Leibler s theory

Fluctuation Effects and Other Extensions of Leibler s Theory. . 274... 

Experimental data (Fig. 43a) indeed are fit by the structure factor of Leibler s theory [43] nicely - the curves shown in this figure from a convolution of Eq. (187) with the resolution function, as shown in the insert (the dashed curve is Eq. (187), the full curve is result of the convolution, for T = 126.3 °C). However, at each temperature both Rg(T) and x(T) are used as adjustable parameters - therefore a nearly perfect fit is possible although the peaks at the different temperatures do not occur at precisely the same q. Thus the agreement shown in Eq. (187) should not be taken as a proof for the accuracy of Leibler s theory -such a proof would require an independent measurement of Rg(T) and y(T). 

A more distinctive test of Leibler s theory has been possible via Monte Carlo simulations [325, 326], as will be discussed in detail in Sect. 5.4, though these simulations are restricted to very short chains only (modelled by self-avoiding walks with N = 16 to N = 60 steps on the simple cubic lattice). We here only anticipate one example (Fig. 44) that also the simulation results can be adjusted perfectly to Leibler s theory if one treats Rg (in Eq. (189) as an effective parameter... 

Fig.44. Collective structure factor S(x,e) dotted vs x = qRg(e,N) for f = 1/2, N = 20 and various choices of the energy kBTe between monomers of different kinds, allowing for a volume fraction , = 0.2 of vacancies on the simple cubic lattice. Curves are a fit to Eq. (187), treating both % and Sg in Eqs. (187) — (189) as adjustable parameters, while the actual gyration radius is used for the normalization of the abscissa. Perpendicular straightline shows the value x = 1.945of Leibler s theory [43]. The symbols denote the choices eN = 0,1,2,3,4 and6 (from bottom to top). From Fried and Binder [325],...

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). 

Leibler s theory outlined an experimental method for testing its conclusions. The structure factor for scattering of radiation by the disordered phase was given by ... 

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. 

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. 5. Structure factor following from different theories. Note that the structure factor following Leibler s theory at xN = 10.495 would diverge Leihler ( N = 10.49) ...

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.

At temperatures at 138.2 and 118.7°C (profiles c and d) the higho -ordo scattering maxima were observed at the positions of -Jl and >/3 relative to that of the first-order scattering maximum, q . Detailed analyses of them indicates the formation of spherical microdomains packed in a body-centered cubic lattice (denoted as bcc-sphere) [19]. Leibler s theory [8] predicts that the disordered state is transformed into the bcc-sphere when the intention parameter between the constituents, x, is increased. In this sense the structure at 157.7 and 177.4°C (profiles a and b) is expected to exhibit the disordered state since x between PS and PI, Xsi decreases with increasing temperature. However die profiles a and b show evidence of the microdomains (possibly spherical microdomains) as mentioned before. Thus we... 

We emphasize at this point that all these pretransitional chain stretching effects found in the simulations are at variance with the simple RPA treatment as embodied in Leibler s theory this simple form of the RPA would mean only that all the ratios shown in Fig. 7.38 should be strictly unity, irrespective of e and N. Thus the simulations reveal rather drastic deviations from RPA, a hitherto somewhat unexpected result, since for dense melts the RPA has been taken as basically exact by many researchers. In this context, it is also important to recall that these deviations from the RPA should not be attributed to the vacancy content of the model (recall that Fig. 7.17(a) has studied this behavior for variable vacancy content, and behavior independent of cj>v for T > Tc was found " ). 

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