A PINN Model for Time-Dependent Boundary Conditions in the Reaction-Diffusion Equation of Li-Ion Intercalation in Hollow Silicon Anodes in Slow and Fast Cycling Conditions Under Potentiostatic Operation

Analytical modeling of Li-ion intercalation in silicon anodes plays a crucial role in comprehensively understanding the various stresses that may lead to battery failure with these high-capacity materials. Recent studies have emphasized the challenges involved in deriving an exact solution for lithium concentration in relation to spatial variables (r) and time (t), particularly through traditional methods that aim to reduce the governing partial differential equation (PDE). This complexity remains apparent even when considering the problem's simplest geometry: spherical polar coordinates. The reaction-diffusion concentrations for lithium intercalation and de-intercalation can be modeled using exponential time-dependent decay functions: C_in(t)=C_in*(1-e^{-(β₁)*(t)}) and C_out(t)=C_out*e^{-(β₂)*(t)} for lithium concentrations at the inner and outer radii of a silicon hollow sphere. These expressions stem from Dummal's theorem, where β₁ and β₂ signify the time constants at their respective radial boundaries. However, the task of simplifying analytical functions can be intricate, especially when confronted with various complex boundary conditions. To address this, our study employed a physics-informed neural network (PINN) model that utilizes a loss function representing the sum of the residuals from the PDE operator (f), initial value functions, and boundary conditions, optimizing the neural network through backward Propagation and novel machine learning techniques. The model architecture, implemented using PyTorch, was built around datasets for the spatial variable r (N_r=400) and temporal variable t (N_t=400), along with training datasets for boundary conditions (N_u=100) and PDE collocation points (N_f=1000) to enforce constraints and calculate residuals. The optimal dataset configuration for the analysis resulted in a size of [400, 400], with a loss parameter (λ) incrementally varied by 10^-4. The neural network employed multiple hidden layers with 32 nodes per layer, trained over 5 epochs. The PyDOE library was used to generate spacetime coordinates (r, t) as inputs for predicting the lithium concentration C(r, t) in the silicon anode material. The model's output was validated against exact solutions obtained for a hollow silicon sphere with inner and outer radii of 300 nm and 500 nm, respectively, achieving an MSE of 4.2×10^-2. The sphere was subjected to various conditions, including [C_out (mol/m³), C_in (mol/m³), β₁ (s^-1), β₂ (s^-1)]=[0.5, 0.5, 0.01, 0.01] and [1, 1, 5x10^-6, 5x10^-6], under potentiostatic operation to explore different scenarios of fast and slow charging and discharging by modifying β and t (ranging from 1 to 400 s). In a particular case where [C_out, C_in , β₁, β₂, r₁, r₂, t, λ, i^th, x]=[0.5, 1, 0.01, 0.01, 300, 500, 300, 0.0008, 5th, 0.4511], the model predicted C(x, t) to be 0.4101. Here, the concentration and space vector were normalized; x was defined as r/(r₂-r₁). At the same parameters, an analytical solution indicated a concentration value of 0.4173. The model was also evaluated with both thinner and thicker silicon hollow spherical anodes to assess the overall lithium concentration profile during battery cycling. This methodology, when combined with the concepts of diffusion-induced stress (DIS) and reaction-induced stress (RIS), aims to determine the optimal sphere shape that minimizes intercalation stress. The approach developed here provides a framework for analyzing the evolution of electrode stress during the charging and discharging cycles of batteries.