Original article Biomath 2 (2013), 1312302, 1–5

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 ISSN 1314-684X

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h t t p : / / w w w . b i o m a t h f o r u m . o r g / b i o m a t h / i n d e x . p h p / b i o m a t h / Biomath Forum

Multiscale Analysis of Ionic Transport in Periodic
Charged Media

Claudia Timofte
Faculty of Physics, University of Bucharest

Bucharest, Romania
claudiatimofte@yahoo.com

Received: 23 September 2013, accepted: 30 December 2013, published: 23 January 2014

Abstract—A macroscopic model for describing the ion
transport in periodic charged porous media is rigorously
derived. Our results can serve as a tool for biophysicists to
analyze the ion transport through protein channels. Also,
such a model is useful for describing the flow of electrons
and holes in a semiconductor device.

Keywords-homogenization; ion transport; the periodic
unfolding method.

I. INTRODUCTION AND SETTING OF THE PROBLEM

The goal of this paper is to obtain the effective beha-
vior of a system of coupled partial differential equations
appearing in the modeling of transport phenomena in
periodic charged porous media or in the modeling of
the flow of holes and electrons in semiconductors. Such
a system is known in the literature as the Poisson-
Nernst-Planck system, for the case of ion flow through
membrane channels, or as the van Roosbroeck model,
for the case of the transport of holes and electrons in
semiconductors. For the physical aspects behind these
models and for a review of the recent literature, we refer
to [8], [13], [14], [15], [16] and [19].

Let us briefly describe the geometry of the problem.
We assume, as it is customary in the literature, that
the porous medium has a periodic microstructure. More
precisely, for n ≥ 2, we consider a smooth bounded
connected open set Ω in Rn, with | ∂Ω |= 0. In what
follows, we shall only consider the natural cases n = 2
or n = 3. We assume that the unit cell Y = (0, 1)n

consists of two smooth parts, the fluid part Yf and,

respectively, the solid part Ys, which are supposed to
be open, nonempty and disjoint sets such that

Y = Yf ∪Ys

and
Yf ∩Ys = Γ.

We suppose that the solid part has a Lipschitz boundary
and does not intersect the boundary of the basic cell Y .
Therefore, the fluid zone is connected.

Let ε < 1 be a real parameter taking values in a
sequence of positive numbers converging to zero. For
each k ∈ Zn, let

Y ks = k + Ys

and
Kε = {k ∈ Zn | εY ks ⊆ Ω}.

We denote by
Ωsε =

⋃
k∈Kε

εY ks

the solid part, by

Ωε = Ω \ Ωsε

the fluid part and we set

θ =
∣∣Y \Ys∣∣ .

It is easy to see that if the solid part is not allowed
to reach the fixed exterior boundary of the domain Ω,
the intersection between the outer boundary ∂Ω and the

Citation: Claudia Timofte, Multiscale Analysis of Ionic Transport in Periodic Charged Media, Biomath 2
(2013), 1312302, http://dx.doi.org/10.11145/j.biomath.2013.12.302

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C Timofte, Multiscale Analysis of Ionic Transport in Periodic Charged Media

interior boundary Γε of the porous medium is empty,
i.e.:

Γε ∩∂Ω = ∅.

In such a periodic porous medium, we shall consider, at
the microscale, the Poisson-Nernst-Planck system, with
suitable boundary and initial conditions. The Nernst-
Planck equations, describing the diffusion in the fluid
phase, are coupled with the Poisson equation cha-
racterizing the electrical field which can influence the
ionic transfer. More precisely, if we denote by [0,T],
with T > 0, the time interval of interest, we shall
be interested in analyzing the asymptotic behavior, as
ε → 0, of the solution of the following system:


−∆ Φε = c+ε − c−ε + D in (0,T) × Ωε,
−∇Φε ·ν = εσεG(Φε) on (0,T) × Γε,
∇Φε ·ν = 0 on (0,T) ×∂Ω,
∂tc
±
ε −∇· (∇c±ε ± c±ε ∇Φε) = F±(c+ε ,c−ε )

in (0,T) × Ωε,
(∇c±ε ± c±ε ∇Φε) ·ν = 0 on (0,T) × Γε,
(∇c±ε ± c±ε ∇Φε) ·ν = 0 on (0,T) ×∂Ω,
c±ε (0,x) = c

±
0 (x) in Ωε.

(1)

Here, ν is the unit outward normal to Ωε, Φε represents
the electrostatic potential, c±ε are the concentrations of
the ions (or the density of electrons and holes in the
particular case of van Roosbroeck model), D ∈ L∞(Ω)
is the given doping profile, F± is a reaction term and
G is a nonlinear function which takes into account the
effect of the electrical double layer phenomenon arising
at the interface Γε.

