De Gennes theories

A simple scaling model of block copolymer micelles was derived by de Gennes (1978). He obtained scaling relations assuming uniformly stretched chains for the core radius, RB, of micelles with association number p.This model can be viewed as a development of the Alexander de Gennes theory (Alexander 1977 de Gennes 1976,1980) for polymer brushes at a planar interface, where the density profile normal to the interface is a step function. In the limit of short coronal (A) chains (crew-cut micelles) de Gennes (1978) predicted... 

As expected, the distribution shifts toward low Rp values with respect to the depth, revealing a profile of crosslinking inside the elastomer. An estimate of Mc, the critical mass between knots, can be derived using De Gennes theory [34] from the measured value of Prrl , the maximum of each distribution curve. 

The simple Alexander—de Gennes theory, which assumed a steplike monomer density in the brush, captured the dependence of the interaction on the physical parameters (length of the polymer, density of grafting, quality of the solvent) and provided a satisfactory approximation for the calculation of the steric repulsion. However, new applications of grafted polymers on surfaces, such as the control of the catalytic selectivities of some chemical reactions by varying the thickness of a brush,4 the prevention of the adsorption of proteins on surfaces (a condition required for biocompatibility),5 or the control of... 

At film thickness larger than twice the adsorption layer thickness this type of force vanishes [248], Therefore, such a mechanism is operative only at Ijtot 2Iii = 21.2 nm, i.e. hw < 28.0 nm (Table 3.5). The solid line in Fig. 3.40 is the best fit of Eq. (3.87). The van der Waals component has no practical influence on the numerical procedure. The fitted value h = 11.1 nm is in good agreement with the value of 10.6 nm used in the three layers model. Thus, de Gennes theory [248] gives a satisfactory description of the steric interactions at film thickness where brush-to-brush contact is realised. 

According to de Gennes theory [1], the repulsive force P h) per unit area between two uncharged brushes of intact thickness at separation h is given by... 

A Landau-de Gennes theory for the Frank constants of long semiflexible worm-Uke chains at low order parameter S gives (Shimada et al. 1988) K = K7, = 0K2 =... 

In the case of good solvent, the disjoining pressure n = -dfidh can be calculated by means of Alexander-de Gennes theory... 

Substituting max into the equation, we obtain the free energy which is a function of the nematic coupling a, the chain rigidity (l/lo) and the temperature T. The transition temperature can be obtained from the form of free energy. The form of free energy is rather complicated. We apply the Landau-de Gennes theory to analyze it. 

Figure 2.17. Free energy vs. order parameter S for various temperatures according to Landau-de Gennes theory.

There exist pre-transition effects in the isotropic phase heralding the I-N phase transition. Such pre-transition effects, which are consistent with the weakly first-order nature of the I-N transition, can be attributed to the development of short-range orientational order, which can be characterized by a position-dependent local orientational order parameter Q(r), where all component indices have been omitted [2]. In the Landau approximation, the spatial correlation function < G(0)G(r) > has the Omstein-Zemike form < G(0)G(r) exp(—r/ )/r, where is the coherence length or the second-rank orientational correlation length. The coherence length is temperature-dependent and the Landau-de Gennes theory predicts... 

For an experimental check-up of the theoretical considerations about liquid-crystalline elastomers in a mechanical field, Fin-kelmann and coworkers [107, 123] studied, in nematic networks, the evolution of the order parameter and of the transition temperature as a function of the stress. The observed results are in full agreement with the predictions of the Landau-de Gennes theory, since an increasing clearing temperature as well as an increasing order parameter are observed with increasing stress. From their results, it was possible to estimate the crosscoupling coefficient U (see Sec. 3.1.1) between the order parameter and the strain of a nematic elastomer [123]. 

The calculated van der Waals interaction is presented with a dashed line and is nearly temperature independent. On the other hand, it can be clearly seen that the total force is temperature dependent, which can only be a consequence of an additional nematic mean-field contribution. The solid line is a sum of the van der Waals and a nematic mean-field force, derived from the Landau-de Gennes theory. The agreement is quantitatively good and gives us the strengths of the two surface coupling coefficients, which are in the case of DMOAP quite large, i.e. wi = 1.4 x 10 " (1 0.4) J/m and W2 = 7x 10 (1 0.3) J/m [13]. 

The simplest model used to explain the temperature dependence of (Ap) is based on the Landau-de Gennes theory of the isotropic phase. Sluckin and Poniewierski added two surface terms to the free energy density [26]... 

Fig. 3.8. Presmectic interaction in 8CB at three different temperatures above Tni A solid line represents a fit with the Landau-de-Gennes theory (3.6).

Macroscopic Models Phenomenological Landau—de Gennes Theory... 

Equations (10.48) and (10.49) give a special form of the Landau-de Gennes theory equation (10.40) with the phenomenological parameters... 

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