CN102661983B - Value iterative fitting method of electrochemical impedance spectrums of porous electrodes - Google Patents

Value iterative fitting method of electrochemical impedance spectrums of porous electrodes Download PDF

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CN102661983B
CN102661983B CN201210126938.9A CN201210126938A CN102661983B CN 102661983 B CN102661983 B CN 102661983B CN 201210126938 A CN201210126938 A CN 201210126938A CN 102661983 B CN102661983 B CN 102661983B
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石威
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南京洁态环保科技有限公司
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Abstract

The invention discloses a value iterative fitting method of electrochemical impedance spectrums of porous electrodes and belongs to the fitting method of electrochemical impedance spectrums. The matrix method or series-parallel method can be selected according to different requirements and the same result can be obtained. The initial value of fitting parameters is selected in a uniform design method so that computing speed and time are greatly improved. Aiming at whether a reaction is reversible and whether a diffusion layer is of a semi-infinite type, a finite type or a block-layer diffusion type, Warburg impedance uses different expression formulas.

Description

A kind of numerical value iterative fitting method of the electrochemical impedance spectroscopy for porous electrode

Technical field

The invention belongs to the approximating method of electrochemical impedance spectroscopy, particularly a kind of numerical value iterative fitting method of the electrochemical impedance spectroscopy for porous electrode.

Background technology

Electrochemical impedance spectroscopy The electrochemical impedance impedance spectroscopy (EIS) or ac impedance spectroscopy are the technology of the Electrode surface electrochemistry mechanism of being used widely.Because this technology is the electrochemical reaction of measuring under stable state situation, apply wider frequency field, the time dependent mean value of measuring-signal within each cycle.

Porous material is the main electrode material of all types of batteries and fuel cell.This is because it provides its huge internal surface area for electrochemical reaction.But along with diminishing of porous material particle, corresponding service efficiency does not improve thereupon.Cause the reason part of this phenomenon to come from the effective usable floor area of galvanochemistry to be subject to the solution resistance control of the accumulation on interface.And this resistance is to be determined by the diffusion in ion hole.On the other hand, be also by the accumulative total contact resistance between each particle, the impacts such as conduction resistance and Faradic resistance.These are all the inherent electrode kinetics obstacles of porous type electrode.

Under normal circumstances, all inside surfaces of a porous type electrode can not all be utilized.Due to the shape in various holes, aperture, hole depth difference, the response time difference of these structures to additional pumping signal.Therefore in aperture, the motion of ion is than slow in macropore, and this just causes ion rate of propagation in porous type electrode different from bulk solution.This difference changes with hole type and size.

Transmission line model, The Transmission Line model (TML) be proposed by De Levie for explaining the electrode process of porous type electrode.According to this model, in the different degree of depth in hole, additional pumping signal is had to different responses.Bath resistance and the electric double layer capacitance of unit distance (from aperture) evenly distribute along hole wall.The Electrode behaviour of a perfect apertures can be described with uniform resistance-capacitance transmission line model.A resistance process increasing continuously at the bottom of this model has been described from aperture to hole well.This model can be used for various dissimilar electrodes: as semi-conducting electrode, and thin layer electrode, fuel cell, porous type battery electrode, polymkeric substance and polymer wrapped electrode etc.Improved transmission line model, for example pore size distribution transmission line model Pore Size Distribution (PSD) TML is used to the impedance spectrum that simulation only relates to the porous type electrode (as graphite, gold etc.) of non-Faraday process.

In this law, use equivalent electrical circuit that transmission line model is corresponding as shown in Figure 1.

Wherein R srepresentation unit length electrolyte solution resistance (Ω/cm), R ctcharge transfer resistance (Ω cm 2), Z wwarburg impedance (Ω cm 2), C d1electric double-layer capacitance (F/cm 2).In figure, Z unit distributes along hole wall, represents the electrolytic process at electrode surface and electrolyte solution interface, and its unit is specific impedance (the Ω cm of unit area 2).The impedance in a hole, Z pore, can be calculated by formula (1):

Z pore = R s Z · coth ( l p R s Z ) - - - ( 1 )

Wherein 1 pin the degree of depth of garden column structure mesopore of equal value.

In the transmission line model of De Levie, the impedance of Z unit is all identical everywhere hole wall.The formula (1) of deriving is therefrom for calculating the impedance in a hole.Thereby available complex nonlinear least square method (CNLLS) is carried out impedance spectrum matching.Such approximating method is the approximating method of the impedance spectrum to porous type electrode commonly used.Such method itself is comprising irrational hypothesis, thereby uses CNLLS fitting process can not reach good fitting precision, and often can not complete the matching in full rate territory.In this law, in order to apply CNLLS method as a comparison, show C with formula (2) swith C brelation:

C s = C b N c - - - ( 2 )

But in the measurement of AC impedance, if pumping signal is just profound ripple electromotive force, corresponding equivalents is porous cylindrical body, electromotive force is along the distribution of hole wall, according to the derivation of De Levie, by following rule (3):

V = V 0 e ( - x · R s Z ) · sin ( ωt ) - - - ( 3 )

Wherein ω angular frequency, V 0wave amplitude, R selectrolyte solution resistance, the impedance of Z unit in Z equivalent electrical circuit, x certain degree of depth at the bottom of from aperture to hole.

