Segment density distribution chains

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) ... 

Hoeve44,45) extended his theory further by considering not only interactions between the train segments but also interactions among the loops, and found that the latter lead to a decrease in the number of possible conformations of adsorbed polymer chains. He assumed that the segment density distribution in any loop is uniformly expanded in one dimension by a factor of at as a result of loop-loop interactions. The volume fraction of segments at a distance z > 6 is then given by... 

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 ... 

While a chain within a branched molecule is by no means Gaussian in this real world (d = 3 note that we are considering the end-to-end distance distribution, and not the segment-density distribution about the center of gravity), by virtue of the perturbation expansion with respect to one can apply the... 

J. M. H. M. Scheutjens and G. J. Fleer (1979) Statistical-theory of the adsorption of interacting chain molecules. 1. Partition-function, segment density distribution, and adsorption-isotherms. J. Phys. Chem. 83, pp. 1619-1635 ibid. (1980) Statistical-theory of the adsorption of interacting chain molecules. 2. Train, loop, and tail size distribution. 84, pp. 178-190 ibid. (1985) Interaction between 2 adsorbed polymer layers. Macromolecules 18, pp. 1882-1900... 

Scheutjens JMHM, Fleer GJ. Statistical theory of the adsorption of interacting chain molecules. I. Partition function, segment density distribution and adsorption isotherms. J Phys Chem 1979 83 1619-1635. 

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... 

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. 

To calculate the form of the segment density distribution in the depletion layers, it is first necessary to calculate Gw(ri,r), the statistical weight of a chain of N links going from ri to r which is subject to a purely external potential U(r) that acts on each segment. 

The structure of the adsorbed layer (conformation of the polymer at the interface) is described in terms of the segment density distribution p z) which is simply the number of segments in each layer in the z-direction from the surface. Calculations by Scheutjens and Fleer (14) showed that for a chain with r = 1000, 0 = 10 and X = 0.5, 38% of the segments are in trains, 55.5% in loops and 6.5% in tails. This theory demonstrates the importance of tails which dominate the total distribution in the outer region of the adsorbed layer. 

Specifically, Edwards results (Equations 3.1-234,-238) hold only within the mean field approximation. Moreover, his calculations a.s.siimed the segment density distribution function g f) as one of ideal chains in external potential over all the range of r (see Equations 3.1-218,-229) though in practice it is valid only at small distances r < f. 

Other lattice polymer efforts have been based on the self-consistent fleld theory of Scheutjens and Fleer (150,151). This approach differs from previously posed statistical theories for chain molecules in that the partition function is expressed in terms of the distribution of chain conformations rather than the distribution of segment densities. The equilibrium distribution of chain (ie model protein) conformations is thus calculable. Quantities predicted using this approach include the force between parallel plates coated with protein (152,153), the adsorption isotherm (154,155), and the segmental density distribution (154-157). 

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