Dynamic Response of Ion Transport in Nanoconfined Electrolytes

Ion transport in nanoconfined electrolytes exhibits nonlinear effects caused by large driving forces and pronounced boundary effects. An improved understanding of these impacts is urgently needed to guide the design of key components of the electrochemical energy systems. Herein, we employ a nonlinear Poisson–Nernst–Planck theory to describe ion transport in nanoconfined electrolytes coupled with two sets of boundary conditions to mimic different cell configurations in experiments. A peculiar nonmonotonic charging behavior is discovered when the electrolyte is placed between a blocking electrode and an electrolyte reservoir, while normal monotonic behaviors are seen when the electrolyte is placed between two blocking electrodes. We reveal that impedance shapes depend on the definition of surface charge and the electrode potential. Particularly, an additional arc can emerge in the intermediate-frequency range at potentials away from the potential of zero charge. The obtained insights are instrumental to experimental characterization of ion transport in nanoconfined electrolytes.


Comparison between 𝑸 𝐄𝐃𝐋 and 𝑸 𝐌 in time space
There are two common definitions of the EDL charge 1,2 , including the total diffuse charge  EDL (), and the electrode surface charge  M (), as shown in FIG S1.  EDL () is calculated from the integration of the net ionic charge from the Helmholtz plane to the middle plane, while  M () is calculated by the Gauss's The  +,mid eq can described by the following approximate analytical expression 3 , where, (c).It means that the DBCC is equivalent to the SBOC when 2 → ∞.Thirdly, compared with the DBCC, the SBOC shows a nonmonotonic charging behavior when the nonlinear PNP theory is used, which is discussed in the main text.

Time evolution of net charge density distribution in the SBOC
where M + is the metal ion, and M the metal atom.The kinetics of this charge transfer reaction can be described using the Frumkin-Butler-Volmer (FBV) theory, where  0, is the rate constant of charge transfer,  +,HP the concentration of M +

Analytical solution of EIS at PZC 4, 5 :
Herein, the ion size effect is ignored, using  = 0, and the linear PNP equation can be simplified to, Applying a perturbation to (S8), we obtain, where the over-tilde marks a quantity in the frequency domain and   is the dimensionless angular frequency with respect to Due to the initial conditions  + 0 =  − 0 = 1,  0 = 0, (S9) can be written as, The (S10) can be solved, where, where  1 ,  2 ,  1 ,  2 ,  1 ,  2 are determined by the boundary conditions.
For the DBCC, at  = 0, ∇ ̃+ + ∇ ̃= 0，we obtain, and ∇ ̃− − ∇ ̃= 0, we obtain, Solving the above two equations, then we can obtain, And  ̃= − ̃M +   ∇ ̃ at  = 0, we can obtain, , then we can obtain, , At  = 2 ̃, ∇ ̃+ + ∇ ̃= 0, we obtain, and ∇ ̃− − ∇ ̃= 0, we obtain, Then we can obtain , . when the EDL charge refers to  ̃EDL , and the impedance is calculated, When the EDL charge refers to  ̃M, and the impedance is calculated, For the SBOC, in the similar way, we can obtain, When the EDL charge refers to  ̃EDL , the impedance is calculated, When the EDL charge refers to  ̃M, and the impedance is calculated, where  nd is the dimensionless form with respect to When the EDL charge refers to the metal surface charge, the impedance reads, In the low frequency range, namely,  → 0,  1 ≈ 1, and tanh(   ⁄ ) ≈ 1.Then Eq.

EIS response of the single blocking closed cell
We consider a case with a non-blocking metal on the right side, namely, the

2 . 3 .
FIG S1.Two definitions of EDL charge, the electrode surface charge  M () by Gauss's law and total diffuse charge  EDL () by integrating the ionic charge from the Helmholtz plane to the middle plane.
FIG S6.(a) Schematic diagram of the single-reactive open cell with one side in contact with a non-blocking electrode, and the other side connected to a reservoir of electrolyte solution; (b) The charging dynamics of nanoconfined electrolytes in terms of the total diffuse charge  EDL () for the nonlinear (line) and linear PNP theory (dash line) at different rate constants of the charge transfer reaction; (c) Regime of nonlinearity of the PNP theory.(d) Regime of non-monotonic EDL charging dynamics.Model parameters are  0 = 1 × 10 −3 mol L −1 ,  ± = 1 × 10 −11 m 2 s −1 ,  HP = 0.3 nm , and the corresponding references values are   ≈ 9.63 nm,   =   2  + ⁄ = 9.27 × 10 −7 s,   =   ⁄ = 25 mV.