Helfand’s theory

Application of Helfand s theory has been limited due to the necessity for numerical analysis (although FORTRAN code to facilitate calculations is provided by Helfand and Wasserman (1982) ).This problem was circumvented with the introduction of the seminal analytical SSL theory by Semenov (1985). 

For the strategies of compatibilization, Helfand s theory provides three important conclusions (1) the chain-ends of both polymers concentrate at the interface, (2) any low molecular weight third component is forced by the thermodynamic forces to the interface, and (3) the interfacial tension coefficient increases with molecular weight up to an asymptotic value. 

From Helfand s theory, it follows that the free energy of a system will be less with a diffuse interface. As a measure of the interphase thickness, the theory assumes the segment between the points of intersection of the inflectional tangent to the density gradient curve with density levels pi and p2 ... 

This result has been shown to be asymptotically correct in the limit xN — °° (Matsen and Bates 1996a Matsen and Whitmore 1996). These exponents differ from those obtained by Helfand and Wasserman, who found d aAr0< 43 0143 (with a geometry-dependent prefactor) and F JkT (dlaN a)2S (Helfand and Wasserman 1976). The difference arises because Helfand s calculations were only carried out numerically to d/aNm 3, whereas in the limit d/aNm 1, the scaling (eqn 2.4) is obtained. As in the Helfand-Wasserman theory, phase boundaries can be computed using Semenov s theory, and are also found to be independent of (Semenov 1985). 

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). 

It is noteworthy that Leibler s (see Eqs 4.20 and 4.22) and Noolandi s theories (see Eq 4.28) predict that the product V Al depends on the binary interaction parameter Thus, the reciprocity between and A1 predicted by Helfand and Tagami for binary systems is not expected to exist in compatibilized binary blends. 

Roe (1975) developed a quasicrystalline lattice model for conditions where Xi2 Xcr (where Xcr is the critical value of the interaction parameter at the phase separation) and for Xn Xcr Xcr- Under the first conditions (high immiscibility), the theory predicted a proportionality between Vj2 and Xn whereas under the secmid (near the phase separation), a proportionality between Vi2 and xn was predicted. By contrast with the previously summarized Helfand and Tagami predictions. Roe s theory indicates that the product V12A/ should be proportional to... 

Such definitive forecasts do not result from the theories of Leibler or Hong and Noolandi. However, the latter theory describes in detail the phase diagram for copolymer — homopolymer mixtures and is therefore pertinent to the X-ray work of Roe Similarly, Leibler s theory provides a detailed description of a microphase separation mechanism and thus is of value in the interpretation of experiments investigating this phenomenon. Small angle neutron scattering data reported to date has been maitily concerned with pure styrene-diene block copolymers which are fully microphase separated and thus examined D, dj, and interfadal layer thickness as a function of molecular weight and composition and therefore comparison has usually been made with the MIA theory of Helfand. 

Fig. 5. Log-log plot of mean sphere size as a function of the molecular weight of the butadiene block of styrene-butadiene block copolymers. Upper dashed line from Helfand NIA theory lower dashed line is a fit to Hashimoto s results. Ref.

Values of domain size and separation were discussed in terms of Helfand s NIA theory, tacitly assuming that the two components of the block copolymer constituted a symmetric polymer pair, i.e. equality of Kuhn statistical step lengths, b, and monomer molar density. Whilst this is approximately true for values of b for styrene and isoprene, the densities are quite different. However, calculations of domain size and domain separation as a function of molecular weight for both styrene and isoprene domains show that both types have the same molecular weight dependence and moreover the difference in the values for either styrene or isoprene domains is negligible. Figure 10... 

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.

These observations are in accord with Helfand s general theory that at thermodynamic equilibrium the interphase in a blend of immiscible polymers... 

Fig. 6 Phase diagram of an SI diblock copolymer in terms of molecular weight vs weight fraction of PS block at 150°C predicted from the Helfand-Wasserman theory, in which S denotes spherical microdomains, C denotes hexagonaUy packed cylindrical microdomains,...

A well-known approximate molecular theory of a fluid at a planar interface is originally due to Helfand, Frisch and Lebowitz [76] and later to Henderson, Abraham and Barker [77] and Perram and White [78]. Consider a binary mixture (A,B) in which one of the species (A) becomes extremely dilute and infinitely large. S.E. [52] show that if the size of species A tends to infinity while the concentration of A tends toward zero, then a consequence of the OZ equation, coupled with the PY equation, is the relation... 

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]...

Work on the SSL has continued since the pioneering work of Helfand et al. Semenov [72] and Likhtman and Semenov [73] developed an analytic theory for the strong segregation regime using the UCA. They predicted that the S/C and L/C phase boundaries for conformationally symmetric diblocks are at... 

Helfand [202, 208] suggested that Roe s work contained a number of assumptions, which made it difficult to appraise the applicability of the theory. Helfand suggested that Roe s lattice theory did not treat the conformational entropy properly by assuming that the chances of going from a cell site to any empty neighboring cell... 

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