Coverage model

RESULTS AND DISCUSSION

3.1.2 Coverage model

Model development

(15) reports CVD graphene to conduct protons atσ=1.6 S/cm² in 0.1M HCl, as measured in Nafion|Graphene|Nafion assemblies in a D-S cell and through calculations representative of the model formalized in 2.3.2. However, that model assumes that each membrane component contributes a series resistance to the equivalent circuit. For the assumption to be valid, the 2D material and Nafion components must be pristine (without defects or pinholes). In the case of CVD graphene, a pristine 2D material component is difficult to grow and non-trivial to transfer into a membrane assembly. In fact, an etching study in (15) indicates that defects are present in the graphene film prior to transfer. As such, a parallel circuit model is proposed here that incorporates potential defects in a 2D material film (Figure 3.2).

Figure 3.2 Equivalent circuit Nafion|2D-material|Nafion membrane, incorporating defects

Assume that defects are filled by Nafion in the hot-press process. Then,

R_{def ect} = R_{N af ion}A_active

A_{def ect} (3.1)

where R_{N af ion} is obtained experimentally and A_{def ect} is the total area of defects in the 2D material. Note that a similar derivation could be performed assuming instead that that defects are filled with solution, which may be appropriate for membranes with varying fabrication procedures.

A_{def ect} is not directly measurable over a large area due to the limited field of view in high-resolution microscopy techniques necessary for imaging defects (e.g. SEM, TEM).

Instead, note that a pristine 2D material sample is impermeable to ions larger thanH⁺(11).

The next smallest ion (in hydrated form) is K⁺ (6). Thus, in KCl (and generalizable to any solution without H⁺ as a cation), R_lattice → ∞ and all K⁺ transport must occur through defects (Figure 3.3).

Figure 3.3 Equivalent circuit for Nafion|2D-material|Nafion membrane in H⁺ free solution, incorporating defects.

Therefore, the value of transport previously calculated as R2D−material is in fact R_{def ect} inH⁺ free environments.

Rearranging Equation 3.1, A_{def ect}=

R_{N af ion}A_active R2D−material

H⁺free environment

(3.2)

is electrolyte dependent, A_{def ect} is a property of the membrane assembly and therefore can be used for normalization across measurements.

Define coverage, C, as the ratio of pristine 2D material area (A_pristine) to total 2D material area (A_total). Assuming that the active area measured in the D-S cell, A_active is representative of the entire membrane,

C = A_pristine

A_total = 1−A_{def ect}

A_active (3.3)

Incorporating Equation 3.2 so as to put Equation 3.3 into directly measurable terms,

C = 1−

Rcell, N af ion|N af ion−Rcell, no membrane

2(Rcell, membrane−Rcell, N af ion|N af ion)

H⁺free environment

(3.4) As coverage is environment independent, it may be a valuable figure in quantifying the quality of the 2D material post membrane fabrication. Further, Equation 3.1 is now solvable with substitution of Equation 3.3:

R_{def ect}= R_{N af ion}

(1−C) (3.5)

or, for direct calculation

R_{def ect} = Rcell, N af ion|N af ion−Rcell, no membrane

2(1−C) (3.6)

Referring to the circuit model in Figure 3.2,

1 R2D−material

= 1

R_{def ect} + 1

R_lattice (3.7)

Rearranging,

R_lattice= R2D−materialR_{def ect} R_{def ect}−R2D−material

(3.8) which is directly calculable by

R_lattice =

(Rcell, membrane−Rcell, N af ion|N af ion)(Rcell, N af ion|N af ion−Rcell, no membrane)(1−C)⁻¹(¹₂) (Rcell, N af ion|N af ion−Rcell, no membrane)(1−C)⁻¹(¹₂)−(Rcell, membrane−Rcell, N af ion|N af ion)

(3.9) Then,

ASR_{def ect}=R_{def ect}A_{def ect}=R_{def ect}A_active(1−C) (3.10)

ASR_lattice =R_latticeA_lattice=R_latticeA_activeC (3.11)

σ_lattice = G_pristine

A_pristine = 1

ASR_lattice (3.12)

σlattice isolates the contributions defect-free regions of a 2D-material and better represent the intrinsic properties of the material as compared to σ2D−material.

Example calculation

The coverage model is now demonstrated with data from (15), who used Luggin capillaries in a D-S cell to minimize IR drop variability between measurements and therefore generate precise data (Table 3.2). The data from Table 3.1 is not used to demonstrate the model due to the IR drop variability in the measurements, where Nafion|2D-material|Nafion containing membranes are not reliably distinguishable from Nafion|Nafion membranes. The example calculation is limited by the methodology of (15), in that the same membranes were not tested in each electrolyte. Rather, each membrane component was converted to the desired from prior to membrane fabrication, and different membrane instances were then tested in each electrolyte. Proper implementation of the model requires testing of the same membrane

3.1.1. Nonetheless, the following analysis assumes that defect area and character is similar across each membrane.

Table 3.2 Raw data from (15). The active area for each membrane was defined by a 5/8" diameter disc, and each electrolyte solution was 0.1M xCl, where x is the neutral form of the cation. The membranes were Nafion|graphene|Nafion assemblies.

Electrolyte cation R_cell,no membrane (Ω) R_cell,Nafion|Nafion (Ω) Rcell,membrane (Ω)

H⁺ 1.08 1.32 1.64

K⁺ 6.6 9.8 56

Coverage was calculated from Equation 3.4, ASR_{def ect} was calculated from Equa- tion 3.1, ASR_lattice was calculated from Equation 3.9, andσ_latticewas calculated from Equa- tions 3.11 and 3.12 (Table 3.3).

Table 3.3 Coverage, resistance, and conductance of Nafion|graphene|Nafion mem- brane.

Coverage ASR_defect (Ωcm²) ASR_lattice (Ωcm²) σ_lattice (S/cm²)

97% 0.2 0.7 1.5

Incorporating the coverage model results in a σ_lattice of 1.5 S/cm² - a nominal de- crease fromσ2D−material due to the high (97%) coverage of pristine graphene across the mem- brane. For comparison, σ_{N af ion} calculated from the same data is 2.1 S/cm² - the graphene lattice is similarly conductive to protons as Nafion 211. σ_lattice here is still three orders of magnitude greater than σ_lattice ≈ σ2D−material ≈ 3 mS/cm² for mechanically exfoliated graphene, where one may assume the entire sample is pristine (4). This discrepancy sug- gests thatH⁺transport through CVD graphene primarily occurs by the Grotthus mechansim and vehicular transport across defects on length scales between the hydrated diameters of K⁺ (6.6Å) andH⁺ (1.0Å as hydronium) (6,18). Additionally, interactions between Nafion

and graphene layers are not included in the model presented here, but may influence ionic conductivity.