analytical solution of the pnp equations in the linear regime at an applied dc-voltage

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THE JOURNAL OF CHEMICAL PHYSICS 134, 154902 (2011)

Analytical solution of the PoissonNernstPlanck equations inthe linear regime at an applied dc-voltage

Anatoly Golovneva) and Steffen Trimperb)Institute of Physics, Martin-Luther-University, D-06120 Halle, Germany(Received 1 December 2010; accepted 24 March 2011; published online 15 April 2011)

The analytical solution of the PoissonNernstPlanck equations is found in the linear regime asresponse to a dc-voltage. In deriving the results a new approach is suggested, which allows to fulfillall initial and boundary conditions and guarantees the absence of Faradaic processes explicitly. Weobtain the spatiotemporal distribution of the electric field and the concentration of the charge carriersvalid in the whole time interval and for an arbitrary initial concentration of ions. A different behaviorin the short- and the long-time regime is observed. The crossover between these regimes is estimated. 2011 American Institute of Physics. [doi:10.1063/1.3580288]

I. INTRODUCTION

Despite polymer electrolytes are used very widely inthe industry, they remain also of interest for discussing fun-damental problems in nonequilibrium statistics. Especially,electrolytes offer an example of a driven system: the diffu-sion of charged particles is influenced by external and in-ternal fields. Such a coupled process is described by thePoissonNernstPlanck (PNP) equations which were formu-lated a long time ago, but until now their exact solution ismissing. The progress has been achieved mainly by numericalsimulations, such as presented in Refs. 1 and 2, using differentboundary conditions3 and different geometries.4 Much effortand different assumptions have been made to speed up the nu-merical calculations.5 The steady state of the PNP equationshas been studied as well in Ref. 6.

The analytical treatment of the dynamical PNP approachis rather complex due to the nonlinearity of the underlyingequations. As discussed in Ref. 7, the exact steady state solu-tion is given in terms of Jacobi elliptic functions. Therefore,it seems to be reasonable that the dynamical solution shouldbe expressed via these functions as well. To avoid difficultiesof dealing with such complicated functions, only the linearregime is considered, see for instance Refs. 8 and 9. The linearregime of the fully nonlinear dynamical equations is charac-terized by small deviations of the particle concentration fromits initial value. Insofar, the results obtained within the linearapproximation should be able to describe the whole nonlinearregime adequately in the initial stage of the evolution.

From the theoretical point of view, the spatiotemporalelectric field distribution and the particle concentrations areof interest, although a direct measurement of these quantitiesseems to be laborious and complicated. For biological sys-tems the concentration and the electric field profiles could beof particular importance as stressed in Ref. 10. In contrastto that, the external current is of particular interest for elec-trolytes. This current is generated in the external circuit by

a)Electronic mail: [email protected])Electronic mail: [email protected].

the motion of ions in the space between the electrodes andcan be extracted from experimental data. Consequently, in or-der to compare theoretical results with experimental obser-vations, one has to calculate the external current. Recently,the problem was considered in detail in a plane geometry forsymmetric binary electrolyte.11 According to that paper thebehavior of the system can be generally influenced by twoparameters: the applied voltage v and the total charge Qtot.This fact suggests to consider four limiting cases: (i) the dif-fusion limited case where both v and Qtot are small, (ii) thedouble layer limited case where Qtot is large and v is small,(iii) the geometry limited case where Qtot is small and v islarge, and finally (iv) the space charge limited case where bothv and Qtot are large. In the regime dominated by a large volt-age v the diffusion is completely suppressed, and therefore itis denoted as the regime without diffusion. Generally, the PNPequations can be simplified in each regime drastically, whichallows under additional assumptions to get explicit analyticalsolutions. The optimal facility to estimate or to check the va-lidity of the assumptions made is to compare the results withan exact solution if available. Another problem concerns thecrossover between the different regimes. In the present paperfor example we are confronted with the case that there appeartwo different solutions for a high and a low initial concentra-tions, respectively. Using our solution we are able to detectwhich of the solutions exhibit the correct crossover behavior.

