Semi-ideal solutions

Figure 2. Schematic semi-ideal discharge curves of Mn02 in 9 mol L 1 and 5 mol L 1 NH4CI2 + 2 mol L l ZnCl2 solutions. IL, range of discharge capacity of commercial alkaline MnO, - Zn R2, range of discharge capacity of commercial Leclanche or zinc chloride cells.

Nj,=N/f is the number of beads per branch or arm). For larger chains, however, the solvent can penetrate in outer regions of the star and the situation within these regions is more Hke a concentrated solution or a semi-dilute solution. These portions of the arms constitute a series of blobs, whose sizes increase in the direction of the arm end. The surface of a sphere of radius r from the star center is occupied by f blobs. Then the blob size is proportional to rf. Most internal blobs are placed in conditions similar to concentrated solutions and, consequently, their squared size is proportional to the number of polymer units inside them as in an ideal chain. This permits one to obtain the density of units inside the blob, as a function of r ... 

The nature of surface adsorption and micelle formation of various mixed FC- and HC-surfactants systems can be conveniently and well investigated by the non-ideal solution theory semi-emplrlcally applied in the surface layer and micelles. The weak "mutual phobic" interaction between FC- and HC-chains has been clearly revealed in the anionic-anionic and nonlonic-nonionic systems as Indicated by the positive values. value cannot be obtained... 

Scamehorn et. al. (20) also presented a simple, semi—empirical method based on ideal solution theory and the concept of reduced adsorption isotherms to predict the mixed adsorption isotherm and admicellar composition from the pure component isotherms. In this work, we present a more general theory, based only on ideal solution theory, and present detailed mixed system data for a binary mixed surfactant system (two members of a homologous series) and use it to test this model. The thermodynamics of admicelle formation is also compared to that of micelle formation for this same system. 

Scamehorn et. al. ( ) also developed a reduced adsorption equation to describe the adsorption of mixtures of anionic surfactants, which are members of homologous series. The equations were semi-empirical and were based on ideal solution theory and the theory of corresponding states. To apply these equations, a critical concentration for each pure component in the mixture is chosen, so that when the equilibrium concentrations of the pure component adsorption isotherms are divided by their critical concentrations, the adsorption isotherms would coincide. The advantage of... 

Under the action of external forces along the axes of <7-measured space appearing in particular at the transition of polymeric star from the ideal solution into the real one, a. o- and spherical conformational space of the pol5mieric star is deformed into the ellipsoid with the semi-axes accordingly to Eq. (30), equilibrium as to (7,. We assume the following variables as a measure of the conformational volume deformation ... 

Put another way, we observe a much smaller change in size than would be expected from the change in refractive index (solvent loss) if the particle were able to respond to solvent removal in the same way as an ideal solution. This suggests that, in most cases, the polymers form a semi-rigid matrix through which trapped residual solvent escapes by diffusion. 

As we discussed in Section 4.6, isolated polymer coils are typical for dilute solutions, where the volumes taken up by the coils do not overlap (Figure 4.7 a). Things change when the polymer concentration exceeds the threshold value c (which is defined by Equation (6.14) for an ideal polymer). In this case we have a semi-dilute solution (Figure 4.7 c). Although the fraction of the volume taken up by the polymer is still rather small, the coils are already highly intermingled. Can we work out what the excluded volume effect does to the coils in this case (i.e., when the polymer concentration c c ) ... 

Since c 0 for X oo, the system for which eq 3.21 and 3.22 hold lacks the dilute regime. If n 0, it corresponds to ideally semi-dilute solutions, and these equations lead to... 

It may be remarked that it is possible for a solution to show zero values of dhiy/dT and dlnyldp without the activity coefficients themselves being equal to unity. Solutions in which both of these conditions are satisfied have been called semi-ideal ] they behave like ideal solutions with regard to the enthalpy and volume, but they have a non-ideal free energy and entropy. 

