電導弛豫法測材料的氧表面交換系數(shù)和體相擴散系數(shù)

1、電導弛豫法測材料的氧表面交換系數(shù)和體相擴散系數(shù)感謝網(wǎng)友yiqieke整理此文背景：鈣鈦礦混合導體氧化物同時具有氧離子導電性和電子導電性'因此已被廣泛地用作固體氧化物燃料電池的陰極和透氧膜。這類材料的性能主要由氧在其表面的交換速率以及氧在其體內(nèi)的擴散速率決定。如果可以提高這兩個速率'體氧化物燃料電池和透氧膜的操作溫度將會降低丿而在較低溫度下工作的燃料電池其材料的老化程度將會有所降低丿并且壽命會更長。因此，如何提高這兩個速率就成為了很多燃料電池和透氧膜材料研究者所感興趣的問題。為此對不同的材料我們需要測量其氧表面交換系數(shù)和體相擴散系數(shù)。目前測定氧表面交換系數(shù)和體相擴散系數(shù)的方法主要

2、有同位素交換法和電導弛豫法（ECR,ElectricalConductivityRelaxation）e由于電導弛豫法所需儀器相對簡單丿故應用廣泛。i理.由于鈣鈦礦混合導體氧化物中存在多價態(tài)的過渡金屬離子（Fe、Co等）'在一定溫度下突然改變氧化物所處環(huán)境的氧分壓會造成過渡金屬離子的變價，同時氧化物中的氧（離子）濃度也發(fā)生變化。若在初始時刻（t=Ot=O）時氧的平衡濃度是Coco,當外界氧分壓發(fā)生突變后，氧在t時刻的濃度為CtCtoCoco到CtCt的變化是通過氧擴散進出氧化物來實現(xiàn)的，這種擴散會引起體系電導率sigmasigma的變化。假定體系電導率和氧濃度之間存在線性關(guān)系丿則Co-

3、CiiiftyoverCt-Ciiifly=sigmao-sigmaiiiftyoversigmat-sigmainftyCO-CinftyoverCt-Cinfty=sigmaO-sigmainftyoversigmat-sigmainfty其中丿sigmaosigmaO為初值時刻（t=Ot=O）電導率sigmatsigmat為氧分壓突變后t時刻的電導率sigmaniftysigmainfty為氧濃度重新達到平衡穩(wěn)定后電導率。為求得氧的表面交換系數(shù)和體相擴散系數(shù)需要對擴散過程進行求解。對擴散過程，F(xiàn)ick第一定律指出擴散通量與濃度梯度成正比，其比例系數(shù)為擴散系數(shù)：mathb£J=-

4、DnablaCmathbfJ=-DnablaC由質(zhì)量守恒定律，paitialCoveipaitialt+nablacdotmathbfJ=OpartialCoverpartialt+nablacdotmathbfj=0進而得到Fick第二定律paitialCoveipartialt-nablacdotDnablaC=OpartialCoverpartialt-nablacdotDnablaC=0若擴散系數(shù)D為常數(shù)丿與濃度無關(guān)'則paitialCoveipartialt=Dnabla2CpartialCoverpartialt=Dnabla2C對于最簡單的一維情況paitialCovei

5、partialt=Dpai1ial2Coveipailial2xpartialCoverpartialt=Dpartial2Coverpartial2x當xin-a?axin-azaffif,其邊界條件為beginsplitJ(a)&=DpartialCoverpartialx|_x=a=KC(a)C(infty)J(-a)&=-DpartialCoverpartialx|_x=-a=-KC(-a)-C(infty)endsplit|beginsplitJ(a)&=DpartialCoverpartialx|_x=a=KC(a)-C(infty)J(-a)&=D

6、partialCoverpartialx|_x=-a=-KC(-a)-C(infty)endsplit其中K為交換系數(shù)。設(shè)初始濃度C(x,O)=CoC(x,O)=CO)利用本征函數(shù)展開丿可將方程的解可寫為C(x,t)-CooverC(infty)-Co=l-sumlimitSHiftyi=i2Lalphacos(alphaix/a)over(alpha2i+Lalpha2+Lalpha)cosalphaiexp(-tovei'taui)C(x,t)-C0overC(infty)-C0=l-sumlimitsi=linfty2Lalphacos(alphaix/a)over(alphai

7、2+Lalpha2+Lalpha)cosalphaiexp(-tovertaui)時間常數(shù)taiii=a2oveiDalpha2itaui=a2overDalphai2z變量alphaalpha滿足方程alphaitan(alphai)=Lalpha=aKovei,Dalphaitan(alphai)二Lalpha=aKoverD實際測量時得到的是濃度的平均值，所以需要求出整個區(qū)間上濃度的平均値beginsplitbarC(x,t)-C_OoverC(infty)-C_0&=1-1over2aintlimitsaAasumlimits_i=lAinfty2L_alphacos(alph

