Semi-dilute

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... 

Since the power 7 is easier to detect in two than in three dimensions, the first MC study [62] sampled a two-dimensional MWD in a range of temperatures (that is, of (L)), so that a change in the degree of interpenetration should trigger a crossover from dilute to semi-dilute regime at some density 0. Evidently, indeed, from Fig. 4, the MWD follows the form of Eq. (16). At 0 one observes a power 7eff 1.300 0.005 which comes closely to the expected one. Above 0 one finds 7eff —> 1, and the distribution (11) becomes relevant. 

An important quantity whieh has been frequently studied is the mean ehain length, (L), and the variation of (L) with the energy J, following Eq. (12), has been neatly eonfirmed [58,65] for dense solutions (melts), whereas at small density the deviations from Eq. (12) are signifieant. This is demonstrated in Fig. 6, where the slopes and nieely eonfirm the expeeted behavior from Eq. (17) in the dilute and semi-dilute regimes. The predieted exponents 0.46 0.01 and 0.50 0.005 ean be reeovered with high preeision. Also, the variation of (L) at the threshold (p, denoted by L, shows a slope equal to... 

For semi-dilute and dense solutions the generalized reptation (or, sHther-ing snake ) algorithm [59] is probably among the most efficient ones. In this method one takes at random an end monomer and tries to add it at the other... 

Under quiescent conditions, polymer solutions are divided into four categories depending on the average distance separating the centers of mass of the molecular coils the dilute, the semi-dilute (or semi-concentrated), the concentrated and the entangled state. 

In the semi-dilute regime, the rate of shear degradation was found to decrease with the polymer concentration [132, 170]. By extrapolation to the dilute regime, it is frequently argued that chain scission should be nonexistent in the absence of entanglements under laminar conditions. No definite proof for this statement has been reported yet and the problem of isolated polymer chain degradation in simple shear flow remains open to further investigation. 

The rheological behavior of storage XGs was characterized by steady and dynamic shear rheometry [104,266]. Tamarind seed XG [266] showed a marked dependence of zero-shear viscosity on concentration in the semi-dilute region, which was similar to that of other stiff neutral polysaccharides, and ascribed to hyper-entanglements. In a later paper [292], the flow properties of XGs from different plant species, namely, suspension-cultured tobacco cells, apple pomace, and tamarind seed, were compared. The three XGs differed in composition and structural features (as mentioned in the former section) and... 

In this work, an experimental study was conducted on gelatin in semi-dilute region in water solution and research the effect of temperature, pH, zeta potential, and ionic strength on hydrodynamic properties by viscometiy, in order to determine the conformational characteristic, and phase transition (Tgei). 

These classical molecular theories may be used to illustrate good agreement with the experimental findings when describing the two extremes of concentration ideally dilute and concentrated polymer solutions (or polymer melts). However, when they are used in the semi-dilute range, they lead to unsatisfactory results. 

Prediction of Rheological Behaviour of Semi-Dilute Polymer Solutions at Finite Rates of Deformation... 

In a 0-solvent no semi-dilute network solution occurs, as free interpenetrability is present with overlapping. Figure 3 reproduces the individual states of solution with respect to the molar mass and the concentration [22]. 

The range of validity of a semi-dilute network lies between the two critical values c (transition between semi-dilute particle solution and semi-dilute net-... 

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-... 

For semi-dilute solutions, two regimes with different slopes are similarly obtained the powers of M, however, can be lower than 1.0 and 3.4. Furthermore, the transition region from the lower to the higher slope is broadened. The critical molar mass, Mc, for polymer solutions is found to be dependent on concentration (decreasing as c increases), although in some cases the variation appears to be very small [60,63]. 

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. 

In semi-dilute solutions, the Rouse theory fails to predict the relaxation time behaviour of the polymeric fluids. This fact is shown in Fig. 11 where the reduced viscosity is plotted against the product (y-AR). For correctly calculated values of A0 a satisfactory standardisation should be obtained independently of the molar mass and concentration of the sample. 

The range of semi-dilute network solutions is characterised by (1) polymer-polymer interactions which lead to a coil shrinkage (2) each blob acts as individual unit with both hydrodynamic and excluded volume effects and (3) for blobs in the same chain all interactions are screened out (the word blob denotes the portion of chain between two entanglements points). In this concentration range the flow characteristics and therefore also the relaxation time behaviour are not solely governed by the molar mass of the sample and its concentration, but also by the thermodynamic quality of the solvent. This leads to a shift factor, hm°d, is a function of the molar mass, concentration and solvent power. 

For concentrated solutions of polystyrene in n-butylbenzene, Graessley [40] has shown that the reduced viscosity r red Cnred=(r ( y)- rls)/(rlo rls)) can be represented on a master curve if it is plotted versus the reduced shear rate (3 ((3= y/ ycnt= y-A0). For semi-dilute solutions a perfect master curve is obtained if (3 is plotted versus a slope corrected for reduced viscosity, T corp as shown in Fig. 16. 

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. 

Below a critical concentration, c, in a thermodynamically good solvent, r 0 can be standardised against the overlap parameter c [r)]. However, for c>c, and in the case of a 0-solvent for parameter c-[r ]>0.7, r 0 is a function of the Bueche parameter, cMw The critical concentration c is found to be Mw and solvent independent, as predicted by Graessley. In the case of semi-dilute polymer solutions the relaxation time and slope in the linear region of the flow are found to be strongly influenced by the nature of polymer-solvent interactions. Taking this into account, it is possible to predict the shear viscosity and the critical shear rate at which shear-induced degradation occurs as a function of Mw c and the solvent power. 

Fig. 2.8.10 (a) Grey scale map of shear taken across gap of 7° cone-and-plate device, for the semi-dilute wormlike micelle system 60 mM cetylpyridinium chloride—100 mM sodium... 

Shear-banded Flow in a Semi-dilute Polymer Solution - T2 Effects... 

Let us remark that relation (6) is given for polymer concentration c lower than the critical overlapping concentration c above which higher terms in c must be considered. In fact, the concentration practically used ( around 10 3 g/cm3) corresponds to the semi-dilute regim for which the behavior is not well known in the case of polyelectrolytes. We have however kept relation (6) by introducing for K a mean apparent value determined from our experiments ( K - 1 )... 

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