Density Functional Theory and Computer Simulation for Block Copolymer Melts and Blends -- Generalization of Ohta-Kawasaki Theory --
The microphase separation of block copolymers are studied theoretically and experimentally. The most widely used theory is the self consistent field (SCF) theory [E. Helfand and Z. R. Wasserman, Macromolecules, 9, 879, 1976; 11, 960, 1978; 13, 994, 1980 ]. Although the SCF agrees with experiments, the computer simulations using the SCF needs large CPU power and memory.
In this work, we studied the density functional (DF) theory for block copolymers, which needs less CPU power and memory than the SCF. The DF for diblock copolymer melts have been studied by Leibler [L. Leibler, Macromolecules, 13, 1602, 1980], Ohta-Kawasaki [T. Ohta and K. Kawasaki, Macromolecules, 19, 2621, 1986] and others, but their theory cannot be applied to general block copolymer systems. We derived the free energy functional for general block copolymer systems. Our theory is the generalization of the Ohta-Kawasaki theory for diblock copolymer melts and also the generalization of the Flory-Huggins-de Gennes theory for homopolymer blends [P. G. de Gennes, J. Chem. Phys., 72, 4756, 1980].
We consider the general block copolymer blends. The block copolymer
species are distinguished by the suffices p,q,... . Each block copolymer
species consists of the several subchains which is distinguished by
the suffices i,j,... (we denote the i-th subchain in the polymer p as (p,i)).
We got the free energy for the system as a functional of
φpi(r), the segment density of the subchain (p,i)
[T. Uneyama and M. Doi, submitted to Macromolecules].
F[{φpi(r)}] =where fpi and bpi are the block ratio and the effective bond length of the subchain (p,i), Ap,ij and Cp,ij are independent of r and determined from the correlation functions for the ideal systems, G(r-r') is the Green function which satisfies -∇2G(r-r') = δ(r-r') and χpi,qj is the Flory-Huggins χ parameter between monomers in the subchain (p,i) and the subchain (q,j). Note that this free energy functional reproduces the Flory-Huggins-de Gennes type free energy functional and the Ohta-Kawasaki type free energy functional.
Σp,ij ∫ dr dr' 2 (fpi fpj)1/2 Ap,ij G(r-r') [φpi(r) φpj(r')]1/2
+ Σp,i ∫ dr fpi Cp,ii φpi(r) ln φpi(r)
+ Σp,i≠j ∫ dr 2 (fpi fpj)1/2 Cp,ij [φpi(r) φpj(r)]1/2
+ Σp,i ∫ dr [bpi2 / 24 φpi(r)] |∇φpi(r) |2
+ Σpi,qj ∫ dr [χpi,qj / 2] φpi(r) φqj(r)
We did the computer simulations for several systems;