Formula (3) shows that electromotive force presents index decreased along hole wall and distributes.This distribution makes active electrolyte ion have different electrochemical reaction speed everywhere at hole wall under a certain frequency: from aperture more away from, speed is slower, corresponding electric current be also more go deep at the bottom of hole less.Therefore impedance also changes along with hole depth.In this law, the conductivity of electrolyte solution is considered to not change with hole depth, i, e, R sit is constant.The diffusion of solution phase is that to cause Z be the reason of the function of hole depth x.The C that this law is derived from theory swith C brelational expression and develop Numerical Iteration Method and calculate Z impedance, and then develop numerical value iteration and fitting method.

Summary of the invention

The object of the invention is to overcome the above problem that prior art exists, a kind of numerical value iterative fitting method of the electrochemical impedance spectroscopy for porous electrode is provided.

For realizing above-mentioned technical purpose, reach above-mentioned technique effect, the present invention is achieved through the following technical solutions:

A kind of numerical value iterative fitting method of the electrochemical impedance spectroscopy for porous electrode comprises the following steps:

Step 1) how much equivalent constructions of selection;

Step 2) selection Z wpattern, described Z wrepresent Warburg impedance;

Step 3) be parametric distribution codomain;

Step 4) uniform designs table;

Step 5) be parameter initialize;

Step 6) establish the sequence number that i is angular frequency, i initial value is 1, described ω irepresent i angular frequency;

Step 7) in a certain angular frequency iunder, use Numerical Iteration Method to calculate Z 1to Z 10;

Step 8) pass through Z 1, Z 2... Z 10, use the interpolate value order in Mathematica to calculate Z (x);

Step 9) Z (x) is transformed to the value of each Z unit, use matrix method or series and parallel connection method to draw Z proe, described Z proerepresent the impedance in a hole in equivalents;

Step 10) Z total=Z proe/ N, described Z totalrepresent resulting impedance in equivalents, described N represents to comprise in equivalents hole count;

Step 11) judge whether phasing degree approaches its experiment value, is, enters next step, no, adjust the first parameter (C d1) and return to step 7, always not, select lower a set of initial parameter value valuation scheme according to uniform designs table, return to step 5;

Step 12) judge whether real part approaches its experiment value, is, enters next step, no, adjust the second parameter (D) and return to step 7, always not, select lower a set of initial parameter value valuation scheme according to uniform designs table, return to step 5;

Step 13) i=i+1, returns to step 6.

Further, the numerical method of loop iteration described in step 7 is calculated Z icomprise the following steps:

Step is a) by C bas C sinitial value;

Step b) is passed through

I F = nF · k · C s γ

σ w = γ R ct | I F | n FC s 2 ωD

Calculate I fand σ w, described C brepresent electrolyte solution bulk concentration, described C srepresent the concentration of electrolyte ion at electrode surface, described I frepresent faraday's current density, described F represents Faraday constant, and described k represents electrochemical reaction constant, and described γ represents electrochemical reaction progression, described σ wrepresent Warburg impedance expression front factor, described R ctrepresent charge transfer resistance, described ω is ω irepresent i angular frequency, described D represents coefficient of diffusion;

Step c) calculates Z w, described Z wrepresent Warburg impedance;

Steps d) pass through

Z = 1 1 R ct + Z w + jω C d 1

Calculate Z, the impedance of Z unit in formula, Z wrepresent Warburg impedance, R ctrepresent charge transfer resistance, C d1represent electric double layer capacitance, j represents imaginary unit, ω ibe i angular frequency;

Step e) is passed through

V = V 0 e ( - x · R s Z ) · sin ( ωt )

Calculate V, in formula, V represents added sine voltage, V 0represent the wave amplitude of V, R srepresent corresponding to the electrolyte solution resistance in transmission line model equivalent circuit, Z is the impedance of Z unit, ω ibe i angular frequency;

Step f) is passed through

k = i 0 [ e - αfη - e ( 1 - α ) fη ] nF · C s γ

Calculate k, in formula, k represents electrochemical reaction constant, and i0 represents exchange current density, and α represents symmetrical factor, and F represents Faraday constant, and η represents overpotential, C srepresent the concentration of electrolyte ion at electrode surface, γ represents electrochemical reaction progression, f=F/RT, and F represents Faraday constant, and R represents gas law constant, and T represents absolute temperature;

