Tangential Velocities at the Intersection Between a Depositing and a Nondepositing Boundary

Shared points (in 2D) or edges (in 3D) between depositing and nondepositing boundaries are handled specifically in the Electrodeposition interfaces.

The deformation in the normal direction of a nondepositing boundary is set to zero at all times. However, for the deformation velocity of the shared points/boundaries in the tangential direction of the nondepositing boundary, special conditions apply. These conditions can be derived expressions by assuming growth or dissolution to occur only in the normal direction of the depositing boundary by addition or removal of spherical particles (for example metal atoms), see Figure 3-1 below.

Figure 3-1: Gray/Black arrows in whole stroke indicate the tangential electrode growth/dissolution velocities, vt, point, at the three-phase intersections between an electrolyte, a deposition or dissolution electrode and a nondepositing material. Dashed arrows are the growth or dissolution velocities, vdep, based on the electrodeposition rate expressions. Note that the tangential velocities depend on both the angle between the depositing surface and the nondepositing surface, as well as the direction of the normal velocity.

In the following, the boundary tangents are denoted by t (pointing from electrolyte to electrode) and the normal by n (pointing in the direction out from the electrolyte domain).

If the angle between the depositing boundary and the nondepositing boundary is larger than π, the tangential velocity is set to zero:

(3-3)

Otherwise (that is, if the angle between the depositing boundary and the nondepositing boundary is less than π), the following expressions are used:

(3-4)

Note that Case 2 above results in a lower velocity in the normal direction of the depositing surface than the deposition velocity, and that this acts toward forming a π/2 angle between the depositing and the nondepositing boundary, a phenomenon observed in experiments (Ref. 1).

In the Electrodeposition Interfaces, on shared points (2D) and edges (3D) between a Nondeforming Boundary and a Deforming Electrode Surface, the velocity of the depositing boundary is set according to the expressions above.

Reference

1. J. Deconinck, “Mathematical Modeling of Electrode Growth,” J. Appl. Electrochem., vol. 24, pp. 212–218, 1994.