Scaling self-consistent field theory

M. Challacombe A simplified density matrix minimization for linear scaling self-consistent field theory, J. Chem. Phys. 110, 2332-2342 (1999). 

This chapter is concerned with the application of liquid state methods to the behavior of polymers at surfaces. The focus is on computer simulation and liquid state theories for the structure of continuous-space or off-lattice models of polymers near surfaces. The first computer simulations of off-lattice models of polymers at surfaces appeared in the late 1980s, and the first theory was reported in 1991. Since then there have been many theoretical and simulation studies on a number of polymer models using a variety of techniques. This chapter does not address or discuss the considerable body of literature on the adsorption of a single chain to a surface, the scaling behavior of polymers confined to narrow spaces, or self-consistent field theories and simulations of lattice models of polymers. The interested reader is instead guided to review articles [9-11] and books [12-15] that cover these topics. 

Scaling theory Self-consistent field theory ... 

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. 

The contribution "Application of Meso-Scale Field-based Models to Predict Stability of Particle Dispersions in Polymer Melts" by Prasanna Jog, Valeriy Ginzburg, Rakesh Srivastava, Jeffrey Weinhold, Shekhar Jain, and Walter Chapman examines and compares Self Consistent Field Theory and interfacial Statistical Associating Fluid Theory for use in predicting the thermodynamic phase behavior of dispersions in polymer melts. Such dispersions are of quite some technological importance in the... 

Experimentally the overall size of the polymer chain can be studied by light scattering and neutron scattering. A great deal of theoretical work is present in the literature which tries to predict the properties of mixtures in terms of their components. The analytical model by Rouse-Zimm [85,86] is one of the earliest works to derive fundamental properties of polymer solutions. Advances were made subsequently in dilute and concentrated solutions using perturbation theory [87], self-consistent field theory [88], and scaling theory [89],... 

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... 

An almost identical conclusion was obtained analytically by way of the self-consistent field theory. Edward [23] showed Pcy j 9/5 for d = 3 in good accord with the numerical calculation mentioned above, together with the mean end-to-end distance that scales as... 

Self-consistent field theory leads to the prediction that the domain spacing, dy for a lamellar semicrystalline diblock, scales as [20] ... 

This scaling law was compared with the results of self-consistent field theory by van der Linden and Leermakers (1992) they found that the profiles did follow a power law over the central region. In the limit of vanishing bulk volume fraction and infinitely long chains the power law exponent did indeed tend towards 2 as predicted by equation (5.2.38), but the corrections for finite relative molecular mass and bulk volume fractions are considerable. For calculations on a cubic lattice they found that the power law exponent a could be represented by... 

To obtain useful theoretical results for the concentration profile, we need to go beyond these simple scaling arguments. Luckily, at least for the situation of relatively dense, strongly stretched, brushes, we can expect self-consistent field theories to work rather well in such a dense brush the basic mean-field assumption that any polymer chain will interact with its neighbours more than it will with itself should be well obeyed. Niunerical mean-field theories of the kind described in chapter 5 are very well suited to this kind of calculation the earliest results, due to Hirz (these results are still unpublished but some were reproduced by Milner et al. (1988)) showed profiles very different in character from those found for adsorbed chains. Rather than a concave concentration profile, the curves were notably convex, with the concentration dropping rather abruptly to zero on the outside of the brush. In fact it turns out that the profiles are rather well described by a parabolic form (see figure 6.7). It soon turned out that there was a remarkably good analytical solution to the self-consistent mean-field equations which provided an explanation for these parabolic profiles. 

The dependence of h on M and a is rather weaker than that predicted by Alexander-de Gennes scaling laws and by the self-consistent field theory of Milner, Witten and Cates (h A self-consistent field theory of... 

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