Step g) pass through

C s = C b · Csch ( 2 k · V D · n · r 0 · R ct · l p ) · Sinh [ 2 k · V D · n · r 0 · R ct · ( l p - x ) ]

Calculate C s, with gained C before svalue compares, if twice of front and back C sthe difference of value is less than 10 -3think C sreach and stablized and enter step h, otherwise entered step b;

Step h) calculates Z w, described Z wrepresent Warburg impedance;

Step I) pass through

Z = 1 1 R ct + Z w + jω C d 1

Calculate Z, in formula, Z is the impedance of Z unit, Z wrepresent Warburg impedance, R ctrepresent charge transfer resistance, C d1represent electric double layer capacitance, j represents imaginary unit, ω ibe i angular frequency.

Further, in described step c and step h, calculate Z wfor semiinfinite diffusion types formula be

Z w=σ w·(1-j)

For wired diffusion types formula be

Z w = σ w · ( sinh ( 2 ω D δ ) + sin ( 2 ω D δ ) cosh ( 2 ω D δ ) + cos ( 2 ω D δ ) - j sinh ( 2 ω D δ ) - sin ( 2 ω D δ ) cosh ( 2 ω D δ ) + cos ( 2 ω D δ ) )

For piece layer diffusion types formula be

Z w = σ w · ( sinh ( 2 ω D δ ) - sin ( 2 ω D δ ) cosh ( 2 ω D δ ) - cos ( 2 ω D δ ) - j sinh ( 2 ω D δ ) + sin ( 2 ω D δ ) cosh ( 2 ω D δ ) - cos ( 2 ω D δ ) )

Z in formula wrepresent Warburg impedance, σ wrepresent Warburg impedance expression front factor, ω ibe i angular frequency, D represents coefficient of diffusion, and δ represents thickness of diffusion layer, and j represents imaginary unit.

Further, in step 9, for comprising by compared with the transmission line model of big unit number, observe base ear Hough principle by electric current in the distribution of each unit, use described matrix method to calculate the electrochemical impedance in a hole; For comprising by compared with the transmission line model of big unit number, by the connection in series-parallel relation of all parts in equivalent electrical circuit, calculate the electrochemical impedance in a hole with described series and parallel connection method.

Further, the first parameter described in step 11 is C d1, adjustment mode is for being greater than its experiment value, C when phasing degree d1=C d1+ C d1/ 100; When phasing degree is less than its experiment value, C d1=C d1-C d1/ 100, C in formula d1represent electric double layer capacitance.

Further, the second parameter described in step 12 is D, and adjustment mode is for being greater than its experiment value, D=D+D/100 when real part; When real part is less than its experiment value, D=D-D/100, in formula, D represents coefficient of diffusion.

The invention has the beneficial effects as follows:

Can, according to different selection matrix method or the series-multiple connection methods of needing, all can obtain identical result.With uniform design, the initial value of fitting parameter is selected, greatly improved speed and the time of computing.Whether reversible corresponding to what react, diffusion layer is semiinfinite, limited or piece layer diffusion types, and different expression formula is applied in Warburg impedance.

Above-mentioned explanation is only the general introduction of technical solution of the present invention, in order to better understand technological means of the present invention, and can be implemented according to the content of instructions, below with preferred embodiment of the present invention and coordinate accompanying drawing to be described in detail as follows.The specific embodiment of the present invention is provided in detail by following examples and accompanying drawing thereof.

Brief description of the drawings

Accompanying drawing described herein is used to provide a further understanding of the present invention, forms the application's a part, and schematic description and description of the present invention is used for explaining the present invention, does not form inappropriate limitation of the present invention.In the accompanying drawings:

Fig. 1 transmission line model equivalent electrical circuit is for numerical value iteration and fitting method;

Fig. 2 calculates V, k, C s, Z w, and the circular chart of Z;

Fig. 3 cycling numerical value method is calculated Z1;

Fig. 4 | C s| value with the variation of hole depth;

Fig. 5 illustrates the relation of Z (x) and hole depth and calculates Z imethod;

The distribution of Fig. 6 electric current in transmission line model equivalent circuit;

The process flow diagram of the electrochemical impedance spectroscopy of Fig. 7 numerical value iteration and fitting method matching porous type electrode;

In Fig. 8 embodiment 1, use the Nyquist figure of CNLLS method match value and experiment value;

In Fig. 9 embodiment 1, use the Nyquist figure of Numerical Iteration Method match value and experiment value;

In Figure 10 embodiment 2, use the Nyquist figure of CNLLS method match value and experiment value;

In Figure 11 embodiment 2, use the Nyquist figure of Numerical Iteration Method match value and experiment value;

In Figure 12 embodiment 3, use the Nyquist figure of CNLLS method match value and experiment value;

In Figure 13 embodiment 3, use the Nyquist figure of Numerical Iteration Method match value and experiment value.

