Overlap concentration estimation

In an extremely important paper, Heuer, et al. used photomicrography to examine electrophoresis of synthetic DNA stars in linear polyacrylamide having an estimated molecular weight of 5-6 MDa(13). Polymer concentrations covered 0.5-10 g/1, the nominal overlap concentration (estimated in several ways) being near 1.6 g/1. Stars were created via the approach of Seeman(22) four short (44 bp) synthetic oligonucleotides, whose complementary sequences cause them to self-assemble into a four-arm star, and whose arms bind X-phage DNA ends, were synthesized, leading to monodisperse star polymers. 

The overlap concentration can be estimated as the concentration at which the number of coils per unit volume, v, times the volume pervaded by a single coil, R, is roughly unity, where Rg = R )q /Vb is the radius of gyration of the polymer coil. Now v = cNa/M, where c is the mass per unit volume of polymer in solution and Na is Avogadro s number. Therefore, in a theta solvent (a solvent at a temperature where polymer excluded-volume interactions are negligible see Section 2.3.1.2), the overlap concentration, c, is given by... 

Problems and Worked Examples 183 Problem 3.3 From Eq. (3-4), estimate the overlap concentration in g/cm for polystyrene of... 

The radius of gyration of a polystyrene molecule (M — 3x lO gmol ) was found to be R = lOlOA. Estimate the overlap concentration c in... 

Estimate the specific viscosity at the overlap concentration using the following equations ... 

Experimental data were obtained for solutions with polymer concentrations varying from w = 0.50% to w = 4.00% or, equivalently, from c = 0.00431 to c = 0.0348 g cm to be compared with the estimated overlap concentration c = 0.0274 g cm-3 [22]. Hence, the experimental results correspond to polystyrene-toluene solutions in the dilute and semidilute solution regime. 

The dilute regime is defined by Cm < where Cm denotes the monomer concentration (per unit volume) and is the concentration where individual chains start to overlap. Clearly, the overlap concentration is reached when the average bulk monomer concentration exceeds the monomer concentration inside a polymer cod. To estimate the overlap concentration c, we simply note that the average monomer concentration inside a coil with radius R bN is given by... 

The latter behavior has been used to estimate the overlap concentration [24]. 

Observations from this viscosity study and previous studies with other gel systems suggest that there is a minimum polymer concentration for gelation for a particular polymer. Below this concentration, IMC may be overwhelming. For the Aldrich polyacrylamide, this minimum concentration appears to fall somewhere between 2000 and 4000 ppm. The minimum concentration is suspected to be related to the critical overlap concentration commonly used to describe the transition between a dilute solution and a semi-dilute solution. The critical overlap concentration for the Aldrich PAAM was estimated from its intrinsic viscosity (5.0 g/dl)... 

Polyphenylene oxide may be alloyed with polystyrene at any ratio. Significant overlap of the aromatic spectral features will be experienced, as noted in Fig. 43. The 760 and 699 cm bands of polystyrene are readily observed, even at low concentrations. Estimation of the polyphenylene oxide content of the alloys with polystyrene may be obtained from the intensity ratio of the 1021 cm (PPO) and 699 cm (PS) bands as follows [73] ... 

Weakly charged polyelectrolyte chains are nonuniformly stretched. The relation between the number of monomers N and chain size Re in salt-free solutions is given by eqn [11]. Thus, the overlap concentration c can be estimated as... 

Figure 4 displays the general behavior of the shear-thickening transition observed with imposed shear rate. The steady shear stress cj(7) and the steady apparent shear viscosity 17(7) are shown as a function of the applied shear rate for the hexadecyl-trimethylammonium/7-toluenesulfonate (CTAT) at c = 0.41 wt. %. For this system, the overlap concentration was estimated at c = 0.5 wt. % and the shear-thickening to be present over the range 0.05-0.8 wt. % [75,103]. 

The radius of gyration for a polystyrene (MW 30 x 10 ) in benzene solution was found to be 1010 A in a semidilute solution. Estimate the overlapping concentration c in grams per cubic centimeters. 

Semidilute In the semidilute regime, the concentration c is approximately equal to c, the overlap concentration. The polymers are still coiled however, the volume is filled with polymers (i.e., every volume R in the solution contains about 1 polymer molecule), and some chain interpenetration occurs. In general, this regime begins at volume fractions of approximately 1%. We can estimate the semidilute concentration threshold in molecules per unit volume as... 

In this study, the percolation threshold of HPMC and NaCMC binary matrices were estimated analyzing the water uptake and the release behavior of the matrices. Ihe HPMC percolation threshold was situated between 29 and 41% (v/v) HPMC for the binary KCl-HPMC matrices, while the excipient percolation threshold for the binary KCl-NaCMC hydrophihc matrices was found between 39 and 54% (v/v) NaCMC. As it can be appreciated, there is a narrow range of overlapping concentrations for the polymers percolation threshold. Thus it suggests that there is a small possibility that these two polymers can show a combined percolation threshold. This hypothesis is confirmed since considerable differences in the Higuchi s slope values (1.416 and 0.49, respectively) were found in two batches of matrices prepared employing very similar volume percentages of both polymers. 

The value of the effective degree of ionization a of a chain is determined as described above for different values of N and Cp in the salt-free case (n+ =0). Eight different values of CpI in the range of 10 —10 have been considered for N = 20, 40, and 60. The results are presented in Figure 6.6b where the single data point corresponds to N = 100. By estimating the overlap concentration C Iq by... 

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