CA5 Calvet, E., Sur les volumes specifiques des systemes nitrocellulose-acetone, les chaleurs de dilution et I etat semi-ideal des solutions acetoniques de nitrocellulose. Bull. Soc. Chim. Belg., 12, 553, 1945. 

Fig. 3.7. Result of a neutron scattering experiment on a semi-dilute solution of a mixture of deuterated and protonated PS (Mw = 1.1 10 ) in CS2 (cw = 0.15 g cm" ). Intensities reflect the structure factor of individual chains. The cross-over from the scattering of an expanded chain to that of an ideal chain at 5/3 is indicated. Data from Farnoux [6]...

Equilibrium Properties of Dilute and Semi-dilute Solutions. Excluded volume causes a polymer to favor expanded states. Large fluctuations, analogous to critical fluctuations, lead to non-ideal power law relations between properties. In the semi-dilute regime excluded volume correlations are screened. Scaling laws relate dilute and semi-dilute exponents. 

The main results for the dynamics of dilute solutions reflect the importance of hydrodynamic interactions each moving monomer in the solvent creates a backflow field which decays very slowly with distance. In a semi-dilute solution, the interference between all these velocity fields induces a screening of the backflow field of a given monomer, which falls off exponentially after a characteristic distance A. This idea was originally proposed by Edwards and Freed we shall briefly summarize their theory for ideal Gaussian chains. 

The last quantity that we discuss is the mean repulsive force / exerted on the wall. For a single chain this is defined taking the derivative of the logarithm of the chain partition function with respect to the position of the wall (in the —z direction). In the case of a semi-infinite system exposed to a dilute solution of polymer chains at polymer density one can equate the pressure on the wall to the pressure in the bulk which is simply given by the ideal gas law The conclusion then is that [74]... 

The rheological behaviour of polymeric solutions is strongly influenced by the conformation of the polymer. In principle one has to deal with three different conformations, namely (1) random coil polymers (2) semi-flexible rod-like macromolecules and (2) rigid rods. It is easily understood that the hydrody-namically effective volume increases in the sequence mentioned, i.e. molecules with an equal degree of polymerisation exhibit drastically larger viscosities in a rod-like conformation than as statistical coil molecules. An experimental parameter, easily determined, for the conformation of a polymer is the exponent a of the Mark-Houwink relationship [25,26]. In the case of coiled polymers a is between 0.5 and 0.9,semi-flexible rods exhibit values between 1 and 1.3, whereas for an ideal rod the intrinsic viscosity is found to be proportional to M2. 

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. 

For the success of the osmotic experiment the availability of a membrane through which solvent but not solute molecules can pass freely is essential. Existing membranes can only be considered as of approximate ideal semi-permeability. The major limitation of the osmotic method is the diffusion of low-molecular-weight species through the membrane. 

Gel electrophoresis is widely used in the routine analysis and separation of many well-known biopolymers such as proteins or nucleic acids. Little has been reported concerning the use of this methodology for the analysis of synthetic polymers, undoubtedly since in many cases these polymers are not soluble in aqueous solution - a medium normally used for electrophoresis. Even for those water-soluble synthetic polymers, the broad molecular weight dispersities usually associated with traditional polymers generally preclude the use of electrophoretic methods. Dendrimers, however, especially those constructed using semi-controlled or controlled structure synthesis (Chapters 8 and 9), possess narrow molecular weight distribution and those that are sufficiently water solubile, usually are ideal analytes for electrophoretic methods. More specifically, poly(amidoamine) (PAMAM) and related dendrimers have been proven amendable to gel electrophoresis, as will be discussed in this chapter. 

A satisfactory description of the state of solution is therefore only obtained in the case of a thermodynamically good solvent by taking five distinct states of solution into consideration, namely ideally-dilute particle solutions, semi-dilute particle solutions, semi-dilute network solution, concentrated particle solution, and concentrated network solution (Fig. 27). 

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