Free energy of interpenetration

Comparison of Theory and Experiment. The expression for the free energy of interpenetration of sterically stabilized particles may be obtained by combining Equations 2, 3 and 6. Using these expressions can be calculated as a function of both... 

Figure A The total free energy of interpenetration of two particles is shown plotted as a function of temperature for atmospheric and 180 bars pressure.

Napper (14) has derived analytical expressions for the free energy of interpenetration and compression of the adsorbed polymer chains as a function of the distance of separation between two particles. The region of dose approach was divided into two domains ... 

The combinatorial contribution The combinatorial free energy of interpenetration is readily understood for it is purely entropic in origin. It arises on interpenetration of the stabilizing... 

In summary, for nonaqueous dispersions, the combinatorial free energy of interpenetration favours stabilization. Both of the corresponding free energies associated with contact dissimilarity and free volume dissimilarity favour flocculation. These conclusions are represented schematically in Fig. 7.2. Since the combinatorial free energy is purely entropic in origin, it is scarcely surprising that nonaqueous sterically stabilized systems are usually found to be entropically stabilized at room temperature and pressure for it is this term that imparts stability. Anticipating the results of the next section, we stress that this does not necessarily imply that all nonaqueous dispersions are entropically stabilized at room temperature. 

Cooling an entropically stabilized dispersion decreases the contribution of the combinatorial entropy of mixing to the overall free energy of interpenetration. Whilst the free volume contribution also decreases significantly in absolute magnitude, the contact dissimilarity term is relatively unaffected by the drop in temperature, as discussed in the preceding section. It is therefore not... 

In summary, for some aqueous dispersions, both the combinatorial and contact dissimilarity contributions to the free energy of interpenetration favour stabilization. Only the free volume contribution apparently promotes flocculation (see Fig. 7.3). 

When the temperature is raised, the contact dissimilarity term, which favours stability, is signiflcantly reduced. Although the combinatorial contribution to the overall free energy of interpenetration increases on heating, it would be expected, by analogy with nonaqueous dispersions, that the absolute magnitude of the free volume term (which favours flocculation) would increase. The net effect of a reduced contact dissimilarity contribution to... 

Repulsive forces between Fe oxide particles can be established by adsorption of suitable polymers such as proteins (Johnson and Matijevic, 1992), starches, non-ionic detergents and polyelectrolytes. Adsorption of such polymers stabilizes the particles at electrolyte concentrations otherwise high enough for coagulation to occur. Such stabilization is termed protective action or steric stabilization. It arises when particles approach each other closely enough for repulsive forces to develop. This repulsion has two sources. 1) The volume restriction effect where the ends of the polymer chains interpenetrate as the particles approach and lose some of their available conformations. This leads to a decrease in the free energy of the system which may be sufficient to produce a large repulsive force between particles. 2) The osmotic effect where the polymer chains from two particles overlap and produce a repulsion which prevents closer approach of the particles. 

Here A is the surface to surface separation between the bare particles and 6 is the thickness of the adsorbed layer. For distances of separation A > 26, the free energy of mixing of the chains is zero. Assuming constant segment density distribution in the adsorbed layers, Evans and Napper (15) derived the following expression for the free energy in the interpenetration domain, which is due only to the mixing of the chains ... 

When the concentration of the free polymer is set equal to zero, the situation corresponds to pure steric stabilization. The free energy of interaction due to the interpenetration of the adsorbed polymer chains has a range of 26, where 6 is the thickness of the adsorbed layer. This free energy is proportional to the quantity (0.5 - x), where x is the Flory interaction parameter for the polymer-solvent system. Thus, a repulsive potential is expected between two particles when x < 0.5 and this repulsion is absent when x = 0.5. For this reason, it was suggested [25] that instabilities in sterically stabilized dispersions occur for x > 0.5, hence for theta or worse-than-theta conditions. However, the correlation with the theta point only holds when the molecular weight of the added polymer is sufficiently high... 

According to the free volume theory, the total free energy of close approach (or flocculation), AGr, in the interpenetrational domain is the sum of three free energy contributions... 

The interpenetrational domain. For parallel flat plates separated by a distance h, the interpenetrational domain is defined by Ls h<2Ls. The excess free energy of mixing at separatij n h may be expressed as... 

The interpenetrational-plus-compressional domain. In the interpenetrational-plus-compressional domain, defined by hexcess free energy of mixing is given by... 

In real polymer network the effects of excluded volume and chain entanglements should be taken into account. In 1977 Hory [26] formulated the constrained junction model of real networks. According to this theory fluctuations of junctiOTs are affected by chains interpenetration, and as the result the elastic free energy is a sum of the elastic free energy of the phantom network AAph (given by Eq. (5.78)) and the free energy of constraints AA ... 

A key difference between elecfrostatic and steric stabilization is the significant influence that the solvent quality and the temperature can have in steric stabilization. As we saw above, interpenetration of the polymer chains gives rise to a mixing effect. At a certain temperature, referred to as the (theta) temperature, the interpenetration of the polymer chains does not lead to a change in the free energy of mixing (AGmix = 0), and a system of the polymer dissolved in the solvent behaves like an ideal solution. The solvent is referred to as a 0 solvent. 

At temperatures greater than the 0 temperature, the interpenetration of the polymer chains leads to an increase in the free energy of mixing (AGmix>0), that is to say that the polymer chains prefer contact with the solvent rather than contact with one another. The solvent is referred to now as a good solvent (or a better than 0 solvent). The repulsion between the chains leads to stability of the suspension. Below the temperature, the polymer chains prefer contact with one another rather than contact with the solvent (AGmix < 0), and in this case the solvent is... 

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