The Self-Consistent Field Theory

In the presence of interactions between the connected segments of a single chain, aforementioned simple diffusion or random walks get affected and the walks are no more random. However, the intricate coupling of the different components such as monomers, solvent, or small ions in the case of polyelectrolytes via the interaction potentials complicates the theoretical analysis. In order to decouple different components, the conformations of the chain can be envisioned as the walks in the presence of fields, which arise solely due to the fact that there are interactions present in the system. This physical argument is the basis of the use ofcertain field theoretical transformations such as Hubbard-Stratonovich [60] transformation, which is well known in the field theory. So, the conformational characteristics of a polymer chain in the presence of different kinds of intrachain interactions can be described once the fields are known. In general, an exact computation of these fields is almost an impossible task. That is the reason theoretical developments resort to certain approximations for computing these fields, which work well for most of the practical purposes. Once these fields are known, the physical properties can be described in terms of these fields. It was shown by Edwards [50] that the similar analysis can be carried out for systems with many chains, where interchain interactions also affect the properties in addition to intrachain interactions. 

Recently, the field theory developed for neutral polymers has been extended to describe various polyelectrolytic systems in the absence/presence of externally added salt ions. The theory has been used to investigate the micro- and macrophase separation in polyelectrolyte systems [61-63], adsorption of polyelectrolytes on to the charged surfaces [64, 65], polyelectrolyte brushes [66, 67], confinement effects [14], counterion adsorption [15], translocation of polyelectrolytes (R. Kumar and M. Muthukumar, unpublished), and the assembly of single stranded RNA viruses (J. Wang, R. Kumar, and M. Muthukumar, unpublished). In this chapter, we review the general methodology behind the SCFT for polyelectrolytes. 

A general background on the field theoretical formalism for polyelectrolytes is presented in Section 6.4.1. Details of the commonlyused transformations in order to switch from a particle to the field description are presented in Section 6.4.2. DiEFerent kinds of charge distributions along the polydectrolyte chain and the well-known saddle-point approximation for computing the free energy are described in Sections 6.4.3 and 6.4.4, respectively. Numerical techniques to solve the nonlinear set of equations and one-loop expansions to go beyond the well-known saddle-point approximation are presented in Sections 6.4.5 and 6.4.6, respectively. 

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The general fomi of the expansion is dictated by very general synnnetry considerations the specific coefficients for the example of a polymer blend can be derived from the self-consistent field theory. For a... 

The simplest model of polymers comprises random and self-avoiding walks on lattices [11,45,46]. These models are used in analytical studies [2,4], in particular in the numerical implementation of the self-consistent field theory [4] and in studies of adsorption of polymers [35,47-50] and melts confined between walls [24,51,52]. 

We close these introductory remarks with a few comments on the methods which are actually used to study these models. They will for the most part be mentioned only very briefly. In the rest of this chapter, we shall focus mainly on computer simulations. Even those will not be explained in detail, for the simple reason that the models are too different and the simulation methods too many. Rather, we refer the reader to the available textbooks on simulation methods, e.g.. Ref. 32-35, and discuss only a few technical aspects here. In the case of atomistically realistic models, simulations are indeed the only possible way to approach these systems. Idealized microscopic models have usually been explored extensively by mean field methods. Even those can become quite involved for complex models, especially for chain models. One particularly popular and successful method to deal with chain molecules has been the self-consistent field theory. In a nutshell, it treats chains as random walks in a position-dependent chemical potential, which depends in turn on the conformational distributions of the chains in... 

The obvious disadvantage of this simple LG model is the necessity to cut off the infinite expansion (26) at some order, while no rigorous justification of doing that can be found. In addition, evaluation of the vertex function for all possible zero combinations of the reciprocal wave vectors becomes very awkward for low symmetries. Instead of evaluating the partition function in the saddle point, the minimization of the free energy can be done within the self-consistent field theory (SCFT) [38 -1]. Using the integral representation of the delta functionals, the total partition function, Z [Eq. (22)], can be written as... 

