Disorders self-consistent mean field

Fig. 2.44 Phase diagram for a conformationally-symmetric diblock copolymer, calculated using self-consistent mean field theory. Regions of stability of disordered, lamellar, gyroid, hexagonal, BCC and close-packed spherical (CPS), phases are indicated (Matsen and Schick 1994 ). All phase transitions are first order, except for the critical point which is marked by a dot.

Fig. 1 Phase diagram of self-assembled structures in AB diblock copolymer melt, predicted by self-consistent mean field theory [31] and confirmed experimentally [33]. The MesoDyn simulations [34, 35] demonstrate morphologies that are predicted theoretically and observed experimentally in thin films of cylinder-forming block copolymers under surface fields or thickness constraints dis disordered phase with no distinct morphology, C perpendicular-oriented and Cy parallel-oriented cylinders, L lamella, PS polystyrene, PL hexagonally perforated lamella phase. Dots with related labels within the areal of the cylinder phase indicate the bulk parameters of the model AB and ABA block copolymers discussed in this work (Table 1). Reprinted from [36], with permission. Copyright 2008 American Chemical Society...

In a more detailed calculation the self-consistent mean field theory reduces the problem of calculating the interactions among polymer chains to that of a single noninteracting polymer chain placed in an external field self-consistent with the composition profiles (26). Again, the primary objective is to compute the free energy and polymer distribution functions near the order-disorder transition. 

Keywords Block copolymer Disordered micelles Fluctuation effect Order-disorder transition Self-consistent mean-field theory... 

A phase diagram computed using self-consistent mean field theory [49,51] is shown in Figure 1.2. This shows the generic sequence of phases accessed just below the order-disorder transition temperature for diblock copolymers of different compositions. The features of phase diagrams for particular systems are different in detail, but qualitatively they are similar, and well accounted for by SCF theory. 

Figure 11. A schematic representation of the mean-field approximation, a central issue in the self-consistent-field theory. The arrows symbolically represent the lipid molecules. The head of the arrow is the hydrophilic part and the line is the hydrophobic tail. On the left a two-dimensional representation of a disordered bilayer is given. One of the lipids has been selected, as shown by the box around it. The same molecule is depicted on the right. The bilayer is depicted schematically by two horizontal lines. The centre of the bilayer is at z = 0. These lines are to guide the eye the membrane thickness is not preassumed, but is the result of the calculations. Both the potential energy felt by the head groups and that of the tail segments are indicated. We note that in the detailed models the self-consistent potential profiles are of course much more detailed than shown in this graph...

Fig. 6.22 Phase diagram for blends of PE and PEP homopolymers (A/j, - 392 and 409 respectively) with a PE-PEP diblock (iVc = 1925) (Bates et al. 1995). Open and filled circles denote experimental phase transitions between ordered and disordered states measured by SANS and rheology respectively. Phase boundaries obtained from self-consistent field calculations are shown as solid lines. The diamond indicates the Lifshitz point (LP), below which an unbinding transition (UT) separates lamellar and two-phase regions in mean field theory.

Self-consistent field theory has recently been employed by Janert and Schick (1996,1997a) to analyse the swelling of diblock lamellar phases with homopolymer. It was shown that a complete unbinding transition, where added homopolymer swells the lamellae, finally leading to a transition to a disordered phase, is predicted by mean field theory. The swelling does not continue without limit. 

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). 

The microphase separation of block copolymers is sometimes called order-disorder transition (ODT). The self-consistent-field theory (SCFT) provides a mean-field method to calculate various geometric shapes of microdomains. Edwards first introduced the SCFT into polymer systems on the basis of making path integrals along chain conformations (Edwards 1965). Helfand applied it to the mean-field description of immiscible polymer blends on the basis of the Gaussian-chain model... 

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