A.J. Bard, L.R. Faulkner - Electrochemical methods - Fundamentals and Applications (794273), страница 96
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Techniques Based on Concepts of ImpedanceEquivalent Circuit of a Cell ( 1 , 4 , 1 3 , 1 4 )In a general sense, an electrochemical cell can be considered simply an impedance to asmall sinusoidal excitation; hence we ought to be able to represent its performance by anequivalent circuit of resistors and capacitors that pass current with the same amplitudeand phase angle that the real cell does under a given excitation. A frequently used circuit,called the Randies equivalent circuit, is shown in Figure 10.1.14a. The parallel elementsare introduced because the total current through the working interface is the sum of distinct contributions from the faradaic process, /f, and double-layer charging, ic.
The double-layer capacitance is nearly a pure capacitance; hence it is represented in the equivalentcircuit by the element Q . The faradaic process cannot be represented by simple linear circuit elements like, R and C, whose values are independent of frequency. It must be considered as a general impedance, Zf. Of course, all of the current must pass through thesolution resistance; therefore R^ is inserted as a series element to represent this effect inthe equivalent circuit.3The faradaic impedance has been considered in the literature in various ways. Figure10.1.14& shows two equivalences that have been made.
The simplest representation is totake the faradaic impedance as a series combination comprising the series resistance, Rs,and the pseudocapacity, Cs.4 An alternative is to separate a pure resistance, Rcb thecharge-transfer resistance (Sections 1.3 and 3.4.3), from another general impedance, Z w ,the Warburg impedance, which represents a kind of resistance to mass transfer. In contrast to RQ and Qi, which are nearly ideal circuit elements, the components of the faradaicimpedance are not ideal, because they change with frequency, w. A given equivalent circuit represents cell performance at a given frequency, but not at other frequencies. In fact,a chief objective of a faradaic impedance experiment is to discover the frequency dependencies of Rs and Cs. Theory is then applied to transform these functions into chemical information.The circuits considered here are based on the simplest electrode processes.
Many others have been devised in order to account for more complex situations, for example, thoseinvolving adsorption of electroreactants, multistep charge transfer, or homogeneousFigure 10.1.14(a) Equivalent circuit ofan electrochemical cell.(b) Subdivision of Zf into Rsand Cs, or into Rci and Zw.3In the faradaic impedance measurements described above (Section 10.1.1), the impedance determined by thebridge is a whole-cell impedance and includes contributions from the counter electrode's interface. Processes atthe counter electrode are not usually of interest; hence the impedance at that interface is intentionally reduced toinsignificance by employing a counter electrode of large area.4In some treatments, # s is called the polarization resistance.
However, that name is applied to other variables inelectrochemistry, so we avoid it here.10.2 Interpretation of the Faradaic Impedance377chemistry. It is important to understand that equivalent circuits drawn for electrochemicalcells are not unique.
Moreover, only in the simplest cases can one identify individual circuit elements with processes that occur in the electrochemical cell. This is especially truefor equivalent circuits that represent more complicated processes, such as, coupled homogeneous reactions or the behavior of adsorbed intermediates. In fact even the simple RaCdcircuit in the absence of a faradaic process at low electrolyte concentration shows frequency dispersion (i.e., variation of R^ and Q with frequency) (15). For specific information, the original or review literature should be consulted (1, 4, 8-14, 16).10.2 INTERPRETATION OF THE FARADAIC IMPEDANCE10.2.1 Characteristics of the Equivalent CircuitThe measurement of the cell characteristics in a bridge yields values of RB and C B that inseries are equivalent to the whole cell impedance, including the contributions from RQand C<j, which are often not of interest in studies focused on the faradaic process.
In general, one desires to separate the faradaic impedance from RQ and C&. It is possible to do soby considering the frequency dependencies of R# and Св, or by evaluating RQ and Qfrom separate experiments in the absence of the electroactive couple.5 Techniques formaking such determinations are considered in Section 10.4. For the moment, let us assume that the faradaic impedance, expressed as the series combination Rs and C s , is evaluable from the total impedance (see Figure 10.1.14).Now consider the behavior of this impedance as a sinusoidal current is forcedthrough it. The total voltage drop isE=iRs + ^-(10.2.1)hence§ = *sf + ^(10.2.2)If the current isi = Ism cot(10.2.3)thenЩ- = (RsIa)) COS cot + (-pA sin cot(10.2.4)atat^This equation is the link we will use to identify Rs and C s in electrochemical terms.
