Semi-dilute solutions at

Let us start with single coils at the 0 point, with a radius / = and an internal concentration d c [eq. (IV. 14)] let us then increase the concentration d . When 4 becomes higher than I c (but still smaller than 1), the coils overlap, and we reach a well-defined semi-dilute regime, which differs strongly from the semi-dilute regime in good solvents discussed in Chapter III. Neutron data on the system polystyrene-cyclohexane near T = 

6 have been taken by the Saclay group, and the related scaling laws have been constructed theoretically by Jannink and Daoud. We summarize the results briefly. 

At short distances this coincides with the pair correlation for a single, ideal chain. At larger distances r it is reduced. The correlation length depends on concentration but is independent of the polymerization index. Scaling suggests 

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The repulsive energy between the blobs should be comparable with the energy level of kT for thermal fluctuations. Accordingly, the osmotic pressure of the semi-dilute solution at the theta point becomes... 

The variation of the zero-stress viscosity (also known as specific viscosity for Newtonian liquids) of HA as a function of the rod-likeness parameter (q) for dilut and semi-diluted solutions at constant temperature is shown in Figure 4.13. 

One can easily extend the above analysis to dilute and semi-dilute solutions of EP [65,66] if one recalls [67] from ordinary polymers that the correlation length for a chain of length / in the dilute limit is given by the size R of the chain oc When chains become so long that they start to overlap at I I (X the correlation length of the chain decreases and reflects... 

Taking into account the relevance of the range of semi-dilute solutions (in which intermolecular interactions and entanglements are of increasing importance) for industrial applications, a more detailed picture of the interrelationships between the solution structure and the rheological properties of these solutions was needed. The nature of entanglements at concentrations above the critical value c leads to the viscoelastic properties observable in shear flow experiments. The viscous part of the flow behaviour of a polymer in solution is usually represented by the zero-shear viscosity, rj0, which depends on the con-... 

A very convenient method for determining c is provided by the t 0-Mw-c relationship. In complete analogy to Bueche, r 0 is also found to correlate in semi-dilute solutions with M3 4. Consequently, the onset of a polymeric network is that point at which the first two terms of Eq. (9) are equal to the third term, which represents the influence of couplings on r 0. 

Viscoelastic properties have been discussed in relation to molar mass, concentration, solvent quality and shear rate. Considering the molecular models presented here, it is possible to describe the flow characteristics of dilute and semi-dilute solutions, as well as in simple shear flow, independent of the molar mass, concentration and thermodynamic quality of the solvent. The derivations can be extended to finite shear, i.e. it is possible to evaluate T) as a function of the shear rate. Furthermore it is now possible to approximate the critical conditions (critical shear rate, critical rate of elongation) at which the onset of mechanical degradation occurs. With these findings it is therefore possible to tune the flow features of a polymeric solution so that it exhibits the desired behaviour under the respective deposit conditions. 

Under good solvent conditions the dynamics of semi-dilute solutions was investigated by NSE using a PDMS/d-benzene system at T = 343 K and various concentrations 0.02 c < 0.25. The critical concentration c as defined by (112) is 0.055. 

The transition from single- to many-chain behavior already becomes obvious qualitatively from a line shape analysis of the NSE spectra (see Fig. 60) [116]. For dilute solutions (c = 0.05) the line shape parameter (3 is equal to about 0.7 for all Q-values, which is characteristic of the Zimm relaxation. In contrast, in semi-dilute solutions (e.g. c = 0.18), ft-values of 0.7 are only found at larger Q-values, whereas P-values of about 1.0, as predicted for collective diffusion [see Eq. (128)] are obtained at small Q-values. A similar observation was reported by [163]. 

The second position assumes that in semi-diluted solutions the polymeric chains are as much strong intertwined that the all thermodynamic values, in particular the osmotic pressure, achieve the limit (at N —>oc) depending only on the concentration of monomeric links, but not on the chain length. 

In semi dilute (c>c, [B] = 0.3-0.5 mole.l- ) or dilute (c< c, [B] = 10-2 mole.l- -) solution, Kp is significantly greater for copolymers than for the model compounds whatever the solvent is. For semi-dilute solution in a given solvent, the complex influences of composition, unit distribution and tacticity do not result in definite trends on K, values, as illustrated in table 6 by some representative Kj data related to keto-2-picolyl structures at 25°C. 

Typical patterns of G and G" produced from frequency sweep experiments are classified into three categories (Figure H3.2.6). A dilute solution of macromolecules shows G < G" at most frequencies. With increasing concentration, interactions among molecules become pronounced in what is called an entangled or semi-dilute solution. An entangled solution has a distinguishing feature of a cross-over point where G = G" or tan(8) = 1. The material is fluid like at a frequency below the cross-over point, but solid like at a frequency above the cross-over point. The frequency at the crossover point is thus also referred to as the relaxa-... 

Fig. 25 Viscoelastic behavior of a semi-dilute solution of DNA. Elastic modulus (G, filled symbols) and viscous modulus (G", open symbols) are plotted, together with their ratio G"IG = tan 5 (solid curve), for a 93 mg/mL DNA solution subjected to a heating-cooling cycle. The entanglement of DNA helices and, at high temperature, of single strands causes the almost monotonous increase of the dynamic moduli. Reproduced with permission from [110]...

Fig. 27 77 - F isotherms (a) and e - F for sample I (PEO-PPO-PEO) on the air/water interface at various temperatures 9 °C ( ), 23 °C, ( ) and 30 °C (A). F indicates the surface mass density where the static elasticity at 9 °C reaches the maximum, not the onset point of semi-dilute solution...

Before discussing theoretical approaches let us review some experimental results on the influence of flow on the phase behavior of polymer solutions and blends. Pioneering work on shear-induced phase changes in polymer solutions was carried out by Silberberg and Kuhn [108] on a polymer mixture of polystyrene (PS) and ethyl cellulose dissolved in benzene a system which displays UCST behavior. They observed shear-dependent depressions of the critical point of as much as 13 K under steady-state shear at rates up to 270 s Similar results on shear-induced homogenization were reported on a 50/50 blend solution of PS and poly(butadiene) (PB) with dioctyl phthalate (DOP) as a solvent under steady-state Couette flow [109, 110], A semi-dilute solution of the mixture containing 3 wt% of total polymer was prepared. The quiescent... 

De Gennes applied his method to study the chain behaviour similar to that used by Wilson (1971) to study magnetic phenomena. He pointed out that at a certain concentration the behaviour of a polymer chain is analogous to the magnetic critical and tricritical phenomena. De Gennes classifies the concentration c into three categories the dilute solution d, the semi-dilute solution c and the concentrated solution c". The concentration c is equivalent to the critical point where the crossover phenomenon occurs from randomness... 

Thus, the transition between poor and good solvent in a semi-dilute solution corresponds to a concentration C = C at which the screening effect begins to... 

The borderline between dilute and semi-dilute solutions intersects the borderline between poor and good solvents at a point the ordinate of which can be chosen as the value z0. Thus, by writing C = C and by comparing (13.2.154) with (13.2.156), we find... 

Figure 15.26 suggests the following comment. In the interval q < q, the function H q) is similar to a Lorentzian, in agreement with (15.4.31) and (15.4.32), and such behaviour is characteristic of the relatively homogeneous structure of semi-dilute solutions. However, in the complementary interval, q > q, the structure looks inhomogeneous, as in dilute solutions. Thus, we find that the boundary between dilute and semi-dilute depends on the value of q at which the observation is made. 

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