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...

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. 

There is no comprehensive theory for crystallization in block copolymers that can account for the configuration of the polymer chain, i.e. extent of chain folding, whether tilted or oriented parallel or perpendicular to the lamellar interface. The self-consistent field theory that has been applied in a restricted model seems to be the most promising approach, if it is as successful for crystallizable block copolymers as it has been for block copolymer melts. The structure of crystallizable block copolymers and the kinetics of crystallization are the subject of Chapter 5. 

Whitmore and Noolandi (1985b) developed the self-consistent field theory to examine micellization in AB diblocks in a blend of AB diblock and A homopolymer solvent . The model was applied specifically to the case of PS-PB diblocks in PB homopolymer for comparison with the results of small-angle neutron scattering (SANS) experiment by Selb et al. (1983). This model is discussed in more detail in Section 3.4.2. 

Survey of Results Obtained with the Self-Consistent Field Theory... 

Chakrabarti et al. [91] used a different measure of the amount of interpenetration but also found that their results agreed with the self-consistent field theory. 

Orbital models are almost invariably the first approximation for the electronic structure of molecules within the Born-Oppenheimer approximation. By considering the motion of each electron in the averaged field of the remaining electrons in the systems in the self-consistent field theory of Hartree and Fock, a model is obtained in which each electron is described by an spin-orbital, a description which is familiar to all chemists. 

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... 

Fig. 3 Cartoon representation of an isolated grafted colloidal sphere (black core) along with the calculated monomer density profile 0(r) (for various core radii, re) [119]. The three solid lines correspond to values of the ratio of the overall particle radius R to the grafted polymer end-to-end distance of 0.5, 2 and 16, from bottom to top, respectively. The dashed line represents the self-consistent field theory calculation for the flat brush limit (r [119]. Note the depletion region...

The substitution of Eqs. (17) into (15) makes the semi-axes of the equilibrium conformational ellipsoid identical and equal to the undeformed Flory coil radius x° = R, . Let us underline two important circumstances. First, SARW statistics leads to the same result, that is, to Eq. (2), that Flory method, which takes into account the effect (repulsion) of the volume interaction between monomer links in the self-consistent field theory. However, as it was explained by De Gennes [6], accuracy of Eq. (2) in Flory method is provided by excellent cancellation of two mistakes top-heavy value of repulsion energy as a result of neglecting of correlations and also top-heavy value of elastic energy, written for ideal polymer chain, that is in Gaussian statistics. Additionally, one must note, that Eq. (2) is only a special case of Eq. (15), which represents conformation of polymer chain in the form of ellipsoid with semi-axes x° 4- allowing to consider this conformation as deformed state of Flory coil. 

Fig. 4.5 Order-order transition from cylinders to double-gyroid and vice versa, a The minimum energy path connecting the two ordered phases is computed using the recently developed string method within the self-consistent field theory (Reprinted with permission from Cheng et al. [29]. Copyright 2013 by the American Physical Society) b Schematic illustration of nucleation and growth (Reprinted with permission from Matsen et al. [27], Copyright 2013 by the American Physical Society)...

Equation (75) expresses the partition function of the many-polymer system in terms of the partition functions of single polymers subjected to external fluctuating fields. The self-consistent field theory approximates this functional integral over the fields by the value of the integrand evaluated at those values of the fields, and Wb, that minimize the functional F[h, 4, vb]. From the definition of F it follows that these functions satisfy the self-consistent equations... 

This is the basic equation of the self-consistent field theory the reader may notice that it has the same form as Schrodinger s equation. This is the basis of the analogy between polymer chain conformations in interacting systems and the properties of interacting electrons. 

Although Xs is referred to as an energy parameter and Ua are adsorption energies it is clear that they are actually viewed as enthalpies by the original authors of the self-consistent field theory of polymer adsorption. This needs to be appreciated in reading the literature since great confusion can arise otherwise. 

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