Preconditioning for a Cahn-Hilliard-Navier-Stokes model for morphology formation in organic solar cells

TL;DR

The paper addresses morphology formation in drying organic solar cell blends by developing a coupled CHNS-AC phase-field model that includes solvent evaporation. A block-structured preconditioning strategy with an accurate Schur-complement approximation and AMG enables parameter-robust, scalable solution of the resulting large saddle-point systems. Numerical results in 1D–3D demonstrate stable GMRES convergence and morphologies that qualitatively capture solvent-driven phase separation and evaporation, validating the approach. This provides a computational tool to explore fabrication-processing conditions and guide optimization of OPV device performance.

Abstract

We present a model for the morphology evolution of printed organic solar cells which occurs during the drying of a mixture of polymer, the non-fullerene acceptor and the solvent. Our model uses a phase field approach coupled to a Navier-Stokes equation describing the macroscopic movement of the fluid. Additionally, we incorporate the evaporation process of the solvent using an Allen-Cahn equation. The model is discretized using a finite-element approach with a semi-implicit discretization in time. The resulting (non)linear systems are coupled and of large dimensionality. We present a preconditioned iterative scheme to solve them robustly with respect to changes in the discretization parameters. We illustrate that the preconditioned solver shows parameter-robust iteration numbers and that the model qualitatively captures the behavior of the film morphology during drying.

Figures (15)

• Figure 1: Sketch of the domain.
• Figure 2: Initial condition in 2D
• Figure 3: Volume fraction fields for the solvent, air and vapor from left to right. The spatial discretization of a 1D domain is chosen $n_y =100$. Moreover, the parameters $\alpha_i = 10^{-8}$, $\beta_i = 10^{-1}$ for $i \in \{ s, a \}$, $\beta_v = 1$, $\gamma_i = 1$ for $i \in \{ s, a, v \}$, and $\delta_v = 1$, the molar size of the fluid $N_s=N_s =1$, the final time $t_{\mathrm{max}} = 2.5$, the Flory–Huggins interaction parameters $\chi_{s,a} = 0$, and $\tau = 10^{-4}$ are used.
• Figure 4: GMRES iteration numbers and runtimes (right) per time step for different spatial discretizations of a 1D domain of Cahn-Hilliard (left) and Navier-Stokes (middle). The longest observed runtime of both equations throughout the first 20 time steps is shown. The other parameters and setup are the same as in Figure \ref{['fig:sa_vol_t']}.
• Figure 5: GMRES iteration numbers and runtimes (right) per time step for different time step $\tau$ of Cahn-Hilliard (left) and Navier-Stokes (middle) with $n_y = 400$ in a 1D domain. The longest observed runtime of both equations throughout the first 20 time steps is shown. The other parameters and setup are the same as in Figure \ref{['fig:sa_vol_t']}.