Concentration and Molar Mass

The concentration is not a true parameter affecting the intrinsic viscosity since the intrinsic viscosity is defined in ideal dilution (c- 0) according to Eq. (4.7). Nevertheless, the influence of the concentration on the reduced viscosity q ed is of great importance for the determination of the intrinsic viscosity. 

At low concentrations, the specific viscosity /]sp of a polymer solution increases linearly with the concentration c. The Huggins Eq. (4.9) was developed to eliminate this concentration dependence. Dividing the specific viscosity by the concentration c gives the reduced viscosity /]red hich should be independent of the concentration. The observed linear relation between the reduced viscosity is caused by intermolecular interactions. The Huggins constant is a measure for these intermolecular interactions (see Chap. 5). The linear relationship between the reduced viscosity q ed the concentration c is only valid at low concentrations. As soon as the concentration is so high that interactions between several polymer coils become important, q ed increases superproportional with the concentration. This behavior can clearly be seen in Fig. 5.1. 

In particular, at high molar masses the linear region is attained only at very low concentrations. The additional interactions already start at concentrations below the critical concentration c (see Chap. 7) where the solution volume is totally filled 

At theta-conditions, the intramolecular interactions of the polymer chain are compensated by the solvation force of the solvent molecules and the polymer coil resumes its unperturbed dimensions ( Dimensions of a real polymer coil in Chap. 8). These theta-solvents correspond to thermodynamically poor solvents. The temperature at which the theta-conditions occur (theta-temperature) is normally close to the precipitation point of the polymer-solvent system. 

Often the retrieval of a suitable solvent for a polymer is not easy. Gnamm and Fuchs showed that the rule of thumb that solvents similar to the polymer show a good solubility cannot be used for all polymer solvent systems [45]. In general, one has to perform practical solution attempts to find a suitable solvent. Especially for complex copolymers with different monomer units no general rule for the solubility can be given. 

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Fig. 5. Double logarithmic plot of zero-shear rate viscosity against concentration and molar mass... 

On the basis of a relationship between T Sp and the dimensionless product c [rj], simple three-term equations can be developed to correlate the zero-shear viscosity with the concentration and molar mass. 

The viscosity level in the range of the Newtonian viscosity r 0 of the flow curve can be determined on the basis of molecular models. For this, just a single point measurement in the zero-shear viscosity range is necessary, when applying the Mark-Houwink relationship. This zero-shear viscosity, q0, depends on the concentration and molar mass of the dissolved polymer for a given solvent, pressure, temperature, molar mass distribution Mw/Mn, i.e. 

A plot of logrjsp (Eq. (9)) versus log (c-[r ]) results in a linear relationship. The unknown quantities Bn and n can be obtained from this linear regression (Fig. 7). A correlation of rj0 with concentration and molar mass can now be achieved using a [rj]=KMa relationship and replacing [q] in Eq. (9) by its molar mass dependent form to give ... 

A peculiar effect of concentration and molar mass on viscosity show solutions of rodlike macromolecules. These phenomena have originally been described by Flory (1956) and Hermans (1962) for polypeptides. They were also observed by Papkov et al. (1974) with polyparabenzamide and by Sokolova et al. (1973) with poly(paraphenylene ter-ephthalamide) (LCP-solutions see also Sect. 16.5 of this chapter). 

To overcome the problems associated with classical SEC of complex polymers, molar mass-sensitive detectors are coupled to the SEC instrument. Since the response of such detectors depends on both concentration and molar mass, they have to be combined with a concentration-sensitive detector. The following types of molar-mass-sensitive detectors are used frequently [25-28] ... 

One of the benefits of coupled SEC-FTIR is the ability to identify directly the individual components separated by chromatography A typical SEC separation of a polymer blend is shown in Fig. 28 [ 142]. Two separate elution peaks 1 and 2 were obtained, indicating that the blend contained at least two components of significantly different molar masses. A quantification of the components with respect to concentration and molar mass, however, cannot be carried out as long as the chemical structure of the components is unknown. 

