Star polymer conformation

For star polymers a value of e = 0.5 has been obtained (1, V7) and studies (18) of model comb polymers indicate a value of 1.5. Other work (191 has suggested that e is near 0.5 at low LCB frequencies. For a random LCB conformation of higher branching frequency an e value between 0.7 and 1.3 might be expected, i.e. somewhere between a star and a comb configuration. 

Poly(macromonomers) with moderately long side chains attached to every few (second) atom along the backbone are very densely branched polymers. When the degree of polymerization of the backbone is low then the poly(macromon-mers) tend to resemble star polymers [39, 40]. When the degree of polymerization is very high the poly(macromonomer) acquires a cylindrical conformation (bottlebrush), due to the stretching and linearization of the backbone [40]. 

Relatively few theoretical studies have been devoted to the conformational characteristics of asymmetric star polymers in solution. Vlahos et al. [63] studied the conformational properties of AnBm miktoarm copolymers in different solvents. Analytical expressions of various conformational averages were obtained from renormalization group calculations at the critical dimensionality d=4 up to the first order of the interaction parameters uA> uB> and uAB between segments of the same or different kind, among them the radii of gyration of the two homopolymer parts < S > (k=An or Bm) and the whole miktoarm chain < /im > > the mean square distance between the centers of mass of the two homopolymer parts A and B < > and the mean square distance between the center of... 

As discussed above, the strong correlation of the counterions to the macroions leads to a remarkable osmotic pressure within the macroions, in this case the polyelectrolyte stars. To decrease this pressure, the arms of the polyelectrolyte stars are strongly stretched in dilute solution in the absence of added salt. However, if salt is added to the solution, the osmotic pressure difference will be reduced greatly, and the stretching of the arms will be diminished dramatically. Under these conditions the size and conformation of the polyelectrolyte stars will be comparable to the solution structure of uncharged star polymers. 

This situation is reminiscent of what has been suggested for the case of the structure of the star polymers with a large number of arms [20]. A numerical computation shows that for star polymer with a compact core, the statistical conformation of the arms does not have the expected scaling law but a stretched conformation which is intermediate between the usual swollen state and the rod conformation. 

The adsorption kinetics, studied by time-ressolved ellipsometry show two processes. At the initial stages the adsorption is diffusion controled. At longer times the polymers must penetrate the barrier formed by the initially adsorpted chains. It was found that the star polymers penetrate this barrier faster than the linear chains, due to the different conformations adopted by the stars. 

A quantitative analysis of counterion localization in a salt-free solution of star-like PEs is carried out on the basis of an exact numerical solution of the corresponding Poisson-Boltzmann (PB) problem (Sect. 5). Here, the conformational degrees of freedom of the flexible branches are accounted for within the Scheutjens-Fleer self-consistent field (SF-SCF) framework. The latter is used to prove and to quantify the applicability of the concept of colloidal charge renormalization to PE stars, that exemplify soft charged colloidal objects. The predictions of analytical and numerical SCF-PB theories are complemented by results of Monte Carlo (MC) and molecular dynamics (MD) simulations. The available experimental data on solution properties of PE star polymers are discussed in the light of theoretical predictions (Sect. 6). 

Finally, an analytical theory of conformations of highly branched PE stars is discussed (Sect. 7). The predictions are critically compared to numerical SCF-PB results. Here we focus on the responsive properties of strongly and weakly dissociating PE star polymers, e.g., their ability to change their conformations in response to a varied ionic strength and pH in solution. Inferior solvent quality triggers conformational transitions in PE star polymers (Sect. 8). Relevant theoretical insights are reviewed and compared to MD simulation results. 

The first theories that implemented a proper balance of intramolecular interactions and conformational elasticity of the branches were developed by Daoud and Cotton [21] and by Zhulina and Birshtein [22-24]. These theories use scaling concepts (the blob model), originally developed by de Gennes and Alexander to describe the structure of semidilute polymer solutions [64] and planar polymer brushes [65, 66]. Here, the monomer-monomer interactions were incorporated on the level of binary or ternary contacts (corresponding to good and theta-solvent conditions, respectively), and both dilute and semidilute solutions of star polymers were considered. Depending on the solvent quality and the intrinsic stiffness of the arms, the branches of a star could be locally swollen, or exhibit Gaussian statistics [22-24]. 

