Tethered chain

When compared to the early models that describe the adhesion of elastomers in the presence of interfacial chains, the central findings of this work, namely, observations b and d above, were quite surprising. In his recent work, de Gennes [102] pointed out that when a surface tethered chain diffuses into an elastomeric... 

Fig. 10. The variation of 6, with E for three different molecular weight polyisoprene tethered chains in a cross-linked polyisoprene lens [23].

A. Halperin, M. Tirrell, T. P. Lodge. Tethered chains in polymer microstructures. Adv Polym Sci 100 21-11, 1992. 

G. S. Grest, M. Murat. Computer simulations of tethered chains. In K. Binder, ed. Monte Carlo and Molecular Dynamics Simulations in Polymer Science. New York Oxford University Press, 1995, pp. 476-578. 

The distinctive properties of densely tethered chains were first noted by Alexander [7] in 1977. His theoretical analysis concerned the end-adsorption of terminally functionalized polymers on a flat surface. Further elaboration by de Gennes [8] and by Cantor [9] stressed the utility of tethered chains to the description of self-assembled block copolymers. The next important step was taken by Daoud and Cotton [10] in 1982 in a model for star polymers. This model generalizes the... 

Fig. 1A-L. Examples of polymer microstructures comprising tethered chains...

Alexander approach to spherical geometries, while making the connection between tethered chains and branched polymers. The internal structure of tethered layers was illuminated by numerical and analytical self-consistent field calculations, and by computer simulations. 

One of the signatures of densely tethered chains is expressed in Eq. 5, namely the linear variation of L with N. This stands in marked contrast with free chains where excluded volume interaction produces, at most, an R N3/5 distortion from the R N1/2 unperturbed dimensions. Tethered layers are stretched and this is the origin of their interesting behavior. 

Fig. 3 a-c. Summary of data from different laboratories, obtained by surface force measurement, on the average layer thickness L as a function of tethered chain length for flat, tethered layers constructed by adsorption of amphiphilic polymers on mica. Adapted from Ref. 21. (a) Data of reference 20 on poly-tert-butylstyrene chains anchored by adsorbing blocks of poly-2-vinylpyridine. (b) Data of references 11 and 12 on polystyrene chains anchored by adsorbing blocks of poly-2-vinylpyridine. (c) Data of references 13 and 14 on polystyrene chains anchored by adsorbing zwitterionic groups [13] or by small adsorbing blocks of polyethyleneoxide [14]... 

The essential idea of the Alexander model, a global balance of interaction and stretching energies, can be applied to other situations involving tethered chains besides the good solvent case. In theta or poor solvents, the interaction term must be modified to account for poorer solvent quality. A simple limit is precisely at the theta point [29, 30] where binary interactions effectively vanish (% = 1/2 or v = 0). The leading term in Fim now accounts for three-body interactions ... 

Tethering may be a reversible or an irreversible process. Irreversible grafting is typically accomplished by chemical bonding. The number of grafted chains is controlled by the number of grafting sites and their functionality, and then ultimately by the extent of the chemical reaction. The reaction kinetics may reflect the potential barrier confronting reactive chains which try to penetrate the tethered layer. Reversible grafting is accomplished via the self-assembly of polymeric surfactants and end-functionalized polymers [59]. In this case, the surface density and all other characteristic dimensions of the structure are controlled by thermodynamic equilibrium, albeit with possible kinetic effects. In this instance, the equilibrium condition involves the penalties due to the deformation of tethered chains. 

A more accurate analysis of this problem incorporating renormalization results, is possible [86], but the essential result is the same, namely that stretched, tethered chains interact less strongly with one another than the same chains in bulk. The appropriate comparison is with a bulk-like system of chains in a brush confined by an impenetrable wall a distance RF (the Flory radius of gyration) from the tethering surface. These confined chains, which are incapable of stretching, assume configurations similar to those of free chains. However, the volume fraction here is q> = N(a/d)2 RF N2/5(a/d)5/3, as opposed to cp = N(a/d)2 L (a/d)4/3 in the unconfined, tethered layer. Consequently, the chain-chain interaction parameter becomes x ab N3/2(a/d)5/2 %ab- Thus, tethered chains tend to mix, or at least resist phase separation, more readily than their bulk counterparts because chain stretching lowers the effective concentration within the layer. The effective interaction parameters can be used in further analysis of phase separation processes... 

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