Patterns of Nanoporous Spherical Packing Emerging under Influence of Curvature and Confinement




Spherical forming system, Diblock copolymers, Celle dynamic simulations, Confinement, Discretization, Polar grid system, Nanoporous membrane


Nanoporous membranes are popular in nanotechnology due to biomedical and industrial applications. During the past decade, experimental, theoretical and computational research into porous membranes and soft materials has opened up new mathematical dimensions. In bulk, diblock copolymers exhibit ordered morphologies such as parallel matrices of lamellae, bicontinuous matrices of gyroids, hexagonal matrices of cylinders and body-centred cubic matrices of spheres. In melt, confinement plays an essential role in tuning the frustration of the diblock copolymer system to predict more nanostructures. These nanostructures depend on the composition of the copolymers, their confining geometries and the degree of structural frustration. An isotropic 9-point stencil for Laplacian is constructed. The discrete finite-difference technique is used in polar grids to discretize the macromolecule of the diblock copolymer system to study spherical patterns to study the effect of curvature and confinement with a well-known and efficient cell dynamic simulation model. Intel FORTRAN (IFORT) codes are generated to run the CDS model and visualisation of simulation results is observed with the help of OPENDX. A comparison of the proposed study with existing experimental and computational studies is also presented.


Bates, F. S. and Fredrickson, G. H. (1999), ‘Block copolymers—designer soft materials’, Physics Today 52(2), pp. 32–38.

Borah, D., Shaw, M., Rasappa, S., Farrell, R., O’Mahony, C., Faulkner, C., Bosea, M., Gleeson, P., Holmes, J. and Morris, M. (2011), ‘Plasma etch technologies for the development of ultra-small feature size transistor devices’, Journal of Physics D: Applied Physics 44(17), p. 174012.

Chenkual, L., Lalchandani, D. S., Padakanti, A. P., Chella, N. and Porwal, P. K. (2023), Synthesis and self-assembly of block copolymers, in ‘Block Co-polymeric Nanocarriers: Design, Concept, and Therapeutic Applications’, Springer, pp. 75–119.

Diaz, J., Pinna, M., Breen, C., Zvelindovsky, A. and Pagonabarraga, I. (2023), ‘Block copolymer nanocomposites under confinement: Effect on frustrated phases’, Macromolecules 56(13), pp. 5010–5021.

Diaz, J., Pinna, M., Zvelindovsky, A. and Pagonabarraga, I. (2022), ‘Nanoparticle anisotropy induces sphere-to-cylinder phase transition in block copolymer melts’, Soft Matter 18(19), pp. 3638–3643.

Doi, M. (2013), Soft matter physics, Oxford University Press, USA.

Feng, H., Lu, X., Wang, W., Kang, N.-G. and Mays, J. W. (2017), ‘Block copolymers: Synthesis, self-assembly, and applications’, Polymers 9(10), p. 494.

Gupta, S. and Chokshi, P. (2020), ‘Diblock copolymer templated self-assembly of grafted nanoparticles under circular pore confinement’, Soft Matter 16(14), pp. 3522–3535.

Hamley, I. (2003), ‘Nanotechnology with soft materials’, Angewandte Chemie International Edition 42(15), pp. 1692–1712.

Herr, D. J. (2011), ‘Directed block copolymer self-assembly for nanoelectronics fabrication’, Journal of Materials Research 26(2), pp. 122–139.

Hsu, N.-W., Nouri, B., Chen, L.-T. and Chen, H.-L. (2020), ‘Hexagonal close-packed sphere phase of conformationally symmetric block copolymer’, Macromolecules 53(21), pp. 9665–9675.

Inayatullah Soomro, I. A., Shah, S. B., Majid, A., Muhammad, R., Hameed, A., Abas, G., Zvelindovsky, A. V., Pinna, M. and Ahmed, W. (2019), ‘Mathematical modelling of cylindrical forming di-block copolymers confined in circular annular pores’, IJCSNS 19(2), p. 16.

Iqbal, M. J., Soomro, I., Bibi, M. and Mallah, R. N. (2023), ‘Morphological investigation of lamellae patterns in diblock copolymers under change of thickness and confinement in polar geometry’, VFAST Transactions on Mathematics 11(2), pp. 174–197.

Juan, Y.-T., Lai, Y.-F., Li, X., Tai, T.-C., Lin, C.-H., Huang, C.-F., Li, B., Shi, A.-C. and Hsueh, H.-Y. (2023), ‘Self-assembly of gyroid-forming diblock copolymers under spherical confinement’, Macromolecules 56(2), pp. 457–469.

