Fredrickson-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... 

Figure 13.3 Dependence of the apparent X parameter on inverse temperature, extracted from fits of the Fredrickson-Helfand fluctuation theory to neutron scattering data for the polystyrenc-polyisoprene diblock copolymers listed. The reference volume v is 1.503 x 10 cm. Note that for most of the samples, a linear relationship between / and 1/ T is observed, consistent with Eq. (13-1). The apparent x values for highly asymmetric diblocks are higher than those for more symmetric ones. (Reprinted with permission from Lin et al., Macromolecules 27 7769. Copyright 1994, American Chemical Society.)-------------...

Figure 13.11 Phase diagram for a diblock copolymer in the weak-segregation limit predicted by (a) the Leibler mean-field theory and (b) the Fredrickson-Helfand fluctuation theory. (From Bates et al., reprinted with permission from J. Chem, Phys. 92 6255, Copyright 1990, American Institute of Physics.)...

Figure 5 shows the structure factor from both mean-field theory (Leibler) and the fluctuation correction (Fredrickson-Helfand) for a symmetric diblock... 

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-... 

Fig. 10. Schematic phase diagram of a semi-infinite block copolymer melt for the special case of a perfectly neutral surface (Hj=0). Variables chosen are the surface interaction enhancement parameter (-a) and the temperature T rescaled by chain length (assuming X l/T the ordinate hence is proportional to %c/%). While according to the Leibler [197] mean-field theory a symmetric diblock copolymer transforms from the disordered phase (DIS) at Tcb oc n in a second-order transition to the lamellar phase (LAM), according to the theory of Fredrickson and Helfand [210] the transition is of first-order and depressed by a relative amount of order N 1/3. In the second-order case, the surface orders before the bulk at a transition temperature T (oc l / ) as soon as a is negative [216], and the enhancement...

For the pair styrene/isoprene, various other dependences of x or temperature have been reported in the literature, which give values of x that at 100°C differ by almost a factor of two, from 0.06 to around 0.14 (see Fig. 13-3) (Lin et al. 1994 Han et al. 1995). These estimates of x ot obtained by direct calorimetric measurements (the energetic effects are too minute to measure), but by fitting the predictions of thermodynamic theories (such as that of Leibler or Fredrickson and Helfand see below) to x-ray or neutron scattering data for diblock copolymers or blends. The values of / thereby obtained are only as accurate as the theories to which the data are fitted. 

Strong-segregation theories predict for the domain size D, the interfacial tension F, and the interfacial width A (Helfand and Wasserman 1982 Bates and Fredrickson 1990 Wang 1994) ... 

Although the gross shape of the phase envelope predicted by the mean-field theory, as well as the regions of lamellar and hexagonal phases, are more or less in agreement with experiments on diblock copolymers (see Fig. 13-4), the predictions of the theory near the critical point at / = 0.5, /(V = 10.5 are incorrect. Fredrickson and Helfand (1987) showed that the second-order transition predicted by the mean-field theory is corrected to SL first-order transition when the effects of fluctuations on the free energy are accounted for using a so-called Brazovskii Hamiltonian (Brazovskii 1975). 

Fig. 43a. Neutron small angle scattering intensity I(q) plotted vs q for three temperatures T above Tmst (main graph), for a polyethylenepropylene(PEP) — polyethylethylene(PEE) diblock copolymer, with f = 0.55, molecular weight Mw — 57.500, polydispersity index Mw/Mn = 1.05. The microphase separation transition occurs for Tmst = 125°C. For further explanations cl Textb Inverse peak intensity I (q ) dotted vs inverse temperature.The full curve is a one-para meter fit to the theoty of Fredrickson and Helfand [58], while Leibler s [43] prediction for the intensity at the transition is marked as mean field theory . From Bates et al. [317]...

Figure 47 shows the qualitative behavior of this free energy density. A crucial feature is that the renormalized distance xR corresponds still to the inverse scattering intensity S-l(q) at q = q. Since xocxocl/T in simple polymers, the nonlinear relation between x and xR then implies a nonlinear relation between xR and 1/T. Thus while Leibler s theory [43] predicts a linear variation of S" (q ) with 1/T (near the temperature where S-1(q ) should vanish for f = 1/2), the fluctuation effects of Helfand and Fredrickson [58] imply a curved variation of S l(q ) with 1/T. Such a curved variation indeed is found both in experimental data [317-323] and simulations [325, 328], see Figs. 43b, 48. Of course, due to finite size problems in the simulation one cannot as yet detect the small jump singularity that signals the mesophase separation transition in the experiment (Fig. 48). 

