Abstract
We consider the inverse problem of finding guiding pattern shapes that result in desired self-assembly morphologies of block copolymer melts. Specifically, we model polymer self-assembly using the self-consistent field theory and derive, in a non-parametric setting, the sensitivity of the dissimilarity between the desired and the actual morphologies to arbitrary perturbations in the guiding pattern shape. The sensitivity is then used for the optimization of the confining pattern shapes such that the dissimilarity between the desired and the actual morphologies is minimized. The efficiency and robustness of the proposed gradient-based algorithm are demonstrated in a number of examples related to templating vertical interconnect accesses (VIA).
Similar content being viewed by others
Data Availability
Data sets generated during the current study are available from the corresponding author on reasonable request.
References
Arias, V., Bochkov, D., Gibou, F.: Poisson equations in irregular domains with Robin boundary conditions-solver with second-order accurate gradients. J. Comput. Phys. 365, 1–6 (2018)
Bayat, E., Egan, R., Bochkov, D., Sauret, A., Gibou, F.: A sharp numerical method for the simulation of Stefan problems with convective effects. J. Comput. Phys. 471, 111627 (2022)
Bertelli, L., Chandrasekaran, S., Gibou, F., Manjunath, B.: On the length and area regularization for multiphase level set segmentation. Int. J. Comput. Vision 90, 267–282 (2010)
Bertelli, L., Sumengen, B., Manjunath, B., Gibou, F.: A variational framework for multiregion pairwise-similarity-based image segmentation. IEEE Trans. Pattern Anal. Mach. Intell. 30(8), 1400–1414 (2008)
Bita, I., Yang, J.K., Jung, Y.S., Ross, C.A., Thomas, E.L., Berggren, K.K.: Graphoepitaxy of self-assembled block copolymers on two-dimensional periodic patterned templates. Science 321(5891), 939–943 (2008)
Bochkov, D., Bagaric, I., Ouaknin, G., Gibou, F.: Equilibrium of free surfaces and nanoparticles in self-consistent field theory of block copolymers. arXiv:2112.08660 (2021)
Bochkov, D., Gibou, F.: Solving Poisson-type equations with Robin boundary conditions on piecewise smooth interfaces. J. Comput. Phys. 376, 1156–1198 (2019)
Bochkov, D., Gibou, F.: PDE-based multidimensional extrapolation of scalar fields over interfaces with kinks and high curvatures. SIAM J. Sci. Comput. 42(4), 2344–2359 (2020)
Bochkov, D., Gibou, F.: Solving elliptic interface problems with jump conditions on Cartesian grids. J. Comput. Phys. 407, 109269 (2020)
Brun, E., Guittet, A., Gibou, F.: A local level-set method using a hash table data structure. J. Comput. Phys. 231(6), 2528–2536 (2012)
Burchard, P., Cheng, L.-T., Merriman, B., Osher, S.: Motion of curves in three spatial dimensions using a level set approach. J. Comput. Phys. 170, 720–741 (2001)
Burger, M., Osher, S.: A survey on level set methods for inverse problems and optimal design. In: CAM Report (04-02) (in Press) (2004)
Caflisch, R.E., Gyure, M.F., Merriman, B., Osher, S., Ratsch, C., Vvedensky, D.D., Zinck, J.J.: Island dynamics and the level set method for epitaxial growth. Appl. Math. Lett. 12, 13 (1999)
Chang, Y.-C., Hou, T., Merriman, B., Osher, S.: Eulerian capturing methods based on a level set formulation for incompressible fluid interfaces. J. Comput. Phys. 124, 449–464 (1996)
Chen, H., Min, C., Gibou, F.: A supra-convergent finite difference scheme for the Poisson and heat equations on irregular domains and non-graded adaptive Cartesian grids. J. Sci. Comput. 31, 19–60 (2007)
Chen, H., Min, C., Gibou, F.: A numerical scheme for the Stefan problem on adaptive Cartesian grids with supralinear convergence rate. J. Comput. Phys. 228(16), 5803–5818 (2009)
Chen, S., Merriman, B., Osher, S., Smereka, P.: A simple level set method for solving Stefan problems. J. Comput. Phys. 135, 8–29 (1997)