8、a_ix/a)over(alphaA2+L_alphaA2+L_alpha)cosalpha_iexp(-tovertau_i)dx&=1-1over2asumlimits_i=lAinfty2L_alphaover(alpha_iA2+L_alphaA2+L_alpha)cosalphaexp(tovertau)&quadquadintlimits_aAacos(alpha_ix/a)dx&=1-sumlimits_i=lAinfty2L_alphaA2overalpha_iA2(alpha_iA2+L_alphaA2+Lalpha)exp(-tovertau)end

9、splity|beginsplitbaC(x,t)-C_0overC(infty)-C_0&=1-1over2aintlimits_aAasumlimits_i=1Ainfty2L_alphacos(alpha_ix/a)over(alpha_iA2+L_alphaA2+L_alpha)cosalphaJexp(-tovertau_i)dx&=11over2asumlimits_i=lAinfty2L_alphaover(alpha_iA2+L_alphaA2+L_alpha)cosalpha_iexp(-tovertau_i)&quadquadintlimits_aA

10、acos(alpha_ix/a)dx&=1-sumlimits_i=lAinfty2L_alphaA2overalpha_iA2(alpha人2+L_alphaA2+L_alpha)exp(-tovertauj)endsplit上式有兩種極限情形:1.KllD/a9Lalphato0KIID/azLalphato07此時tauitaui增長很快'只取第一項，alphaitanalphaisimeqalpha2i=Lalphaalphaltanalphalsimeqalphal2=Lalphaz故|beginsplitbarC(x,t)C_0overC(infty)C_O&

11、;=1-2L_alphaA2overalpha_lA2(alpha_lA2+L_alphaA2+L_alpha)exp(-tovertau_l)&=l-exp(-Ktovera)endsplit|beginsplitbaC(x,t)-C_0overC(infty)-C_0&=2I_alpha人2overalpha_lA2(alpha_lA24-alpha人2+L_alpha)exp(-tovertau_l)&=l-exp(-Ktovera)endsplit2.KggD/a?LalphatoinftyKggD/azLalphatoinfty,此時alphai=(i+lov

12、er2)pialphai=(i+lover2)pi,barC(x,t)-CooveiC(infty)-Co=1-8overpi2sumlimitSHiftyi=o1over(2i+l)2exp(-(2i+l)2pi2Dtover4a2)barC(xzt)-C0overC(infty)-C0=l-8overpi2sumlimitsi=0inftylove(2i+l)2exp(-(2i+!)2pi2Dtover4a2)按這兩種極限情況進行數(shù)據(jù)處理都無法得到氧交換系數(shù)K,還需借助其他的關(guān)系式.上面的方程可以推廣到三維，設(shè)三維區(qū)間長寬高分別為2可2®2c,則barC(x,t)-CooveiC

13、(infty)-Co=1-sumlimitsmftyi=1sumlimitsmftyj=isumliniitSHiftyk=iAijkexp(-tovertauijk)barC(x,t)-COoverC(infty)-CO=l-sumlimitsi=linftysumlimitsj=linftysumlimitsk=1inftyAijkexp(-tovertauijk)beginsplitA_ijk&=8L_alphaA2L_betaA2L_gammaA2overalpha人2betaA2gammakA2(alpha2+Lalpha人2+Lalpha)(beta2+L_beta人2+L

14、_beta)(gammaA2+L_gamma人2+L_gamma)1overtau_ijk&=1overtau_i+lovertauj+lovertau_kends|plit|beginsplitA_ijk&=8L_alphaA2L_betaA2L_gammaA2overalpha人2betajA2gamma_kA2(alpha2+L_alpha人2+L_alpha)(betaJA2+L_betaA2+L_beta)(gamma_iA2+L_gammaA2+L_gamma)1overtau_ijk&=1overtau+1overtauj+lovertau_kendspl

15、ittaui=a2overDalplia2i?taiij=b2overDbeta2j?tauk=c2overDganmia2ktaui=a2overDalphai2ztauj=b2overDbetaj2ztauk=c2overDgammak2alphaitan(alphai)=Lalpha=aKoverD,betajtan(betaj)=Lbeta=bKoverD,ganmiaktan(ganmiak)=Lganmia=cKoverDalphaitan(alphai)=Lalpha=aKoverDzbetajtan(betaj)=Lbeta=bKoverD,gammaktan(gammak)二

