Understanding ion transport is crucial in various fields, from battery technology to biological systems. In the realm of materials science, simulating ion movement, particularly lithium ions, within nanostructures like graphene is vital for designing advanced energy storage devices. This exploration delves into the Equation For Ion Transport as it is applied within numerical simulations to investigate lithium ion behavior in double-layered graphene sheets.
The cornerstone of these simulations is the Nernst-Planck equation, a fundamental equation for ion transport. When coupled with the Poisson equation, it forms the Poisson-Nernst-Planck (PNP) system, a powerful tool for describing electrodiffusion processes. The finite difference method emerges as a robust numerical technique to solve the PNP equation, enabling researchers to simulate complex scenarios. The general algorithm for these simulations typically begins with defining initial and boundary conditions, along with temperature and external electric field parameters. These inputs drive the simulation of flow and force fields within the system.
The Nernst-Planck equation then plays its pivotal role in updating the number density of ions based on these fields. This updated density subsequently refines the velocity and force fields, creating an iterative loop that continues until the numerical outcomes converge, reaching a stable state that represents the system’s equilibrium.
To ensure the simulations are broadly applicable and scalable, non-dimensionalization is often employed. This process involves scaling variables to remove units and reduce the number of independent parameters. Equation (11) in the original study outlines the non-dimensionalization approach used, transforming variables like concentration, time, spatial coordinates, force, and velocity into dimensionless forms.
[
bar{c}=frac{c}{C},,bar{t}=frac{Ut}{ell },,overline{{x}_{i}}=frac{{x}_{i}}{ell },,overline{{F}_{i}}=frac{ell {F}_{i}}{m{U}^{2}},,overline{{u}_{i}}=frac{{u}_{i}}{U}
]By applying this non-dimensionalization, the core ion transport equation, Equation (3) from the source article, is transformed into Equation (12):
[
frac{partial bar{c}}{partial bar{t}}=frac{1}{Pe}{nabla }^{2}bar{c}-nabla cdot (bar{c}bar{{bf{u}}})-{rm{Upsilon }}{bar{{bf{F}}}cdot nabla bar{c}+bar{c}nabla cdot bar{{bf{F}}}},
]Here, key dimensionless parameters emerge: the Peclet number (Pe) and ${rm{Upsilon }}$, which represents the ratio between convection and thermal energy. The Peclet number, defined as (Pe=(Uell )/D), quantifies the relative importance of advection to diffusion in the ion transport process. ${rm{Upsilon }}$, given by ({rm{Upsilon }}=(DmU)/({k}_{B}Tell )), provides a measure of the convective influence relative to thermal energy within the system.
This simulation framework was applied to investigate lithium ion transport within double-layered graphene sheets of varying separations (6 Å, 8 Å, and 10 Å). Periodic boundary conditions were set for the x and z directions, while Dirichlet boundary conditions were applied in the y direction, effectively confining the ions between the graphene layers. The spatial grid size was meticulously chosen to match the approximate size of a lithium ion, and an initial uniform concentration was set. Simulations were run at 300K under varying external electric field strengths (1e4, 8e5, and 1e8 NC−1) to analyze their impact on ion distribution and transport.
Figure 2 illustrates the normalized concentrations in the mid-z plane after the systems reached stable states. These contour plots visually represent the configurations of lithium ions within the double-layered graphene for different separations and electric field strengths.
For the smallest separation (6 Å), even under low electric fields, double layers of ions form near the graphene sheets. As more ions are introduced, an oscillation between these double layers and a completely filled cavity is observed. Under higher electric fields, the cavity becomes entirely filled, indicating the strong influence of graphene’s molecular forces in trapping ions at this separation.
In contrast, at a larger separation of 10 Å, double layers form initially at low electric fields. However, as the electric field strength increases, multi-layers spontaneously emerge. These multi-layers exhibit oscillations in the middle due to the interplay of attractive and repulsive forces between ionic layers and the graphene sheets. At very high electric fields, thinner double layers reappear. The repulsive forces between ion layers become dominant, overpowering the weaker ion-graphene interactions in the central region, leading to ions being pulled towards the graphene surfaces and hindering multi-layer formation. Interestingly, even in these cases, residual multi-layers with lower concentrations can be observed in the central region, as depicted in Figure 3.
Figure 3 provides a closer look at these residual ions, highlighting the nuanced behavior of ion distribution even under strong electric fields and larger graphene separations.
Image of Residual ions in the central cavity under high electric field
To bridge the gap between the 6 Å and 10 Å separations, simulations were also conducted at 8 Å separation. The results showed a transitional behavior. At low fields, the ion distribution resembled the 6 Å case. In mid-field conditions, the behavior mirrored the 10 Å scenario. However, under high electric fields, a notable amount of ionic residuals persisted in the channel’s center, indicating a more complex interplay of forces at this intermediate separation.
To generalize these observations, Figure 4 illustrates the effect of external electric fields on the relative concentration of lithium ions for all three separations. This figure plots the normalized concentration per unit volume against varying electric field strengths, providing a clear overview of the system’s response.
The relative concentration is zero when no external field is applied. For fields below 0.6e6 NC−1, thinner double layers form across all separations, with the 6 Å separation exhibiting lower concentration due to its smaller cavity volume. Around 0.6e6 NC−1, a transition occurs: multi-layers emerge for 8 Å and 10 Å, while 6 Å shows oscillations between double layers and a filled cavity. At even higher fields, the 6 Å cavity fills completely, while thicker double layers with fluctuating ions in between develop for 8 Å and 10 Å.
In conclusion, simulations employing the equation for ion transport, specifically the Nernst-Planck equation within the PNP framework, reveal the intricate dynamics of lithium ions in double-layered graphene. The study highlights the significant influence of both graphene separation and external electric fields on ion distribution and concentration. The transition from thinner double layers to multi-layers and back to thicker double layers as separation and field strength vary underscores the complex interplay of forces at the nanoscale. These findings are crucial for optimizing the design of graphene-based materials for advanced energy storage and other applications where controlled ion transport is paramount.
