Analytical self-consistent field theory

So, in the analytical self-consistent field theory, the brush height has the same ftmctional dependence as that predicted by the simple energy balance argument that led to equation (6.1.11). 

Figure 6.9. Volume fraction profiles of an end-grafted polystyrene brush, of relative molecular mass 105 000, imder various solvent conditions (O, toluene at 21 C and cyclohexane at A, 53.4 °C , 31.5 °C o, 21.4 °C and A, 14.6 "C), deduced from neutron reflectivity measurements (all the solvents are deuterated). Toluene at 21 °C is a good solvent and the solid line is the classical parabolic profile. The theta temperature for d-cyclohexane is 34 °C and the dashed line is the elliptical profile predicted by analytical self-consistent field theory for theta conditions. After Karim et al. (1994).

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]. 

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],... 

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... 

The relativistic theory and computation of atomic structures and processes has therefore attained some sort of maturity and the various codes now available are widely used. Those mentioned so far were based on ideas originating from Hartree and his students [28], and have been developed in much the same way as the non-relativistic self-consistent field theory recorded in [28-30]. All these methods rely on the numerical solution, using finite differences, of the coupled differential equations for radial orbital wave-functions of the self-consistent field. This makes them unsuitable for the study of molecules, for which it is preferable to expand the radial amplitudes in a suitably chosen set of analytic functions. This nonrelativistic matrix Hartree-Fock method, as it is often termed, was pioneered by Hall and Lennard-Jones [31], Hall [32,33] and Roothaan [34,35], and it was Roothaan s students, Synek [36] and Kim [37] who were the first to attempt to solve the corresponding matrix Dirac-Hartree-Fock equations. Kim was able to obtain solutions for the ground state of neon in 1967, but at the expense of some numerical instability, and it seemed at the time that the matrix Dirac-Hartree-Fock scheme would not be a serious competitor to the finite difference codes. 

As mentioned earlier, studies of simple linear surfactants in a solvent (i.e, those without any third component) allow one to examine the sufficiency of coarse-grained lattice models for predicting the aggregation behavior of micelles and to examine the limits of applicability of analytical lattice approximations such as quasi-chemical theory or self-consistent field theory (in the case of polymers). The results available from the simulations for the structure and shapes of micelles, the polydispersity, and the cmc show that the lattice approach can be used reliably to obtain such information qualitatively as well as quantitatively. The results are generally consistent with what one would expect from mass-action models and other theoretical techniques as well as from experiments. For example. Desplat and Care [31] report micellization results (the cmc and micellar size) for the surfactant h ti (for a temperature of = ksT/tts = /(-ts = 1-18 and... 

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. 

Wijmans, C. M., Scheutjens, J. M. H. M., Zhulina, E. B. 1992. Self-consistent field-theories for polymer brushes—lattice calculations and an asymptotic analytical description. Macromolecules 25 2657-2665. 

The various response tensors are identified as terms in these series and are calculated using numerical derivatives of the energy. This method is easily implemented at any level of theory. Analytic derivative methods have been implemented using self-consistent-field (SCF) methods for a, ft and y, using multiconfiguration SCF (MCSCF) methods for ft and using second-order perturbation theory (MP2) for y". The response properties can also be determined in terms of sum-over-states formulation, which is derived from a perturbation theory treatment of the field operator — [iE, which in the static limit is equivalent to the results obtained by SCF finite field or analytic derivative methods. 

Analytical self-consistent mean field theories were developed independently by Zhulina el al.29 30 and Milner et al.31,32 They are based on the assumption that for large stretchings of the grafted chains with respect to their Gaussian dimension, one can approximate the set of conformations of a stretched grafted chain by a set of most likely trajectories, and predict for such cases a parabolic density profile. In the calculations of the interactions between the two brushes, the interdigitation between the chains was ignored. 

Another kind of self-consistent theory used a continuum diffusion representation to describe the distribution of segments.21 23 The segments were assumed to be subjected to an external potential of a mean field. An analytical approximation of the latter self-consistent theory was suggested by Milner et al. (the MWC model)24 on the basis of the observation that at high stretching the partition function of the brush is dominated by the classical path as the most probable distribution. Under this assumption, it was found that the self-consistent field is parabolic and leads to a parabolic distribution of the monomer density. Similar theories for polyelectrolyte brushes25 27 also adopted the parabolic distribution approximation. 

A quantitative analysis of counterion localization in a salt-free solution of star-like PEs is carried out on the basis of an exact numerical solution of the corresponding Poisson-Boltzmann (PB) problem (Sect. 5). Here, the conformational degrees of freedom of the flexible branches are accounted for within the Scheutjens-Fleer self-consistent field (SF-SCF) framework. The latter is used to prove and to quantify the applicability of the concept of colloidal charge renormalization to PE stars, that exemplify soft charged colloidal objects. The predictions of analytical and numerical SCF-PB theories are complemented by results of Monte Carlo (MC) and molecular dynamics (MD) simulations. The available experimental data on solution properties of PE star polymers are discussed in the light of theoretical predictions (Sect. 6). 

Abstract We present an overview of statistical thermodynamic theories that describe the self-assembly of amphiphilic ionic/hydrophobic diblock copolymers in dilute solution. Block copolymers with both strongly and weakly dissociating (pH-sensitive) ionic blocks are considered. We focus mostly on structural and morphological transitions that occur in self-assembled aggregates as a response to varied environmental conditions (ionic strength and pH in the solution). Analytical theory is complemented by a numerical self-consistent field approach. Theoretical predictions are compared to selected experimental data on micellization of ionic/hydrophobic diblock copolymers in aqueous solutions. 

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