Self-consistent field theory length

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].

There is a substantial body of theoretical work on micellization in block copolymers. The simplest approaches are the scaling theories, which account quite successfully for the scaling of block copolymer dimensions with length of the constituent blocks. Rather detailed mean field theories have also been developed, of which the most advanced at present is the self-consistent field theory, in its lattice and continuum guises. These theories are reviewed in depth in Chapter 3. A limited amount of work has been performed on the kinetics of micellization, although this is largely an unexplored field. Micelle formation at the liquid-air interface has been investigated experimentally, and a number of types of surface micelles have been identified. In addition, adsorption of block copolymers at liquid interfaces has attracted considerable attention. This work is also summarized in Chapter 3. 

Proposed Scaling in Self-consistent Field Theory.—March and Parr have proposed an extension of the 1 fZ expansion for atoms, analogous to rearrangement (48), for the case of diatomic molecules. Introducing a scaled length X=RZ, they note first that from the above discussion of the bare Coulomb field one has... 

Recent theoretical efforts to address the phase behavior of associating polymer blends have involved self-consistent field theory (SCFT) and lattice cluster theory (LCT). The SCFT framework was applied to enumerate all possible linear reaction products for blends of self-complementary and heterocomplementary telechelics. Mesophase regions were identified. Dudowicz and Freed reformulated the LCT to model solvent-telechelic polymer blends and identified several trends, including an enhancement of miscibility as the molar mass of the associating polymer was increased [92, 93]. This trend was explained by unbalanced entropy and enthalpy changes that occur with increasing chain length. 

Figure 1. Theoretically calculated adsorption isotherm for a polymer of 100 segments adsorbing from a theta solvent (x=0.5). The isotherm was obtained from the self-consistent field theory of Scheutjens and Fleer. Two arrows indicate the values of the concentration difference Cb-Cs between bulk and surface zone one for adsorption (high cb ) and one for desorption (cb =0). The length of the arrows is a measure of the rate of the corresponding processes.

The dynamic self-consistent field theory has been widely used in the form of MESODYN [97]. This scheme has been extended to study the effect of shear on phase separation or microstructure formation, and to investigate the morphologies of block copolymers in thin films. In many practical applications, however, rather severe numerical approximations (e.g., very large discretization in space or contour length) have been invoked, that make a quantitative comparison to the the original model of the SCF theory difficult, and only the qualitative behavior could be captured. 

To summarize, the example of homopolymer/copolymer mixtmes demonstrates nicely how field-theoretic simulations can be used to study non-trivial fluctuation effects in polymer blends within the Gaussian chain model The main advantage of these simulations is that they can be combined in a natural way with standard self-consistent field calculations. As mentioned earlier, the self-consistent field theory is one of the most powerful methods for the theoretical description of polymer blends, and it is often accurate on a quantitative level hi many regions of the parameter space, fluctuations are irrelevant for large chain lengths (large Jf) and simulations are not necessary. Field-theoretic simulations are well suited to complement self-consistent field theories in those parameter regions where fluctuation effects become important. 

Fig. 58 Mean-field density profiles obtained fiom self-consistent field theory simulations. A- versus B-rich domains are displayed for a blend of A- and B-homopolymers (a) and for AB-diblock-copolymer melts (b, c). In each case, all A-, and B-blocks contain equal numbers of monomers. Here, spherical confinement is implemented by blending either A- and B-homopolymws (a), or AB-diblock-copolymers (b, c) with C-homopolymers. The C-homopolymers act as a very bad solvent, thus enforcing the formation of A-, and B-rich spherical domains. In this case, the geometry of the confined polymer phases is studied in two dimensions. Whether Janus (a), core-shell (b), or onion (c) particles form depends on the number of monomers per block, and the interactirais between different monomer species. From (a) to (c), the length of A-, and B-sequences steadily decreases the sequences in (a) are roughly four times as long as in (b), and are about 15 times as long as in (c). To form Janus particles, the A-C versus B-C inlmactions need to be equal. To form layered structures, there has to be a significant difference...

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