The main purpose of our paper is to find the solution ofthe PNP equations, which fulfills all initial and boundary con-ditions. To that aim we offer a new approach to treat the PNPequations and solve this set of equations in case of an applieddc-voltage. Especially, we demonstrate that the discrepancyin the time constant obtained experimentally and numerically,see Ref. 12 or alternatively, by solving the linearized PNPequations13 can be traced back that the absence of Faradaicprocesses was not taken into account. With other words, thisadditional constraint was disregarded sufficiently in the be-fore cited theoretical papers. In the present one we considerthe restrictions posed by avoiding of Faradaic processes andshow how this fact can be included in the mathematical for-mulation of the problem. For a detailed discussion of the

0021-9606/2011/134(15)/154902/6/$30.00 2011 American Institute of Physics134, 154902-1

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154902-2 A. Golovnev and S. Trimper J. Chem. Phys. 134, 154902 (2011)

influence of Faradaic processes we refer to Ref. 14. Our analy-sis is concentrated on the linear regime which is characterizedby a low applied voltage v . Otherwise, our calculation is validfor arbitrary initial concentrations of the ions. Furthermore,no limitations are imposed for the time regime.

II. POISSONNERNSTPLANCK EQUATION INTHE LINEAR REGIME

To be specific we consider a symmetric binary electrolyteplaced between two flat infinite electrodes located at x = L .The charge of each ion is ze, where z is the valence of theion and e is the elementary charge. The set of PNP equationsreads

C(x, t) = F (x, t)

D(

C (x, t) ze

kB TC(x, t)(x, t)

),

(x, t) = ze (C+(x, t) C(x, t)) . (1)Here C(x, t) are the concentrations of the positive and neg-ative ions, (x, t) is the potential related to the electric field,F(x, t) are the fluxes of the ions, is the permittivity of themedium, T is the temperature, and kB is the Boltzmann con-stant. The prime denotes the spatial derivative whereas the dotstands for the time derivative. The first two equations describethe transport of the positive and negative ions, the last equa-tion is the Poisson equation. As it was shown in Ref. 13, thePNP are simplified in the linear regime and read

C(x, t) = D(

C (x, t) ze

kB T(x, t)

),

(x, t) = ze (C+(x, t) C(x, t)) . (2)The linear regime is characterized by assuming a small de-viation of the concentration field from its initial value = C+(x, t = 0) = C(x, t = 0). Introducing the charge con-centration (x, t),

(x, t) ze (C+(x, t) C(x, t)) , (3)and summing the transport equations, Eq. (3) is reduced to asingle equation,

(x, t) = D( (x, t) 2(x, t)), (4)where is the inverse Debye screening length D ,

=

2z2e2

kB T. (5)

Equation (4) is called the DebyeFalkenhagen equation. Thesolution of that equation leads immediately to the particleconcentrations according to

C(x, t) = (x, t)2ze . (6)

Some properties of the solution within the linear regime hadbeen already discussed in Ref. 13. The main problem occur-ring is that the charging time is in conflict with experimentaldata. This contradiction arises from the fact that one of theboundary conditions, naturally realized in the experiment, is

not taken into account. To illustrate this fact let us remindthat the system is locked between two electrodes. Becauseso-called Faradaic processes are neglected, the particles arenot allowed to move through the electrodes. This is the miss-ing constraint. The particle fluxes should tend to zero at theelectrodes, i.e., F(x = L , t) = 0. This additional condi-tion makes the problem more difficult, especially from themathematical point of view. So, we should clarify this pointin more detail. Whereas the basic equation is formulated forthe charge density (x, t), see Eq. (4), the boundary conditionis related to the potential (x, t),

(x = L , t) (x = L , t) = v . (7)This condition means the applied voltage is constant andequal to v . The additional constraint, imposed on the particlefluxes F(x, t), reads