Figure 1. Emission anistropy (r) of excimer for solutions of polystyrene of varying concentration and molar mass.

Polymer solutions can be classified into five regimes (dilute, semidilute not entangled, semidilute entangled, concentrated not entangled, and concentrated entangled) [22] according to the polymer concentration and molar mass however, it is much common to classify them into only... 

It is often useful to monitor simultaneously not only concentration and molar mass but also composition of eluting mactomolecules. Detectors that for example combine a differential refiactometer with the ultraviolet photometer, and the flowthrough light scattering monitor or the flow-through viscometer are comercially available. 

The combined detectors, which simultaneously monitor not only concentration and molar mass but also chemical composition of eluting macromolecules became quite popular in modem SEC. Most of combined detectors include a differential refractometer, an ultraviolet photometer, a flow-through light scattering monitor and sometimes also a flow-through viscometer. 

At (ft—> 0 equation [7.2.37] gives a linear dependence of the relative vapor pressure, P°v(/Pvo on the solvent volume concentration with the angle coefficient exp(l+x). At 1 solution obeys the Raul s law. Note that the value of < )i in [7.2.37] is temperature-dependent due to difference in thermal expansion coefficients of components. The % value for a given solvent depends on the concentration and molar mass of a polymer as well as on temperature. However, to a first approximation, these features may be ignored. Usually % varies within the range 0.2 - 0.5. For example, for solutions of polyethylene, natural rubber, and polystyrene in toluene % = 0.28,0.393 and 0.456, correspondingly. 

The parameter K was assumed to be independent of the concentration but dependent of the molar mass. The determination of K was carried out with a single viscosity measurement. In reality the Fikentscher K depends on the concentration and shows only small changes with the molar mass for high molar mass polymers [44]. It should be used only in a small and known concentration and molar mass range, for which the K value was determined. In all other cases, the intrinsic viscosity should be used as a characteristic value for polymer solvent systems. 

The intrinsic viscosity [q] increases with the molar mass (see Concentration and molar mass in Chap. 5) even at the same concentration and for the same solvent. The resulting expansion of the polymer coils leads to an increase of the interactions and therefore to an increasing slope KnX[qY. Since the Huggins constant Kii is independent of the molar mass for a specific polymer solvent system, the increasing intrinsic viscosity solely causes the increasing slope. This is shown in Fig. 5.4 for sodium poly(styrene sulfonate) of different molar masses at a constant concentration. 

In a dilute polymer solution, the dependence of the viscosity on the concentration is almost linear, that is, the intrinsic viscosity is independent of the concentration. However, as the concentration of polymer in solution increases (above the overlap concentration, c, when polymer coils get entangled with each other), the dependence on the concentration is not linear. At high molar masses and concentrations, the dependence of viscosity on concentration and molar mass can be expressed by the following equation [53] ... 

P.T. Callaghan and D.N. Pinder, "Self-Diffusion of Random-Coil Polystyrene Determined by Pulsed Field Gradient Nuclear Magnetic Resonance Dependence on Concentration and Molar Mass,"... 

P. T. Callaghan andD. N. Finder. Self-diffusion ofrandom-coil polystyrene determined by pulsed field gradient nuclear magnetic resonance Dependence on concentration and molar mass. Macromolecules, 14 (1981), 1334-1340. 

Koene RS, Mandel M (1983) Scaling relations for aqueous polyelectrolyte salt solutions. 1. Quasi elastic light scattering as a function of polyelectrolyte concentration and molar mass. 

The Nemst equation can also be expressed in another form let us suppose that the pressure on the anode and cathode are essentially the same and the system pressure can be taken p. Then the pu2,POi andpHjO are expressed as ip, n- p and n-ip where i, 2 and 3 are constants that depend on concentration and molar mass of hydrogen, oxygen and water, respectively. Equation (2.67) then becomes... 

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