Polymeric Self-Consistent Field Theory

Instead of the MC and MD methods using explicit particles, another method, that is, polymeric self-consistent field theory (SCFT) proposed by Edwards, is often used to study the phase separation of block copolymers. In SCFT, a polymer chain is treated as a Gaussian string, which is exposed to a set of effective chemical potentials ( ). The chemical potentials are used instead of the actual interactions between different components. Importantly, the relation between the external potentials and the concentration field ((/ ) is bijective. 

In the melting system of diblock AB copolymer, the free energy of such a system per volume (in units of k T) equals 

The density of each component is calculated by the following equation. 

is the volume fraction of A block in diblock copolymer. To study the dynamics of phase separation, the polymeric external potential dynamics (EPD) method can be employed, which was proposed by Maurits and Fraaije [23] in dynamic density functional theory (DDFT) method (bead-string model). In EPD, the monomer concentration is a conserved quantity, and the polymer dynamics is inherently of Rouse type. The external dynamical equation in terms of the potential field m,- is expressed as 

Here /i, = SF/Scpi and D is a constant diffusion coefficient of copolymer. /, is a Gaussian distributed random noise corresponding to the thermal fluctuations in the experiment and fulfills  

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Self-consistent field theory has been applied to analyse the phase behaviour of binary blends of diblocks by Shi and Noolandi (1994,1995), Matsen (1995a) and Matsen and Bates (1995). Mixtures of long and short diblocks were considered by Shi and Noolandi (1994) and Matsen (1995a), whilst Shi and Noolandi (1995) and Matsen and Bates (1995) calculated phase diagrams for blends of diblocks with equal degrees of polymerization but different composition. 

Roan, J.-R. Kawakatsu, T. Self-consistent-field theory for interacting polymeric assemblies. I. Formulation, implementation, and benchmark tests. J. Chem. Phys., 2002, 116,7283-7294. 

Keywords Polymer blends Self-consistent field theory External potential dynamics Field-theoretic polymer simulations Polymeric microemulsion Polymer dynamics... 

Kane and Spontak developed a self-consistent field theory for lamellar ABC triblock copolymers [171] based on Semenov s approach for diblock copolymers [143]. They also described the scaling behavior of the periodicity L with the degree of polymerization N, which was found to be similar to diblock copolymers (L N ). The periodicity of an ABC triblock copolymer was found to be slightly larger than the periodicity of an AC-diblock copolymer with the same overall degree of polymerization. The theoretical results confirm systematic SAXS and SANS studies by Mogi et al. on lamellar I-S-VP block copolymers [172],... 

A brush-type theory was developed by DiMarzio et al. [168] and a self-consistent field theory by Whitmore and Noolandi [169]. The latter approach predicts a scaling for the overall domain spacing d I V- ATAa (where N is the total degree of polymerization and ATa is that of the amorphous block) that is in good agreement with experimental results [170], as detailed elsewhere... 

The method developed in this book is also used to provide input parameters for composite models which can be used to predict the thermoelastic and transport properties of multiphase materials. The prediction of the morphologies and properties of such materials is a very active area of research at the frontiers of materials modeling. The prediction of morphology will be discussed in Chapter 19, with emphasis on the rapidly improving advanced methods to predict thermodynamic equilibrium phase diagrams (such as self-consistent mean field theory) and to predict the dynamic pathway by which the morphology evolves (such as mesoscale simulation methods). Chapter 20 will focus on both analytical (closed-form) equations and numerical simulation methods to predict the thermoelastic properties, mechanical properties under large deformation, and transport properties of multiphase polymeric systems. 

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