Cheng, L.T., Liu, H., Osher, S.: Computational high-frequency wave propagation using the level set method, with applications to the semi-classical limit of Schrödinger equations. Commun. Math. Sci. 1, 593–621 (2003)
Chowdhury, R., Egan, R., Bochkov, D., Gibou, F.: Efficient calculation of fully resolved electrostatics around large biomolecules. J. Comput. Phys. 448, 110718 (2022)
Darling, S.: Directing the self-assembly of block copolymers. Prog. Polym. Sci. 32(10), 1152–1204 (2007)
Detrixhe, M., Doubeck, M., Moehlis, J., Gibou, F.: A fast Eulerian approach for computation of global isochrons in high dimensions. SIAM J. Appl. Dyn. Syst. 15(3), 1501–1527 (2016)
Detrixhe, M., Gibou, F.: Hybrid massively parallel fast sweeping method for static Hamilton-Jacobi equations. J. Comput. Phys. 322, 199–223 (2016)
Detrixhe, M., Gibou, F., Min, C.: A parallel fast sweeping method for the Eikonal equation. J. Comput. Phys. 237, 46–55 (2013)
Du Chéné, A., Min, C., Gibou, F.: Second-order accurate computation of curvatures in a level set framework using novel high-order reinitialization schemes. J. Sci. Comput. 35, 114–131 (2008)
Egan, R., Gibou, F.: Geometric discretization of the multidimensional Dirac delta distribution-application to the Poisson equation with singular source terms. J. Comput. Phys. 346, 71–90 (2017)
Egan, R., Gibou, F.: Fast and scalable algorithms for constructing solvent-excluded surfaces of large biomolecules. J. Comput. Phys. 374, 91–120 (2018)
Egan, R., Gibou, F.: xGFM: recovering convergence of fluxes in the ghost fluid method. J. Comput. Phys. 409, 109351 (2020)
Enright, D., Nguyen, D., Gibou, F., Fedkiw, R.: Using the particle level set method and a second order accurate pressure boundary condition for free surface flows. In: Fluids Engineering Division Summer Meeting, 36975, 337–342 (2003)
Fedkiw, R.P., Aslam, T., Merriman, B., Osher, S.: A non-oscillatory Eulerian approach to interfaces in multimaterial flows (the ghost fluid method). J. Comput. Phys. 152, 457–492 (1999)
Fedkiw, R.P., Sapiro, G., Shu, C.-W.: Shock capturing, level sets, and PDE based methods in computer vision and image processing: a review of Osher’s contributions. J. Comput. Phys. 185(2), 309–341 (2003)
Fredrickson, G.: The Equilibrium Theory of Inhomogeneous Polymers, vol. 134. Oxford University Press, Oxford (2006)
Gibou, F., Chen, L., Nguyen, D., Banerjee, S.: A level set based sharp interface method for the multiphase incompressible Navier-Stokes equations with phase change. J. Comput. Phys. 222(2), 536–555 (2007)
Gibou, F., Fedkiw, R., Caflisch, R., Osher, S.: A level set approach for the numerical simulation of dendritic growth. J. Sci. Comput. 19, 183–199 (2003)
Gibou, F., Fedkiw, R., Osher, S.: A review of level-set methods and some recent applications. J. Comput. Phys. 353, 82–109 (2018)
Gibou, F., Hyde, D., Fedkiw, R.: Sharp interface approaches and deep learning techniques for multiphase flows. J. Comput. Phys. 380, 442–463 (2019)
Gibou, F., Min, C.: Efficient symmetric positive definite second-order accurate monolithic solver for fluid/solid interactions. J. Comput. Phys. 231(8), 3246–3263 (2012)
Gibou, F., Min, C., Fedkiw, R.: High resolution sharp computational methods for elliptic and parabolic problems in complex geometries. J. Sci. Comput. 54, 369–413 (2013)
Gibou, F., Ratsch, C., Caflisch, R.: Capture numbers in rate equations and scaling laws for epitaxial growth. Phys. Rev. B 67(15), 155403 (2003)
Gibou, F., Ratsch, C., Gyure, M., Chen, S., Caflisch, R.: Rate equations and capture numbers with implicit islands correlations. Phys. Rev. B 63(11), 115401 (2001)
Guittet, A., Lepilliez, M., Tanguy, S., Gibou, F.: Solving elliptic problems with discontinuities on irregular domains-the Voronoi interface method. J. Comput. Phys. 298, 747–765 (2015)
