A nonlocal theory of heat transfer and micro-phase separation of nanostructured copolymers

Abstract

Directed self-assembly of block copolymer (BCP) has shown promise as a nano-patterning technology for various applications such as microelectronics, semiconductor manufacturing and biotechnology. In this paper, we present a thermodynamically consistent nonlocal model of phase transformation and heat transfer for BCP via the continuum theory of mixtures. Our model is developed based on the mass and energy balances of each phase and a set of microforce balances to describe the dynamics of microphase separation during self-assembly and the size and memory effects of heat conduction in BCP thin films. We derive constitutive relations for coupling the phase field and temperature through upscaling the self-consistent field theory of polymers, and present the connections between the derived nonlocal heat conduction model and the phonon Boltzmann transport equation. We provide a finite element solution of the proposed model that includes unconditionally stable time-stepping schemes, and we perform various numerical experiments that showcase different features of the model. Furthermore, we conduct three-dimensional numerical experiments of laser zone annealing process to qualitatively investigate the effect of laser velocity on the self-assembly behavior of BCP thin films. Our results show that the proposed model can simulate essential features of thermally induced phase responses of BCPs, consistent with experimental observations and high-fidelity simulations.