F(x = L , t) = 0. (8)This condition is not exactly fulfilled by the solution pre-sented in Ref. 13. Mathematically, one has to deal with threedifferent functions, the charge density (x, t), the potential(x, t), and the particle fluxes F(x, t), where both F(x, t)and (x, t) are related to (x, t) in a nontrivial manner in-volving an integration. In that manner the constraint is for-mulated as an integro-differential equation which complicatesthe problem drastically. To avoid such a situation we proposeanother way, namely, by incorporating the constraint for theflux Eq. (8) into the initial conditions. In other words, the ba-sic equation Eq. (4) is reformulated in such a manner thatEq. (8) is satisfied automatically. To that purpose let us con-sider the flux F(x, t),

F(x, t) = DC (x, t) ze(x, t), (9)where is mobility of ions which is related to the diffusiv-ity via the Einstein relation D = kB T . Using Eq. (3) onecan express the particle concentrations C(x, t) by the chargedensity (x, t), what alters Eq. (9) into

F(x, t) = D( 1

2ze (x, t) ze

kB TE(x, t)

). (10)

Here E(x, t) is the electric field which can be expressed by thepotential due to E(x, t) = (x, t). According to the Pois-son equation E(x, t) is related to the charge density (x, t).In terms of the function y(x, t), defined by the relation,

y(x, t) =

(x, t)dx,

the Poisson equation reads

E(x, t) = y(x, t) + f (t). (11)By definition, the function y(x, t) is a symmetric oney(x, t) = y(x, t) and satisfies the DebyeFalkenhagenequation,

y(x, t) = D( y(x, t) 2 y(x, t)). (12)Inserting y(x, t) from Eq. (11) into Eq. (10) we get

F(x, t) = D2ze y(x, t) Dze

kB T[y(x, t) + f (t)].

(13)


154902-3 Solution of the PNP equations J. Chem. Phys. 134, 154902 (2011)

Using Eq. (12) it results

F(x, t) = D2ze (y(x, t) 2 f (t)). (14)

In Ref. 13 the function f (t) was considered as an arbitraryone. In the present paper this function will be fixed by thecondition F(x = L , t) = 0 leading to

f (t) = y(x = L , t)2

. (15)

As long as the function y(x, t) is obtained, the charge densityand the electric field are found according to

(x, t) = y(x, t) (16)and

E(x, t) = y(x, t) + 2 y(x = L , t). (17)Hence, the remaining task is to solve Eq. (12). The boundarycondition follows immediately from Eq. (17), L

Ly(x, t)dx + 2L2 y(x = L , t) = v . (18)

Now the constraint Eq. (8) is fulfilled automatically.

III. SOLUTION OF THE PNP EQUATIONSIn this section we discuss the solution of the reformu-

lated PNP equations. Because Eq. (18) is sufficiently com-plex, it seems to be appropriate to reformulate the problem interms of the electric field. Inserting y(x, t) from Eq. (17) intoEq. (12) it follows:

E(x, t) 2 y(x = L , t)= D(E (x, t) 2 E(x, t) + y(x = L , t)). (19)

Hereafter, for the sake of simplicity of the notation the time isscaled by t tD . This allows to eliminate the diffusion coef-ficient from the equations. Consequently, the time derivativechanges its dimension. The final result will be given in termsof the unscaled time and the diffusion coefficient will appearexplicitly. According to Eq. (17), Eq. (19) turns into

E(x, t) = E (x, t) 2 E(x, t) + E(x = L , t), (20)which is the basic equation of the reformulated problem. Forx = L one gets

E (x = L , t) = 2 E(x = L , t). (21)Because E(x, t) is a symmetric function, one needs to con-sider the electric field only at one point, for instance x = L .Equation (21) is valid at every time and can be solved by theseparation of variables. Because Eq. (21) is a second-orderdifferential equation, its solution includes two arbitrary func-tions of time denoted as E1(t) and E2(t). The general solutionof Eq. (21) reads

E(x = L , t) = E1(t) exp((x + L))+ E2(t) exp((x L)), (22)

where the first term on the right-hand side represents the influ-ence of the left electrode, and the second term represents the

influence of the right electrode. Notice that the further calcu-lations are not restricted to small . In case of L 1, onlyone of these terms is relevant which simplifies the calculationonly marginally. Remark that the notation used in the last twoequations should be interpreted in the following sense. Thefunction E(x = L , t) is considered in a very small vicinityof the point x = L , the size of which is not important, as itwill be shown below.