We assume that

σε = σ
(x
ε

)
,

with σ(y) being a Y -periodic, bounded, smooth real
function such that σ(y) ≥ δ > 0, and G is a conti-
nuously differentiable function, monotonously increasing
and such that G(0) = 0. Also, we shall suppose that there
exist C ≥ 0 and r, with 0 ≤ r ≤ n/(n− 2) for n = 3
and 0 ≤ r < ∞ for n = 2, such that

| G′(s) |≤ C(1+ | s |r−1), ∀s ∈ R.

Let us notice that this hypothesis concerning the
smoothness of the nonlinearity G can be relaxed by using
a regularization technique, such as Yosida approximation
(see [18]). Also, the results of this paper can be obtained,
under our assumptions, without imposing any growth
condition (see [17]).

In practical applications, based on the Gouy-Chapman
theory, one can use the Grahame equation (see [2], [8]
and [9]) in which

G(s) = K1 sinh(K2s), K1,K2 > 0.

For the case of lower potentials, sinh(x) can be expanded
in a power series of the form

sinh(x) = x +
x3

3!
+ ...

and one can use the approximations sinh x ≈ x or
sinh x ≈ x + x3/3!.

Concerning the reaction terms, we deal, as in [13],
with the linear case in which

F±(c+ε ,c
−
ε ) = ∓(c

+
ε − c

−
ε ),

but we can also address by our techniques the more
general case in which

F±(c+ε ,c
−
ε ) = ∓(a

εc+ε − b
εc−ε ),

with
aε(x) = a

(x
ε

)
, bε(x) = b

(x
ε

)
,

where a(y) and b(y) are Y -periodic, bounded, smooth
real functions such that a(y) ≥ a0 > 0, b(y) ≥ b0 > 0.

For the case of other nonlinear reaction rates F± and
more general functions G, see [6], [10], [11] and [18].

We assume that the initial data are non-negative and
bounded independently of ε and∫

Ωε

(c+0 − c
−
0 + D)dx = ε

∫
Γε

σεG(Φε)ds. (2)

We also assume that the potential Φε has zero mean value
in Ωε.

Let us mention that, for simplifying the notation, we
have suppressed in system (1) some constant physical
relevant parameters.

We consider here only two oppositely charged species,
i.e. positively and negatively charged particles, with
concentrations c±ε , but all the results can be generalized
for the case of N species.

For the case in which we consider different scalings
in (1), see [13] and [18]. Also, let us remark that we
can treat the case in which the electrostatic potential is
defined all over the domain Ω, with suitable transmission
conditions at the interface Γε (see, for instance, [8] or
[16]).

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C Timofte, Multiscale Analysis of Ionic Transport in Periodic Charged Media

From the Nernst-Planck equation, it is easy to see that
the total mass

M =

∫
Ωε

(c+ε + c
−
ε )dx

is conserved and suitable physical equilibrium conditions
hold true, at the microscale and, also, at the macroscale
(see, for details, [8] and [18]).

Using similar arguments as in [8] or [13], we can
prove the well posedness of problem (1) in suitable func-
tion spaces and we can obtain proper energy estimates.

The high complexity of the geometry and of the go-
verning equations implies that an asymptotic procedure
becomes necessary for describing the solution of such a
problem.

Using the periodic unfolding method recently intro-
duced by D. Cioranescu, A. Damlamian, G. Griso, P.
Donato and R. Zaki (see [3], [5] and [4]), we can
prove that the asymptotic behavior of the solution of
our problem is governed by a new coupled system of
equations (see (3)-(5)). In particular, the evolution of the
macroscopic electrostatic potential is governed by a new
law, similar to Grahame’s law (see [8] and [9]).

An advantage of this approach is that we can avoid
the use of extension operators and, therefore, we can deal
in a rigorous manner with media which are less regular
than those usually considered in the literature (composite
materials and biological tissues are highly heterogeneous
media with not very smooth interfaces, in general).

Similar problems have been considered, using diffe-
rent techniques, in [8], [13] or [15]. As already men-
tioned, our approach is based on a new method, i.e.
the periodic unfolding method, which allows us to
consider very general heterogeneous media. Another
novelty brought by our paper consists in dealing with a
general nonlinear boundary condition for the electrostatic
potential and with more general reaction terms.

The rest of the paper is organized as follows: in
Section 2, we formulate our main convergence result,
while Section 3 is devoted to the proof of this result. The
paper ends with some conclusions and a few references.

II. THE MAIN RESULT

Using the periodic unfolding method, we are allowed
to pass to the limit in the weak formulation of problem
(1) and to obtain the effective behavior of the solution
of our microscopic model.