Wewill find that the response of the electrode process to the current stimulus, (10.2.3), willalso give dE/dt having the form of (10.2.4). That is, sine and cosine terms will appear;thus Rs and C s can be identified by equating the coefficients of those terms in the electricaland chemical equations.10.2.2 Properties of the Chemical System (1,4,13,14)For our standard system, О + ne <=^ R, with both О and R soluble, we can writeE = E[U C o (0, t\ CR(0, 0]5However, О and R must not affect R^ and Q appreciably if separate experiments are used.(Ю.2.5)378 I Chapter 10.
Techniques Based on Concepts of Impedancehence,dE=(dE\di+Гdt[б>Со(0, Ojdi ) dtdE 1 dCo(0, t) +, t)dtdt(10.2.6)ordEdtJt1dCo(0, t)dtLdCR(0, t)dt(10.2.7)where(10.2.8)C o (0,0,C R (0,0dCo(0, t)(10.2.9)i,CR(0,f)dE(10.2.10)/?C R (0, t) i,Co(0,t)Obtaining an expression for dEldt depends on our ability to evaluate the six factors on theright of (10.2.7). The three parameters Rcb /3Q, and /3R depend specifically on the kineticproperties of the electrode reaction.
Special cases will be considered later. The remainingthree factors [the derivatives of /, CQ and C R ] can be evaluated generally for current flowaccording to (10.2.3). One of them is trivial:didt(10.2.11)= I(x) COS (t)tThe others are evaluated by considering mass transfer.6Assuming semi-infinite linear diffusion with initial conditions CQ{X, 0) = CQ andCR(x, 0) = CR, we can write from our experience in Section 8.2.1 thatCo(0, s) = -p-(10.2.12)(10.2.13)nFAD^s1'2Inversion by convolution givesC o (0, 0 =CR(0, 0 =in1/97n~du4(t-u)du(10.2.14)(10.2.15)Note that the equivalent impedance was analyzed just above in terms of current as it is usually defined for circuitanalysis. That is, a positive change in E causes a positive change in /.
On the other hand, the electrochemicalcurrent convention followed elsewhere in this book denotes cathodic currents as positive; hence a negative changein E causes a positive change in i. If we adhere to this convention now, confusion will reign when we try to makecomparisons between the electrical equivalents and the chemical systems. We must have a common basis for thecurrent. Since the interpretations of the measurements are closely linked to electronic circuit analysis, it isadvantageous to adopt the electronic convention. For this chapter, then, we take an anodic current as positive.This expedient will turn out not to cause much trouble, because we never really follow the instantaneoussign of the current in ac experiments. Instead, we measure the amplitude of the sinusoidal component and itsphase angle with respect to the sinusoidal potential.
Of course the phase angle would depend on our choice ofcurrent convention, but it is advantageous even here to take the electronic custom, because the electronicdevices used to measure phase angle are based on it.37910.2 Interpretation of the Faradaic ImpedanceFrom (10.2.3), we can substitute for i(t - u)\ hence the problem becomes one of evaluating the integral common to both of these relations.We begin with the trigonometric identity:sin co(t — u) = sin cot cos сои - cos cot sin сои(10.2.16)which implies thatsin( )du=7§.п_jшl/IJulocos шVm^idu(10.2.17)Now let us consider the range of times in which we are interested.
Before the current isturned on, the surface concentrations are CQ and CR, and after a few cycles we can expectthem to reach a steady state in which they cycle repeatedly through constant patterns. Wecan be sure of this point because no net electrolysis takes place in any full cycle of currentflow. Our interest is not in the transition from initial conditions to steady state, but in thesteady state itself.
The two integrals on the right side of (10.2.17) embody the transition period. Because um appears in their denominators, the integrands are appreciable only at shorttimes. After a few cycles, each integral must reach a constant value characteristic of thesteady state. We can obtain it by letting the integration limits go to infinity:Г/ sin co(t - и)f °° cos сои ,f °° sin сои ,т^du = I sin cot—— du — IT cos cot—ТТГ- duШ112J SteadyиhU•>0 UV2state(10.2.18)It is easy to show that both integrals on the right side of (10.2.18) are equal to (тг/2со)т;hence we have by substitution into (10.2.14) and (10.2.15)C o (0, f) = CQ+-rjz (sin cot - cos cot)nFA(2Doco) 'CR(0, t) = CR— (sin cot - cos cot)nFA(2DRco)m(10.2.19)(10.2.20)Now we can evaluate the derivatives of the surface concentrations as required above:7It)/T\i/2(sin cot + cos cot)dCR(0,i)If со \(10.2.21)m10.2.3 Identification of Rs and C sBy substitution of (10.2.11), (10.2.21), and (10.2.22) into (10.2.7), we obtain\"ZL = ( Rct + - ^ - Ico cos cot + Iacom sin cot(10.2.23)со I7In general, we ought to consider the current as i = i^c + I sin cot, where i^ is steady or varies only slowly withtime.
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