Mathematical Topics in Block Copolymers
Open Access DepositedThis thesis delves into mathematical topics in block copolymers. We first study the diblock copolymers. A diblock copolymer molecule is a linear sub-chain of A-monomers grafted covalently to another sub-chain of B-monomers. The free energy of a diblock copolymer system in the Ohta-Kawasaki theorytakes the form \[ {\cal J}_{OK} (\Omega) = {\cal P}_D (\Omega) + \frac{\gamma}{2} \int_\Omega (-\Delta)^{-1}(\chi_\Omega - \omega)(x) \, dx. \] Here a perimeter term in the problem gives a growth force while the Laplace operator, $(-\Delta)^{-1}$, provides an inhibition force. The operator $(-\Delta)^{-1}$ is chosen largely for convenience. However a careful examination of the derivation of the Ohta-Kawasaki theory from the first principles shows that a more complex long range operator should be used. As a first step to study more general long range inhibitory forces, we propose to consider $(-\Delta)^{-s}$ with $s > 0$ as a long range interaction operator, so that the free energy functional changes to \[ {\cal J}_D(\Omega)={\cal P}_{D}(\Omega) +\frac{\gamma}{2}\int_{\Omega} (-\Delta)^{-s}(\chi_{\Omega}-\omega)dx. \] Two phenomena, splintering and coarsening, which are sources of morphological instability, are investigated. Splintering, which means large pieces of structures break into small pieces, is favored by the inhibition force and opposed by the growth force. On the other hand coarsening, which means large pieces increase in size and small pieces shrink and vanish, is favored by the growth force and opposed by the inhibition force. Several thresholds in terms of the parameters of the problem for these phenomena are found. Subsequently, the focus shifts to triblock copolymers. A triblock copolymer system is studied on a flat torus, the quotient space of the complex plane by a lattice, denoted by $\mathbb{C}/\Lambda$. The free energy of a triblock copolymer system is defined as \[ {\cal J}_\Lambda(\Omega_1, \Omega_2) = \frac{1}{2} \sum_{j=1}^3 {\cal P}_{\mathbb{C}/\Lambda} (\Omega_j) + \sum_{j,k=1}^2 \frac{\gamma_{jk}}{2}\int_{\mathbb{C}/\Lambda} \nabla I_\Lambda(\Omega_j)(z) \cdot \nabla I_\Lambda(\Omega_k)(z) \, dA(z). \] Here we treat two of the three monomer types $\Omega_1,\Omega_2$ in a triblock copolymer as species and view the third type $\Omega_3= D \setminus (\Omega_1 \cup \Omega_2)$ as the surrounding environment, dependent on the two species. This way a triblock copolymer is a two-species interacting system. The free energy of this system admits disc-disc like stationary points. The relative displacement of the disc centers in a stationary point is related to Green's function of the Laplace operator on the flat torus. When restricted to disc-disc configurations with relative displacements equal to half periods, the free energy is minimized with respect to the lattice and its half periods. The resulting optimal lattice depends on a single parameter. As this parameter varies, the optimal lattice may be rectangular, square, rhombic, or hexagonal. This is in sharp contrast to single species systems where optimal lattices are always hexagonal.
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