Helfand theory

Another approach [64,338] combines the Hartree fluctuation corrections of the Fredrickson-Helfand theory [58] with contributions from multiple harmonics in the concentration expansion, chosen compatible with the considered... 

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]... 

The interfacial thickness, 1, of immiscible polymer pairs was described on the basis of Helfand theory [41] by the following equation [42]... 

Helfand E and Tagami Y 1972 Theory of the interfaee between immiseible polymers J. Polym. Sc/. Polym. Lett. 9 741 Helfand E and Tagami Y 1972 J. Chem. Phys. 56 3592... 

Helfand E., Block copolymer theory. I. Statistical thermodynamics of the microphases, ACS Poly. Prep., 14, 970, 1973. 

Helfand E. and Wasserman Z.R., Block copolymers theory. 6. Cyfindrical domain. Macromolecules, 13, 994, 1980. 

Helfand,E. Theory of the kinetics of conformational transitions in polymers. J. Chem. Phys. 54,4651-4661 (1971). 

Application of Helfand s theory has been limited due to the necessity for numerical analysis (although FORTRAN code to facilitate calculations is provided by Helfand and Wasserman (1982) ).This problem was circumvented with the introduction of the seminal analytical SSL theory by Semenov (1985). 

Fig. 2.38 Phase diagram computed using the strong segregation limit theory of Helfand and Wasserman (1982) for the poly(ethylene oxide)-poly(butylene oxide) (PEO-PBO) diblock system. Because the ratio of statistical segment lengths aPB0/ 1, the phase diagram is asymmetric about/= 0.5 (Hamley 1997).

This result has been shown to be asymptotically correct in the limit xN — °° (Matsen and Bates 1996a Matsen and Whitmore 1996). These exponents differ from those obtained by Helfand and Wasserman, who found d aAr0< 43 0143 (with a geometry-dependent prefactor) and F JkT (dlaN a)2S (Helfand and Wasserman 1976). The difference arises because Helfand s calculations were only carried out numerically to d/aNm 3, whereas in the limit d/aNm 1, the scaling (eqn 2.4) is obtained. As in the Helfand-Wasserman theory, phase boundaries can be computed using Semenov s theory, and are also found to be independent of (Semenov 1985). 

Theory for block copolymer rheology is still in its infancy. There are no models that can predict the rheological behaviour of a block copolymer from microscopic parameters. Fredrickson and Helfand (1988) considered fluctuation effects on the low frequency linear viscoelastic properties of block copolymers in the disordered melt near the ODT. They found that long-wavelength transverse momentum fluctuations couple only to compositional order parameter fluctua-... 

Helfand, E., and Y. Tagami. 1972. Theory of the interface between immiscible polymers. J Chem Phys 57 1812. 

Several attempts have been made to explain theoretically the effects of flow on the phase behavior of polymer solutions [112,115-118,123,124]. This has been done by modification of the mean-field free energy. The key point is to include properly the elastic energy of deformation produced by flow. A more rigorous approach originates from Helfand et al. [125, 126] and Onuki [127, 128] who proposed hydrodynamic theories for the dynamics of concentration fluctuations in the presence of flow coupled with a linear stability analysis. 

A well-known approximate molecular theory of a fluid at a planar interface is originally due to Helfand, Frisch and Lebowitz [76] and later to Henderson, Abraham and Barker [77] and Perram and White [78]. Consider a binary mixture (A,B) in which one of the species (A) becomes extremely dilute and infinitely large. S.E. [52] show that if the size of species A tends to infinity while the concentration of A tends toward zero, then a consequence of the OZ equation, coupled with the PY equation, is the relation... 

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