Embodiment

Below with reference to the accompanying drawings and in conjunction with the embodiments, describe the present invention in detail.

1. numerical value iteration and fitting method has following supposition:

(1) all porous circles of equal value have the hole (radius r of same size size 0, degree of depth l p), these holes are all arranged parallel to each other, and total number of perforations is N.

(2) electrode material does not have resistance.

(3) in equivalent electrical circuit, there is identical height each unit in equivalent circular cylinder.

(4) electrode process comprises faraday and non-Faraday process.

(5) just profound ripple electromotive force is as disturbing signal.

(6) diffusion adopts planar diffusion pattern.

2.C swith C bthe derivation of relationship and to Warburg impedance Z wthe impact of expression formula

Because electromotive force is the function of hole depth x along hole wall from aperture to the variation at the bottom of hole, electroactive ion spreads from bulk solution to hole wall according to Fick's second law of diffusion:

∂ C s ∂ t = D ∂ 2 C s ∂ x 2 - 2 I f nF r 0 C s γ - - - ( 4 )

Separate this equation, boundary condition is C s(0)=C band C s(1 p)=0, the C solving sexpression formula is:

C s = C b · Csch ( 2 k · V D · n · r 0 · R ct · l p ) · Sinh [ 2 k · V D · n · r 0 · R ct · ( l p - x ) ] - - - ( 5 )

K electrochemical reaction constant in formula, V excitation potential or overpotential, D is coefficient of diffusion, n electrochemical reaction metastatic electron number, R ctcharge transfer resistance, r 0and l prespectively equivalents pore radius and hole depth.Equation (4) first carries out Laplace conversion to be made it become frequency field from time domain then to solve formula (5).

For a non-reversible reaction, Z wpreposition coefficient be:

σ w = γ R ct | I F | n FC s 2 ωD - - - ( 6 )

I F = i 0 [ e - αfη - e ( 1 - α ) fη ] = n Fv r = nF · k · C s γ - - - ( 7 )

I in formula 0for exchange current density, wherein R is gas law constant, and T is Kelvin temperature, and n is electron transfer number, and F is Faraday constant.

γ electrochemical reaction progression in formula, | I f| be in Fig. 1, to pass through R ct-Z wthe size of the Faradic electricity current density of branch, C sthe electroactive electrolyte ion of d is at the concentration of hole surface, C bthe concentration of electroactive electrolyte ion in bulk solution.

Then basis is to different diffusion types, to the Warburg impedance (Z of irreversible electrochemical reaction w) can calculate acquisition:

To semiinfinite diffusion types:

Z w=σ w·(1-j)?????????????(8)

To limited diffusion types:

Z w = σ w · ( sinh ( 2 ω D δ ) + sin ( 2 ω D δ ) cosh ( 2 ω D δ ) + cos ( 2 ω D δ ) - j sinh ( 2 ω D δ ) - sin ( 2 ω D δ ) cosh ( 2 ω D δ ) + cos ( 2 ω D δ ) ) - - - ( 9 )

To piece layer diffusion types:

Z w = σ w · ( sinh ( 2 ω D δ ) - sin ( 2 ω D δ ) cosh ( 2 ω D δ ) - cos ( 2 ω D δ ) - j sinh ( 2 ω D δ ) + sin ( 2 ω D δ ) cosh ( 2 ω D δ ) - cos ( 2 ω D δ ) ) - - - ( 10 )

Wherein ω is angular frequency, and D is coefficient of diffusion, and δ is diffusion length, σ wz wpreposition coefficient.Its reversible or irreversible character according to reaction has different expression formulas.

When formula (5) is updated to formula (6), σ wjust become hole depth, the function of x.So Z walso become the function of x-with Z.

The numerical method of 3.Z

In Fig. 1, the impedance of Z unit can be calculated by formula (11):

Z = 1 1 R ct + Z w + jω C d 1 - - - ( 11 )

Formula will calculate Z, C in (11) d1, R ct, and Z wvalue must be known.To an irreversible electrochemical reaction, its Z wcan calculate from formula (8) to (10).But formula (5) can not be directly used in C scalculating.

In formula (5), C sby η (V) and k, determine.K is the reaction constant of electrochemical reaction, and it is calculated and obtained by formula (12).Formula (12) is from the Butler-Volmer derivation of equation:

k = i 0 [ e - αfη - e ( 1 - α ) fη ] nF · C s γ - - - ( 12 )

α symmetrical factor in formula, f=F/RT.

