Self-consistent fields polymer chains

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].

Alexander approach to spherical geometries, while making the connection between tethered chains and branched polymers. The internal structure of tethered layers was illuminated by numerical and analytical self-consistent field calculations, and by computer simulations. 

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. 

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. 

A self-consistent field theory (SCFT) for micelle formation of block copolymers in selective solvents was developed by Yuan el at. (1992). They emphasized the importance of treating the isolated chain at the same level of theoretical approximation at the micelle, in contrast to earlier approaches. This was achieved by modifying the Edwards diffusion equation for the excluded volume of polymers in solution to the case of block copolymers, with one block in a poor solvent. The results of the continuum model were compared to experimental results for PS-PI diblocks in hexadecane, which is a selective solvent for PI and satisfactory agreement was obtained. 

Eq. (10) represents the self-consistent field equation for the local segment density of the polymer chains subject to an external electrical potential ip, a van der Waals interaction with the plates —UkT and an excluded volume interaction. Eq. (11) is a modified Poisson-Boltzmann equation in which the first term accounts for the charges of the small ions of the salt, the second term for the charges of the polyelectrolyte chains and the third one for the charges of the ions dissociated from the polyelectrolyte molecules. 

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],... 

The conformations of the molecules in the polymer layer and the resulting steric interaction energy can be calculated by means of a numerical self-consistent field model. The free energy of the polymer layers then is minimized by considering all possible conformations (including adsorbed segments) of the chains. We will not discuss the theory because it can rarely... 

While 8 hence does not depend on the chain length N, the period D depends on both N and %. The theoretical analysis of this problem [41, 42, 56, 57, 59, 313-316] is rather complicated and requires somewhat restrictive assumptions, such as the self-consistent field - approach [56] for polymers. We shall not describe these theories here, but restrict ourselves to a scaling-type plausibility argument, using the fact that the interfacial tension between fully segregated phases is also depending on the x parameter only but not on chain length [305-313]... 

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