Segment density distribution functions

The S parameter is a function of the segment density distribution of the stabilizing chains. The conformation, and hence the segment density distribution function of polymers at interfaces, has been the subject of intensive experimental and theoretical work and is a subject of much debate (1). Since we are only interested in qualitative and not quantitative predictions, we choose the simplest distribution function, namely the constant segment density function, which leads to an S function of the form (11) ... 

Consider now the segment density distribution function in the x-direction normal to the impenetrable interface. This we shall denote by p x). At its upper limit (which occurs at x Rg), the segment density p(x) can be approximated by the product of the average segment concentration inside the coils multiplied by the fraction of the wall area occupied by coils. The multiplicand is roughly N/Rg ) whereas the multiplier is (vRf /P). Thus... 

Fig. 4.16. Schematic representation of the segment density distribution functions for tails (a) low surface coverage (b) high surface coverage (after de Cranes, I9g0).

Fig. 9.7. The distance dependence of the nomalized segment density distribution function for 1, an exponential function, 2, a radial Gaussian function and 3, a constant segment density function (after Smitham and Napper, 1979).

Neither of these two approaches is very satisfactory. The confoi. -nation of a polymer molecule at an interface, which determines the segment density distribution function, is critically dependent upon a manifold of factors that include ... 

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... 

Fig. 10.2. The segment density distribution function for two flat plates stabilized by tails for an excluded volume parameter (u=5) according to Dolan and Edwards (1975). The dashed line is for zero excluded volume ( =0).

Meier solved the diffusion equation with the appropriate boundary conditions. Curiously, as first pointed out by Di Marzio (1965), this corresponds to the placement of adsorbing barriers at jc=0 and x=d, even though the physical surface at x=0 corresponds to an impenetrable reflective surface. Meier obtained rather unwieldy expressions for the segment density distribution functions, which will not be reproduced here. Results were, however, obtained for both low and high surface coverages. 

Hesselink (1969) has argued that Meier s derivation for tails incorporates a procedural error that invalidates his quantitative results for the mixing free energy. Meier formulated a probability distribution function WJ x,d), which describes the probability that the terminal (A th) bond of the tail lies at a distance in th ange xiox+dx from the surface when the plate separation is d. The segment density distribution function was then evaluated by summing the end-to-end probability functions for all subchains of k bonds over the total number of bonds n... 

The mixing free energy term depends critically upon the mode of attachment since this determines the segment density distribution functions. For identical isolated tails, Hesselink et al. derived the following expression... 

Flat plates. The foregoing segment density distribution functions allow the... 

Other extensions of the Dolan and Edwards approach Levine et al. (1978) have developed an elegant matrix procedure to evaluate both the segment density distribution functions and the steric interaction for tails, loops and bridges that are subject to excluded volume considerations. 

The constant segment density model is, of course, only an approximation at best. It would be expected that in general the segment density would be a function of the distance from the surface of the particle. The precise form adopted by the segment density distribution function should depend upon the steric layer properties. These properties will be determined by such factors as the chemical nature of the surface and the polymer, the quality of the solvency of the dispersion medium, the surface coverage, and the mechanism of attachment of the polymer chains to the surface. Some of these expectations have been confirmed by the recent experimental determinations of the segment density distribution functions for several different systems. 

The fraction of stabilizing segments that is bound to the surface can be measured by microcalorimetry (Eisenlauer et al., 1977), IR (Peyser et al., 1967), ESR (Robb and Smith, 1977) and pulsed nmr (Cosgrove and Barnett, 1981) spectroscopy. The only method to-date that has allowed the segment density distribution function to be determined is small angle neutron scattering. Because of its likely importance in the future, this method will now be considered in detail. 

Two representative types of steric stabilizer have been studied by Cosgrove et al. (1982). The first was a partially hydrolyzed (88%) sample of poIy(vinyl alcohol). It was, of course, a copolymer containing short blocks (e.g. of average length 6 repeat units) of poly(vinyl acetate). The second sample was poly(oxyethylene) that was terminally anchored to the surface of the latex particles. The normalized segment density distribution functions for these two samples are shown in Fig. 12.1a and 12.1b. 

Fig. 12.t. The normalized segment density distribution functions determined by neutron scattering (a) poly(vinyl alcohol) with the arrow showing the hydrodynamic thickness (b) poly(oxyethylene) where curves 1 and 2 give the theoretical (according to Hesselink, 1969) and experimental results respectively (after Cosgrove et al., 1982). 

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... 

Exponential distribution function. Smitham and Napper (1979) showed that for an exponential segment density distribution function... 

The elastic repulsion calculated in this fashion is a rather sensitive function of the assumed form of the segment density distribution function. It is likely to be in error if only because of the unphysical nature of the model employed. Since the elastic term rises so steeply, however, its precise functional form is rarely of great moment. 

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