The basic equation, Eq. (20), is supplemented by the fol-lowing conditions: L

LE(x, t)dx = v, (23)

E(x, t = 0) = v2L

, (24)

E(x, t = ) = v2

cosh(x)sinh(L) . (25)

Equation (23) coincides with Eq. (7). Equation (24) followsfrom Eq. (23) and the fact that at the initial moment thecharge distribution is homogeneous and the electric field isconstant. Equation (25) represents the steady state solution.For a discussion of the steady state solution see also Ref. 7.The prefactor in Eq. (25) is chosen in order to satisfy Eq. (23).Using Eq. (23) and integrating Eq. (20) from L to L onefinds

0 = L

LE (x, t)dx 2v + 2L E(x = L , t) (26)

or

E (x = L , t) + L E(x = L , t) = 2v

2. (27)

Inserting Eq. (22) in the last equation yields[ E1(t) E2(t) exp(2L)]

+ L[ E1(t) + E2(t) exp(2L)] = 2v

2. (28)

Notice that this equation is valid only at the point x = L .The further solution is searched in the form E1,2(t) = B1,2 A1,2 exp(t). This leads to four equations,

B1 B2 exp(2L) = v2 , A2 exp(2L) + L A1 + L A2 exp(2L) A1 = 0,B1 A1 + (B2 A2) exp(2L) = v2L ,

B1 + B2 exp(2L) = v2 coth(L). (29)This system is solved easily. Taking into account the initialcondition, Eq. (24), and using Eq. (22) one can formulate theboundary condition for the electric field as

E(x = L , t) = v2

( coth(L)

A exp(t)) , with = L

(30)



and

A = L coth(L) 1L

. (31)

Equation (30) reflects the condition imposed by Eq. (23). Thisresult demonstrates clearly one of the advantages of our treat-ment, namely, that the quite complicated constraint given byan integro-differential equation, Eq. (18), is reduced to thesimple exponential boundary condition of Eq. (30).

In the discussion after Eq. (22) it was stated that thefunction E(x = L , t) should be considered in a small vicin-ity of the point x = L , whose size is not important. Thisstatement is now justified by the fact that the final result ofEq. (30) is formulated exclusively at the point x = L anddoes not depend on the coordinate.

In a view of Eq. (30), the basic equation Eq. (20) can berewritten as

E(x, t) = E (x, t) 2 E(x, t) + v2A exp(t).

(32)In this equation the inhomogeneity represents the initialconditions.

To proceed it is convenient to introduce a new functionw(x, t) according to

w(x, t) = L coth(L) 1L(L 1) exp(t) +

cosh(x)sinh(L)

2v

E(x, t). (33)

The function w(x, t) has to satisfy following equations andconditions:

w(x, t) = w (x, t) 2w(x, t) with

w(x, t = 0) = cosh(x)sinh(L) +

coth(L) 1L 1 ,

w(x, t = ) = 0, (34)

w(x = L , t) = L coth(L) 1L 1 exp(t).

Performing Laplace transformation w(x, t) w(x, z) onegets

w (x, z) 2w(x, z) = zw(x, z) cosh(x)sinh(L)

coth(L) 1L 1 (35)

w(x = L , z) = L coth(L) 1L 1

1z + .

The general symmetric solution of this set of equationsreads

w(x, z) = cosh(x)z sinh(L) +

coth(L) 1(L 1)(2 + z)

[coth(L)

z

+ coth(L) 1(L 1)(2 + z) L coth(L) 1(L 1)(z + )

]

cosh(

2 + zx)cosh(2 + zL) . (36)

FIG. 1. Contour of integration.

