Solution semidilute

The observation that branches A and B in Fig. 6.25 merge at large Q is consistent with the predictions for and T since 6ti and 18.84 deviate from 16 by less than 15% and statistical errors of the experiment and systematic uncertainties in methods to extract the cumulant exceed this difference. In [325] for both the collective concentration fluctuations and the local Zimm modes the observed rates are too slow by a factor of 2 if compared to the predictions with T (the solvent viscosity) and (the correlation length) as obtained from the SANS data. It is suggested that this discrepancy may be removed by the introduction of an effective viscosity qf that replaces the plain solvent viscosity Finally at very low Q, i.e. 1, branch C should level at the centre of mass 

When the temperature is increased above the 0-temperature the enhanced local viscosity at higher concentration drops. The variation is the same as observed for the macroscopic viscosity [329]. In [328] polyisoprene (PI) and PS in different solvents have also been investigated and the authors observe that the slopes of the concentration dependence of the scaled local viscosities for PS and PI have a ratio of 1.7, which matches the value of the concentration ratio either on the Kuhn length (1.6) or the persistence length (1.7) for the two polymers. 

The final conclusion from the observed concentration dependence is that under 0-conditions friction can occur between monomers, not only in the same blob but also between monomers belonging to any polymer in the solution. However, the underlying mechanism for this property has not yet been unravelled. 

A rescaling of D(Q) with respect to a length scale ls=(d Qy the mean distance between two successive self-knots where dj=2Q A is one adjustable parameter leads to the good superposition shown in Fig. 6.27. 

An analogous PS solution (M = 1,030 kg/mol) in cyclohexane at one concentration (0.052 g/cm ) has been investigated by Brulet et al. [330]. However, the NSE experiments were performed under zero average contrast, i.e. the 

So far in this review, we have confined our attention to dense melts, where we found good agreement to the reptation model. For short times, however, not all the data fit to the Rouse model perfectly. One way to examine this in more detail is to study crossover from solution to melt in the free draining limit as a function of density. Experimentally this certainly is not possible, because of the effects of hydrodynamics, which influence the dynamics very strongly. The bond fluctuation algorithm was used because at the relatively low densities of interest the MD is not as suitable.  

The absence of any hydrodynamic interaction allows one directly to ask how the entanglement length scales as a function of density. Long-range hydrodynamic interactions in real solutions make this problem more complicated. One would like to know how Ng scales with the static excluded 

The entanglement length, tube diameter dr and are taken from Ref. 124. The monomeric friction coefficient is taken from the Vogel-Fulcher form of PE from Ref 74, since there is httle difference between PEB-2 and PE. Pearson et al2 fit their data for PE to the form C = exp(B/a(7 - To)) with 

One can then plot the diffusion constant versus N/Ne for different as was done in Fig. 4.6. However the data were found not to scale at all. Similar to the problem with the bond length, the monomeric friction C, and the mobility fV C) are dependent. Paul et al. used the mobility W, determined from the mean-square displacements of the monomers 

Again one finds, as for the melts, that Hess s form of the diffusion crossover, which considers an enhanced constraint release mechanism, fits the data well, giving iVe( ) 500(4 / )  

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A. Khatory, F. Lequeux, F. Kern, S. J. Candau. Linear and nonlinear viscoelasticity of semidilute solutions of wormlike micelles at high-salt content. Langmuir 9 1456-1464, 1993. 

For example, at MW = 4 X 10, c = 12 g/liter, and at MW = 5 X 10, c " = 62 g/liter. A polymer solution with concentration c > c is called a semidilute solution because mass concentration is low yet repulsive interactions between solutes are strong. Thermodynamics, viscoelasticity, and diffusion properties of semidilute polymer solutions have been studied extensively since the 1960s. 

A. Intermicellar Interactions in Semidilute Solutions of Water-Containing Reversed Micelles... 

Fang, L Brown, W, Decay Time Distributions from Dynamic Light Scattering for Aqueous Poly(vinyl alcohol) Gels and Semidilute Solutions, Macromolecules 23, 3284, 1990. 

Recently the wall-PRISM theory has been used to investigate the forces between hydrophobic surfaces immersed in polyelectrolyte solutions [98], Polyelectrolyte solutions display strong peaks at low wavevectors in the static structure factor, which is a manifestation of liquid-like order on long lengths-cales. Consequently, the force between surfaces confining polyelectrolyte solutions is an oscillatory function of their separation. The wall-PRISM theory predicts oscillatory forces in salt-free solutions with a period of oscillation that scales with concentration as p 1/3 and p 1/2 in dilute and semidilute solutions, respectively. This behavior is explained in terms of liquid-like ordering in the bulk solution which results in liquid-like layering when the solution is confined between surfaces. In the presence of added salt the theory predicts the possibility of a predominantly attractive force under some conditions. These predictions are in accord with available experiments [99,100]. 

The second virial coefficient is not a universal quantity but depends on the primary chemical structure and the resulting topology of their architecture. It also depends on the conformation of the macromolecules in solution. However, once these individual (i.e., non-universal) characteristics are known, the data can be used as scaling parameters for the description of semidilute solutions. Such scaling has been very successful in the past with flexible linear chains [4, 18]. It also leads for branched macromolecules to a number of universality classes which are related to the various topological classes [9-11,19]. These conclusions will be outlined in the section on semidilute solutions. 

In scenario (c) corresponding to semidilute solutions, polyelectrolyte chains interpenetrate. Under these circumstances, there are three kinds of screening. The electrostatic interaction, excluded volume interaction, and the hydro-dynamic interaction between any two segments of a labeled polyelectrolyte chain are all screened by interpenetrating chains. Each of these three interactions is associated with a screening length, namely, and These screening... 

In infinitely dilute silutions, l is proportional to N and and consequently the radius of gyration Rg is proportional to N and N, respectively, in low-salt (k 0) and high salt KRg > 1) limits as already pointed out. In semidilute solutions, l is proportional to and and consequently Rg is... 

As shown in Ref. 48, l is intimately related to the static correlation length In infinitely dilute solutions is proportional to Rg. In semidilute solutions, is proportional to and respectively, in low and high salt limits. In... 

Analogous to the derivation of the effective Langevin equation for the chain in dilute solutions, we get in semidilute solutions... 

The scattering function g k) is a function of static correlation length as given by Eqs. (225)-(227). For semidilute solutions at high salt concentrations, Dc follows from Eqs. (226) and (282) in the —> 0 limit. 

Dependencies of Viscosity and Modulus in Unentangled Dilute and Semidilute Solutions on c, N, and tn for Low and High Salt Conditions"... 

Therefore in this Rouse regime of unentangled semidilute solutions where hydrodynamic interaction is screened, both the reduced viscosity and reduced modulus decrease with increase in polymer concentration in salt free solutions... 

Thus in salt-free semidilute solutions, the fast diffusion coefficient is expected to be independent of both N and c, although the polyelectrolyte concentration is higher than the overlap concentration. This remarkable result is in agreement with experimental data [31, 33, 34] discussed in the Introduction. Upon addition of salt, Df decreases from this value as given by the above formulas. 

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