From formula (3), η (V) is determined by Z, and Z is by Z wdetermine by formula (11).Z wagain by C sdetermine C from formula (6) to (10) sdetermined by formula (5) by k and η (V), k is by η (V) and C simultaneously sdetermine by formula (12), again, η (V) is determined by formula (3) by Z ..., such calculating has caused loop nesting, as shown in Figure 2.

For a hole depth x, ω under a certain angular frequency, when reaching after stable state, electroactive electrolyte ion has a stable value.From then on principle is set out, and can develop a Numerical Iteration Method for calculating in a certain hole depth, the value under a certain angular frequency.As initial, C bbe treated as C sinitial value.Then by formula (6), formula (7) (8)/(9) and formula (10) calculate Z w.Then using formula (11) calculates Z.After obtaining Z, calculate V by formula (3), then calculate k from V by formula (12).Calculate and obtain the gained C that circulates for the first time by formula (5) when k value is known svalue.Use this C svalue is as the initial value C that circulated for the second time that circulates again next time svalue.If twice of front and back C sthe difference of value is less than 10 -3, think C sreach stationary value.Otherwise once circulate again, until C sthe requirement of stationary value reaches.The process flow diagram of this process is as Fig. 3.

4.C sconstringent proof

Calculate final C at this Numerical Iteration Method svalue is not to get by a formula.Needing proves under a certain angular frequency, at a certain depth of hole wall, its C svalue has and only has a stable end value.In this law, as former and later two C svalue differs and is no more than 10 -3shi Ze thinks C sreach stationary value.In most cases need at least three C of circulation sjust can reach stable.Therefore need the more degree of depth C proving at hole wall scan reach stationary value.

Fig. 4 shown | C s| value is in the variation at ten hole depth places.For certain a bit, its value tends to gradually an end value after cycle calculations.The direction of arrow points cycle calculations income value.At the bottom of from aperture to hole | C s| diminish gradually.This is that electromotive force reduces to cause faradic currents also to reduce because of the increase along with the degree of depth.Electrochemical reaction speed is corresponding reducing also, therefore | and C s| diminish.In Fig. 4, after the each cycle calculations of arrow points | C s| change direction.To all ten o'clock from 0.11 pto 0.951 p, shown identical variation tendency.This just illustrates that each point has its final stationary value.From 0.11 pto 0.951 p, each point | C s| final stationary value diminish gradually.This meets analysis above.

Except ten points shown in Fig. 4, remaining point, comprises 0.151 p, 0.251 p, 0.351 p, 0.451 p, 0.551 p, 0.651 p, 0.751 p, 0.851 p, 0.131 p, 0.231 p, 0.331 p, 0.431 p, 0.531 p, 0.631 p, 0.731 p, 0.831 p, 0.931 p, 0.181 p, 0.281 p, 0.381 p, 0.481 p, 0.581 p, 0.681 p, 0.781 p, 0.881 p, and 0.981 p, they | C s| final stationary value has also been carried out calculating them and has also all been met identical rule.From above each point | C s| calculating show, we can think by after numerical value iterative computation, all hole depths are in a final stable C under a certain angular frequency s.

5. structure and the Z of numerical value Z (x) icalculating

Z is the impedance of the Z unit in equivalent circuit.It is hole depth as previously mentioned, the function of x.Under a certain ω, select ten depth points along hole wall: 0.11 p, 0.21 p, 0.31 p, 0.41 p, 0.51 p, 0.61 p, 0.71 p, 0.81 p, 0.91 p, and 0.951 pcalculate Z resistance value separately.Use " Interpolation " (interpolation) order of Mathematica software package, the function changing with x of Z:

Z(x)=Interpolation[{Z1,0.11 p},{Z2,0.21 p},{Z?3,0.31 p},{Z4,0.41 p},{Z5,0.51 p},{Z6,0.61 p},{Z7,0.71 p},{Z8,0.81 p},{Z9,0.91 p},{Z10,0.951 p}]

The principle of work of interpolate value order is to carry out polynomial curve fitting between continuous point.The technological document of Wolfram company (developer of Mathematica) describes the principle of work of interpolate value order in detail.

Z (x) be one with x continually varying function.To each Z unit, it is worth Z i, by Z i+1and Z idifference obtain.Footmark i is the sequence number of Z unit, from aperture to hole at the bottom of.Calculating Z from Z (x) i, first x will become the sequence number of Z unit.As shown in Figure 5, after Z (x) obtains by the Interpolation order of operation Mathematica, the value of i Z is from Z i+1-Z iobtain.I is the sequence number of Z unit.

