Polyelectrolyte solutions osmotic pressure

Experimentally, the fraction of free counterions in salt-free polyelectrolyte solutions is believed to give the main contribution to the system osmotic pressure. In the framework of the two-state models, the osmotic pressure is equal to the osmotic pressure of the free counterions. The more accurate analysis of the counterion effect on the solution osmotic pressure can be done in the framework of the Poisson-Boltzmann approach and its two-zone model simplification. In order to obtain an expression for the osmotic pressure in the framework of the two-zone model, one has to know the counterion concentration at the outer boundary of the spherical region. This requires knowledge of the electrostatic potential within the spherical zone. However, one can avoid solving the nonlinear Poisson-Boltzmann equation and use the relation between the local pressure P(r) and the electric field (r). To obtain this relation, one has to combine the differential form of the Gauss law ... 

These results show more clearly than Fq. (8.126)-of which they are special cases-the effect of charge and indifferent electrolyte concentration on the osmotic pressure of the solution. In terms of the determination of molecular weight of a polyelectrolyte by osmometry. ... 

If 0.6 N lithium bromide is added to the solution of the polyelectrolyte and also to the solvent on the opposite side of the osmometer membrane, the lowermost set of points in Fig. 145 (lower and left scales) is observed. The anion concentration inside and outside the coil is now so similar that there is little tendency for the bromide ions belonging to the polymer to migrate outside the coil. Hence the osmotic pressure behaves normally in the sense that each poly electrolyte molecule contributes essentially only one osmotic unit. The izjc intercept is lower than that for the parent poly-(vinylpyridine) owing to the increase in molecular weight through addition of a molecule of butyl bromide to each unit. 

The precipitation of polyelectrolytes by the addition of multivalent counterions may be explained in these terms. When there are no multivalent ions in solution there is a strong repulsive force between polyions and the osmotic pressure is large. The solubility of polyions is a result of these repulsive forces. 

The nion term is simply an expression for the osmotic pressure generated across a semipermeable membrane effectively, the gel serves as a membrane which restricts the polyelectrolytes to one phase, while small ions can readily redistribute between phases. Assuming that the ions form an ideal solution, the expression for nion becomes simply... 

If salt is present in the solution, counterions as well as co-ions do penetrate into the brush, which leads to additional screening of the Coulomb repulsion inside the brush. The amount of this screening, and the stretching of the polyelectrolyte chains, are now also controlled by the bulk salt concentration. Since the additional salt screening weakens the swelling of the brush caused by the counterion osmotic pressure, salt leads to a brush con-... 

In the polyelectrolyte regime, due to the presence of low-molecular salt, the osmotic pressure of ions becomes less pronounced because the concentration of salt within the network turns out to be less than the concentration of salt in the outer solution n [27]. As the concentration ns grows, the amplitude of the jump of the dependence a(x) decreases and the jump shifts to the region of better solvents (Fig. 2, curve 2). At some critical value of n, the jump on the curve a(x) disappears, i.e. collapse of the network becomes smooth (Fig. 2, curve 3). Under the subsequent increase of n, the curve a(x) becomes closer and closer to the swelling curve of corresponding neutral network (Fig. 2, curves 4). 

Another viable method to compare experiments and theories are simulations of either the cell model with one or more infinite rods present or to take a solution of finite semi-flexible polyelectrolytes. These will of course capture all correlations and ionic finite size effects on the basis of the RPM, and are therefore a good method to check how far simple potentials will suffice to reproduce experimental results. In Sect. 4.2, we shall in particular compare simulations and results obtained with the DHHC local density functional theory to osmotic pressure data. This comparison will demonstrate to what extent the PB cell model, and furthermore the whole coarse grained RPM approach can be expected to hold, and on which level one starts to see solvation effects and other molecular details present under experimental conditions. 

The osmotic pressure of a solution of non-interacting molecules or ions at the zero concentration limit is determined by the total molar concentration of solute species. In this hypothetical state, a solution of a polyelectrolyte may contain highly charged polyions at concentration c and monovalent counterions at a concentration where c is the molar concentration of the polyelectrolyte and z is the number of counterions per polyion. Since the total molar concentration is c (z +1) c z for z l, the ideal osmotic pressure is... 

The osmotic pressure ti of a polyelectrolyte solution is complex, since it depends largely on the amount of salt added. In order to determine the molecular weight of the polyelectrolyte by measuring the osmotic pressure it is important to add a small amount of salt and plot nfc against the concentration c at a constant salt concentration. 

Theoretical discussion of the osmotic pressure of polyelectrolytes has been made by two methods, one using the Donnan equilibrium and the other the McMillan and Mayer theory. Both methods are equivalent but in order to obatin explicitly the osmotic pressure we should know in the former case the activities of component systems and in the latter case the potential of average force between the solute molecules. 

Gum arabic (acacia) has been used in pharmacy as an emulsifier. It is a polyelectrolyte whose solutions are highly viscous owing to the branched stmcture of the macromolecular chains its adhesive properties are also believed to be due to, or in some way related to, this branched stmcture. Molecular weights of between 200 000 and 250 000 (MJ have been determined by osmotic pressure, values between 250 000 and 3 x 10 by sedimentation and diffusion, and values of 10 by light scattering, which also points to the shape of the molecules as short stiff spirals with numerous side-chains. Arabic acid prepared from commercial gum arabic by precipitation is a moderately strong acid whose aqueous solutions have a pH of 2.2-2.7. It has a higher viscosity than its salts, but emulsions prepared with arabic acid cream are not as stable as those made with its salts. 

Before presenting numerical results, it is worth summarizing the main characteristics of the experimental results for the osmotic pressure of polyelectrolyte solutions [9, 17, 18, 57, 107], The measured osmotic coefficients most often exhibit strong negative deviations from ideality. The measured values are a) lower than it was predicted by the cylindrical cell model theory, b) rather (but not completely) insensitive to the nature of the counterions, and c) also insensitive to the polyelectrolyte concentration in a dilute regime and/or for... 

Takahashi A, Kato T, Nagasawa M. The osmotic pressure of polyelectrolyte in neutral salt solutions. J Phys Chem 1970 74 944-946. 

Polyelectrolytes also show the Donnan effect. Charged macromolecules always are accompanied by counterions, i.e., small ions of opposite charge, in order to make the solution electroneutral. In the presence of salt, also the distribution of coions (small ions of the same charge) is affected, and relatively more so for a lower salt concentration. These phenomena are most readily observed when the polymer solution is contained within a semipermeable membrane, where the osmotic pressure is greatly affected by the Donnan effect. However, the effect also occurs for a polyelectrolyte in solution this implies, for instance, that it will be difficult to remove all ions of a certain species, even if exchanged for other ions of the same charge sign. 

Concentrated polymer solutions show strong nonideality. This is, for instance, observed in the osmotic pressure being very much higher than would follow from the molar concentration. The main variables are the [j value and the volume fraction of polymer, and for polyelectrolytes also charge and ionic strength. 

---