Polymers density profiles

The opposite limiting case of a PE star with a small number p -c of arms in a salt-free solution was considered in [121], In the latter case, the counterions can be disregarded and the Poisson equation allowed for an exact numerical solution for the polymer density profile, which confirmed the uniform stretching of the arms in the interior region of the star. The LEA may be applied for analysis of conformations of stars with a small number of arms in salt-added solution, provided the bulk Debye length ro is smaller than the overall size of the star [28]. 

The polymer density profile, Cp(r), and the radius of the star, R, are determined from the minimization of the free energy, (36), while taking the conservation of the number of monomer units, (2) into account as a constraint. This leads to ... 

When (42) is used in the derivation of the polymer density profile, the profile appears independent of the degree of polymerization N of the arms. The degree of polymerization N only determines the cut-off distance for the profile via the normalization condition, (2). This implies that the local conformations of the arms at any distance r < / are independent of N. In particular, the elastic tension in the chains at any distance r [Pg.32]

Alternatively, implementing (40) and (41) leads to a quite different picture for the star structure. Here, the elastic tension in the arms is determined by the local monomer-monomer repulsion only at the edge of the corona, r = I . At r < 7 the arms are stretched more strongly, due to an excess pulling force exerted by the terminal parts of the arms. Therefore, the polymer density profile Cp r,N,R) and the chemical potential A(A,7 ) depend explicitly on N (or the star size R) [123]. 

Closed analytical expressions for the polymer density profiles Cp(r) can be obtained only in certain limiting cases (asymptotic regimes), when the free energy density can be presented as a power law function of the polymer concentration, /int Cp(/ ) Cp(r). The density profiles have the simplest form when they are presented in reduced variables, r/R and Cp(r)/Cp(7 ). 

By simultaneously solving (40) and (41), one gets for the polymer density profile ... 

We, therefore, find a logarithmic correction to the polymer density profile predicted earlier, Cp(r) r, which corresponds to a uniform radial stretching of the arms. 

The polymer density profile of ideal chains next to a hard sphere for arbitrary size ratio q was first ealeulated by Taniguchi et al. [125] and later independently by Eisenriegler et al. [126]. Eisenriegler also considered the pair interaction between two colloids for Rg< R [127] and for Rg R [128], as well as the interaction between a sphere and a flat wall due to ideal chains [129]. Depletion of excluded volume polymer chains at a wall and near a sphere was considered by Hanke et al. [130]. One of their results is that the ratio /Rg at a flat plate, which is 1.13 for ideal chains [118, 119], is slightly smaller (1.07) for excluded-volume chains. 

For the calculation of the form factors of unperturbed central regions of the stars, one has to take into account that (1) the central-symmetrical unperturbed polymer density profile c(r)... 

G. Smith, and W. Hamilton. Determination of end-adsorbed polymer density profiles by neutron reflectometry. 25(1) 434—439,1992. 

Because these structure factors depend on the particular geometry of the substrate, there is no simple relationship between them and the polymer density profile (z). There is, however, one limiting case of interest, when the surface of the substrate is sharp and well-defined and when its curvature is smaller than the scattering vector q, then the surface can be considered as flat at the scale q . In this regime, the partial structure factors reach an asymptotic limit depending only on the structure of the DOlvmer layer and on the area per unit volume of the solid phase,... 

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