To invert this result one has to take the following integral:

w(x, t) = 12 i

a+iai

exp(zt)w(x, z)dz, (37)

where the real number a is sufficiently large so that all poles ofw(x, z) lie on the left-hand side from the line along which theintegral is taken. The integral will be evaluated in the frame oftheory of functions of a complex variable. Following the con-ventional procedure we extend the variable z onto the wholecomplex plane and choose the contour of integration in a wayto enclose all poles as shown in Fig. 1. The integral over thewhole contour is a sum of the integral over the arc C , whichtends to zero, and the integral along the vertical line, whichwe are interested in. According to the Cauchy residue theo-rem the integral over the whole contour is proportional tothe sum of all residues lying within the contour. To find thepoles of w(x, z) one has to find the zeros of the denominatorof the right-hand side of Eq. (36). Three poles are obvious:z = 0, z = 2, and z = . The other zeros are determinedby the equation,

cosh(

2 + zL) = 0, (38)which leads to

2 + zL = i (2 + n) , where n is the inte-

ger running from to . Hence the corresponding polesare

zn = 1L2(

2+ n

)2 2 + i0.

The poles are located on the negative part of the real axis asdepicted in Fig. 1, but n should run from 0 to , otherwisesome poles would be counted twice. The expansion of the hy-perbolic cosine in the vicinity of the point zn yields

cosh(

2 + zL) (1)n L2

(1 + 2n) (z zn) + . (39)

This expansion allows the integration of w(x, t) and per-forming the inverse Laplace transformation. The first termof w(x, t) leads to the Heaviside step function which will be


154902-5 Solution of the PNP equations J. Chem. Phys. 134, 154902 (2011)

omitted because we consider only t > 0. Collecting the con-tributions from all terms of w(x, t) and returning to unscaledvariables we get finally for the electric field,

E(x, t) = v2

{cosh(x)sinh(L)

L coth(L) 1L 1

(

cosh(2 x)cosh(2 L)

1L

)eDt

e2 Dt S(x, t)}

. (40)

Here we have introduced

S(x, t) =

n=0(1)n exp

(

2

4L2(1 + 2n)2 Dt

)

cos(

2L(1 + 2n)x

)(1 + 2n)

{coth(L)

P2n + 2L2

+ coth(L)1P2n (L1)

+ L coth(L)1(L1)(L2L2P2n )}

,

(41)with Pn = 2 + n. From Eq. (6) and the Poisson equationwe end up with the particle concentrations,

C(x, t)

= 1 v2

ze

kB T

{sinh(x)sinh(L)

L coth(L) 1L 1

1 1L

sinh(2 x)cosh(2 L)e

Dt

+

e2 Dt S(x, t)

}. (42)

Equations (40)(42) are the solution of the PNP equations inthe linear regime when a dc-voltage is applied.

IV. DISCUSSION

In this section we want to discuss the results for the elec-tric field and for the particle concentration in detail. The ex-pression for the electric field, Eq. (40), consists of two partscomprising the first two terms on the right-hand side and anadditional part containing the sum S(x, t). As demonstratedbelow both terms are relevant on different time scales denotedas = LDD and 1 =

2DD , respectively. The same statement is

valid for the concentrations. The solutions are characterizedby a set of parameters, namely, the initial concentration ofthe ions , the diffusion coefficient D, the temperature T , andthe valence of the ions z. Introducing dimensionless variablesC(x, t) C(x,t) , t Dt , v v zekB T these parameters canbe scaled off, compare also Ref. 13. The remaining depen-dence on the applied voltage v becomes quite trivial: it entersboth Eqs. (40) and (42) only as a prefactor. Therefore, chang-ing of the applied voltage claims only a rescaling of the axis.The same is true for the initial concentration which appearsonly on the left-hand side of Eq. (42). As the consequence theDebey screening length D = 1 and the size of the systemL have a strong impact on the evolution. As stressed in the

