Interpenetrational domain

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 ... 

In order to apply the above procedure to determine the conditions of phase separation, we have chosen the system of polyisobutene-stabilized silica particles with polystyrene as the free polymer dissolved in cyclohexane. The system temperature is chosen to be the 8 temperature for the polystyrene-cyclohexane system (34.5°C), corresponding to the experimental conditions of deHek and Vrij (1). The pertinent parameters required for the calculation of the contribution of the adsorbed layers to the total interaction potential are a = 48 nm, u, =0.18 nm3, 5 = 5 nm, Xi = 0.47(32), X2 = 0.10(32), v = 0.10, and up = 2.36 nm3. It can be seen from Fig. 2 that these forces are repulsive, with very large positive values for the potential energy at small distances of separation and falling off to zero at separation distances of the order of 25, where 6 is the thickness of the adsorbed layer. At the distance of separation 5, the expressions for the interpenetration domain and the interpenetration plus compression domain give the same value for the free energy, indicating a continuous transition from one domain to the other. 

FIGURE 10.17 Schematic representation of the three domains of close approach for sterically stabilized flat plates (a) noninteraction domain, h> L -, (b) interpenetrational domain, h < 2L (c) compressional domain, h < L. ... 

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... 

Fig. 9.8. The distance dependence of the steric interaction function in the interpenetrational domain for two spheres with different segment density distribution functions I, exponential function 2, radial Gaus n function 3, constant segment density function (after Smitham and Napper, 1979).

Hesselink etal. 91 ) have preferred the designation osmotic for the major free energy change in the interpenetrational domain. This terminology accurately portrays the consequences of the steric interaction. The interaction of the steric layers generates a difference in the chemical potential between molecules of the dispersion medium in the interactional zone and those in the bulk dispersion medium. This chemical potential difference could, of course, be related to an excess osmotic pressure. 

Whilst Osmond et al. (1975) are strictly correct in drawing attention to the existence of an elastic free energy in the interpenetrational domain, the magnitude of this term is likely to be trivially small. The reason for this is that the segmental volume fraction for most polymer chains is usually quite small (< 0-05, say) and so the volume from which the incoming polymer chains are excluded is also relatively small. This conclusion is corroborated by the results of Dolan and Edwards (1975) that are reproduced in Fig. 10.2. Here the polymer segment density distribution functions both in the presence and... 

We conclude that no serious error is inflicted upon the theoretical predictions in the interpenetrational domain by ignoring the minor conformational readjustments that the polymer chains undergo in the presence of the opposing chains. It follows that we may write without serious error... 

Everett, following Evans and Napper (1973a), distinguished between three domains of close approach (i) h > 2Ls, when no direct interaction between the steric layers is possible (ii) Lssteric layers can occur and (Hi) h < Ls, when not only the interpenetration of the two steric layers can occur but also the compression of the layers by the close approach of the opposing plates takes place. Everett confined his attention to the interpenetrational domain (ii). 

A. The interpenetrational domain. If Lspolymer chains interpenetrate but are not compressed by the opposing surface. The expression for the segment density integral becomes... 

In the interpenetrational domain (specified by d nl )% the normalized segment densities can be approximated by ihe following analytical expressions (Hesselink, 1977)... 

The foregoing expressions reduce to simple analytic forms for d> (nF) (i.e. in the interpenetrational domain). Only the leading terms, corresponding to m=0, 1, need be considered. as the summations converge rapidly. Expansion of the logarithms then yields for tails the simple exponential form... 

One disadvantage of the approach of Fischer (1958) and Ottewill and Walker (1968) is that, as noted above, the formulae proposed by these authors are only valid for the mixing of constant segment density steric layers in the interpenetrational domain. Once the interpenetrational-plus-compressional domain is entered, no allowance is made for the elastic contribution to the free energy. The elastic interactions can become important, even paramount, when the minimum distance of separation between the surfaces of the particles (Hq) is less than the barrier layer thickness. An additional defect of the Fischer approach is apparent in this domain the overlap volume is decreased below that given by equation (12.8) because part of it is occupied by the cores of the particles. Both the Fischer and the Ottewill and Walker theories disregard this decrease in volume. Implicit in their formulae is the notion that the solid cores become equivalent to the steric barriers. This is, of course, quite unphysical. 

The normalized segment density distribution functions in the interpenetrational domain (Lssteric layer) for the constant segment density model Ph = p = p h = 1/i-s since Jo (,dx= 1. This leads to... 

The expression for the distance dependence of the steric repulsion obtained in this way in the interpenetrational domain is... 

The values of ct/Ls for different fractions of segments renormalized can be determined by the integration procedure described above. Some typical values are given in Table 12.2. In the interpenetrational domain, this form of the distribution function yields... 

Constant segment density model. The way by which the Deijaguin integration is performed for spheres is amply illustrated by the constant segment density model. In the interpenetration domain, equation (12.43) given above leads to... 

The radial Gaussian distribution function. The radial Gaussian distribution function in the interpenetrational domain gives for spheres... 

It is apparent from Fig. 12.5 that the measured distance dependence of the osmotic pressure is well described by the theoretical equations in the interpenetrational domain (/fo 9nm). On closer approach, however, the predicted repulsion rose too steeply, due presumably to an overestimation of the elastic repulsion. Better agreement between theory and experiment could be achieved by arbitrarily softening the elastic repulsion this was accomplished by replacing in equation (12.55) by (5o(3- o)V4. 

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... 

Free volume theory enables the relative magnitudes of the various components of steric stabilization to be calculated for nonaqueous dispersions. The basis of these theoretical predictions is the thermodynamic factor (i - y i), where X = interaction parameter. The way by which this factor enters the steric interaction is exemplified by the interpenetrational domain for spheres... 

This temperature is important because the function [i—z(T)] must reach its maximum value at r . Consequently, corresponds, at least approximately, to the temperature at which the steric repulsion in the in-terpenetrational domain is maximal. This provides a relatively simple criterion by which the steric repulsion in the interpenetrational domain can be... 

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