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To focus on the occupancy prole, we shall again write the occupancy as from the ensemble average,

φ [V] -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 I [pA] -30 -20 -10 0 10 20 30 χ=0.5, A=exp[0.6] χ=0.75, A=exp[4.6] χ=0.25, A=exp[4.6] Approximation Approximation (A) φ [V] -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 I [pA] -30 -20 -10 0 10 20 30 χ=0.5, A=exp[0.6] χ=0.75, A=exp[-4.4] χ=0.25, A=exp[-4.4] Approximation Approximation (B)

Figure 7.9: Plots A, and B compare rectied current (and its approximations in dashed curves) against symmetrical current under standard parameters. The approximations are only derived for the small voltage and rectied regime, and so beyond this are not valid. This results in exponentially increasing current, and hence we have limited to the current [+30,-30]pA. The approximations other- wise closely agree with the numerical current, thereby conrming how to observe rectication.

hni= 2P({2}) + 3P({3}). (7.45)

If we recall the probabilities can be simplied using the diusion-limit of the tran- sition rates and so occupancy can be written as,

hni= 4D

c_/L2_{+ (Γ}L

23+ ΓR23)

2Dc_/L2 . (7.46)

There are three established domains to investigate: rst the equilibrium/linear response, second general non-equilibrium conditions and nally far from equilib- rium limiting conditions. The rst and the latter are easy to identify because we know that at equilibrium we exactly recover the GCE probabilities. At limiting non-equilibrium conditions such as a large voltage drop and zero concentration drop (or anything in between) the probabilities converge to1/2.

To discuss this second domain we need to investigate the nature of the rates. If we reintroduce the constant Ab _{(dened earlier), then we can rewrite the occupancy} as,

hni= 2 + 2 +A

L_e−(1−χ)qφ/kT _+AR_e−(0−χ)qφ/kT

2 + 2AL_e−(1−χ)qφ/kT _{+ 2A}R_e−(0−χ)qφ/kT _{+ 2A}L_AR_e−(1−2χ)qφ/kT. (7.47)

which reduces to the GCE occupancy under equilibrium conditions becauseAL=

AR and we can factor terms. To simplify this expression it can be approximated

by removing either left or right voltage terms depending on the voltage domain. Thus if we consider thatφ is positive, then the occupancy becomes,

hni= 2 + 2 +A

R_e−(0−χ)qφ/kT

2 + 2AR_e−(0−χ)qφ/kT _{+ 2A}L_AR_e−(1−2χ)qφ/kT. (7.48) which is a step function vs. φ. If this condition is not met then the terms: AL_e−(1−χ)qφ/kT _{and A}L_AR_e−(1−2χ)qφ/kT _{are always very small and the occupancy} is xed at 2.5. In gure 7.10 we plot this occupancy relationship for dierent symmetrical values of Ab _{where dashed line is the approximation and solid line} is the full expression given by equation (7.47). The approximation clearly breaks down when close to equilibrium when the energy barrier (or well) is small. This is because the contribution from the cancelled term is non-negligible here.

φ [V] 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 h n i 2 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3 A=exp(+1) A=exp(-1) A=exp(+4) A=exp(-4)

Figure 7.10: Comparison of occupancy vs. +φ for a range of A values. The

To discuss how occupancy behavesvs. concentration we should collect and explic-

itly show all concentration terms. Thus to simplify we shall introduce the constant

Bb,

Bb = exp(∆E −qz∆φb−∆¯µ0)/kT

, (7.49)

analogous toAb introduced earlier. The occupancy now can be written as,

hni= 2 + 2x

2_e2∆¯µD/kT _+x(BL_+BR_)e∆¯µD/kT

2x2_e2∆¯µD/kT + 2x(BL+BR)e∆¯µD/kT + 2BLBR, (7.50)

where ∆¯µD is the Debye-Hückel contribution. Thus in the large concentration limit the occupancy reduces converges to 3. The functional dependence on the RHS of the expression is our eective adsorption isotherm,

Θ = 2x

2_e2∆¯µD/kT _+x(BL_+BR_)e∆¯µD/kT

2x2_e2∆¯µD/kT + 2x(BL+BR)e∆¯µD/kT + 2BLBR, (7.51)

and is plottedvs.concentration in gure 7.11. It produces a saturating functionvs.

concentration in quasi-Langmuir form, due to the inclusion of the Debye-Hückel interaction term. This can be seen from equation (7.51) because in the quasi equilibrium limitBL≈BR=Be and therefore the isotherm can be reduced to its

equilibrium form.

If we consider the limit that probabilities equal each other and are therefore 1/2, we can establish the condition,

ΓL₀₁+ ΓR₀₁ = ΓL₁₀+ ΓR₁₀. (7.52)

If we revisit our master equations then we can also establish an additional condi- tion,

ΓL₁₀+ ΓL₀₁ = ΓR₁₀+ ΓR₀₁, (7.53)

and hence if we sum these we nd: ΓL

01 = ΓR10 and ΓR01 = ΓL10. From inspection

of the rates this condition is always established if Ab _{∼ 1, otherwise it requires} an applied voltage. Maximal current also requires a large applied voltage and so the rates take their limiting form0orDc_/L2 and therefore current is immediately

±qDc_/(2L2₎. Concentration [M] 0 0.5 1 1.5 2 Θ 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 ∆_µ 0 = 5 kT ∆_µ 0 = 0 kT ∆_µ 0 = -5 kT Θ_× 200 Θ_× 20

Figure 7.11: Eect of∆¯µ0 on the adsorption isotherm calculated at 10mV. The

orange and yellow curves were multiplied by 20 and 200 respectively.