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THENUMERICALSOLUTIONOF

THETIME-DEPENDENT

NERNST-PLANCKEQUATIONS

H.COHENandJ.W.COOLEY

1.INTRODUCTION

145brought to you by COREView metadata, citation and similar papers at core.ac.ukprovided by Elsevier - Publisher Connector

theseresultsinthespiritofprovidingamorecompleteviewof thetime-dependent characteroftheNernst-Plancksystem. canbedonenumericallywithoutdifficulty. themembraneduringthistransition? inAppendix1): i[Fd(1.1) (1.2) ax acj*=--a$-(1.3)ata e47O2(1.4)

BIOPHYsIcALJOURNALVOLUME51965146

renttobe

I=F(J-E-5)+E2atIO"M(1.5)

thelasttermbeingthedisplacementcurrent.2 Using

U=,V=,zj(1.6)

ii e109aaE=ZFE(U+P)+RTc(U-F).(1.7)4762aiax i=_[i/axcC*EJ(1.1)'

E=-d(1.2)'ax

dCji_ac(.3)atax axp=S(C4-Cj)=-C-(1.4)' aEI-E(U+V)+a(UV)(1.7) =T.(1.8) where

T=E109

47CrFC(O)+/7cj

isanelectricrelaxationtimeand

Td=12pcI0RTIF

X=1.65x10-10.

2.THENUMERICALRESULTSFORK=0

C-=Z,C,++YjC-,thenforK=0

C=[C(1)-C(O)]x+C(O)(2.1)

E=GIC(2.2)

(x)=C(1)'C(O)nC(O)(2.3)

BIOPHYSICALJoURNALVOLUME51965148

E(x,t)=+dx()(2.4)u+vU+V

UandVaretheweightedmobilities

U=12N,+CN,++IAH+CH+(2.5)

V=1CI-Ccl-

inEisthenfoundfromequation(2.4)tobe

AIE(x,+0)-E(x,-0)=+;,(2.6)

whereU+Vhavethesteady-statevaluesatt=-0. -5

3fO1tg|JItI00.040n0AIQ

FiGuRElbConcentrationsforI(-O)x10-cm

=0.5,I(+O)=0amp/cmi,tinseconds.

Thesecondtermofequation(2.4),

a(U-V)El(x,t)=dx(2.7)U+v,(27

BIoPHYsIcALJOURNALVOLUME51965lSO

j326- 24\
.22- I-

54£

0.2~0.40.6.0

22
16 14120
M 10 al%6 4 2 0

Oo.t0.4M0.

XloomID

_,0*0.6.~~~~~~~~~~~I1o lSl obtainedbyintegratingequation(2.4),is -*(t)=I-R(t)+'I'mf(t)(2.8) where

R(t)=1dx(2.9)

'I'emf(t)=fEl(x,t)dx.(2.10) .14 tel~~ ~ ~~~~ma

0020.40A0iLW

BIOPHYSICALJOURNALVOLUMES1965152

uI23u tsec.

20temfi-emf

0- -20_ -40 -60 -80 100_

120_-*,

140
160_
180_
0.5

0i23012

tsec.I2FiGuRE5Membranevoltage-I',and

FIGURE6Membranevoltage-I',and'Irt

=f(U.-V.)/(U+V)dxinmillivolts,re- sponsetostep-functioncurrentI=0,-0.5,0 amp/cm2. 360
340
320
300
280
260
240
220
200
E180 160
140
120
100
80
60
40
20 0.5

00051-temf_IW*'emf

7,n-I,,

E-1 -1 -1 -1 -1

OLUtrloi-H

153
cases. exceptforthinsurfacelayers.

BIOPHYSICALJOURNALVOLUME51965154

APPENDIX1

NUMERICALVALUESOFCONSTANTSANDUNITS

t=lTd x=X/l

Cji=CZ(ciEi'/f+0

E/(RT)

I=10-2cm

2MNa.=0.000456CMvoltsec.

2I.H=0.003242ltM

2PuC,=0.0006i76CM=0.00676voltsec.

