Interaction parameter from osmotic pressure

The thermodynamic parameters 1/) and K introduced above, pertaining to polymer-solvent interactions in dilute solutions, may be determined from thermodynamic studies of dilute solutions of the polymer, e.g., from osmotic pressure or turbidity measurements at different temperatures. These parameters may also be determined, at least in principle, from viscosity measurements on polymer solutions (see Frictional Propcitics of Polymers). The parameter ij, which is a measure of the entropy of mixing, appears to be related to the spatial or geometrical character of the solvent. For those solvents having cyclic structures, which are relatively compact and symmetrical (e.g., benzene, toluene, and cyclohexane), xp has relatively higher values than for the less symmetrical acyclic solvents capable of assuming a number of different configurations. Cyclic solvents are thus more favorable... 

GA Ogawa, E., Yamaguchi, N., and Shima, M., Estimation of the interaction parameter between polystyrene and poly(p-chlorostyrene) from osmotic pressure measurements, Polym. J., 18, 903, 1986. 

Although an estimate of the required interaction parameters can be made, e.g. from osmotic pressure or light scattering measurements [3], the significance of these bulk values in respect of an oriented adsorbed layer is somewhat doubtful, especially in emulsion systems when the hydrophobic portion of the surfactant (which contributes to the interaction parameter in bulk water) will be embedded in the oil phase. However, the value of these theoretical approaches is that they give guidelines for sensible estimates of the likely effects of changing individual parameters. 

In the above equations, % is the polymer-solvent interaction parameter, kB the Boltzmann constant, T the absolute temperature, NA Avogadro s number, v, the molar volume of solvent. If the network is made up only of neutral monomers, 7im and 7te sum up to generate the total osmotic pressure it . On the other hand, when ionizable species are contained in the network, the osmotic pressure due to ions must be considered. The contribution from counter ions [5,10,11] is... 

Fig. 1. Experimental values for the osmotic pressure as a function of separation distance from Ref. [13] (stars water circles 1 M KCl triangles 1 M KBr) are compared with calculations based on the simple equations ((1), (3) and (9)), with parameters reported in Ref. [13] (note that in Ref. [13] it was suggested that hydration interaction increases upon addition of salt) Ah = 1.6 x 10s N/m2, H = 2.1 A, H = 9.2 x KT21 J. b = 39 A, Kc = 5.8 x 10 2(1 J, An = 1.06 A 2, >.fl = 6.0 A (Line 1). Even for H = 0 and the rest of the parameters as before (Line 2), the repulsion at short separations is weaker than in the experiment. This points out that either hydration and/or undulation forces must increase upon addition of electrolyte.

Fig. 5. Experimental values for the osmotic pressure as a function of separation distance from Ref. [13] (stars water circles 1 M KC1 triangles 1 M KBr) are compared with calculations for constant parameter values for hydration (A a = 1.6 x 108 N/m2, >.// =2.1 A) and van der Waals (H =9.2 x 10 2. 1 and / 59 A) interactions, but various values for A/.

Rudin s aim was to predict the size of dissolved polymer molecules and the colligative properties of polymer solutions (hydrodynamic volume, second virial coefficient, interaction parameter, osmotic pressure, etc) from viscometric data (average molar mass, intrinsic viscosity, etc.). 

Many molecular parameters, such as ionization, molecular and electronic structure, size, and stereochemistry, will influence the basic interaction between a solute and a solvent. The addition of any substance to water results in altered properties for this substance and for water itself. Solutes cause a change in water properties because the hydrate envelopes that are formed around dissolved molecules are more organized and therefore more stable than the flickering clusters of free water. The properties of solutions that depend on solute and its concentration are different from those of pure water. The differences can be seen in such phenomena as the freezing point depression, boiling point elevation, and increased osmotic pressure of solutions. 

It is to be expected from Eq. (4.52). that measurements of the osmotic pressures of the solutions of the same polymer in different solvents should yield plots with a common intercept (at c = 0) but different slopes (see Fig. 4.6), since the second virial coefficient, which reflects polymer-solvent interactions, will be different in solvents of differing solvent power. For example, the second virial coefficient can be related to the Flory-Huggins interaction parameter x (see p. 162) by... 

