Pincus regime

Fig. 4 Log-log plot of scaled extension versus force curve, ( X)IL)Cy versus ftf/k T)Cx, for moderately stiff chains (q, = 0.1), both in d = 2 (a) and in d = 3 (b). Contotn lengths from L = 100 up to L = 25,600 are included. The two straight lines have the theoretical slopes appropriate for the linear response (for small /) and for the nonlinear Pincus regime.

The various crossover predictions and the numerical evidence that we have obtained for these crossovers are described in detail in two long papers [20, 22] here we show only two examples that illustrate the crossover from the linear response to the Pincus regimes, both in d = 2 and d = 3 dimensions (Fig. 4). We stress that the widely used interpolation formula for the force versus extension curve quoted in Eq. (2) does not include the Pincus regime. 

Fig. 7 Scaling theory phase diagram at v2//B3< 1 in logarithmic scales with quasi-neutral (NB), osmotic (OsB), Pincus (PB) and collapsed (CB) brush regimes...

The concentration profile was calculated by de Gennes and Pincus (3) and by Eisenriegler et al. (4), who found three regimes as a function of the distance z to the surface. 

What is the elastic energy of a coil in this regime In the harmonic approximation, it would be proportional to (A — 1) (or equivalently to A, since A > 1). However, for coils in good solvents, the harmonic approximation is not very good the shape, and the monomer repulsions, change with A. This has been analyzed by Pincus (1976). 

The analysis of Rabin and Alexander was elaborated by Barrat" in order to ex-plain shear induced swelling of brushes immersed in good solvents. The stretching of brushes in poor solvents was considered by Halperin and Zhulina. Ross and Pincus considered the role of lateral inhomogeneities in this problem. Finally, in this regime shear is expected to induce brush shrinkage as was shown by Williams. ... 

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