Segment density-distance distribution

In order to render the expression for d AFa) in a usable form, it remains to evaluate pk and pi. We have already pointed out that the average segment density of a molecule will be greatest at the center of gravity and that it will decrease smoothly as the distance 5 (Fig. 114,a) from the center is increased. While the distribution will not be exactly a Gaussian function of s, it may be so represented without introducing an appreciable error in our final result, which can be shown to be insensitive to the exact form assumed for the radial dependence of the segment density. Hence we may let... 

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 mixture of sterically stabilized colloidal particles, solvent, and free polymer molecules in solution is considered. When two particles approach one another during a Brownian collision, the interaction potential between the two depends not only on the distance of separation between them, but also on various parameters, such as the thickness and the segment density distribution of the adsorbed layer, the concentration and the molecular weight of the free polymer. The various types of forces that are expected lo contribute to the interaction potential are (i) forces due to the presence of the adsorbed polymer, (ii) forces due to the presence of the free polymer, and (iii) van der Waals forces. It is assumed here that there are no electrostatic forces. A brief account of the nature of these forces as... 

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 ... 

Figure 3.22. Segment density profiles for amphiphiles in monolayers , Q mean field A. O, Monte Carlo. The normalized density distribution normal to the surface is givenas a function of distance, characterized by the layer number, at two Indicated values of the surface coverage, T= 300 14 chain elements, (Redrawn from Wang ind Rice, loc. cit.)...

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... 

Using the above theory it is possible to derive an expression for the free energy of mixing of two hydrated layers (assuming a uniform segment density distribution in each layer) surrounding two spherical particles as a function of the separation distance h between the particles [3, 4]. 

Generally, X-ray scattering reflects periodical fluctuations of the electron density within the sample [2, 3, 4], At a typical wavelength of 0.15 nm, which corresponds to the widely used CuK radiation, WAXS is normally caused by intra- and intermolecu-lar distances. In liquid or amorphous polymer materials the WAXS is determined by the distance distribution of the chain segments of the macromolecules, while in crystalline-ordered polymers the maxima of the WAXS reflections are determined by the distances of certain netplanes. The angular positions 20 of the reflection maxima... 

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).

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 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 foregoing thermodynamic concepts can be illustrated in a qualitative, yet revealing, fashion as follows. Consider two parallel flat interfaces separated by an infinite distance. Suppose, arbitrarily, the segment density distribution increases linearly with distance from the interface until it reaches the bulk value at a distance W, as shown in Fig. 17.18a. If is the bulk polymer concentration (in appropriately chosen units), the surface excess (in its simplest form) at each interface is simply Wc-ifl — Wci = — Wc p.. For two interfaces, the surface excess is thus - Wc-. ... 

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. 

Figure 5.18. Distribution function of segment density f r) for the normalized end-to-end distance, r = hl h yf (o), and distribution function of segment density versu.s distance r, 47rr /(r) (6). Numbers at the curves relate to the crossover parameter C (C = 0 corresponds to a Gaussian chain, C oo corresponds to a self-avoiding walk) (Oono and Freed, 1982) [Reprinted with permission from Y.Oono, K.F.Fiwed. J. Phys. A Math. Gen. 15 (1982) I93I-I95U. Institute of Physics Publishing Ltd.]...

The inhomogeneous structure was monitored by the energy transfer method. The rate of energy transfer between a donor molecule and an acceptor molecule is determined by the distance of separation in a nanometer dimension. Therefore, the inhomogeneity of segment density results in faster decay of donor fluorescence compared with homogeneous distribution. To observe this phenomenon, Pe and Eo dyes were introduced into the PMMA... 

Figure 2.41. Probability distribution for the distance r of other segments from a given segment in a Gaussian chain. The segment density autocorrelation function p(r)p(0))/p multi-phed by is plotted as a function of r/Rg. Short-distance and long-distance asymptotes are indicated.

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