Introduction and following,11 one has to distinguish four lim-iting cases with different values of those parameters. Becauseour results are valid in the linear regime, we discuss the casescharacterized by small applied voltage. They are denoted asdouble layer limited case with L D and diffusion limitedcase with L D . To discriminate these two cases the nor-malized total charge Qtot = 4L is introduced, whereas thefactor 4 is due to the fact that L is a half of the distance be-tween the electrodes. Let us start with the case L 1. Inthis limit 1, and therefore the term with the sum S(x, t)in Eqs. (40) and (42) can be neglected. The first two terms inEq. (40) can be further simplified and then they will coincidewith the solution found in Ref. 11. However in that case thesolution does not satisfy the initial condition. Therefore thissolution is valid only in the long time limit as it was found inRef. 8. According to our findings this long-time regime startsactually immediately after the short time 1.

In the opposite case L 1 the solution is dominated bythe sum term S(x, t). In Ref. 11 the solution is expressed viaa different sum. We performed the calculations for L = 0.5and L = 0.2. In both cases the agreement with the resultspresented in Ref. 11 is quite well. Generally in both limit-ing cases, L 1 and L 1, our results coincide withthose given in Ref. 11. Different to Ref. 11, we have madeno assumptions concerning the efficiency of the screening ofthe electric field. Therefore our results are also able to de-scribe the crossover between the limiting cases. ChangingL the charge profile changes smoothly everywhere apartfrom one point. For L > 1 the charge excess close to oneelectrode and hence the charge deficiency close to the otherone spreads into the bulk, but the mid-plane concentrationC(x 0, t) remains constant. For L < 1 the charge excessand the charge deficiency are not well separated and over-lap each other making the charge profile more complicated.For L = 1 there is a singularity of the electric field. Whilethe sum term in Eq. (40) is regular; the second term givesrise to the singularity due to the prefactor (L 1)1. Thecondition in Eq. (23) is definitely violated if E(x, t) .The physical situation behind this singular point deserves fur-ther analysis. Otherwise the special point L = 1 is charac-terized that the thickness of the system is accurately 2D . For

FIG. 2. Normalized concentration C0 = C+(s, t)/ near the electrode. = 1 107 m1, L = 5 105 m, s is a distance from the electrode.



that case the charge excess and the charge deficiency meeteach other exactly at the mid-plane. Notice that according toRef. 11 the mentioned point corresponds to Qtot = 4 and of-fers no special behavior.

Another interesting feature can be observed if L > 1,but not sufficiently larger. In this case the long- and the short-time regimes are not very well separated due to 1 . Atthe beginning of the time evolution there occur oscillationsof the charge concentration as it is shown in Fig. 2. Di-rectly after switching on the voltage, the particles start toform a narrow layer close to the electrode. Since the parti-cles situated within the bulk reach the electrodes after a de-lay time, the charge profile becomes nonmonotonous. Whenthe time proceeds, the profile tends to an exponential form.The crossover from the short-time regime to the long-timeregime means that the function becomes monotonous. Thehigher the L is, the less pronounced the oscillations appear.A further problem is to find the external current. Although ac-cording to Ramos theorem, see Ref. 15, there is the relationJ (t) = E(x = L , t), it is a nontrivial problem to extracta transparent expression for the external current. The maindifficulties come from the sum term S(x, t) because one can-not interchange differentiation and summation.

Let us summarize that the main result of our paper wasto present a solution of the dynamical PNP equations in thelinear regime taking into account all initial and boundary con-ditions exactly. In previous papers on this subject,8, 9, 11, 13 thecondition imposed by Eq. (8) was not satisfied explicitly. Inthe present paper we have demonstrated in detail how onecan ensure that the flux satisfies the condition F(x = L , t)= 0.

ACKNOWLEDGMENTS

One of us (A.G.) acknowledges support by the Interna-tional Max Planck Research School for Science and Tech-nology of Nanostructures in Halle, Germany. We benefitfrom discussions with Thomas Thurn-Albrecht (UniversityHalle).

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analytical solution of the pnp equations in the linear regime at an applied dc-voltage

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