Gennes formula

Benoit et al. [11-12] and Akcasu et al. [13-15] have extended de Gennes formula to describe multicomponent polymer systems. Their results are reproduced in Appendix A in a matrix form (following Akcasu [13-15]). Consider a number of components (noted A, B, etc.) with degrees of polymerization NA, etc., volume fractions A, etc., monomer volumes vA, etc. Some of these components could be block copolymers. Having one of the components (called matrix component) as a homopolymer simplifies the calculations. The main result is ... 

Fig. 11. SANS from deuterated polystyrene/polyfvinyl methyl ether) at equal compositions (50%/50% weight fractions). Experimental data fnmcro op.c cross, on nd fits to the de Gennes formula are plotted for four temperatures 60 C, 100 C, 110 C and 120 C (from bottom to...

This generalization of de Gennes formula to multicomponent homopolymer and copolymer blends can describe a wide variety of situations. After a slight generalization (described in Appendix B) it can also, describe the case of pure copolymer mixtures whereby a copolymer has to be shared between M and R. Using Appendix A, one could obtain this limit (pure copolymer mixtures) by... 

The de Gennes formulae [28, 29] establish a relation between the elastic coefficients and the scattering intensities. Small thermal director fluctuations can be expressed in terms of two eigenmodes, the splay-bend mode Srii and the twist-bend mode Sn2. The equipartition theorem gives the intensities... 

Consider a binary polymer blend [43] of deuterated polystyrene, PSD, (Mw = 1.95 x 10s g/mole, Mw/Mn = 1.02) and poly(vinyl methyl ether), PVME, (Mw = 1.59 x 10s g/mole, Mw/M = 1.3) with a composition of 48.4% PSD (volume fraction). SANS data were taken at various temperatures ranging from ambient to 160°C. De Gennes s RPA formula ... 

According to the Flory excluded volume theory and the de Gennes screening effect, the exponent of the molecular dimension is given by the Isaacson-Lubensky formula ... 

We can assume that the formula remains true when the moderately long chains constitute a polymer melt and, in this way, we recover a result announced by de Gennes.52... 

In this formula, C(r) is the average number of monomers per unit volume, vc is the volume defining the two-body interaction, and we the hyper-volume defining the three-body interaction. Thus, the initial calculation given by Flory can be generalized as was shown by Flory himself and by de Gennes (1975).7... 

Formula (15.5.4) can also be derived by using the self-consistent approximation introduced by de Gennes.25... 

Since the B-C bonds in RB2C2 do not require any electron transfer from rare earth metals, all the rare earth diborodicarbides are expected to have three conduction electrons per formula unit. Therefore, it is reasonable to assume that the main magnetic interaction in RB2C2 is the f-f indirect exchange via conduction electrons (the RKKY interaction), as was subsequently verified by the relation of the values with the de Gennes factors (Sakai et al. 1982a). 

This formula was also derived by Jannink and de Gennes [20] and Daoud et al. [1] by using different methods but invoking the random phase approximation. To the author the theory of Benoit and Benmouna appears most transparent. 

Using this approximation, de Gennes (1979) obtained a formula for the structural scattering factor in the Pl- -P2 mixture... 

The experimental classroom demonstration of the circular polarization of the reflected light beam is much easier so it is pertinent to explain its theoretical origin. The simplest explanation of this phenomenon has been proposed by de Gennes [7]. As a starting point he took the formula... 

Figure 5.5 A typical pancake. A PDMS microdroplet, whose viscosity = 5,600 cP (Mp = 52 kg/mole]. Ip = 1.08 (fractionated sample], and y = 21.37 mN/m. Substrate Silicon wafer grafted with hexamethyldlsllazane critical surface tension close to 21.5 mN/m. Pancake thickness 7.2 nm, corresponding to de Gennes s formula (Eq. 5.11a] if yt = 21.43 mN/m, with the Maxwell construction shown on the right. Initial spreading parameter, 0.06 mN/m baseline, silica layer lateral resolution, 30 pm. Profile recorded 26 days after drop deposition. - ...

Chapters 1 and 2 introduce the main phases and basic properties of liquid crystals and other anisotropic fluids, such as soaps, foams, mono-layers, fluid membranes and fibers. These chapters do not include difficult mafliematical formulas and are probably suitable for imdergraduates or for other professionals, such as K-12 teachers. Chapter 3 describes the nature of phase transitions based on the phenomenological Landau-de Gennes theories, and on the self-consistent mean-field theories that use concepts in statistical physics. 

If one introduces the intensity scattered per chain ij q) = ((q) one sees that the contribution of the x parameter can be neglected. This formula shows, as already pointed out by de Gennes, that the intensity starts from zero at q = o and reaches a plateau when q R > since the quantity B2(q)/PJ[q) vanishes for large q. 

The total electric energy density Weiec arising when a fixed voltage is maintained and applied on external conductors is given by (see the formulae 10.05.4 and 10.08.1 in Guggenheim [120] and de Gennes and Frost [110, p.l34])... 

If the friction coefficient for motion of entangled polymer chains along the tube is Cp.t = as assumed by de Gennes, then the self-diffusion coefficient for reptation can be computed using Einstein s formula ... 

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