Solvents theta

The phase diagram for a polymer solution is shown in Fig. 5.1. Here the attention is focused on the case where A — 0 and. 6 0 in Eq. (4.31), which is the most common case for polymer solutions. The 0-temperature sep-arates the poor solvent (bottom) half of the diagram from the good solvent (top) half. At this special temperature (T=9) the interaction parameter X=l/2 and the excluded volume is zero [see Eq. (4.72)]  

The net excluded volume at the -temperature is zero because of the exact cancelation between the constant steric repulsion between monomers (the b ) and the solvent-mediated attraction between monomers (the -2xb ). At the -temperature, the chains have nearly ideal con-formations at all concentrations  

Subtle deviations from ideal chain statistics are caused by three-body monomer interactions (see Section 3.3.2.2) at the i9-temperature, leading to logarithmic corrections to scaling. 

At very low concentrations, the polymers exist as isolated coils that are very far apart. The concentration increases moving from left to right in Fig. 5.1. At T=(9, there is a special concentration that equals the concentration inside the pervaded volume of the coil. This is the overlap concentration for -solvent [see Eq. (1.21)] ------------------------------ 

Polymer solutions with volume fractions p p0 are called dilute -solutions. Above (f g, linear chains interpenetrate each other. At volume fractions above overlap (0 0 ) at T—9, polymer solutions are called semidilute 9. This name originates from the fact that at low volume frac-tions 0 C 1 the solution consists of mostly solvent, but many solution properties are dictated by overlapping chains. 

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Electrolyte Effect on Polymer Solution Rheology. As salt concentration in an aqueous poly(1-amidoethylene) solution increases, the resulting brine becomes a more Theta-solvent for the polymer and the polymer coil compresses(47) This effect is particularly pronounced for partially hydrolyzed poly(l-amidoethylene). The... 

Comparison of the limiting viscosity numbers determined in deionized water with those determined in 1 molar sodium nitrate shows a 20 per cent decrease in copolymer intrinsic viscosity in the saline solution. These results are consistent with previous studies using aqueous saline solutions as theta solvents for 2-propenamide polymers(47) Degree of hydrolysis controls the value of limiting viscosity number for the hydrolyzed copolymers in distilled water. 

The plateau adsorbances at constant molecular weight increased linearly with the square root of NaCl concentration. For the same NaCl concentration the adsorbance was nearly independent of the molecular weight. The thickness of the adsorbed layer was approximately proportional to the square root of the molecular weight for the Theta solvent (4.17 M NaCl). For good solvents of lower NaCl concentrations the exponent of the molecular weight dependence of the thickness was less than 0.5. At the same adsorbance and molecular weight the cube of the expansion factor at, defined by the ratio of the thicknesses for good solvent and for Theta solvent, was proportional to the inverse square root of NaCl concentration. 

Expansion of Thickness of the Adsorbed Layer. In the low salt concentration the large thickness compared with the case of the Theta solvent (4.17 M NaCl) is considered to be due to the electrostatic repulsion, i.e., the excluded volume effect of the adsorbed NaPSS chains. Usually, the expansion factor at, defined by the ratio of the thickness in good solvent and that in the Theta solvent, is used to quantitatively evaluate the excluded volume effect for the adsorbed polymers. 

If the polymer concentration increases so that the number of high order bead-bead interactions is significant, c>>c =p, (when c is expressed as the polymer volume fraction. Op), the fluctuations in the polymer density becomes small, the system can be treated by mean-field theory, and the ideal model is applicable at all distance ranges, independent of the solvent quaUty and concentration. These systems are denoted as concentrated solutions. A similar description appHes to a theta solvent, but in this case, the chains within the blobs remain pseudoideal so that =N (c/c ) and Rg=N, i.e., the global chain size is always in-... 

The simultaneous agreement of exponents in Eqs. (12) and (14) characterizes the crossover condition. Then it is derived that the validity of Eq. (14) corresponds to N(j>>f 3 . This means that for an athermal solvent, where 3=1, the intermediate region governed by Eq. (12) disappears, while for a theta solvent Eq. (14) is not applicable. 

Fig.9a-c. Scaled distribution function for the center-to-end distances of stars of f=3,10 and 50 arms (a is the repulsive distance range of the intramolecular potential) T=4 /kg corresponds to a good solvent T=3e/kg corresponds to a theta solvent T=2e/kg (lower temperatures correspond to the curves on the left). Solid curves Simulation data dashed lines Gaussian functions. Reprinted with permission from [131]. Copyright (1994) American Chemical Society... 

