Polyions spherical

Random coil conformations can range from the spherical contracted state to the fully extended cylindrical or rod-like form. The conformation adopted depends on the charge on the polyion and the effect of the counterions. When the charge is low the conformation is that of a contracted random coil. As the charge increases the chains extend under the influence of mutually repulsive forces to a rod-like form (Jacobsen, 1962). Thus, as a weak polyelectrolyte acid is neutralized, its conformation changes from that of a compact random coil to an extended chain. For example poly(acrylic acid), degree of polymerization 1000, adopts a spherical form with a radius of 20 nm at low pH. As neutralization proceeds the polyion first extends spherically and then becomes rod-like with a maximum extension of 250 nm (Oosawa, 1971). These pH-dependent conformational changes are important to the chemistry of polyelectrolyte cements. 

Conversely, conformation affects the binding of counterions to polyions (Jacobsen, 1962). In the compact spherical conformation some ionized groups on polymer chains will be inaccessible for ion binding. 

Several models of polyelectrolyte thermodynamics have been proposed and have been recently reviewed [84], Historically, two general approaches [84,85] have been used to model polyelectrolyte thermodynamics, spherical and chain models. In the former approach the coiled polyions are treated as spherical domains with charge density distributed continuously within the sphere... 

Choi, J.S., Lee, E.J., Choi, Y.H., Jeong, Y.J. and Park, J.S. (1999) Poly(ethylene glycol)-block-poly(L-lysine) dendrimer novel linear polymer/dendrimer block copolymer forming a spherical water-soluble polyionic complex with DNA. Bioconjug. Chem., 10, 62-65. 

A generalization of the cell model to the case of dilute solution of polyions was recently formulated by Deshkovski et al. [47]. For dilute solutions of polyions, it is fruitful to partition the volume around each poly ion into two regions. One of the regions (outer) has spherical symmetry, while the other (inner) region has cylindrical symmetry [47]. 

The second virial coefficient of polyelectrolytes is treated theoretically either by applying the theory for charged spherical colloids with a correction for the chain character of the polyion or as an extension of the theory of the second virial coefficient for nonionic linear polymers. An example of an extension of the theory of the second virial coefficient for nonionic linear polymers to polyelectrolytes is the above-mentioned Yamakawa approach of using perturbation theory of excluded volume to calculate A2 [42],... 

A theoretical analysis of the effect of counterion localization in a dilute solution of weakly charged branched polyions of different topologies [31-33] and ionic microgels [34, 35], was performed on the basis of a cell model, similar to that used here for a star-like PE. The elastic term in the free energy that accounts for the conformational entropy of a uniformly swollen branched macromolecule, has to be specified depending on the polyion topology. The shape of the cell might also be modified. For example, in the case of a molecular PE brush, a cylindrical instead of spherical cell should be used. 

The coupling between the ionization of an annealing polyion and its conformation is expected for other branched macroions as well. Recently, this effect was unambiguously demonstrated for thennoresponsive spherical star-like micelles of diblock copolymers with a polybasic (PDMAEMA) corona [134]. Due to the... 

Synthesis macrosurfactants, polysoaps, polyelectrolytes as building blocks preparation of spherical, cylindrical, multicompartment, and schizophrenic micelles, polymer vesicles, polyion complexes bottom-up self-assembly stimuli-sensitive colloids... 

An important complication arises here because the intermolecular structure factor as introduced in Eq. (2.13) has now become a function of the form factor P(q), i.e., the distribution of polyions depends on their mutual orientation and their shape and vice versa. It is only in the case of spherical polyions that S(q) and P(q) are separable by the use of center-of-mass coordinates. For rod-like polyions the mutual orientation and the spatial distribution are correlated, and for flexible polyions the chain conformation and the spatial distribution of chains depend on each other. Assuming weak interactions, several approximations were introduced to separate form- and structure factor. However, for strong, long-range electrostatic interactions intra- and intermolecular correlations cannot yet be properly separated [28]. This is an important limitation to all current theories except for Monte Carlo simulations. 

Large spherical polyions are usually treated as an effective one-component system where the interaction between the polyions is given by a hard sphere potential plus a repulsive screened Coulomb potential (DLVO model) [31]. The screening of the polyion interactions is entirely due to the charges and concentrations of counterions and salt ions. As a result, the polyions interact via an effective charge Zeff or an effective surface potential. The value of z f depends on how the correlations between the polyions themselves and between polyions and counterions are theoretically formulated. All models discussed so far lead to an effective interaction in terms of screening arguments. A more detailed theory is required to consider the small ions in the system explicitly. Different approaches... 

In contrast to spherical ions, a proper evaluation of the static structure factor for rod-like polyions by means of pair distribution functions is not available so far. As mentioned earlier, the difficulty arises from the coupling of the form factor and structure factor, i.e., the orientation of the rods depends on the position and orientation of the neighbouring rods. The distribution function for rod-like particles may be written as g(r, Ui, Uj) which indudes both the rod-to-rod distance r and the mutual orientation of two rods given by the vectors Ui and Uj [35]. From this distribution function the averaged distribution function g(r), the analog to g(r) in the case of spherical ions, and the structure factor S(q) are principally accessible. 

The viscosity of spherical polyions has also been studied by Ise et al. [108, 109]. A peak of the reduced viscosity was observed as a function of concentration similar to flexible polyions. The significance of this observation will be discussed in Sect. 2.3.3. 

A solution of the TMV was also studied under variation of the ionic strength (denoted as series 1-3 in Fig. 7), which was simply assumed to be proportional to the conductivity of the solutions. The addition of salt leads to a decrease in the peak height and to a shift of the peak position to larger q-values. This is in agreement with studies on spherical polyions where rearrangements in the nearest neighbour shell upon weakening interactions are the cause of the shift in the peak position (see Sect. 2.1.2). 

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