperturbed /-g, then the ratio of hydrodynamic volumes will be equal to a and its intrinsic viscosity will be related to [ /] by... 

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... 

If the sample is polydisperse, it is possible to extract four -averages of the polymer radius distribution, and ... 

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... 

Nowadays, nanoparticles have been widely used as fillers and compatibilizers. They exert certain effect on the miscibility of blends. Ginzburg applied a simple theory to study the effect of nanoparticles on the miscibility of PVA/PMMA blends and compared theoretical and experimental results for the same system with fillers and without fillers (Ginzburg 2005) when nanoparticle radius is smaller than polymer radius of g5n ation, the addition of nanoparticles increases the critical value of Xn and stabilizes the homogeneity (Fig. 10.38). 

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... 

A master list of solubility parameters for 166 resins and polymers is shown in Table 5.1. These values were derived from solubility studies, polymer swelling studies and estimated using the group contribution theory. This list is also contained in the spreadsheet data files discussed in Chapter 19. These values can be used to calculate the solvent-polymer radius of interaction values using Equation 5.1. 

The above equation allows the calculation of the polymer radius at infinite dilution by knowing the intrinsic viscosity of the polymer solution [78]. 

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. 

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