Fig. 13a,b. Scaled bead profiles for star chains of different values of f=4,10,20,50 (higher f, more extended profile) a good solvent b theta solvent. According to Eqs. (11) and (13), the profiles should correspond to universal straight lines with slopes -4/3 (case a) and -1 (case b). Reprinted with permission from [131]. Copyright (1994) American Chemical Society... 

Fig.22a,b. Bead density profiles in a brush of grafted chains in a theta solvent with N=200, and two different values of the grafting point densities. The smooth curves correspond to the elliptical theoretical prediction. Reprinted with permission from [199]. Copyright (1993) American Chemical Society... 

The new information necessary to make this approach quantitative is the dependence of the effective entanglement molecular weight on the concentration, (j) of unrelaxed segments. This is known from experiments on dilution of polymer melts by theta-solvents to be approximately which corre-... 

In particular it has been conjectured that the terminal relaxation of star polymers might be the most sensitive test of the dilution exponent P in Go mean value of nearer 2.3 [32]. A physically reasonable scahng assumption for the density of topological entanglements in a melt of Gaussian chains leads to a value of 7/3 [31]. 

Then we address the dynamics of diblock copolymer melts. There we discuss the single chain dynamics, the collective dynamics as well as the dynamics of the interfaces in microphase separated systems. The next degree of complication is reached when we discuss the dynamic of gels (Chap. 6.3) and that of polymer aggregates like micelles or polymers with complex architecture such as stars and dendrimers. Chapter 6.5 addresses the first measurements on a rubbery electrolyte. Some new results on polymer solutions are discussed in Chap. 6.6 with particular emphasis on theta solvents and hydrodynamic screening. Chapter 6.7 finally addresses experiments that have been performed on biological macromolecules. 

For a theta solvent (V2 = 0) the relevant interaction is described by the third virial coefficient using a simple Alexander approach similar to the one leading to Eq. 13, the brush height is predicted to vary with the grafting density as h pa in agreement with computer simulations [65]. 

A theoretical expression for the concentration dependence of the polymer diffusion coefficient is derived. The final result is shown to describe experimental results for polystyrene at theta conditions within experimental errors without adjustable parameters. The basic theoretical expression is applied to theta solvents and good solvents and to polymer gels and polyelectrolytes. 

Note 2 The solvent involved is often referred to as theta solvent or 0 solvent . 

Staudinger showed that the intrinsic viscosity or LVN of a solution ([tj]) is related to the molecular weight of the polymer. The present form of this relationship was developed by Mark-Houwink (and is known as the Mark Houwink equation), in which the proportionality constant K is characteristic of the polymer and solvent, and the exponential a is a function of the shape of the polymer in a solution. For theta solvents, the value of a is 0.5. This value, which is actually a measure of the interaction of the solvent and polymer, increases as the coil expands, and the value is between 1.8 and 2.0 for rigid polymer chains extended to their full contour length and zero for spheres. When a is 1.0, the Mark Houwink equation (3.26) becomes the Staudinger viscosity equation. 

What is the value of the exponent a in the Mark-Houwink equation for polymers in theta solvents ... 

If c and dc/dx are known as a function of x and the measurement is carried out in a theta solvent, the molecular weight M of monodisperse polymers can now be calculated precisely. If the solvent is not a theta solvent, the obtained value of M is an apparent molecular weight from which the true value can be calculated upon plotting 1/M vs. c and extrapolation to c —> 0. For polydisperse samples, one has to insert the average of dc/dx in the above equation, and the thus calculated molecular weight represents a weight-average,... 

An alternative approach for determining the molecular weight of a polymer in theta solvents includes the determination of the polymer s concentration at the meniscus (c ,) and at the bottom ic, ) (or alternatively at two other positions Xi and X2) in the cell. These two outstanding positions have a distance of x ix ) and Xh(x2), respectively, from the center of rotation. Then, one obtains the weight-average molecular weight of a polydisperse polymer sample via the equation ... 

As demonstrated by numerous experiments, temperature does not well influence the exclusion processes (compare Equation 16.6) in eluents, which are thermodynamically good solvents for polymers. In this case, temperature dependence of intrinsic viscosity [ii] and, correspondingly, also of polymer hydrodynamic volume [p] M on temperature is not pronounced. The situation is changed in poor and even theta solvents (Section 16.2.2), where [p] extensively responds to temperature changes. 

Relations between g and go are semi empirical and approximate (19,20). It is assumed that g is independent of solvent conditions and that a theta solvent for a linear polymer is also a theta solvent for its branched analogues. Neither of these assumptions is well founded (19). In practical applications, exponential relations between g and go of the form... 

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