Guittet, A., Poignard, C., Gibou, F.: A Voronoi interface approach to cell aggregate electropermeabilization. J. Comput. Phys. 332, 143–159 (2017)
Hannon, A.F., Ding, Y., Bai, W., Ross, C.A., Alexander-Katz, A.: Optimizing topographical templates for directed self-assembly of block copolymers via inverse design simulations. Nano Lett. 14(1), 318–325 (2014)
Hannon, A.F., Gotrik, K.W., Ross, C.A., Alexander-Katz, A.: Inverse design of topographical templates for directed self-assembly of block copolymers. ACS Macro Lett. 2(3), 251–255 (2013)
Helgadóttir, Á., Gibou, F.: A Poisson-Boltzmann solver on irregular domains with Neumann or Robin boundary conditions on non-graded adaptive grid. J. Comput. Phys. 230(10), 3830–3848 (2011)
Hou, T.Y., Li, Z., Osher, S., Zhao, H.: A hybrid method for moving interface problems with application to the Hele-Shaw flow. J. Comput. Phys. 134(2), 236–252 (1997)
Hu, H., Gopinadhan, M., Osuji, C.O.: Directed self-assembly of block copolymers: a tutorial review of strategies for enabling nanotechnology with soft matter. Soft Matter 10(22), 3867–3889 (2014)
Jeong, S.-J., Kim, J.Y., Kim, B.H., Moon, H.-S., Kim, S.O.: Directed self-assembly of block copolymers for next generation nanolithography. Mater. Today 16(12), 468–476 (2013)
Jin, S., Osher, S.: A level set method for computing multivalued solutions to quasi-linear hyperbolic equations and Hamilton-Jacobi equations. Commun. Math. Sci. 1(3), 575–591 (2003)
Langavant, C.C., Guittet, A., Theillard, M., Temprano-Coleto, F., Gibou, F.: Level-set simulations of soluble surfactant driven flows. J. Comput. Phys. 348, 271–297 (2017)
Larios-Cárdenas, L.Á., Gibou, F.: A deep learning approach for the computation of curvature in the level-set method. SIAM J. Sci. Comput. 43(3), 1754–1779 (2021)
Larios-Cárdenas, L.Á., Gibou, F.: Error-correcting neural networks for two-dimensional curvature computation in the level-set method. J. Sci. Comput. 93(1), 6 (2022)
Latypov, A.: Computational solution of inverse directed self-assembly problem. In: Alternative Lithographic Technologies V, vol. 8680, p. 86800. International Society for Optics and Photonics (2013)
Lepilliez, M., Popescu, E.R., Gibou, F., Tanguy, S.: On two-phase flow solvers in irregular domains with contact line. J. Comput. Phys. 321, 1217–1251 (2016)
Liu, C.-C., Ramírez-Hernández, A., Han, E., Craig, G.S., Tada, Y., Yoshida, H., Kang, H., Ji, S., Gopalan, P., Pablo, J.J., Nealey, P.F.: Chemical patterns for directed self-assembly of lamellae-forming block copolymers with density multiplication of features. Macromolecules 46(4), 1415–1424 (2013)
Losasso, F., Fedkiw, R., Osher, S.: Spatially adaptive techniques for level set methods and incompressible flow. Comput. Fluids 35, 995–1010 (2006)
Losasso, F., Gibou, F., Fedkiw, R.: Simulating water and smoke with an octree data structure. ACM Transactions on Graphics 23(3), 457–462 (2004)
Losasso, F., Talton, J., Kwatra, N., Fedkiw, R.: Two-way coupled SPH and particle level set fluid simulation. IEEE Transactions on Visualization and Computer Graphics 14(4), 797–804 (2008)
Merriman, B., Bence, J., Osher, S.: Motion of multiple junctions: a level set approach. J. Comput. Phys. 112, 334–363 (1994)
Min, C., Gibou, F.: A second order accurate level set method on non-graded adaptive Cartesian grids. J. Comput. Phys. 225(1), 300–321 (2007)
Min, C., Gibou, F.: Geometric integration over irregular domains with application to level-set methods. J. Comput. Phys. 226(2), 1432–1443 (2007)
Min, C., Gibou, F.: Robust second-order accurate discretizations of the multi-dimensional Heaviside and Dirac delta functions. J. Comput. Phys. 227(22), 9686–9695 (2008)
Min, C., Gibou, F., Ceniceros, H.D.: A supra-convergent finite difference scheme for the variable coefficient Poisson equation on non-graded grids. J. Comput. Phys. 218(1), 123–140 (2006)