6.Z poreand Z totalcalculating

In Fig. 1, the impedance (Z in a hole pore) be by R swith Z determine.Because Z is the function of hole depth x, formula (1) can not be used for calculating Z pore.As the resistance value Z of each Z unit iwith solution resistance R sknown, the impedance in a hole can nationality matrix method or series-multiple connection method calculating acquisition.In this law, the equivalents of porous type electrode is the right cylinder of porous.There is identical size in all holes, between them, are arranged in parallel.When obtaining after the impedance in a hole, the impedance of whole right cylinder or whole porous type electrode is calculated by formula (13):

Z total = Z pore N - - - ( 13 )

N right cylinder central hole structure of equal value number in formula.Because be to be arranged in parallel between them, so resulting impedance is N/mono-in single hole.

6.1. matrix method

As Fig. 6 has shown the distribution of electric current in the each unit of equivalent circuit.

By the electric current (I of each unit 2, I 3..., I n) meet base ear Hough principle (Kirchoff ' s law).R in the each unit of formula (14) s, Z iand I irelation:

V = ( n R s + Z n ) I 1 - R s ( I 2 + I 3 + . . . + I n ) 0 = ( Z 1 + Z 2 + R s ) I 2 - R s I 1 - Z 2 I 3 0 = ( Z 2 + Z 3 + R s ) I 3 - R s I 1 - Z 2 I 2 - Z 3 I 4 0 = ( Z 3 + Z 4 + R s ) I 4 - R s I 1 - Z 3 I 3 - Z 4 I 5 . . . . 0 = ( Z n - 3 + Z n - 2 + R s ) I n - 2 - R s I 1 - Z n - 3 I n - 3 - Z n - 2 I n - 1 0 = ( Z n - 2 + Z n - 1 + R s ) I n - 1 - R s I 1 - Z n - 2 I n - 2 - Z n - 1 I n 0 = ( Z n - 1 + Z n + R s ) I n - R s I 1 - Z n - 1 I n - 1 - ( Z n + R s ) I 1 - - - ( 14 )

Formula (15) is that the linear matrix of formula (14) is described:

( I i ) n × n × ( R s / Z i ) n × 1 = V 1 0 0 0 0 n × 1 - - - ( 15 )

Separate this matrix, I 1to I ncan obtain the impedance Z in a hole pore, can calculate thus:

Z pore = V 1 I 1 - - - ( 16 )

The impedance of whole electrode can be calculated and be obtained by formula (13).

This method need to be longer operation time.But can obtain the distribution of electric current at unit, this can be used for the calculating of penetrating percentage and the penetrating degree of depth.

6.2. series-multiple connection method

This method is carried out computing impedance by the connection in series-parallel relation of all parts in equivalent electrical circuit.In Fig. 6, Z nwith R sin series connection.The resulting impedance S of these two n=Z n+ R s, again with Z n-1, in state in parallel.Above resulting impedance S n-1=1/ (1/S n+ Z n-1), again with R sin series connection ... by such process, the resulting impedance of equivalent circuit can be calculated acquisition.The advantage of this method is that computing time is short.

If the impedance spectrum of matching and experiment value difference are larger, alternative plan has: or the scope of change initial parameter value, or adopt different Z wpattern, or use difform equivalents etc. until reach fitting precision.These steps can be by flowcharting, if Fig. 7 is exactly the process flow diagram of numerical value iteration and fitting method.Write Mathematica6.0 program according to this figure and carried out impedance spectrum matching.

Here contrasted application CNLLS and Numerical Iteration Method to three electrolytic manganese dioxide electrodes, one is full of the matching of the electrochemical impedance spectroscopy of the negative plate of electric sulfuric acid-excide battery.In two kinds of methods, irreversible-limited diffusion types is as the type of Warburg impedance Z w.

Embodiment 1

The matching of the electrochemical impedance spectroscopy to three electrolytic manganese dioxide electrodes

A. use CNLLS method

As Fig. 8, the Nyquist figure of three EMD, full rate territory match value vs.Trial value, applies irreversible-limited diffusion Z w.Solid icon represents experiment value; Hollow icon represents match value.CNLLS is as the Fitting Calculation method; EMD1: ■ experiment value, mouthful match value; EMD2: ▲ experiment value, △ match value; EMD3: ● experiment value, zero match value.

B. use Numerical Iteration Method

As Fig. 9, the Nyquist figure of three EMD, full rate territory match value vs.Trial value, applies irreversible-limited diffusion Z w.Solid icon represents experiment value; Hollow icon represents match value; Numerical Iteration Method is as approximating method; EMD1: ■ experiment value, mouthful match value; EMD2: ▲ experiment value, △ match value; EMD3: ● experiment value, zero match value.

Embodiment 2

To being full of one by one the matching of electrochemical impedance spectroscopy of negative plate of electric sulfuric acid-excide battery

A. use CNLLS method

As Figure 10, one is full of the Nyquist figure of electric sulfuric acid-excide battery negative pole, full rate territory match value vs.Trial value, applies irreversible-limited diffusion Z w.CNLLS is as the Fitting Calculation method; Battery capacity: 7Ah; I=0; ◆ experiment value, ■ match value.