Beyond the overlap concentration threshold, c>c = pN/lP, star polymers form a semidilute solution. Because of the fact that the arms in a star are stretched, the scaling theory [24] predicts that the properties of semidilute solutions of star polymers are distinctively different from those of linear polymers. When the polymer concentration c > c, a semidilute solution is envisioned as a system of closely packed and virtually non-interpenetrating (segregated) polymer stars. A further increase in polymer concentration leads to a progressive contraction of the coronae of the individual stars. This contraction results in an increase in the conformational entropy of the partially stretched star arms. 

To analyze the effects of ionic strength and pH of the solution on the conformations of PE stars, we switch from the canonical cell model (where the number of ions was fixed) to the partially open ensemble. In the latter model, (a) one central star polymer occupies a spherical volume within radius R, and (b) the chemical potentials of all mobile ions are set equal to those in the bulk of the solution (infinite reservoir). 

Relatively few theoretical studies have been devoted to the conformational characteristics of asymmetric star polymers in solution. The conformational properties of A B miktoarm star copolymers in different solvents were studied by renormalization group calculations. Analytical expressions of various conformational averages were obtained at the critical dimensionality d = 4 up to the first order of the interaction parameters... 

Three capture its ability to reveal qualitative structural information in a visual format. The applications discussed show how peak displacement of the tripledetector chromatograms reflects polymer polydispersity (dextran), how detector response can relate to aggregation (chitosan), and how peak area differences can indicate a change in polymer chemical composition (PS vs. PS-star). The M-H plots (dextran, brominated PS) give information about polymer conformational changes, structural differences, and branching distributions. ... 

For star polymers, an increase in the complexity of the matrix schemes may be required with rectangular matrices for branch points. This is justified if the property of interest is concerned with the conformations of the bridge itself, as for example with the optical activity of the cystine residue disulphide bond, or when interest is directed to the deviations from random flight statistics shown by short-branched stars of homo- and sequential copoly-peptides and by disordered proteins cross-linked by a disulphide bond. Simpler schemes are sufficient for the study of the large effect of helix-coil transition upon the charac-... 

Problem 3.6 Use Eq. 3.55 to calculate / H.siar of a star polymer consisting of A arms that have a conformation of a Gaussian chain with Ni segments of length b. What is ga = (/ H,starMH,im) where is for a linear chain of rij Ni segments Compare with defined as the ratio of the mean square radii of gyration for the same pair of polymers (Eq. 1.85). 

All these observed characteristics of viscoelasticity for star polymers are natural consequences of the tube model. As suggested by the sketch in Fig. 3.49, the presence of even one long branch would surely suppress reptation [53]. There is no longer any direction for the star to move freely into new positions and conformations, and accordingly relaxation and diffusion must occur by some other motion. The Pearson-Helfand theory for stars based on tube-length fluctuations alone [72]... 

Eqs. (8.2) and (8.3) is that the effective interaction between two sites on the polymer depends on their instantaneous position, and only on the entire macromolecule conformation in an average (implicit) sense via the direct and collective pair correlations. This simplification does not preclude describing situations of broken conformational symmetry, such as polymer collapse, solvated electron localization, or spatially inhomogeneous conformational characteristics such as occur in star polymers (see Section IX)... 

Constrained Polymers. The conformation of polymers constrained in various ways, for example, grafted to a flat surface ( brush ), adsorbed on a spherical colloidal particle, or tethered to a central branch point as in star polymers. All such problems involve potentially large nonideal conformational effects and also introduce additional complications associated with site inequivalence within the PRISM formalism. Progress for star polymers is briefly described in the next section. 

TroUsas, M., Atthoff, B., Wiirsch, A. et al. (2000a) Constitutional isomers of dendrimer-like star polymers Design, synthesis, and conformational and structural properties. Macromolecules, 33,6423-6438. 

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