End distribution

Even from those first remarks it is evident that our knowledge of polymers at surfaces and interfaces depends largely on analytical techniques. They should yield information on chemical composition, density, roughness, chain conformation, end distribution etc. across the interface with subnanometer resolution. In Sect. 2... 

While thin polymer films may be very smooth and homogeneous, the chain conformation may be largely distorted due to the influence of the interfaces. Since the size of the polymer molecules is comparable to the film thickness those effects may play a significant role with ultra-thin polymer films. Several recent theoretical treatments are available [136-144,127,128] based on Monte Carlo [137-141,127, 128], molecular dynamics [142], variable density [143], cooperative motion [144], and bond fluctuation [136] model calculations. The distortion of the chain conformation near the interface, the segment orientation distribution, end distribution etc. are calculated as a function of film thickness and distance from the surface. In the limit of two-dimensional systems chains segregate and specific power laws are predicted [136, 137]. In 2D-blends of polymers a particular microdomain morphology may be expected [139]. Experiments on polymers in this area are presently, however, not available on a molecular level. Indications of order on an... 

Thus we may retain the root-mean-square end-to-end distance as a measure of the size of the random-coiling polymer chain, and the parameter jS required to characterize the spatial distribution of polymer segments (not to be confused with the end-to-end distribution) can be calculated from It should be noted that the r used here... 

End to end distribution function of a chain length with n segments... 

To the bitter end distribution of defense compounds within plants... 

The end-to-end distribution of short polymer molecules (represented by a RIS model that includes long-range interactions through a hard-sphere potential is calculated by means of a Monte-Carlo method. The model predictions are contrasted with experimental data of the equilibrium cyclization constants. 

It is more difficult to use the Monte Carlo method to estimate the form of the probability distribution since the tails of the distribution correspond to rare events. Mazur20 attempted to fit the three-dimensional end-to-end distribution by an expression of the form... 

In most cases we will assume Gaussian statistics for the subchains connecting two units j and k in the molecule. These two points have then an end-to-end distribution of... 

For a loop in which the first segment starts at the interface (z = 0) and the i-th segment first returns to the surface, Hoeve43) derived the probability P(z, k i) of finding its k-th segment at the distance z by use of a method analogous to that employed in deriving the end-to-end distribution of free chains. His formula is... 

The simplest example is provided by the end-to-end distribution of a given chain (m = 1, for definiteness)... 

Now the part of the chain between two successive heavy dots representing segments j,f represents a random walk going from ly to iy in j ] steps. Integrating over the internal segment coordinates of this random walk we, according to Eq. (3.41), find the end-to-end distribution (47rF i) P(j(rj — iy j — j) of this part of the chain, Thus the first order contribution (4.7) takes the form... 

The end-to-end distribution for a chain in a solution of arbitrary concentre tion takes the form... 

Consider the e-expansion (12.27) of the renormalized end-to-end distribution. It contains the constant b and thus depends on our renormalization scheme. This dependence can be eliminated by replacing the chain length, which is a microscopic parameter, by the end-to-end distance... 

The conLguration of the polymers in the micelle has also been studied within the framework of the Daoud-Cotton model (Daoud and Cotton, 1982) for star polymers. However, this model assumes that all the chain ends lie at the outer surface of the corona, therefore overlooking the possibility of a chain-end distribution Marques (1997). 

Fig. 7. Free end distribution for brushes of length AT=100 in a shear flow as a function of the height z above the grafting surface and a distance r from the grafting point. The shear rate y=0 (solid line), 7.2X10-4 (dotted line), 3.6X10-3 (short dashed line), and 7.2X10 3t 1 (long dashed line). Results are for pacr2=0.03. From ref. [63].

Bishop and Clarke employed Brownian dynamics to study the end-to-end distribution function of two-dimensional linear chains in different regimes excluded volume, theta condition, and collapsed. The results show that Mazur s function fits the first regime while the collapsed condition is satisfactorily represented by a Gaussian. However, the changeover in behavior between these two conditions appears to occut at temperatures that are well above the expected theta point. 

M. Bishop and J. H. R. Clarke,/. Chem. Phys., 91, 6345 (1989). Brownian Dynamics Study of the End-to-End Distribution Function of Two-Dimensional Linear Chains in Different Regimes. 

The molecular-weight dependence has also been investigated in our simulations. For a constant thickness of d, = 18a the end-to-end distribution of the terminal subchains, dependent on the molecular weight of the chain, was... 

Calculations of the end-to-end distribution have been performed by many authors but, in general, these calculations are not very useful, owing to a lack of precision in the results, or to a prejudice, for instance, when authors have represented their results in unacceptable analytic forms. We can, however, mention a quite comprehensive study of the subject by Mazur and McCrackin.25 Among more recent work, we note articles by Havlin and Ben Avraham,33 by Aragao de Carvalho and Caracciolo34, and by Webman, Lebowitz, and Kalos.35 These new studies confirm the previously known results... 

V Vu w(a,p,y) w(N, r) W t) wq volume of a polymer segment. 6.1.1.3 scattering volume. 1.2.2 unit cell volume. 3.3.1 crystallite orientation distribution function. 3.6.3 end-to-end distribution of a Gaussian chain. 5.2.1 [5.12] slit-length weighting function. 5.6.1 constant value of W(t) with infinite slit approximation. 5.6.3... 

Hie simjdest case of reactions between functicmal grou vdiidi are atuated apart one from another are the reactions of cyclization, i.e., the reactions of end groups of one maciomolecule. The interaction between such groui is possible onfy if they are dose to one another. The prcAiability of such proximity can be calculated uang the end-to-end distribution function. 

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