The first steps toward such a theory of blend flow behavior were proposed by Helfand and Fredrickson [1989], then by Doi and Onuki [1992]. A greatly simplified constitutive equation for immiscible 1 1 mixture of two Newtonian fluids having the same viscosity and density was also derived [Doi and Ohta, 1991]. The derivation considered time evolution of the area and orientation of the interface in flow, as well as the interfacial tension effects. The relation predicted scaling behavior for the stress and the velocity gradient tensors ... 

Theory of Theories viscosity - concentration Helfand Fredrickson,... 

In the original analysis, it was understood that this theory should not be applied near the critical point. Mean-field theories ignore concentration fluctuations at distances other than q = q. Near the critical point, concentration fluctuations on very large length scales become increasingly important. A modiflcation to this theory that includes concentration fluctuations has been developed (Fredrickson and Helfand, 1987). A critical point is not predicted by the fluctuation theory, rather a first-order phase transition is predicted for all compositions. A molecular weight dependence is found for /A, which delineates the ordered from the disordered phase shown below for the symmetric diblock copolymer ... 

The self-consistent field theory phase diagram is also likely to be inaccurate at low relative molecular mass, because, like any mean-field theory, it neglects fluctuations. The effect of fluctuations is to stabilise the disordered phase somewhat (Fredrickson and Helfand 1987) in addition the seeond-order transition predicted for the symmetrical diblock is replaced by a first-order transition and, for asymmetrical diblocks, there are first-order transitions directly from the disordered into the hexagonal and lamellar phases. In addition it seems likely that fluctuations tend to stabilise high symmetry states such as the gyroid (Bates et al. 1994). 

Fredrickson GH, Helfand E (1987) Fluctuation effects in the theory of micro-phase separation in block copolymers. J Chem Phys 87 697-705... 

Experiments of Roe et al. [26] and Hashimoto et al. [27] demonstrated that scattering experiments on disordered block copolymers may also be used to determine x I parameters, using the RPA theory of Leibler [9]. In a subsequent paper, Fredrickson and Helfand showed that fluctuation corrections to the RPA are important in block copolymer melts [28]. When available, x parameters obtained from block copolymer melts are reported after fluctuation corrections have been incorporated, k values obtained from block copolymers are often [29,30] but not always [31] larger than those obtained in homopolymer blends. 

For the symmetric diblock copolymer a second-order transition between lamellar and disordered phase was predicted, while at all other compositions a first-order transition between disordered state and a body-centered cubic phase of spherical domains formed by the minority component was predicted, which changes into hexagonally packed cylinders and finally into lamellae upon further increasing xN. It has already been noted by Leibler s that his approach does not include fluctuation effects, which become important for finite degrees of polymerization (74). Fredrickson and Helfand accounted for this problem by modifying Leibler s theory in the following way (90) ... 

Fig. 6. Phase diagram of a diblock copolymer according to Leibler s theory (left) and including fluctuation corrections according to Fredrickson and Helfand (right). From Ref. 91. Copyright (1990) American Institute of Physics.

Fredrickson and Helfand developed a weak segregation theory with fluctuations [77], starting by mapping the Leibler free energy functional onto one treated earlier by Brazovskii [78]. They used the UCA and treated the three classical phases. This has since been extended to the G phase [79,80] and to triblocks [81], and modified to include a multihannonic approximation for the density profiles, although not for the G phase [82]. 

The mean-field theory of Leibler agrees very well with experimental observations based on X-ray and neutron scattering when obtained relatively far from the microphase separation temperature (MST). In the vicinity of the MST, however, mean-field treatment is less accurate. Both, Leibler and Fredrickson and Helfand noted that the effective Hamiltonian appropriate for diblock copolymers is in the Brazovskii-universality class [15,16]. Based on the Hartree treatment used in the Brazovskii theory, Fredrickson and HeUand found that the structure factor can still be written as the mean-field expression Eq. (7.101), but with renormalized values % and N [16]. The fluctuation renormalization makes the order parameter nonlinear in x, and thereby nonlinear in T". This is shown in the experimental example given in Figure 7.13. 

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