Theorem 1. The solution (Φε, c+ε ,c
−
ε ) of system (1)

converges, as ε → 0, to the unique solution (Φ, c+,c−)

of the following macroscopic problem in (0,T) × Ω:

−div (D0 ∇Φ) +

1

| Yf |
σ0G = c

+ − c− + D,

∂ c±

∂ t
− div(D0 ∇c± ±D0 c±∇Φ) = F±0 ,

(3)

with the boundary conditions on (0,T) ×∂Ω:{
D0 ∇Φ ·ν = 0,
(D0 ∇c± ±D0 c±∇Φ) ·ν = 0 (4)

and the initial conditions

c±(0,x) = c±0 (x), ∀x ∈ Ω. (5)

Here,

σ0 =

∫
Γ

σ(y)ds,

F±0 (c
+,c−) = ∓(c+ − c−)

and D0 = (d0ij) is the homogenized matrix, defined as
follows:

d0ij =
1

|Yf|

∫
Yf

(
δij +

∂χj
∂yi

(y)

)
dy ,

in terms of the functions χj, j = 1, ...,n, solutions of
the cell problems


χj ∈ H1per(Yf ) ,

∫
Yf

χj = 0,

−∆ χj = 0 in Yf,
(∇χj + ej) ·ν = 0 on Γ,

(6)

where ei, 1 ≤ i ≤ n, are the vectors of the canonical
basis in Rn.

III. PROOF OF THE MAIN RESULT

We shall only sketch the proof of our main conver-
gence result. For details, we refer to [18].

Let us consider now the equivalent variational formu-
lation of problem (1):

Find (Φε, c+ε , c
−
ε ), with


Φε ∈ L∞(0,T; H1(Ωε)),
c±ε ∈ L∞(0,T; L2(Ωε)) ∩L2(0,T; H1(Ωε)),
∂tc
±
ε ∈ L2(0,T; (H1(Ωε))

′
)

(7)

such that, for any t > 0 and for any ϕ1, ϕ2 ∈ H1(Ωε),
(Φε, c

+
ε , c

−
ε ) satisfy:∫

Ωε

∇Φε ·∇ϕ1 dx−
∫
Γε

∇Φε ·νϕ1 dσ =

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C Timofte, Multiscale Analysis of Ionic Transport in Periodic Charged Media

∫
Ωε

(c+ε − c
−
ε + D) ϕ1 dx, (8)

〈∂tc±ε , ϕ2〉(H1)′ ,H1 +
∫
Ωε

(∇c±ε ± c
±
ε ∇Φε) ·∇ϕ2dx =

∫
Ωε

F±(c+ε ,c
−
ε )ϕ2 dx (9)

and
c±ε (0,x) = c

±
0 (x) in Ωε. (10)

There exists a unique weak solution (Φε, c+ε , c
−
ε ) of

problem (8)-(10) (see [8], [13] or [18]).
Moreover, exactly like in [13], we can prove that the

concentration fields are non-negative, i.e. are bounded
from below uniformly in ε. Also, the concentration fields
are bounded from above uniformly in ε.

Under the above hypotheses, by standard techniques,
we can show that there exists a constant C ∈ R+, inde-
pendent of ε, such that the following a priori estimates
hold true:

‖Φε‖L2((0,T )×Ωε) + ‖∇Φε‖L2((0,T )×Ωε) ≤ C

max
0≤t≤T

‖c−ε ‖L2(Ωε) + max
0≤t≤T

‖c+ε ‖L2(Ωε)+

‖∇c−ε ‖L2((0,T )×Ωε) + ‖∇c
+
ε ‖L2((0,T )×Ωε)+

‖∂tc−ε ‖L2(0,T ;(H1(Ωε))′ ) + ‖∂tc
+
ε ‖L2(0,T ;(H1(Ωε))′ ) ≤ C.

As already mentioned, we are interested in obtaining
the limit behavior, as ε → 0, of the solution (Φε, c+ε , c−ε )
of problem (8)-(10). Our approach is based on the
periodic unfolding method introduced by D. Cioranescu,
A. Damlamian, G. Griso, P. Donato and R. Zaki (see [3]
and [5]). This approach has the advantage that we do not
need to use extension operators like in [8] or [13].

Using the properties of the unfolding operator Tε intro-
duced in [3] and [5] and the above a priori estimates, we
can easily prove that there exist Φ ∈ L2(0,T; H1(Ω)),
Φ̂ ∈ L2((0,T) × Ω; H1per(Yf )), c± ∈ L2(0,T; H1(Ω)),
ĉ± ∈ L2((0,T) × Ω; H1per(Yf )), such that, up to a
subsequence,

Tε(Φε) ⇀ Φ weakly in L2((0,T) × Ω; H1(Yf )),

Tε(∇Φε) ⇀ ∇Φ +∇yΦ̂ weakly in L2((0,T)×Ω×Yf ),

Tε(c±ε ) → c
± strongly in L2((0,T) × Ω; H1(Yf )),

Tε(∇c±ε ) ⇀ ∇c
±+∇yĉ± weakly in L2((0,T)×Ω×Yf ).