Nevertheless, from a physical point of view, it is better to use other parameters. First, instead of choosing z to define the interaction, we can use the osmotic parameter g which is related to it. In this connection, let us recall that the function g = g(z) was studied in detail at the beginning of this chapter, Section 1.1. Secondly, instead of choosing the size S1/2 of the non-interacting polymer as the length scale, we can use the size x = (R2/d)112 of the isolated polymer in solution [the variable x is defined by (13.1.1)]. Thus, the osmotic pressure can be expressed in the form... 

As discussed in Chapter 2 of this Handbook, the osmotic pressure can provide one of the most direct methods of determination of interaction parameter. The estimation of these interaction parameters is applied to PS/PVME blends in toluene and ethyl benzene respectively [Shiomi et al., 1985]. It is observed that in toluene varied with composition from -0.044 to h-0.0093, while in ethyl benzene it increased with PVME content from -0.06 to 0.027. 

Equation 1.15 may be regarded as the defining equation for the parameter y, since Ago can be evaluated by use of such standard techniques as osmotic pressure, light scattering, and sedimentation equilibrium. In the classic FH theory, y is taken as the strength of the polymer-solvent interaction. However, in the Koningsveld-Staverman formalism, it has no such meaning but is an empirical function which absorbs all the deviations of Ago in an actual quasibinary solution from that in the reference FH solution. 

The reference temperature is not the true theta temperature at which the second virial coefficient of the osmotic pressure vanishes. The latter lies far below due to H-bonding and hydrophobic interaction in addition to the van der Waals interaction in the background. The parameters related to the strength of hydration, such as no, yn, were taken from Section 6.4 for PEO, and Section 6.5 for PNIPAM. 

Vapour pressure depression and membrane osmometry are the most common methods to determine the polyer-solvent interaction parameter. The latter method will be described briefly. In a membrane osmometer a dilute polymer solution has been separated from pure solvent by means of a membrane. The membrane is penneable for solvent molecules but not for polymer molecules. Due to a chemical potential difference solvent molecules will diffuse from the diluted phase to the concentrated phase and this results in a pressure increase which is called the osmotic pressure ti (see also section VI - 2 for a more detaUed description of osmosis). The osmotic pressure is given by... 

Another method of reducing creaming or sedimentation is to induce weak flocculation in the emulsion system. This may be achieved by controlling some parameters of the system, such as electrolyte concentration, adsorbed layer thickness and droplet size. These weakly flocculated emulsions are discussed in the next section. Alternatively, weak flocculation may be produced by addition of a free (non-adsorbing) polymer. Above a critical concentration of the added polymer, polymer-polymer interaction becomes favourable as a result of polymer coil overlap and the polymer chains are squeezed out from between the droplets. This results in a polymer-free zone between the droplets, and weak attraction occurs as a result of the higher osmotic pressure of the polymer solution outside the droplets. This phenomenon is usually referred to as depletion flocculation [59] and can be applied for structuring emulsions and hence reduction of creaming or sedimentation. 

Measurements performed to determine the molar masses of polymers yield - as a valuable byproduct - information on the pair interaction between the macromolecules [30]. The composition dependence of the osmotic pressure Tiosm observed via membrane osmometry is directly related to the chemical potential of the solvent [cf. (14) of Sect. 2] and provides the second osmotic virial coefficient A2, from which Xo> Ihe Flory-Huggins interaction parameter in the limit of high dilution becomes accessible [cf. (15)]. Such data are particularly valuable because they can be measured with higher accuracy than the x values for concentrated polymer solutions and because they represent a solid starting point for the sometimes very complex function xiV )- In principle, membrane osmometry can also be operated with polymer solutions of different composition in the two chambers (differential osmometry) to gain data for higher polymer concentrations however, little use is made of this option. 

The power of a solvent can be determined from the Huggins interaction parameter, xi> which represents the polymer solvent interactions. The parameter can be calculated experimentally from vapor pressure or osmotic measurements using Eq. (16) ... 

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