Mirzadeh, M., Gibou, F.: A conservative discretization of the Poisson-Nernst-Planck equations on adaptive Cartesian grids. J. Comput. Phys. 274, 633–653 (2014)
Mirzadeh, M., Guittet, A., Burstedde, C., Gibou, F.: Parallel level-set methods on adaptive tree-based grids. J. Comput. Phys. 322, 345–364 (2016)
Mirzadeh, M., Theillard, M., Gibou, F.: A second-order discretization of the nonlinear Poisson-Boltzmann equation over irregular geometries using non-graded adaptive Cartesian grids. J. Comput. Phys. 230(5), 2125–2140 (2011)
Mistani, P., Guittet, A., Bochkov, D., Schneider, J., Margetis, D., Ratsch, C., Gibou, F.: The island dynamics model on parallel quadtree grids. J. Comput. Phys. 361, 150–166 (2018)
Mistani, P., Guittet, A., Poignard, C., Gibou, F.: A parallel Voronoi-based approach for mesoscale simulations of cell aggregate electropermeabilization. J. Comput. Phys. 380, 48–64 (2019)
Mulder, W., Osher, S., Sethian, J.: Computing interface motion in compressible gas dynamics. J. Comput. Phys. 100, 209–228 (1992)
Ng, Y.T., Chen, H., Min, C., Gibou, F.: Guidelines for Poisson solvers on irregular domains with Dirichlet boundary conditions using the ghost fluid method. J. Sci. Comput. 41, 300–320 (2009)
Ng, Y.T., Min, C., Gibou, F.: An efficient fluid-solid coupling algorithm for single-phase flows. J. Comput. Phys. 228(23), 8807–8829 (2009)
Nguyen, D., Gibou, F., Fedkiw, R.: A fully conservative ghost fluid method and stiff detonation waves. In: 12th Int. Detonation Symposium, San Diego, CA (2002)
Ohta, T., Kawasaki, K.: Equilibrium morphology of block copolymer melts. Macromolecules 19(10), 2621–2632 (1986)
Osher, S., Cheng, L.-T., Kang, M., Shim, H., Tsai, Y.-H.: Geometric optics in a phase-space-based level set and Eulerian framework. J. Comput. Phys. 179, 622–648 (2002)
Osher, S., Fedkiw, R.P.: Level set methods: an overview and some recent results. J. Comput. Phys. 169(2), 463–502 (2001)
Osher, S., Fedkiw, R.P.: Level Set Methods and Dynamic Implicit Surfaces, vol. 153. Springer, New York (2003)
Osher, S., Paragios, N.: Geometric Level Set Methods in Imaging, Vision, and Graphics. Springer, New York (2003)
Osher, S., Santosa, F.: Level set methods for optimization problems involving geometry and constraints: frequencies of a two-density inhomogeneous drum. J. Comput. Phys. 171, 272–288 (2001)
Osher, S., Sethian, J.A.: F ronts propagating with curvature dependent speed: algorithms based on Hamilton-Jacobi formulations. J. Comput. Phys. 79(1), 12–49 (1988)
Ouaknin, G., Laachi, N., Delaney, K., Fredrickson, G., Gibou, F.: Shape optimization for DSA. In: Alternative Lithographic Technologies VIII, 9777, 97770. International Society for Optics and Photonics (2016)
Ouaknin, G., Laachi, N., Delaney, K., Fredrickson, G.H., Gibou, F.: Self-consistent field theory simulations of polymers on arbitrary domains. J. Comput. Phys. 327, 168–185 (2016)
Ouaknin, G.Y., Laachi, N., Delaney, K., Fredrickson, G.H., Gibou, F.: Level-set strategy for inverse DSA-lithography. J. Comput. Phys. 375, 1159–1178 (2018)
Pakravan, S., Mistani, P.A., Aragon-Calvo, M.A., Gibou, F.: Solving inverse-PDE problems with physics-aware neural networks. J. Comput. Phys. 440, 110414 (2021)
Papac, J., Gibou, F., Ratsch, C.: Efficient symmetric discretization for the Poisson, heat and Stefan-type problems with Robin boundary conditions. J. Comput. Phys. 229(3), 875–889 (2010)
Papac, J., Helgadottir, A., Ratsch, C., Gibou, F.: A level set approach for diffusion and Stefan-type problems with Robin boundary conditions on quadtree/octree adaptive Cartesian grids. J. Comput. Phys. 233, 241–261 (2013)
Papac, J., Margetis, D., Gibou, F., Ratsch, C.: Island-dynamics model for mound formation: effect of a step-edge barrier. Phys. Rev. E 90(2), 022404 (2014)