For proving Theorem 1, let us take, first, in the Poisson
equation (8), the test function

ϕ1(t,x) = ψ0(t,x) + εψ1(t,x,
x

ε
),

with
ψ0 ∈D((0,T); C∞(Ω))

and
ψ1 ∈D((0,T) × Ω; H1per(Yf )).

Unfolding each term by using the operator Tε and
passing to the limit with ε → 0, we obtain (see, for
details, [18]):

T∫
0

∫
Ω×Yf

(∇Φ(t,x)+

∇yΦ̂(t,x,y)) (∇ψ0(t,x) + ∇yψ1(t,x,y)) dxdy dt+

σ0

T∫
0

∫
Ω

Gψ0(t,x) dxdt =

T∫
0

∫
Ω×Yf

(c+(t,x) − c−(t,x)+

D(x))ψ0(t,x)dxdy dt. (11)

Then, by density, it follows that (11) holds true for
any ψ0 ∈ L2(0,T; H1(Ω)) and ψ1 ∈ L2((0,T) ×
Ω; H1per(Yf )).

Taking ψ0(t,x) = 0, we obtain

−∆y Φ̂(t,x,y) = 0 in (0,T) × Ω ×Yf,
∇y Φ̂ ·ν = −∇x Φ(t,x) ·ν on (0,T) × Ω × Γ,
Φ̂(t,x,y) periodic in y.

By linearity, we get

Φ̂(t, x, y) =

n∑
j=1

χj(y)
∂Φ

∂xj
(t, x), (12)

where χj, j = 1, n, are the solutions of the cell
problems (6).

Taking ψ1(t,x,y) = 0, integrating with respect to x
and using (12), we easily get the macroscopic problem
for the electrostatic potential Φ.

Now, taking in the Nernst-Planck equation (9) the test
function

ϕ2(t,x) = ψ0(t,x) + εψ1(t,x,
x

ε
),

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C Timofte, Multiscale Analysis of Ionic Transport in Periodic Charged Media

with
ψ0 ∈D((0,T); C∞(Ω))

and
ψ1 ∈D((0,T) × Ω; H1per(Yf )),

unfolding each term by using the operator Tε and passing
to the limit with ε → 0, we get (see [18])

−
T∫

0

∫
Ω×Yf

(c±(t,x)) ∂tψ0(t,x) dxdy dt+

T∫
0

∫
Ω×Yf

(∇c±(t,x) + ∇yĉ±(t,x,y))(∇xψ0(t,x)+

∇yψ1(t,x,y)) dxdy dt =

T∫
0

∫
Ω×Yf

F±0 (c
+,c−)ψ0(t,x) dxdy dt. (13)

By standard density arguments, we see that (13) holds
true for any ψ0 ∈ L2(0,T; H1(Ω)) and ψ1 ∈ L2((0,T)×
Ω; H1per(Yf )).

Taking, first, ψ0(t,x) = 0, and, then, ψ1(t,x,y) =
0, we obtain exactly the macroscopic problem for the
concentrations c±.

Since Φ and c± are uniquely determined (see [13]
and [18]), the whole sequences of microscopic solutions
converge to a solution of the unfolded limit problem and
this completes the proof of Theorem 1.

IV. CONCLUSION

Using the periodic unfolding method, the macroscopic
behavior of the solution of a system of equations de-
scribing the ion transport in periodic charged media is
analyzed.

Our model is relevant for studying the ion transport
through protein channels or the flow of electrons and
holes in a semiconductor device.

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Biomath 2 (2013), 1312302, http://dx.doi.org/10.11145/j.biomath.2013.12.302 Page 5 of 5

http://dx.doi.org/10.1137/080713148
http://dx.doi.org/10.1137/100817942
http://dx.doi.org/10.1007/s10958-011-0443-2
http://dx.doi.org/10.1007/s10659-013-9427-4
http://dx.doi.org/10.1021/cr60130a002
http://dx.doi.org/10.1016/j.jmaa.2012.01.052
http://dx.doi.org/10.4310/CMS.2011.v9.n3.a3
http://dx.doi.org/10.1007/s11565-007-0018-9
http://dx.doi.org/10.1002/j.1538-7305.1950.tb03653.x
http://dx.doi.org/10.11145/j.biomath.2013.12.302

	Introduction and setting of the problem
	The main result
	Proof of the main result
	Conclusion
	References