B。Use Numerical Iteration Method

As Figure 11, one is full of the Nyquist figure of electric sulfuric acid-excide battery negative pole, full rate territory match value vs.Trial value, applies irreversible-limited diffusion Z w.Numerical Iteration Method is as the Fitting Calculation method; ◆ experiment value, ■ match value.

Embodiment 3

To a matching that is full of the electrochemical impedance spectroscopy of electric sulfuric acid-excide battery

A. use CNLLS method

As Figure 12, one is full of the Nyquist figure of electric sulfuric acid-excide battery, full rate territory match value vs.Trial value, applies irreversible-limited diffusion Z w.CNLLS is as the Fitting Calculation method; Battery capacity: 7Ah; I=0; ◆ experiment value, ■ match value.

B. use Numerical Iteration Method

As Figure 13, one is full of the Nyquist figure of electric sulfuric acid-excide battery, full rate territory match value vs.Trial value, applies irreversible-limited diffusion Z w.Numerical Iteration Method is as the Fitting Calculation method; Battery capacity: 7Ah; I=0; ◆ experiment value, ■ match value.

When these two kinds of approximating methods of contrast, use Numerical Iteration Method can obtain than using the better fitting precision of CNLLS method and matching frequency range.

The foregoing is only the preferred embodiments of the present invention, be not limited to the present invention, for a person skilled in the art, the present invention can have various modifications and variations.Within the spirit and principles in the present invention all, any amendment of doing, be equal to replacement, improvement etc., within all should being included in protection scope of the present invention.

Claims (7)

