Leibler theory

Hence the Leibler theory [43] indeed predicts a second order transition for f = 1/2, while for f + 1/2 where the third order term is present, a first order transition is predicted, S-1(q ) = 0 then only yields the limit of metastability of the disordered phase ( spinodal curve ). Thus using the higher order terms in Eq. (184) to actually compute the free energies of various candidates for the ordered structure, one finds which phase has the lowest free energy, and in this way the phase diagram shown in Fig. 42 (left part) has resulted [43]. 

Landau theory 188, 268, 275 Leibler theory 183, 263-286 Leonard-Jones fluids 242 Lifshitz-Slyozov type growth 258... 

The WSL theory developed by Leibler has been shown to be incorrect because of deviations from the fundamental underlying mean-field assumption. Figure 13.14 shows experimental results for a poly(ethylene-propylene/ethylethylene) (PEP-PEE) diblock copolymer that has been fit to the predictions of the Leibler theory without any adjustable parameters, since the ODT and / were calculated from rheological measurements (Bates et al., 1990). This mean-field theory does not qualitatively describe the behavior of this material. Other experiments have indicated that the RPA approximation (Sttihn and Stickel, 1992) and the Gaussian coil assumption (Bates and Hartney, 1985 Holzer et al., 1991) are inaccurate near the ODT. 

L. Leibler, Theory of Microphase Separation in Block Copolymers," Macromoiecuies. 13,1602-1617 (1980). 

Figure 6.15. The interfacial excess of a deuterated styrene-isopiene diblock, with styrene block relative molecular mass 1000 000 and isoprene block relative molecular mass 10000. The matrix was polystyrene of relative molecular mass 2890000 (o), the excess at the surface of the film and ( ), the excess at the silicon substrate. Solid lines are the predictions of the Leibler theory, whereas the dashed lines are calculated from self-consistent field theory. After Budkowski et al. (1995).

L. Leibler, Theory of microphase separation in block copolymers. Macromolecules, 13,... 

For the analytically tractable thread polymer model, and the R-MMSA or R-MPY/HTA closure approximations, k = 0 values of the direct correlation functions are precisely the same in the long chain limit as found for polymer blends in Sect. 8. In particular, for the symmetric block coprdymer, the R-MMSA closure yields [67,86] the mean field result Xinc = 3Co- Thus, within the symmetric thread idealization and the incompres fe approximation ctf Eq. (9.5), PRISM/R-MMSA theory reduces to Leibler theory for all compositions and block architectures [67,86]. 

Leibler Theory for Strongly Incompatible Systems and Its Modification Diblock Copolymer Additives... 

The phase diagram for diblock copolymers was first derived in the Mean Field Approximation by Leibler [126]. He found that for these materials the same succession of phase transitions occurs with increase of temperature (i.e., lamellae, cylinders, spheres, and disordered state) as calculated phenomenologically (without taking into account the material structure) by Alexander and McTaque [131], However, while according to the Leibler theory a continuous transition between the disordered and the lamellar state could be expected only for a symmetrically composed diblock copolymer at a critical value of %N = 10.495, experiments showed this transition to be discrete and also to occur between asymmetrically composed diblock copolymers within a certain composition range. % is the seg-... 

In addition, the actual interaction potentials are not explicitly considered, their effect is simply lumped into some effective interaction parameters SAAt AB and esB between the respective nearest neighbor pairs. Also the Leibler theory of mesophase ordering in block copolymers may be thought... 

A further consequence results when we note that x should simply be proportional to e = tAs/ksT for this model one then predicts that should simply decrease linearly with the product eN, and then determines the microphase separation transition temperature. In this section we have redefined the meaning of e, in order to have a notation consistent with the original literature.Figure 7.37 demonstrates that there is pronounced curvature on a plot of NS n q ) versus eN, rather than a linear decrease. This behavior is reminiscent of both experimental findings and theoretical predictions due to Fredrickson and Helfand, who have taken fluctuation corrections to the Leibler theory... 

Structure factor 1/S(q ), where q denotes the wavevector for which S q) has a maximum. A modified version of Leibler theory [115] relates / and S q ) through the expression... 

Leibler [13] developed a statistical thermodynamic theory for the microphase separation of an AB-type diblock copolymer on the basis of random phase approximation (RPA) [43], The Leibler theory predicts the spinodal temperature (7].) and Todt of microphase-separated AB-type diblock copolymers, as well as the type of microdomain structure (body-centered cubic, hexagonally packed cylinders or lamellae). This theory is believed to be valid for WSL, whereas the H-W theory is only valid for SSL due to its use of NIA. 

Fig. 7 Phase diagrams of an AB-type diblock copolymer in terms of vs the volume fraction (/) of block A, predicted from the Leibler theory, in which curve (a) denotes the phase transition from the disordered state to spherical microdomains, curve (b) denotes the phase tiansition from spherical microdomains to cyhndrical microdomains, curve (c) denotes the phase tiansition from cylindrical microdomains to lamellar microdomains, and curve (d) denotes the spinodal curve %N)s. (Reprinted from Han et al. [41], Copyright 1989, with permission from the American Chemical Society)...

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