Block Copolymers in the Weak Segregation Limit

For this problem already the simple mean field approximation becomes rather involved [197,213]. Therefore, we describe here only an approach, which is even more simplified, appropriate for wavenumbers q near the characteristic wavenumber q, but strictly correct neither for q— 0 nor for large q the spirit of our approach is similar to the long wavelength approximation encountered in the mean field theory of blends, Eq. (7). That is, we write the effective free energy functional as an expansion in powers of t t and include terms (Vv /)2 as well as (V2 /)2, as in the related problem of lamellar phases of microemulsions [232,233],namely [234] 

From the random phase approximation [200] the characteristic wavenumber q of lamellar ordering can be calculated [ 197], 

Note that on the mean field level at %c a second-order transition is predicted for f=H2, while taking fluctuations into account renders the transition first order [192,210,211], as also found experimentally [231]. Although the nature of surface effects on this transition is quite different for the first-order case [6] than for the second-order case [12], we discuss mostly the second-order case here. The constant pc in Eq. (43) is the density of the chains (pc=l/N if a Flory-Huggins lattice with lattice spacing unity is invoked), and the constants e0 and u0 can be derived [197] from the random phase approximation as 

Assuming now a solution of the form of a simple standing wave, 

When we now consider a thin film of thickness D, Eq. (41) must be supplemented by boundary conditions of the same type as in the polymer blend case, Eqs. (7) and (10), i.e. we add a (bare) surface free energy contribution to the free energy that accounts for preferential attraction of one kind of monomers to the walls, missing neighbors in the pairwise interactions, and possible changes in the pairwise interactions near the surface. As in the blend case, this surface contribution is taken locally at the walls only and expanded to second order in the local order parameter /(z). Per unit area of the wall, this free energy is written as 

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Theoretical descriptions of block copolymers in the weak segregation limit were first given by Leibler and Erukhimovich. °... 

The expressions 2.7-2.12 which define the Leibler structure factor have been widely used to interpret scattering data from block copolymers (Bates and Fredrickson 1990 Mori et al. 1996 Rosedale et al. 1995 Schwahn et al. 1996 Stiihn et al. 1992 Wolff et al. 1993). The structure factor calculated for a diblock with / = 0.25 is shown in Fig. 2.39 for different degrees of segregation JV. Due to the Gaussian conformation assumed for the chains (Leibler 1980), the domain spacing in the weak segregation limit is expected to scale as d Nm. 

In a seminal paper, Leibler [43] presented the first mean-field-like theory of the ODT transition and the phase diagram of block copolymer melts in the weak segregation limit. This work still is the basis for more elaborate theories [58-64] and for the discussion of recent experiments (e.g. [317-323]). As shown in Fig. 42, the quantitative details of the resulting predictions are still subject of current research, but nevertheless we try to sketch this theory here, since this derivation gives a good insight into the relevant physical aspects of this problem. 

In block copolymer/homopolymer blends there is some evidence for lamellar systems (Quan and Koberstein, Macromolecules. 1987) that the homopolymer is localized in the center of the microdomain. This effect would lead to the kind of diffusion resistance that you have observed, that is, a greater resistance than that caused simply by dilution by the second component. Secondly, the solubility of the homopolymer in the microdomains falls off rapidly when its MW is about 1/2 that of the like copolymer sequence. In your case, however, you still get diffusion up to the case of equal MW s, This may be related to your conclusion that your system behaves as if it is in the weak segregation limit. 

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