CNa(0)=10-4CNa(l)=10-5mol-ions/cm3

CH(0)=l0-5,C.(I)=10-4mol-ions/cm3

Ccl(o)=11X10-5,Cc(l)=11X10`5mol-ions/cm3

R=8.3703coul-volts/mol°K

T=288°K

RT/F=0.025volt

c=2.99776X1010cm/sec.oresu/abcoul

F=96,500coul/farad

e=79=dielectricconstantofwater

Mobility

Concentration

Electricpotential

Electricfield

Ionflux

Current:,c,-=0.676X10-3cm!/voltsec.

:C+(O)=11X10-5mol-ions/cm8 :RT/F=25mv :RT/Fl=2.5volts/cm :f=gCcC(O).(RT/FI)=18.59X10-8cmol-eon :a=fXF=0.01793935amp/cm2

ThevaluefoundforKiS

K=r,=1.6476X1010Td

APPENDIX2

C,O()=C-if(O),

theexpressionsfortheconcentrationsare where

Csi(x)=eX)Csio-ie*Zd'Ai0

*=;,i*(Cio-Ci*Ie*'t(xdx dC=FC*E=A*dx

A*=/(A.3)

(A.4)

BIOPHYSICALJouRNALVOLUME51965(A.1)

(A.2)CiI'=Ci'(1) 156
dC-KE=A++A--a(A.5)dxdxand

Kd72-CE=A+-A-=-G.(A.6)

G!a+j a _o+VI(U1-Vo)e*11(A.7)e*-1I

C(x)=a.x+3+KE,(A.8)

withboundaryconditionsonCgiving 2 =a-2(El-Eo2)

PM=Eo2,

7ThisrelationwasderivedbyGoldman(9).

0coE=Enu,+e-h(x)/,Eenv,.(A.10)

n-0n-0 \dhx/ax+(A.11) h+(O)=0(A.12) h-(l)=0,

000000

E=s6e"un+eA+(z)/Eenv+eh(X)/eeeWs(A.13)

n-0n-0n-0with h+(x)=3a[(ax+13)3/2_13/2] (A.14) h-(x)=-2[(aI+1)3/2_(ax+C)3/2] uO=G/(ax+1),(A.15)oforderc: ul=0,(A.16)oforderc': (d2uo/dx2)+(Eo2+2(-Eo3)x)uo--uo(A.17)

BIOPHYSICALJOURNALVOLUME51965158

dv1+d2h+/dx20dw,d2hjXd2(A20) dx2dh+/dx1=dx2dh-/dx withtheconditions =MFduo/dx=Gab-5/2Ldh+/dxj..o '()-[dhduoldxl-Ga(a+#)5/2LI)dh/dxj Thus,

Theasymptoticrepresentationthenbecomes

GGEo=,El=a!+

canbewritten -'\Gae-xfC(A.23)

LettingK=k/llandx=x/l,thiscanbewritten

_512e(A.24)

8/K=1.2835X10-7cm(A.25)

E 0.7 05 -00-0.40.6ID

APPENDIX3

NUMERICALMETHODSFORTRANSIENTSOLUTIONS

4ji=;Li((ciFC,*'EFCE')(A.26)

E=[I+U'-V']/(U+V)(A.27)where

n1+U=EjC(A.28) j-1

V=pi-C,-.(A.29)j-l

f'(x)=(f(xi+)-f(x--1))12h f"(x)=(f(x,.i)+f(x,+i)-2f(x,))/h2.

BIoPHysicALJOURNALVOLUME51965160

*(Xi)=-Cik(xi+)2Ci,1(x,)+Cjk*(xi-1)

Cjik+l(xi)=Cik,(x,)+bteCjk*(X,),

eik(xi)=12(Ck(Xi)-Cj+I*(X,)).(A.33)

Receivedforpublication,July28,1964.

REFERENCES

162BIOPHYSICALJOURNALVOLUME51965

quotesdbs_dbs17.pdfusesText_23

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