Diblock Copolymer Melt in Spherical Unit Cells of Higher Dimensionalities

Abstract

Using self-consistent eld theory in spherical unit cells of various dimensionality, D = 1, 2, 3, and 4, we calculate phase diagram of a diblock, A-b-B, copolymer melt in 4-dimensional space, d = 4. The phase diagram is parameterized by the chain composition, f , and incompatibility between A and B, quantied by the product χN . We predict 4 stable nanophases: layers, cylinders, 3D spherical cells, and 4D spherical cells. We also calculate orderdisorder and orderorder transition lines. In the strong segregation limit, that is for large χN , the orderorder transition compositions are determined by the strong segregation theory in its simplest form, for D = 1, 2, 3, and 4.

Phase Diagram of Diblock Copolymer Melt in Dimension d = 5

“…In the strong segregation limit, that is for large χN , the OOT compositions, f L/C , f C/S 3 , f S 3 /S 4 and f S 4 /S 5 are determined by the strong segregation theory. While f L/C , f C/S 3 , and f S 3 /S 4 are close to the corresponding extrapolations from the self-consistent field theory, as shown known in the previous study [20], the f S 4 /S 5 extrapolation does not agree with the SST predictions. We find that the S 5 nanophase is stable in a narrow strip between ordered S 4 nanophase and the disordered phase.…”

“…In previous work [20] we calculated the phase diagram of the diblock copolymer melt for d = 4, and managed to aswer the following questions:…”

“…Obviously the solution depends on radius, R, and dimensionality, D = 1, 2, 3, 4 and 5, corresponding to 5 different nanophases, shown in Table I. We use the Crank-Nicholson scheme [20] to solve iteratively the modified diffusion equations (eqs 15 and 16) in their radial form, until the self-consistency condition is met, obtaining the saddle point fields, φ A (r), φ B (r), W A (r) and W B (r) for a given R and D. In the MF approximation, the free energy functional becomes the free energy, and therefore we calculate the reduced free energy (per chain in k B T units) by substituting the saddle point fields into eq 7:…”

“…as used in referenced [9] and [20], where f 0 is the extrapolation to the strong segregation limit, and g 0 is a fitting parameter. The resultant limits, f 0 L/C , f 0 C/S 3 , f 0 S 3 /S 4 , as determined in ref 20 and f 0 S 4 /S 5 (determined in this paper) are compared to the SST f 's, as shown in Table III.…”

Monte Carlo simulations and self-consistent field theory applied to calculations of density profiles in A1BA2 triblock copolymer melts

“…Self-consistent field theory was successfully applied to the block copolymer melts [8,14,15]. It enables a relatively fast and accurate calculation of phase diagrams corresponding to diagrams obtained experimentally.…”