Weak segregation

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). 

The weak-segregation phase diagram predicted by the FH theory for N = lO is compared to the Leibler mean-field diagram in Fig. 13-11. In the FH theory, at / = 0.5, the ordering transition to a lamellar phase occurs when 

there is an increasing upward shift in (xA )odt from the Leibler value of 10.5 as the molecular weight decreases, the Leibler result is recovered in the limit of infinite molecular weight. Since the FH theory is a perturbative one that assumes the shift in (xjV)odt lo be small, the theory should work best for high-molecular-weight polymers N k, 10 ) and small values of x (X 0.01). 

As N decreases, (x A)odt exceeds 10.5 to a greater degree, and there is an increasing range of X values (and hence of temperatures) over which fluctuation effects are important. This 

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Figure B3.6.5. Phase diagram of a ternary polymer blend consisting of two homopolymers, A and B, and a synnnetric AB diblock copolymer as calculated by self-consistent field theory. All species have the same chain length A and the figure displays a cut tlirough the phase prism at%N= 11 (which corresponds to weak segregation). The phase diagram contains two homopolymer-rich phases A and B, a synnnetric lamellar phase L and asynnnetric lamellar phases, which are rich in the A component or rich in the B component ig, respectively. From Janert and Schick [68].

Weakly segregated systems, Todt > Tc > Tg with soft confinement. In this case, crystallization often occurs with little morphological constraint, enabling a breakout from the ordered melt MD structure and the crystallization overwrites any previous melt structure, usually forming lamellar structures and, in many cases, spherulites depending on the composition [10-18],... 

Weakly segregated systems, Todt > Tc < Tg with hard confinement. In this case, the crystallization of the semicrystalline block can overwhelm the microphase segregation of the MD structures even though the amorphous block is glassy at the crystallization temperature, because of the weak segregation strength [19]. 

Near the ODT, the composition profile of ordered microstructures is approximately sinusoidal (Fig. 2.1).The phase behaviour in this regime, where the blocks are weakly segregated, can then be modelled using Landau-Ginzburg theory, where the mean field free energy is expanded with reference to the average composition profile. The order parameter for A/B block copolymers may be defined as (Leibler 1980)... 

The phase diagram for weakly segregated diblocks was first computed within the Landau mean field approximation by Leibler (1980). Because it has proved to be one of the most influential theories for microphase separation in block copolymers, an outline of its essential features is given here.The reader is referred to the original paper by Leibler (1980) for a complete account of the theory. 

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. 

Fig. 2.40 Phase diagram for diblock copolymers in the weak segregation limit (Leibler 1980).

Computer simulations of a range of properties of block copolymer micelles have been performed by Mattice and co-workers.These simulations have been based on bead models for copolymer chains on a cubic lattice. Types of allowed moves for bead chains are illustrated in Fig. 3.27. The formation of micelles by diblock copolymers under weak segregation conditions was simulated with pairwise interactions between A and B beads and between the A bead and vacant sites occupied by solvent, S (Wang et al. 19936). This leads to the formation of micelles with a B core. The cmc was found to depend strongly on fVB and % = x.w = %AS. In the range 3 < (xlz)N < 6, where z is the lattice constant, the cmc was found to be exponentially dependent onIt was found than in the micelles the insoluble block is slightly collapsed, and that the soluble block becomes stretched as Na increases, with [Pg.178]

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