1. for a numerical value iterative fitting method for the electrochemical impedance spectroscopy of porous electrode, it is characterized in that, comprise the following steps:
Step 1) how much equivalent constructions of selection;
Step 2) selection Z wpattern, described Z wrepresent Warburg impedance;
Step 3) be parametric distribution codomain;
Step 4) uniform designs table;
Step 5) be parameter initialize;
Step 6) establish the sequence number that i is angular frequency, i initial value is 1, described ω irepresent i angular frequency;
Step 7) in a certain angular frequency iunder, use Numerical Iteration Method to calculate Z 1to Z 10;
Step 8) pass through Z 1, Z 2... Z 10, use the interpolate value order in Mathematica to calculate Z (x);
Step 9) Z (x) is transformed to the value of each Z unit, re-use matrix method or series and parallel connection method draws Z pore, described Z porerepresent the impedance in a hole in equivalents;
Step 10) Z total=Z pore/ N, described Z totalrepresent resulting impedance in equivalents, described N represents the hole count comprising in equivalents;
Step 11) judge whether phasing degree approaches its experiment value, is, enters next step, no, adjust the first parameter (C d1) and return to step 7, always not, select lower a set of initial parameter value valuation scheme according to uniform designs table, return to step 5;
Step 12) judge whether real part approaches its experiment value, is, enters next step, no, adjust the second parameter (D) and return to step 7, always not, select lower a set of initial parameter value valuation scheme according to uniform designs table, return to step 5;
Step 13) i=i+1, returns to step 6.
2. the cycling numerical value iterative fitting method of the electrochemical impedance spectroscopy for porous electrode according to claim 1, it is characterized in that, because excitation potential is the function of hole depth x along hole wall from aperture to the variation at the bottom of hole, active ion spreads from bulk solution to hole wall according to Fick's second law of diffusion: separate this equation, boundary condition is C s(0)=C band C s(1 p)=0, the C solving sexpression formula is: C s = C b · Cosh ( 2 k · V D · n · r 0 · R ct · l p ) · Sinh [ 2 k · V D · n · r 0 · R ct · ( l p - x ) ] ;
K electrochemical reaction constant in formula, V excitation potential or overpotential, D is coefficient of diffusion, n electrochemical reaction metastatic electron number, R ctcharge transfer resistance, r 0and l prespectively equivalents pore radius and hole depth; Fick's second law of diffusion equation first carries out Laplace conversion to be made it become frequency field from time domain then to solve C sformula.
3. the cycling numerical value iterative fitting method of the electrochemical impedance spectroscopy for porous electrode according to claim 1, is characterized in that, Numerical Iteration Method described in step 7 calculates Z icomprise the following steps:
Step is a) by C bas C sinitial value;
Step b) is passed through
I F = nF · k · C s γ
σ w = γ R ct | I F | nFC s 2 ω i D ;
, calculate I fand σ w, described C brepresent electrolyte solution bulk concentration, described C srepresent the concentration of electrolyte ion at electrode surface, described γ represents electrochemical reaction progression, described in represent C sγ power, described I frepresent faraday's current density, described F represents Faraday constant, and described k represents electrochemical reaction constant, described σ wrepresent Warburg impedance expression front factor, described R ctrepresent charge transfer resistance, described ω irepresent i angular frequency, described D represents coefficient of diffusion;
Step c) calculates Z w, described Z wrepresent Warburg impedance;
Steps d) pass through
Z = 1 1 R ct + Z w + j ω i C d 1
Calculate Z, in formula, Z is the impedance of Z unit, Z wrepresent Warburg impedance, R ctrepresent charge transfer resistance, C d1represent electric double layer capacitance, j represents imaginary unit, ω ibe i angular frequency;
Step e) is passed through
V = V 0 e ( - x · R s Z ) · sin ( ω i t )
Calculate V, in formula, V represents added sine voltage, V 0represent the wave amplitude of V, R srepresent corresponding to the electrolyte solution resistance in transmission line model equivalent circuit, Z is the impedance of Z unit, ω ibe i angular frequency;
Step f) is passed through
k = i 0 [ e - αfη - e ( 1 - α ) fη ] nF · C s γ
Calculate k, in formula, k represents electrochemical reaction constant, i 0represent exchange current density, α represents symmetrical factor, and F represents Faraday constant, and η represents overpotential, C srepresent the concentration of electrolyte ion at electrode surface, described γ represents electrochemical reaction progression, described in represent C sγ power, f=F/RT, F represents Faraday constant, R represents gas law constant, T represents absolute temperature;
Step g) pass through
C s = C b · Csch ( 2 k · V D · n · r 0 · R ct · l p ) · Sinh [ 2 k · V D · n · r 0 · R ct · ( l p - x ) ]
Calculate C s, with gained C before svalue compares, if twice of front and back C sthe difference of value is less than 10 -3think C sreach and stablized and enter step h, otherwise with gained C senter step b;
Step h) calculates Z w, described Z wrepresent Warburg impedance;
Step I) pass through
Z = 1 1 R ct + Z w + j ω i C d 1
Calculate Z, in formula, Z is the impedance of Z unit, Z wrepresent Warburg impedance, R ctrepresent charge transfer resistance, C d1represent electric double layer capacitance, j represents imaginary unit, ω ibe i angular frequency.
4. the numerical value iterative fitting method of the electrochemical impedance spectroscopy for porous electrode according to claim 3, is characterized in that: in described step c and step h, calculate Z wfor semiinfinite diffusion types formula be:
Z w=σ w·(1-j)
For wired diffusion types formula be
Z w = σ w · ( sinh ( 2 ω D δ ) + sin ( 2 ω D δ ) cosh ( 2 ω D δ ) + cos ( 2 ω D δ ) - j sinh ( 2 ω D δ ) - sin ( 2 ω D δ ) cosh ( 2 ω D δ ) + cos ( 2 ω D δ ) )
For piece layer diffusion types formula be
Z w = σ w · ( sinh ( 2 ω D δ ) - sin ( 2 ω D δ ) cosh ( 2 ω D δ ) - cos ( 2 ω D δ ) - j sinh ( 2 ω D δ ) + sin ( 2 ω D δ ) cosh ( 2 ω D δ ) - cos ( 2 ω D δ ) )
Z in formula wrepresent Warburg impedance, σ wrepresent Warburg impedance expression front factor, ω is ω i, representing i angular frequency, D represents coefficient of diffusion, and δ represents thickness of diffusion layer, and j represents imaginary unit.
5. the numerical value iterative fitting method of the electrochemical impedance spectroscopy for porous electrode according to claim 1, it is characterized in that: in step 9 for comprising by compared with the transmission line model of big unit number, observe base ear Hough principle by electric current in the distribution of each unit, use described matrix method to calculate the electrochemical impedance in a hole; For comprising by compared with the transmission line model of big unit number, by the connection in series-parallel relation of all parts in equivalent electrical circuit, calculate the electrochemical impedance in a hole with described series and parallel connection method.
6. the numerical value iterative fitting method of the electrochemical impedance spectroscopy for porous electrode according to claim 1, is characterized in that: the first parameter described in step 11 is C d1, adjustment mode is for being greater than its experiment value, C when phasing degree d1=C d1+ C d1/ 100; When phasing degree is less than its experiment value, C d1=C d1-C d1/ 100, C in formula d1represent electric double layer capacitance.
7. the numerical value iterative fitting method of the electrochemical impedance spectroscopy for porous electrode according to claim 1, is characterized in that: the second parameter described in step 12 is D, and adjustment mode is for being greater than its experiment value, D=D+D/100 when real part; When real part is less than its experiment value, D=D-D/100, in formula, D represents coefficient of diffusion.
CN201210126938.9A 2012-04-27 2012-04-27 Value iterative fitting method of electrochemical impedance spectrums of porous electrodes CN102661983B (en)

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