Pearl-necklace

Several studies have demonstrated the successful incoriDoration of [60]fullerene into polymeric stmctures by following two general concepts (i) in-chain addition, so called pearl necklace type polymers or (ii) on-chain addition pendant polymers. Pendant copolymers emerge predominantly from the controlled mono- and multiple functionalization of the fullerene core with different amine-, azide-, ethylene propylene terjDolymer, polystyrene, poly(oxyethylene) and poly(oxypropylene) precursors [63,64,65,66,62 and 66]. On the other hand, (-CggPd-) polymers of the pearl necklace type were fonned via the periodic linkage of [60]fullerene and Pd monomer units after their initial reaction with thep-xy y ene diradical [69,70 and 71]. 

In the presence of the polyelectrolyte polyallylamine hydrochloride (PAAN), the formation of a pearl-necklace structure between AOT-reversed micelles and PAAN was... 

The focus of this chapter is on an intermediate class of models, a picture of which is shown in Fig. 1. The polymer molecule is a string of beads that interact via simple site-site interaction potentials. The simplest model is the freely jointed hard-sphere chain model where each molecule consists of a pearl necklace of tangent hard spheres of diameter a. There are no additional bending or torsional potentials. The next level of complexity is when a stiffness is introduced that is a function of the bond angle. In the semiflexible chain model, each molecule consists of a string of hard spheres with an additional bending potential, EB = kBTe( 1 + cos 0), where kB is Boltzmann s constant, T is... 

The affinity of Cgo towards carbon nucleophiles has been used to synthesize polymer-bound Cgo [120] as well as surface-bound Cjq [121]. Polymers involving G q [54, 68, 69] are of considerable interest as (1) the fullerene properties can be combined with those of specific polymers, (2) suitable fullerene polymers should be spin-coatable, solvent-castable or melt-extrudable and (3) fullerene-containing polymers as well as surface-bound Cgo layers are expected to have remarkable electronic, magnetic, mechanical, optical or catalytic properties [54]. Some prototypes of polymers or solids containing the covalently bound Cjq moiety are possible (Figure 3.11) [68,122] fullerene pendant systems la with Cjq on the side chain of a polymer (on-chain type or charm bracelet ) [123] or on the surface of a solid Ib [121], in-chain polymers II with the fullerene as a part of the main chain ( pearl necklace ) [123], dendritic systems III, starburst or cross-link type IV or end-chain type polymers V that are terminated by a fullerene unit For III and IV, one-, two-and three-dimensional variants can be considered. In addition, combinations of all of these types are possible. 

Eq. (2.14) is identical in form to that derived by Debye (21) in his "pearl-necklace in shear model, where a Stokes law molecular friction factor was also assumed. 

Fig. 3.31 Pearl necklace morphology formed by micelles of PSam-PAA in gelled aqueous solution (Zhang et at. 1996). The structure was induced by the addition of 20mM HC1.

Kirkwood and Riseman (1948) did not encounter this problem, because they used the bead-rod or, in other words, pearl-necklace model of macromolecule (Kramers 1946), in which A is a number of Kuhn s stiff segments, so that N present the length of the macromolecule. 

The next subsections describe the properties of the pearl-necklace structure and the elongation of the pearl-necklace polymer chain by an external force. We will then present numerical simulations of single polyelectrolyte chains in a poor solvent. 

It decreases with the solvent quality and crosses over to the radius of a Gaussian polyelectrolyte for T (f-ls/b)1/3. In the pearl-necklace structure most of the polymer mass and charge belongs to the pearls but the size of the chain is dominated by the stretched strands. 

If the pearl-necklace structure contains only a few pearls, there are always pearls at the end of the chains and these pearls are slightly larger than the inner pearls. This can be proved by doing an explicit calculation of the local electrostatic potential along the necklace very similar to that done in the following section on annealed polyelectrolytes. 

Conformations of single polyelectrolyte chains can be probed by external forces [67, 68]. An interesting question which arises from the formation of the pearl necklace structure is the mechanical response of the chain to an external force. The simplest mean field result can be derived from the free energy proposed by [64], However, their free energy can by strongly modified to take the electrostatic interactions between the different structural ele-... 

The Gibbs free energy provides also a useful investigation concerning the stability of the pearl necklace structure as it was mentioned in the preceding subsection. In Fig. 12 it is shown that the relatively flat minima can be easily disturbed by moderate forces. The shift of the minima is reached over a saddle point. 

Fig. 13 Form-factor Spq) for pearl-necklaces (solid) form-factor of the whole chain, (dashed) form-factor due to intra pearl scattering, (dotted) fits. The marked region is enlarged in the inset...

Parameter values were chosen so that Ub models a stiff covalent bond, whereas the repulsive portion of ii j, approximates a hard sphere potential of diameter au, which was set equal to the bond length a so that the chain becomes the familiar pearl necklace model. 

Figure 5. The results of computer simulations of the variation of the average end force/acting on the chains in a pearl necklace network. The system consisted of 20 identical chains, each with Nb — 14 bonds of length a, 13 free with one tethered. For the latter, the fixed end-to-end distance r — 6.62a. The reduced temperature was T —4.

The conformation parameter a (=A/Af, where Af is A of a hypothetical chain with free internal rotation) for cellulose and its derivatives lies between 2.8-7.5 2 119,120) and the characteristic ratio ( = A2Mb//2, where Ax is the asymptotic value of A at infinite molecular weight, Mb is the mean molecular weight per skeletal bond, and / the mean bond length) is in the range 19-115. These unexpectedly large values of a and Cffi suggest that the molecules of cellulose and its derivatives behave as semi-flexible or even inflexible chains. For inflexible polymers, analysis of dilute solution properties by the pearl necklace model becomes theoretically inadequate. Thus, the applicability of this model to cellulose and its derivatives in solution should be carefully examined. 

The qnD values of cellulose and its derivatives lie between 3 and 25 nm and are larger than those of typical vinyltype polymer ( 1 nm), but markedly smaller than those of typical stiff chain polymers, such as DNA (Table 14)67). Thus, the chains of cellulose and its derivatives can be considered as semi-flexible. It may be concluded that both the pearl-necklace chain and the wormlike chain models are adequately applicable to these polymers. 

---