Solution 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... 

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. 

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. 

The paper is organized in the following way In Section 2, the principles of quasi-elastic neutron scattering are introduced, and the method of NSE is shortly outlined. Section 3 deals with the polymer dynamics in dense environments, addressing in particular the influence and origin of entanglements. In Section 4, polymer networks are treated. Section 5 reports on the dynamics of linear homo- and block copolymers, of cyclic and star-shaped polymers in dilute and semi-dilute solutions, respectively. Finally, Section 6 summarizes the conclusions and gives an outlook. 

For non-interacting, incompressible polymer systems the dynamic structure factors of Eq. (3) may be significantly simplified. The sums, which in Eq. (3) have to be carried out over all atoms or in the small Q limit over all monomers and solvent molecules in the sample, may be restricted to only one average chain yielding so-called form factors. With the exception of semi-dilute solutions in the following, we will always use this restriction. Thus, S(Q, t) and Sinc(Q, t) will be understood as dynamic structure factors of single chains. Under these circumstances the normalized, so-called macroscopic coherent cross section (scattering per unit volume) follows as... 

Based on the analogy between polymer solutions and magnetic systems [4,101], static scaling considerations were also applied to develop a phase diagram, where the reduced temperature x = (T — 0)/0 (0 0-temperature) and the monomer concentration c enter as variables [102,103]. This phase diagram covers 0- and good solvent conditions for dilute and semi-dilute solutions. The latter will be treated in detail below. 

A comparison with Burchard s first cumulant calculations shows qualitative agreement, in particular with respect to the position of the minimum. Quantitatively, however, important differences are obvious. Both the sharpness as well as the amplitude of the phenomenon are underestimated. These deviations may originate from an overestimation of the hydrodynamic interaction between segments. Since a star of high f internally compromises a semi-dilute solution, the back-flow field of solvent molecules will be partly screened [40,117]. Thus, the effects of hydrodynamic interaction, which in general eases the renormalization effects owing to S(Q) [152], are expected to be weaker than assumed in the cumulant calculations and thus the minimum should be more pronounced than calculated. Furthermore, since for Gaussian chains the relaxation rate decreases... 

Since the transition from dilute to semi-dilute solutions exhibits the features of a second-order phase transition, the characteristic properties of the single- chain statics and dynamics observed in dilute solutions on all intramolecular length scales, are expected to be valid in semi-dilute solutions on length scales r < (c), whereas for r > E,(c) the collective properties should prevail [90]. 

It is generally accepted that in semi-dilute solutions under good solvent conditions both the excluded volume interactions and the hydrodynamic interactions are screened owing to the presence of other chains [4,5,103], With respect to the correlation lengths (c) and H(c) there is no consensus as to whether these quantities have to be equal [11] or in general would be different [160],... 

The hydrodynamic interaction is introduced in the Zimm model as a pure intrachain effect. The molecular treatment of its screening owing to presence of other chains requires the solution of a complicated many-body problem [11, 160-164], In some cases, this problem can be overcome by a phenomenological approach [40,117], based on the Zimm model and on the additional assumption that the average hydrodynamic interaction in semi-dilute solutions is still of the same form as in the dilute case. 

NSE Results from Semi-Dilute Solutions of Linear Homopolymers Transition from Single Chain to Collective Dynamics... 

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 observation of single-chain dynamics in semi-dilute solutions requires contrast matching and labelling. In the case of PDMS this can be achieved using... 

To summarize, the following conclusions arise from the NSE investigation on semi-dilute solutions of homopolymers ... 

Semi-dilute Solutions of Polymer Blends and Block Copolymers... 

These models retain the form of the nonbonded interaction used in the chemically realistic modeling, i.e., they use either an interaction of the Lennard-Jones or of the exponential-6 type. The repulsive parts of these potentials generate the necessary local excluded volume, whereas the attractive long-range parts can be used to model varying solvent quality for dilute or semi-dilute solutions and to generate a reasonable equation-of-state behavior for polymeric melts. 

It was discovered, that the expression (1) for the description of n into diluted and semi-diluted solutions required of different values of virial coefficient A. In particular, for the estimation of A in a field of diluted solutions it would be better to accept the whole... 

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. 

Power law depentanizer f(c/c ) = const(c/c )m is postulated for the semi-diluted solution (c/c 1), in which the unknown index m accordingly to the second position of the Scaling method is from independence n on the length of a chain. This leads to the value m = 1/(3 v— l), that is m = 4/5 for d= 3-dimensional space. That is why the expression (6) is as follow... 

However, let note, that the assumption about independence of the osmotic pressure of semi-diluted solutions on the length of a chain is not physically definitely well-founded per se it is equivalent to position that the system of strongly intertwined chains is thermodynamically equivalent to the system of gaped monomeric links of the same concentration. Therefore, both Flory-Huggins method and Scaling method do not take into account the conformation constituent of free energy of polymeric chains. 

Accordingly to (19) the osmotic compressibility dlt / dc into diluted solutions does not depend on the concentration of macromolecules (dft / dc = RT) on the contrary, in semi-diluted solutions it becomes (as it follows from (25)) as linear function of relative concentration ... 

In that way, the thermodynamic approach with the use of conformational term of chemical potential of macromolecules permitted to obtain the expressions for osmotic pressure of semi-diluted and concentrated solutions in more general form than proposed ones in the methods of self-consistent field and scaling. It was shown, that only the osmotic pressure of semi-diluted solutions does not depend on free energy of the macromolecules conformation whereas the contribution of the last one into the osmotic pressure of semi-diluted and concentrated solutions is prelevant. 

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