Semidilute unentangled solutions

Consistent with the fact that the longest relaxation time of the Zimm model is shorter than the Rouse model, the subdiffusive monomer motion of the Zimm model [(Eq. (8.70)] is always faster than in the Rouse model [Eq. (8.58)] with the same monomer relaxation time tq. This is demonstrated in Fig. 8.8, where the mean-square monomer displacements predicted by the Rouse and Zimm models are compared. Each model exhibits subdiffusive motion on length scales smaller than the size of the chain, but motion becomes diffusive on larger scales, corresponding to times longer than the longest relaxation time.  

Time dependence of the mean-square monomer displacements predicted by the Rouse and Zimm models on logarithmic scales. 

In Section 5.3, the static correlation length was defined for semidilute solutions. This correlation length separates single-chain (dilute-like) conformations at shorter length scales (r 0 from many-chain (melt-like) statistics at longer length scales (for r 0. The concentration correlation blob of size contains g monomers of a chain, with conformation similar to dilute solutions  

The exponent j/= 1/2 in -solvents and z/ 0.588 in good solvents. The correlation volumes are densely packed, so the volume fraction within 

The correlation length decreases with increasing concentration [Eq. (5.23)]  

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Note that if the polymer in dilute solution were highly extended with exponent i> > 2/3, the relaxation time in unentangled semidilute solutions would be predicted to decrease with increasing concentration. This is actually observed for semidilute unentangled solutions of charged poly-mers, called polyelectrolytes, which have u=lin dilute solutions because... 

There are three time scales important for the stress relaxation modulus in semidilute unentangled solutions. The shortest time scale is the relaxation time of a monomer [Eq. (8.20)]. The intermediate time scale is the... 

The time dependence of the stress relaxation modulus in semidilute unentangled solution is sketched in Fig. 8.10. Experimental verification of Rouse dynamics for frequencies smaller than 1/r was shown in Fig. 8.5, for a semidilute unentangled polyelectrolyte solution. 

The polymer contribution to viscosity in semidilute unentangled solutions is obtained by integrating the stress relaxation modulus over time iEq. (7-117)].------------------------------------------------------------... 

The correlation length in semidilute solution can be experimentally determined by measuring the diffusion coefficient of very dilute colloidal spheres of various sizes, provided that the spheres do not interact with the polymers. Consider diffusion of a non-interacting sphere in a semidilute unentangled solution. 

For (p>(pe, entanglement effects control chain dynamics and the reptation model must be used as described below. Between the overlap concentration and the entanglement concentration (cp <solution model of Section 8.5 describes dynamics. The width of this semidilute unentangled regime is given by the ratio of Eqs (9.33) and (9.32) ... 

Tnteractions are not important. The dynamics on these intermediate scales (for r < t< Te) are described by the Rouse model with stress relaxation modulus similar to the Rouse result for unentangled solutions [Eq. (8.90) with the long time limit the Rouse time of an entanglement strand Tg]. At Te, the stress relaxation modulus has decayed to the plateau modulus Gg[kT per entanglement strand, Eq. [(9.37), see Fig. 9.9)]. The ratio of osmotic pressure and plateau modulus at any concentration in semidilute solution -in athermal solvents is proportional to the number of Kuhn monomers in ... 

Figure 1.6 Physical representation of the three solution regimes [a] dilute, [b] semidilute unentangled, and [c] semidilute entangled. Reprinted with permission from Ref 34, Copyright 2005, Elsevier Ltd. All rights reserved.

Relationships between the viscosity and concentration in semidilute regimes of linear homopolymers of PMMA in DMF was investigated for electrospun nanofibers (Fig. 1.7]. The plot of the zero shear viscosity with the C/C distinctly separated into different solution regimes, viz. dilute C/C < 1], semidilute unentangled 1 3]. 

This regime of unentangled semidilute polyelectrolyte solutions has much wider concentration range than similar regime in solutions of neutral polymers (open circles in Figure 41). This concentration dependence of specific viscosity is general for salt-free polyelectrolyte solutions. 

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... 

Stress relaxation modulus of an unentangled semidilute solution of chains with tV= 10 monomers at volume fraction ( > = 0.1 in an athermal solvent (logarithmic scales). 

Estimate the time dependence of the mean-square displacement of a monomer in an unentangled semidilute solution. 

Thus, the unentangled semidilute regime in salt-free solutions could be 3-4 decades wide 10 physical reason for this unusually wide unentangled regime is the... 

Polyelectrolyte chains in semidilute regime follow unentangled dynamics in a much wider concentration range and the crossover to the entangled dynamics occurs further away from the chain s overlap concentration than in solutions of uncharged polymers. 

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