Peng, D., Merriman, B., Osher, S., Zhao, H., Kang, M.: A PDE-based fast local level set method. J. Comput. Phys. 155, 410–438 (1999)
Petersen, M., Ratsch, C., Caflisch, R.E., Zangwill, A.: A level set approach to reversible epitaxial growth. Phys. Rev. E 64, 061602 (2001)
Ratsch, C., Gyure, M., Caflisch, R., Gibou, F., Petersen, M., Kang, M., Garcia, J., Vvedensky, D.: Level-set method for island dynamics in epitaxial growth. Phys. Rev. B 65(19), 195403 (2002)
Ruiz, R., Kang, H., Detcheverry, F.A., Dobisz, E., Kercher, D.S., Albrecht, T.R., Pablo, J.J., Nealey, P.F.: Density multiplication and improved lithography by directed block copolymer assembly. Science 321(5891), 936–939 (2008)
Rycroft, C.H., Gibou, F.: Simulations of a stretching bar using a plasticity model from the shear transformation zone theory. J. Comput. Phys. 231(5), 2155–2179 (2012)
Sethian, J.A.: Level Set Methods and Fast Marching Methods: Evolving Interfaces in Computational Geometry, Fluid Mechanics, Computer Vision, and Materials Science, vol. 3. Cambridge University Press, Cambridge (1999)
Shim, S., Shin, Y.: Mask optimization for directed self-assembly lithography: inverse DSA and inverse lithography. In: 2016 21st Asia and South Pacific Design Automation Conference (ASP-DAC), pp. 83–88. IEEE (2016)
Smereka, P.: Semi-implicit level set methods for curvature and surface diffusion motion. J. Sci. Comput. 19, 439–456 (2003)
Sussman, M., Fatemi, E., Smereka, P., Osher, S.: An improved level set method for incompressible two-phase flows. Comput. Fluids 27, 663–680 (1998)
Sussman, M., Smereka, P., Osher, S.: A level set approach for computing solutions to incompressible two-phase flow. J. Comput. Phys. 114, 146–159 (1994)
Theillard, M., Djodom, L.F., Vié, J.-L., Gibou, F.: A second-order sharp numerical method for solving the linear elasticity equations on irregular domains and adaptive grids-application to shape optimization. J. Comput. Phys. 233, 430–448 (2013)
Theillard, M., Gibou, F., Pollock, T.: A sharp computational method for the simulation of the solidification of binary alloys. J. Sci. Comput. 63, 330–354 (2015)
Theillard, M., Gibou, F., Saintillan, D.: Sharp numerical simulation of incompressible two-phase flows. J. Comput. Phys. 391, 91–118 (2019)
Tiron, R., Gharbi, A., Argoud, M., Chevalier, X., Belledent, J., Barros, P.P., Servin, I., Navarro, C., Cunge, G., Barnola, S., Pain, L., Asai, M., Pieczulewski, C.: The potential of block copolymer’s directed self-assembly for contact hole shrink and contact multiplication. In: Alternative Lithographic Technologies V, vol. 8680, p. 868012. International Society for Optics and Photonics (2013)
Tsai, Y.-H., Osher, S.: Level set methods and their applications in image science. Commun. Math. Sci. 1(4), 623–656 (2003)
Zhao, H.-K., Chan, T., Merriman, B., Osher, S.: A variational level set approach to multiphase motion. J. Comput. Phys. 127, 179–195 (1996)
Zhao, H.-K., Osher, S., Fedkiw, R.: Fast surface reconstruction using the level set method. In: 1st IEEE Wrkshp. on Variational and Level Set Meth., 8th Int. Conf. on Comput. Vis., pp. 194–202. IEEE (2001)
Acknowledgements
The authors would like to acknowledge Gaddiel Ouaknin for useful discussions. This research was supported by the NSF DMS 1620471.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of Interest
The authors declare no conflict of interest.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Bochkov, D., Gibou, F. A Non-parametric Gradient-Based Shape Optimization Approach for Solving Inverse Problems in Directed Self-Assembly of Block Copolymers. Commun. Appl. Math. Comput. (2024). https://doi.org/10.1007/s42967-024-00394-x
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s42967-024-00394-x
Keywords
- Block copolymers
- Directed self-assembly
- Inverse design
- Shape optimization
- Vertical interconnect accesses (VIA)