16、Lgamma二cKoveD實驗：測量條狀測試樣品的尺寸。用四探針電導法測定長方體樣品的電導率sigmaosigmaO撚后瞬間改變樣品所在氣氛的氧分壓（比如從0,21atm變到O.Olatm）并記錄固定時間間隔（如10s）處的電導率值sigmatsigmato當重新達到平衡后丿記錄其電導率值sigmainftysigmainfty<,實驗結(jié)果與數(shù)據(jù)處理:利用所測signiao?signiainfty,(t?sigmat)sigmaOzsigmainfty,(t,sigmat)-jf：S(t)=sigmao-sigmainftyoversigmat-sigmainftyS(t)=sigmaO

17、-sigmainftyoversigmat-sigmainfty,對S(t)S(t)進行最小二乘擬合'便可求出氧表面交換系數(shù)K和體相擴散系數(shù)Do這是一個典型的非線性擬合問題利用MatLab編寫代碼時須注意alpha?beta?ganmiaalphazbeta,gamma需要在每一步擬合時進行求解丿這可借助于fsolve函數(shù)實現(xiàn)。F面是某次實驗所得數(shù)據(jù)及擬合結(jié)果。t/sSexpSealScal-Sexp0.000.0000000000.0000137400.00001374010.000.2727631840.2760196180.00325643420.000.4309697810.4

18、354180720.00444829130.000.5510566860.549928006-0.00112868040.000.6434920010.637478253-0.00601374850.000.7039304760.7063251420.00239466660.000.7619988150.761261325-0.00073749070.000.8159194150.805470105-0.01044931080.000.8453486080.841235939-0.00411266990.000.8779379810.870273973-0.007664008100.000.8

19、868259920.8939079680.007081976110.000.9097373100.9131778370.003440527120.000.9245506620.9289101610.004359499130.000.9310685360.9417672500.010698714140.000.9510171830.9522827860.001265603150.000.9553624330.9608885380.005526105160.000.9703732960.967934862-0.002438434170.000.9723484100.9737066810.00135

20、8271180.000.9691882280.9784361010.009247873190.000.9774837050.9823124570.004828752200.000.9940746590.985490353-0.008584306210.000.9853841600.9880961460.002711986220.000.9869642500.9902331790.003268929230.000.9980248860.991986016-0.006038870240.000.9964447960.993423897-0.003020899250.000.9940746590.9

21、946035280.000528869260.000.9917045230.9955713720.003866849270.001.0003950230.996365509-0.004029514參考代碼:12345678K*-Afunctiontofitl(andDoFECRJicunLIJerkidnQ2013-08-19:-DemoFunctionECROclearall;clc678901111122globalahcglobalNdattexpSexpSealNIaua=0.71/2;b=0.25/2;c=0Ji/2;%shouldbe1/2ofthesanplesizeK0=3.3

22、E-3;D0=1.3E-h;%initialualueofKD>NlGUJie；%naximumnumberfortheSUM,shoulebelarge>texp,Sexp-textread('Sexp.dat','%F')；>Ndat-size(texp);>Seal-zeros(Hdat);>LB-0,0；UB=inf,inf;lsqoptions=optinset(*lsqnonlin');lsqoptlons.TolFun=le-12；lsqoptlons.Tolx=le-12;lsqoptrions.Displa

23、y=iter-:lsqoptrionsLR=on;enoughNATLABdefaultIsle-6-defaultisle-6X-off/final*:dis卩l(xiāng)ayeacnIteratlonlsqoptlons.LlnASparchTppp=quadcunic;or*cublcpoly*lsqoptions.NaxFunEuals=-2000;9o123h-52332233PO=KO/DODO;fittedparametersareK/DandDPfit=lsqnonlin(Q(P)ObjFun(P),PO,LB9UBylsqoptions);>Fprintf('8.12fn

24、D8.12Fn-,PFit(1)*Pfit(2),Pfit(2);6789012333344A-參if?%entl>FileID-FopenCSfit.dat1/w-);Fprintf(fileID,%8s%14s14s%145nB,Bf/Sexp*/Seal',1SealSexpB);>fori-1:Hdat;7fprintf(filelD.'塔82F缶14.9F先仙9f,r.texp(i),Sexp(i).ScaL(i),Scal(i)-Sexp(i);endplot(texp.Sexp.o*ttexp.Scal);functionObjFun=-ObjFun(P)iglobalabc1globalLaLbLc1globalNdattexpSexpSealMleu>Kd-P(1);D-P(2);>La=a*Kd;Lb=b*Kd;Lc=c*Kd;58596061626364656667686970i=1:Nleu;X6=i-0.9fi-0.9,l-0.9*pi；Xsol=fsolve(QxTan,X09.H>loptinset('ftlgorithn','