Impedance Spectroscopy

Lecture 5. Impedance spectroscopy Last time we talked about the non-linear response to DC voltage, and today we will discuss about the AC linear response at frequency ω. A simplified model of voltage response of galvanic cell can be written as V(I,Q)=V0(Q)-η(I,Q) If we consider small perturbations around a reference state (constant current), voltage, current and charge can be expressed as ΔV=V-Vref ΔI=I-Iref ΔQ=Q-Qref Since we are assuming small changes (perturbation), we can use taylor expansion to express voltage, ∆V ≃ where ∆! ! !!"# R !"# = ∆I = + R !"# ∆I …(1) !!"# = !! !! ! ,! !"# !"# !! !! ! ,! !"# !"# =− = !!! !! ! ,! !"# !"# !! !! ! ,! !"# !"# − !! !! ! ,! !"# !"# , C!"# = differencial  capacitance ,  R !"# = differencial  resistance ∂∆Q ∂t Consider alternating-current (AC) sinusoidal perturbations at frequency ω and introduce complex amplitudes for the perturbations (to induce phase lag): ΔV=Re(Vei t)= Vcos(ωt) ; chose t to fit real number for ΔV, V:real number) ω 1 ΔI= Re(Iei t)=⏐I⏐cos(ωt+ϕ) ω ΔQ= Re(Qei t) , where I =iωQ since ∆I = ω !∆! !! ************************************************************************ ************ Note: ! = ! + !" = !! !"   ! !" = !"#$ + !"!#$ ,          ! = ! ! + ! ! =⏐z⏐, ! = arg  (!) ************************************************************************ *********** From equation (1), we have our series RC circuit and for the same ω, time dependence disappears: !≃ ! ! + R !"# iω! = + R!! iωC! C!"# To characterize AC linear response, we use following variables, Impedance, ! = ! ! = ! ! e!!! , (ϕ= angle of phase lag of current behind voltage) this expression is complex generalization of resistance for linear AC current. ! ! ! ! Admittance,  ! = = this expression is the generalization of conductance. Complex capacitance, ! = ! ! To express the total impedance of circuit, it is importance to separate the circuit into parallel part and series part. To start with simple examples, let’s consider two simple extremes of circuit, parallel and series. 2 (1) Series RC circuit ! ! = R !"# + !!! ˜! = 1 + !"# !!!!"# ! = !!!!! ˜! = !"# !!"# ! ! = !!!!!!"# ! ˜! = !"# !"# ! !˜! !˜! !!!˜! ! !!!˜! (dimensionless form : ˜ω=ωR !"# C!"# , ˜! = ! !!"#  , ˜! = YR !"#  , ˜! = ! !!"# ) There are two ways to represent “impedance spectra”. First one is Nyquist Plot, representing Z(ω) in complex plane. Second one is Bode plot and it is beneficial to scale view. (a) Nyquist plot (series RC circuit)                            ! = ! ! e!!! , ˜! = 1 + (b) Bode plot (series RC circuit) ˜! = ˜!ט! ∗ = 1+ 1 i˜ω 1− 3 1 =   1 + ˜ω!! i˜ω ! !˜! =1−! ! ˜! ! = − arg( ˜!) = −!"#!! !"˜! 1 = !"#!! !"˜! ˜ω (2) Parallel RC circuit                                                          ˜! = ! !!!˜!                      ˜! = 1 + i˜ω                      ˜! = 1 + i ˜ω (a) Nyquist plot (parallel RC circuit) 4 (b) Bode plot (parallel RC circuit) ************************************************************************ ************ Note: Using Kirchoff’s circuit rules, it is easy to see that impedances can be combined just like resistances. ! = !! + !! dominated by larger impedance ! ! • = ! !! ! +   dominated by smaller impedance !! Later, we will see that electrochemical double layers can be modeled as parallel RC circuit elements if Faradaic reactions occur. ************************************************************************ ************ By combining impedances in series and parallel, we can analyze more complicated circuits. (1) 5 (2) C2= in series geometrical capacitance (dielectric response of dipoles) with the circuit above, representing the response of mobile ions. In nyquist plot, there are two separated semicircles if there is a separation of RC time scales R2C2 (fast electrode double layer relaxation)<