Karayianni, M. and Pispas, S. (2021), ‘Block copolymer solution self-assembly: Recent advances, emerging trends, and applications’, Journal of Polymer Science 59(17), pp. 1874–1898.

Khaksar, E., Golshan, M., Roghani-Mamaqani, H. and Salami-Kalajahi, M. (2023), ‘Confinement effect of blocks on the morphology of composite particles in co-assembly of block copolymers/homopolymers’, Polyolefins Journal 10(3), pp. 137–147.

Kim, M. P. and Yi, G.-R. (2015), ‘Nanostructured colloidal particles by confined self-assembly of block copolymers in evaporative droplets’, Frontiers in Materials 2, p. 45.

Latif, S., Mallah, R. and Soomro, I. (2021), ‘Discretization of Laplacian operator in polar coordinates system on 9-point stencil with mixed PDE’s derivative approximation using finite difference method’, Journal of Mathematical Sciences & Computational Mathematics 2(3), pp. 387–394.

Lee, H., Kim, J. and Park, M. J. (2024), ‘Exploration of complex nanostructures in block copolymers’, Physical Review Materials 8(2), p. 020302.

Ly, D. Q. and Makatsoris, C. (2019), ‘Effects of the homopolymer molecular weight on a diblock copolymer in a 3d spherical confinement’, BMC chemistry 13, pp. 1–9.

Mallah, R. and Soomro, I. (2022), ‘Comparative study of numerical approximation schemes for Laplacian operator in polar mesh system on 9-points stencil including mixed partial derivative by finite difference method’, Journal of Mathematical Sciences & Computational Mathematics 3(4), pp. 516–525.

Mallah, R., Soomro, I., Ahmed, A., Muhammad, D., Latif, S. and Ali, I. (2023), ‘Simulation of 13 points Laplacian operator in cylindrical mesh system by using explicit finite difference technique’, VFAST Transactions on Mathematics 11(1), pp. 84–95.

Pinna, M. and Zvelindovsky, A. (2012), ‘Large scale simulation of block copolymers with cell dynamics’, The European Physical Journal B 85, pp. 1–18.Singh, J., Gupta, S., & Chokshi, P. (2024). "Confinement-Driven Self-Assembly of Diblock Copolymers within Non-Uniform Cylindrical Nanopores." Soft Matter.

Soomro, I., Mallah, R., Iqbal, M. J., Ahmed, W., & Ghafoor, A. (2023). "Discretization of Laplacian Operator on 19-Point Stencil Using Cylindrical Mesh System with Explicit Finite Difference Scheme." Pakistan Journal of Engineering, Technology & Science, 11(2), 1–10.

Sun, M., Zhang, Z., Li, Y., Li, W., Liao, Q., & Qin, L. (2022). "Phase Behavior of ABA Star Polymers and Nanoparticles Confined within a Sphere with a Soft Inner Surface." Polymers, 14(8), 1610.

Tenneti, A., Ackerman, D. M., & Ganapathysubramanian, B. (2020). "Equilibrium Microstructures of Diblock Copolymers under 3D Confinement." Computational Materials Science, 174, 109453.

Wang, Z., Valenzuela, C., Wu, J., Chen, Y., Wang, L., & Feng, W. (2022). "Bioinspired Freeze-Tolerant Soft Materials: Design, Properties, and Applications." Small, 18(37), 2201597.

Wu, J., Chen, S.-T., Li, S.-B., Liu, L.-M., Wang, X.-H., & Lang, W.-C. (2023). "Simulation of Surface-Induced Morphology Transition and Phase Diagram of Linear Triblock Copolymers under Spherical Confinement." Chinese Journal of Polymer Science, 41(1), 166–178.

Xie, J., & Shi, A.-C. (2023). "Phase Behavior of Binary Blends of Diblock Copolymers: Progress and Opportunities." Langmuir, 39(33), 11491–11509.

Zvelindovsky, A., & Sevink, G. (2003). "Sphere Morphology of Block Copolymer Systems under Shear." Europhysics Letters, 62(3), 370.




How to Cite

Muhammad Javed Iqbal, Inayatullah Soomro, & Usama Gulzar. (2024). Patterns of Nanoporous Spherical Packing Emerging under Influence of Curvature and Confinement . VFAST Transactions on Mathematics, 12(1), 121–136.