Fig. 1 Introduction to the electrolyte with immobilized anions. (A) An illustration of the electrolyte. The large blue spheres denote the anions immobilized by tethering to a solid matrix. The small green and red spheres indicate the mobile anions and cations. (B and C) An approximate solution to the base state problem at steady state for (B) high current densities and (C) low current densities, as shown in a previous work (25). Above the critical current density Jcr = 4DcFC0(1 – 2Ca,f0)/L, the migration region that is devoid of mobile anions forms near the metal electrode. The same solution is valid even with the inclusion of pressure gradient–driven transport, because pressure in the separator is uniform in the base state. (D and E) Base state solution to the steady-state transport problem: (D) cation concentration and (E) electric potential. The fraction of immobilized anions Ca,f0 is 0.1, yielding a critical current density Jcr of 3.2DcFC0/L. The profiles at the critical current density are indicated by the solid black lines. (F) Current density–voltage characteristics for various fractions of immobilized anions.
Fig. 2 Linear stability at large current densities. (A and B) Base state (A) and perturbed state (B) of the problem. (C and D) Growth rate versus wave number plots for (C) Ca,f0 = 0.1 with varying Gs, and (D) Gs = 1 MPa with varying Ca,f0. The black line in (C) corresponds to the result with elasticity excluded and only the effect of surface tension being considered. (E and F) Critical wave number (E) and growth rate (F) of the most unstable mode as a function of separator shear modulus for various fractions of immobilized anions.
Fig. 3 Linear stability at small current densities. (A and B) Base state (A) and perturbed state (B) of the problem. (C and D) Growth rate versus wave number plots for (C) Ca,f0 = 0.1 with varying Gs, and (D) Gs = 1 MPa with varying Ca,f0. The black line in (C) corresponds to the result with elasticity excluded and only the effect of surface tension being considered. (E and F) Critical wave number (E) and growth rate (F) of the most unstable mode as a function of separator shear modulus for various fractions of immobilized anions.
Fig. 4 State diagram of the dominant mechanism in stability. (A and B) Dominant mechanism in the stability of electrodeposition at various dimensionless wave numbers and separator moduli for (A) small current densities and (B) large current densities. The shaded region is unstable. See text for definitions of the various dimensionless groups.
Fig. 5 Marginal stability of deposition. Relation between separator modulus and current density at varying fixed anion fractions for which the deposition is stable for all modes. (A) Marginal stability result for all current densities for parameter values mentioned in the paper. The deposition is fully stable at any point to the right of the corresponding curve. (B) Critical value of Eo at small current densities as a function of the separator Poisson ratio. (C) Marginal stability curves for Ca,f0 = 0.1 and νs = 0.33 for various values of cation partial molar volume. The partial molar volumes of the mobile anion (va,m) and the metal (vm) are held constant at 1.78 × 10−4 and 1.3 × 10−5 m3/mol, respectively. (D) Marginal stability for Ca,f0 = 0.1, vc = −8 × 10−6 m3/mol, va,m = 1.78 × 10−4 m3/mol, and vm = 1.3 × 10−5 m3/mol for various values of separator Poisson ratio.
- Table 1 Transport mechanisms and their stabilizing behavior based on parameter values used in the paper.
Term number Term in Eq. 18 Transport mechanism Destabilizing/stabilizing 1 kJ Electric flux due to surface deformation Destabilizing 2 
Change of reaction
equilibrium due
to electrode pressure
caused by:(i) Separator modulus Stabilizing (ii) Surface tension Stabilizing 3 Change in ionic conductivity in migration region due
to separator compressionDestabilizing 4 Pressure-driven flux in
diffusion region.
Pressure caused by:(i) Separator modulus Stabilizing (ii) Surface tension Destabilizing
Supplementary Materials
Supplementary material for this article is available at http://advances.sciencemag.org/cgi/content/full/2/7/e1600320/DC1
Perturbed equations at high current densities
Perturbed equations at low current densities
fig. S1. Critical wave number versus current density.
table S1. List of symbols, subscripts, and superscripts.
Additional Files
Supplementary Materials
This PDF file includes:
- Perturbed equations at high current densities
- Perturbed equations at low current densities
- fig. S1. Critical wave number versus current density.
- table S1. List of symbols, subscripts, and superscripts.
Files in this Data Supplement:
