Transverse ion diffusion inside fixed charge narrow pores
The diffusion dynamics of ions inside a planar slit with a high constant surface charge density was studied by solving the Smoluchowski/HypernnetedChain equation for the mean first passage times. The behavior of local properties, such as the potential of mean force, is correlated with the time i...
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Інститут фізики конденсованих систем НАН України
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| Цитувати: | Transverse ion diffusion inside fixed charge narrow pores / B. Sulbaran, W. Olivares-Rivas, P.J. Colmenares // Condensed Matter Physics. — 2005. — Т. 8, № 2(42). — С. 303–316. — Бібліогр.: 26 назв. — англ. |
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irk-123456789-1195502017-06-08T03:03:11Z Transverse ion diffusion inside fixed charge narrow pores Sulbaran, B. Olivares-Rivas, W. Colmenares, P.J. The diffusion dynamics of ions inside a planar slit with a high constant surface charge density was studied by solving the Smoluchowski/HypernnetedChain equation for the mean first passage times. The behavior of local properties, such as the potential of mean force, is correlated with the time it takes for an ion to get adsorbed at the surface, and also with macroscopic properties like the salt rejection coefficient and the apparent diffusion constant in porous media, as a function of pore size. We show that the width of the walls has an important effect on the transverse ion diffusion process. Дифузійна динаміка іонів всередині плоскої щілини з великою постійною густиною поверхневого заряду досліджувалась шляхом розв’язку супер-ланцюжкового рівняння Смолуховського для середніх часів перших переходів. Поведінка локальних властивостей, таких як потенціал середньої сили, корелює з часом, необхідним для іону бути адсорбованим на поверхні, а також з макроскопічними властивостями, такими як коефіцієнт несприйняття солі та константою уявної дифузії в пористих середовищах як функція розміру пори. Ми показуємо, що ширина стінок має суттєвий вплив на поперечний іонний дифузійний процес. 2005 Article Transverse ion diffusion inside fixed charge narrow pores / B. Sulbaran, W. Olivares-Rivas, P.J. Colmenares // Condensed Matter Physics. — 2005. — Т. 8, № 2(42). — С. 303–316. — Бібліогр.: 26 назв. — англ. 1607-324X PACS: 05.20.Jj, 82.65.+r DOI:10.5488/CMP.8.2.303 http://dspace.nbuv.gov.ua/handle/123456789/119550 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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English |
| description |
The diffusion dynamics of ions inside a planar slit with a high constant surface
charge density was studied by solving the Smoluchowski/HypernnetedChain
equation for the mean first passage times. The behavior of local
properties, such as the potential of mean force, is correlated with the time
it takes for an ion to get adsorbed at the surface, and also with macroscopic
properties like the salt rejection coefficient and the apparent diffusion constant
in porous media, as a function of pore size. We show that the width of
the walls has an important effect on the transverse ion diffusion process. |
| format |
Article |
| author |
Sulbaran, B. Olivares-Rivas, W. Colmenares, P.J. |
| spellingShingle |
Sulbaran, B. Olivares-Rivas, W. Colmenares, P.J. Transverse ion diffusion inside fixed charge narrow pores Condensed Matter Physics |
| author_facet |
Sulbaran, B. Olivares-Rivas, W. Colmenares, P.J. |
| author_sort |
Sulbaran, B. |
| title |
Transverse ion diffusion inside fixed charge narrow pores |
| title_short |
Transverse ion diffusion inside fixed charge narrow pores |
| title_full |
Transverse ion diffusion inside fixed charge narrow pores |
| title_fullStr |
Transverse ion diffusion inside fixed charge narrow pores |
| title_full_unstemmed |
Transverse ion diffusion inside fixed charge narrow pores |
| title_sort |
transverse ion diffusion inside fixed charge narrow pores |
| publisher |
Інститут фізики конденсованих систем НАН України |
| publishDate |
2005 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/119550 |
| citation_txt |
Transverse ion diffusion inside fixed charge narrow pores / B. Sulbaran, W. Olivares-Rivas, P.J. Colmenares // Condensed Matter Physics. — 2005. — Т. 8, № 2(42). — С. 303–316. — Бібліогр.: 26 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT sulbaranb transverseiondiffusioninsidefixedchargenarrowpores AT olivaresrivasw transverseiondiffusioninsidefixedchargenarrowpores AT colmenarespj transverseiondiffusioninsidefixedchargenarrowpores |
| first_indexed |
2025-07-08T16:09:32Z |
| last_indexed |
2025-07-08T16:09:32Z |
| _version_ |
1837095688701214720 |
| fulltext |
Condensed Matter Physics, 2005, Vol. 8, No. 2(42), pp. 303–316
Transverse ion diffusion inside fixed
charge narrow pores
B.Sulbarán 1,2 , W.Olivares-Rivas 1 , P.J.Colmenares 1
1 Grupo de Quı́mica Teórica Quimicofı́sica de Fluidos y Fenómenos
Interfaciales (QUIFFIS), Departamento de Quı́mica,
Facultad de Ciencias, Universidad de Los Andes, Mérida, Venezuela
2 Departamento de Investigación, Area de Ciencias Aplicadas,
Instituto Universitario Tecnológico de Ejido (IUTE), Ejido, Venezuela
Received November 9, 2004
The diffusion dynamics of ions inside a planar slit with a high constant sur-
face charge density was studied by solving the Smoluchowski/Hypernneted-
Chain equation for the mean first passage times. The behavior of local
properties, such as the potential of mean force, is correlated with the time
it takes for an ion to get adsorbed at the surface, and also with macroscopic
properties like the salt rejection coefficient and the apparent diffusion con-
stant in porous media, as a function of pore size. We show that the width of
the walls has an important effect on the transverse ion diffusion process.
Key words: ion diffusion, charged micro-pores, mean first passage time,
apparent transverse diffusion constant
PACS: 05.20.Jj, 82.65.+r
1. Introduction
The development of reliable techniques for analyzing and visualizing the solid
surface in a liquid-solid interface, where catalytic reactions, ion exchange or ad-
sorption might be occurring, has focused the attention into the surface processes,
with great success. On the other hand, processes taking place in the liquid region
are still being interpreted using classical theories, many times beyond their limit
of validity. Most of the fundamental discussions or interpretations of experimental
results on interfacial processes, such as ion diffusion and exchange, electrochemistry,
surface chemistry and heterogeneous liquid-solid catalysis, are based on the a priori
assumptions that ion concentration and charges are small, surface zeta potentials
are constant and low, ions are point charges and that membranes, pores and its
walls are infinitely wide. We have shown in the past that, when those restrictions
are overcome, important nonlinear effects become significant [1–3].
The diffusion of ions in concentrated bulk solutions is by itself a difficult prob-
c© B.Sulbarán, W.Olivares-Rivas, P.J.Colmenares 303
B.Sulbarán, W.Olivares-Rivas, P.J.Colmenares
lem, since long range electrostatic correlations are coupled with volume exclusion
and other short range dispersion interactions, modulated by the polar solvent [4,5].
Inside a charged micro pore, the electrostatic field created by the charged walls sets
up a non-homogeneous distribution of the mobile ions in solution. Therefore, the
transverse diffusion (perpendicular to the surface) is further complicated by such
non-homogeneous environment near the interface. The lateral diffusion (parallel to
the interface) is affected to a lesser degree by the charged walls since the average
lateral forces are null [6].
In a previous work [7], hereafter referred to as paper I, we discussed the ion-
membrane interactions from a statistical mechanics point of view. The ion dynamics
in the vicinity of an isolated charged membrane of finite width was described by
solving the Smoluchowski equation for the Mean First Passage Time of co-ions and
counter-ions, τj(x). That is, we calculated the mean time it takes for and ion at a
given position x away from a membrane to reach its absorbing surface. There we
showed that the use of the classical theory (the point ion Modified Gouy-Chapman
theory, MGC) for the density profiles leads to a monotonic behavior of τj(x) as a
function of x, while the use of more accurate profiles, such as the integral equation
HNC/MSA, gives rise to oscillations about the values of a free diffusing particle,
particularly for high ionic and surface charges.
In this work we examine the diffusion dynamics of the ions of an aqueous solution
enclosed in narrow slit pores bearing a fixed surface charge density σ in the inter-
nal face of the walls. Particular attention will be given to the effect of the ion-ion
correlation across pore walls of finite thickness on the diffusion process.
2. Theory
To simulate a slit pore we consider two planar slabs of width d, bearing a con-
stant surface charge density σ and separated by a distance L, immersed in a prim-
itive model ionic solution of charged hard spheres of finite diameter a. The static
equilibrium properties of this system has been solved by us under a variety of ion
and surface charges, salt concentration, pore sizes and wall widths conditions [1,2].
Details of the model and theory used are given in [2,3] and in paper I [7].
The ions are free to move in the y and z directions parallel to the walls, but are
confined to the transverse diffusion x-domain. The most common model boundary
condition is to fix the electrostatic potential or even the so-called zeta potential.
However, most real systems have a fixed, self regulated and discrete surface charge
which is well represented by a smeared surface charge density σ. Even though the
surface charge densities in the inner wall faces, σin, and in the outer wall faces,
σout, could have any arbitrary fixed value, we have chosen here to let σin = σ and
σout = 0. In this way, the solution outside the pore is in front of a hard wall that has
no interaction with the solution inside the pore. This will allow us to clearly observe
the correlation between the solutions inside and outside the pore [1,2].
The ions are assumed to move according to a Langevine-like dynamics [3]. In the
presence of an electrical double layer, besides the hydrodynamic drag force and the
304
Transverse ion diffusion
white noise force resulting from the collisions of the particles with ion j, a net force
−∇Wj(x, t) also acts on the ions of species j, where Wj(x, t) is the total potential of
mean force. By definition, −βWj(x) = ln[ρj(x)/ρj ], where ρj(x) is the local density
profile inside the pore and ρj is the ion density in the bulk electrolyte solution.
In the so-called Instantaneous Relaxation Approximation (IRA), the time de-
pendent correlation effects are neglected under the assumption that, as a central ion
moves inside a charged pore, the other confined ions adjust their spatial configura-
tion instantaneously. That is to say, the electrical double layer relaxes on a much
faster time scale than the diffusion process. The IRA is then equivalent to the as-
sumption that a given central ion would diffuse against the barrier of the one-particle
equilibrium potential of mean force set up by the charged walls, Wj(x, t) = Wj(x).
A direct molecular dynamics test of the goodness of this approximation was carried
out by Jönsson et al [6,8].
We shall focus our discussion on the determination and analysis of the mean time
taken by a given ion j to reach one of the walls, namely, the Mean First Passage Time
(MFPT), tMFP
j (x). To be comprehensive, we briefly give here the basic concepts.
Let Pj(y, t | x, 0) be the conditional probability of finding an ion j in position
y at a time t, given that it started to diffuse from a position x at the initial time
t = 0. Under the IRA approximation, the backward Fokker-Plank equation can be
written [9]
∂Pj(y, t | x, 0)
∂t
= −Dj
dβWj(x)
dx
∂Pj(y, t | x, 0)
∂y
+Dj
∂2Pj(y, t | x, 0)
∂y2
, (1)
where Dj is the bulk diffusion constant. The symmetry in the x-domain perpen-
dicular to the surfaces ensures a reflecting plane located at x = 0, a mid distance
h = (L− a)/2 from the plane of the closest approach to the pore walls.
The probability of an ion to be located at any point within the diffusion domain
[−h, h] without having reached the walls, given that it was at the initial position x
is the so-called survival probability Gj(x, t):
Gj(x, t) =
∫ h
−h
Pj(y, t | x, 0)dy. (2)
In terms of the survival probability, the probability density for the first passage times
is −∂Gj(x, t)/∂t. The first moment of this distribution is the Mean First Passage
Time, tMFP
j (x)
tMFP
j (x) = −
∫
∞
0
∂Gj(x, t)
∂t
tdt =
∫
∞
0
Gj(x, t)dt. (3)
When equation (1) is integrated over the position and then over time, according to
equations (2) and (3), we obtain, under the IRA approach, an ordinary differential
equation which is satisfied by the MFPT, and referred to as the Smoluchowski
equation:
d2τj(y)
dy2
−
dβWj(y)
dy
dτj(y)
dy
= −1, (4)
305
B.Sulbarán, W.Olivares-Rivas, P.J.Colmenares
where we have defined the dimensionless MFPT τj(y) = DtMFP
j /h2, and have scaled
the positions as y = x/h.
This equation can be solved analytically for several absorbing or reflecting math-
ematical boundary conditions [3,9–12]. A boundary is said to be absorbing if the
diffusion process terminates when the particle reaches that boundary. Then at an
absorbing surface τj(x) = 0. Physical adsorption, ion exchange or a chemical reac-
tion would make a surface a mathematical absorbing boundary. The mathematical
terminology reflecting boundary is used if, when the particle reaches a point on that
plane, it is transferred to a point within the diffusing domain. In a pore with equal
surface charge densities on the opposing walls, the symmetry mid-plane is a reflecti-
ng boundary by definition. At the mid-plane x = 0 the mean first passage time must
satisfy dτj(x)/dx = 0. Then, it would be equivalent to consider an entire diffusing
domain between two absorbing surfaces at x = −h and x = h, or half the domain
between the reflecting midplane plane at x = 0 and the absorbing plane at the plane
of closest approach x = h.
For a symmetrical pore with absorbing walls we then have [11,12]
τj(x) =
∫ 1
x
h
dy
ρj(y)
∫ y
0
ρj(z)dz . (5)
In the absence of correlations, the potential of mean force vanishes, Wj(y) = 0 and
ρj(y) = ρj. Thus, from equation (5), we get the dimensionless free diffusing particle
limit as τ 0
j (x) = (1 − (x/h)2)/2. From this result, the mean first passage time for
diffusion from the midpoint of the pore at x = 0 to a wall at x = h is tB = h2/2D,
the well known one dimension brownian motion result.
The local ion concentrations change markedly in the neighborhood of a charged
interface in order to adjust the chemical potential change under the effect of the
electrostatic field and the ion-ion correlations. This is well described with the gen-
eralized expression for the chemical potential µj of ionic species as a local property
in a non-homogeneous solution next to a charged surface [13].
βµj(r) = βµ0
j + ln ρj(r) + βvj(r) − Cj(r), (6)
where ρj(r) is the local particle density profile for species j at a point r in space,
β = 1/kT , k is Boltzmann´ s constant, T is the temperature, and vj(r) is the bare
external electrostatic potential acting on the ions of species j in solution, excerpted
by the hard charged pore walls. In equation (6), µ0
j is the chemical potential in the
ideal gas reference state, βµ0
j = ln Λ3
j , Λj = h̄(mjkT/2π)−1/2, h̄ is Planck’s constant,
mj is the ion mass and Cj(r) is the one particle correlation function.
At equilibrium, the local chemical potential µj(r) is a constant everywhere. The
position dependent ionic profile or local density inside a micro-pore, ρj(r), in equilib-
rium with a reservoir solution with bulk densities ρj , is expressed in the convenient
form
ρj(r) = ρje
−β[ejψ(r)+Jj(r)], (7)
where the potential terms collected in the exponent give the total potential of mean
force, βWj(r) = − ln(ρj(r)/ρj). The potential Jj(r) is obtained by solving the exact
306
Transverse ion diffusion
relationship obtained by Lovett, Mou and Buff [14] and Wertheim [15], known as
the LMBW equation,
∇1Jj(r1) = −kT
∑
i
∫
∇2ρi(r2)
(
Cij(r1, r2) + β
eiej
εr12
)
dr2 , (8)
where Cij(r1, r2) is the two particle direct correlation function.
The mean electrostatic potential ψ(x) is obtained by integration of Poisson’s
equation. The equilibrium profile is given by the Boltzmann factor ρje
−βejψ(x) on-
ly when the short range correlation potential term Jj(x) is neglected, giving the
Poisson-Boltzmann (PB) or Gouy-Chapman (GC) theory. Inclusion of a plane of
closest approach is referred to as the Modified Gouy-Chapman theory (MGC). The
MGC theory takes into account the short range ion-interface repulsion, but neglects
the ion-ion correlations and ionic size effects. To take these effects into account, one
has to solve equations (8) and (7) for ρi(r).
One can choose to solve the singlet LMBW equation, equations (8) to (7) [17,18]
or, equivalently, the solution of the HNC/MSA singlet integral equation, [16,19,20].
The simplest anzatz for Cij(r1, r2) is to take it as the bulk two particle direct cor-
relation function in the MSA approximation, which is known analytically. Such ap-
proximation corresponds to the singlet HNC/MSA approximation solved by Lozada-
Cassou et al [16] for a number of geometries of different complex colloidal particles.
Improvement can be obtained from Monte Carlo (MC) numerical simulations [21,24],
functional density theory [25], or using a better approximation for Cij(r1, r2) in the
LMBW equation [26].
Therefore, it is fare to say that at equilibrium the ion profiles near a planar
interface, the mean electrostatic potential and therefore the total potential of mean
force acting on the ions approaching the interface are well known.
3. Results and discussion
It is well established that the MGC theory is only valid for monovalent ions at
low ionic strengths, below 0.01 M and low surface charges, of the order of 500 Å2
per site, which fortunately are the conditions of many important systems. However,
for concentrated solutions and high surface areas, the HNC/MSA is in excellent
agreement with the computer Monte Carlo and molecular dynamics simulations
experiments [24], but important differences were found between the MGC and the
formal theories [2].
As discussed in paper I for a single membrane, the mean potentials and parti-
cle density profiles strongly depend on the surface charge density. For charges as
high as 66.67 Å2 (corresponding to about 0.24 C/ M2), as found in phospholipid
membranes,a monotonic behavior was obtained with a moderate agreement of the
MGC and HNC/MSA theories. But for extremely high charge densities as 29.45 Å2
(corresponding to 0.544 C/M2) and a 2:2 electrolyte, a net charge inversion is ob-
tained in the diffuse ionic double layer. We have used here the rather large charge
of 0.544 C/M2 used in the physics literature to allow comparisons and to clearly
307
B.Sulbarán, W.Olivares-Rivas, P.J.Colmenares
illustrate the resulting nonlinear effects, with the understanding that this represents
an extreme situation.
We have chosen to vary the inside surface charge density σ and to let the outside
wall of the pore to be a rigid uncharged surface. Changing the width d increases
the effective distances between the ions in the solution outside as well as the surface
charge and ions inside. As d goes from infinity to zero the ions outside see the
external wall changing from a hard plane to a fully charged wall. The resulting
non-uniform distribution of co-ions and counter-ions on the outside of the pore is
viewed by the ions inside the pore as an effective charge, which decreases the field
of σ and attenuates the potential of mean force inside the pore as d decreases. Since
the outside charge is zero, the changes on the potential of mean force Wj(x) inside
the slit pore, as d changes, are then clearly a consequence of the ion-ion correlations
across the wall. Having a non-zero charge density in the outside walls would simply
introduce additional double layer profile changes on both sides [2,23].
0 0,2 0,4 0,6 0,8 1
X/h
-2
0
2
4
6
W
+
(x
)
d=0
d=1000a
L=5aL=3a
β
L=10a
d=0
d=1000a
d=1000a
d=0
A) Co-ions
____ HNC
- - - - MGC
0 0,2 0,4 0,6 0,8 1
X/h
-5
-4
-3
-2
-1
0
1
2
3
W
-(
x
)
d=0
d=1000a
β
L=3a L=5a
d=0
L=10ad=1000a
d=1000a
d=0
B) Counter-ions
____ HNC
- - - - MCG
Figure 1. The reduced total potential of mean force βWi(x) acting on an ion i as a
function of the reduced distance x/h to the pore midplane: A) co-ion, B) counter-
ion. The system represents a primitive model of a planar pore, for e/σ = 66.67 Å2
and ρb = 0.971 M of a 2–2 electrolyte at 298 K and ε = 78.5. The dashed curves
are the MGC results for d = a. The solid curves are the HNC/MSA singlet results
for several values of the pore walls width d, as indicated. The surface charge on
the outside wall face, σout was set equal to zero.
The one membrane system studied in paper I [7] is equivalent to consider an
infinitely wide pore with thin walls. Here we show the results for finite planar slit
pores with wall to wall L distances of the order of a few ionic diameters. In figure 1A
we show the reduced potential of mean force profile βW+(x) for doubly charged co-
ions, for several values of the slit pore size L, 10, 5 and 3 ionic diameters a and wall
width d of 0 and 1000 diameters, as indicated. The RPM solution has ρb = 0.971 M
of a 2–2 electrolyte at 298 K and ε = 78.5. Here x = 0 is at the mid-plane of the
pore and x = h is located at the inner plane of the closest approach to the walls.
βW+(x) represents the potential barrier where the ions diffuse. In figure 1A, the
308
Transverse ion diffusion
dashed lines show the typically repulsive potential predicted by the MGC approx-
imation for walls with d = a. Though not shown in the figure, changing the value
of d produces only minor changes on the MGC curves. The solid lines show the
HNC/MSA theory predictions, which includes the entropy effects due to volume
exclusion and ion-ion correlations.
It can be seen in figure 1A that, while close to the positive wall the positive
co-ions feel the strong wall repulsion, there are regions within the pore where the
positive ions are attracted by the positive wall. So, as a co-ion diffuses towards
the surface, it sees an effective potential well. For very large pores this potential
oscillates from repulsive to attractive. But, as the pore becomes very narrow, the
interaction with both walls shifts the minimum of the well towards the middle of the
pore. As the pore walls become thinner the potential wells become less pronounced.
Changing the width d increases the effective distances between the ions in the
solution inside the pore and the surface charge and ions on the outside. Since the
outside surface of the pore in figure 1A is a rigid uncharged surface, the changes on
the potential profiles on the inner side are a consequence of the ion-ion correlations
across the walls. The non-uniform distribution of co-ions and counter-ions on the
outer solution is viewed by the ions on inner solution as an effective diffuse charge,
which adds up to σin, increasing the mean electrostatic potential ψ(x) as d decreases
and, therefore, increasing the total potential of mean force Wj(x).
This effect of ion-ion correlation across the walls was first discussed by us when
studying a solution confined in a slit pore of size L separated from infinitely large
pores on both sides by walls of width d, bearing the same fixed charge σ = σin = σout
[1,2]. We observed that, for thin walls, breaking the symmetry of the system by
decreasing the pore size and therefore changing the ions environment inside the pore,
produced important changes on the particle density and potential profiles on the
solution outside the pore. In order to keep the chemical potential constant, the ions
compensate the symmetry breakage with the breakage of the local electroneutrality.
The effect is enhanced by the presence of high charge asymmetric electrolites [22].
This process had long been recognized as the co-ion exclusion or salt rejection. One
can also break the symmetry of a wall by having different surface charges on both
sides. This case was studied with the same computer code by our collaborators [23].
The effect has been further studied and confirmed by Monte Carlo simulations [24].
In figure 1B we show the reduced potential of mean force βW−(x) profile for the
negative counter-ions for the same system of figure 1A. The MGC approximation
(dashed lines) gives an attractive potential everywhere, which is simply the negative
of the co-ion potential plotted in figure 1A. On the other hand, the HNC/MSA
predicts a large attraction very close to the plane of the closest approach, but, for
this high σ repulsive potential barrier is created at lower distances.
In figure 2a, we show the mean first passage time τ+(x) of a co-ion as a function
of the initial reduced position x/h for different pore sizes L and very wide walls
d = 1000a. The plotted values correspond to the ratio τ+(x)/τ 0(x) where τ 0(x) is
the limiting mean first passage time of a free diffusing particle with no interactions
for the reflecting/absorbing boundary condition used here, τ 0(x) = (1− (x/h)2)/2.
309
B.Sulbarán, W.Olivares-Rivas, P.J.Colmenares
0 0,2 0,4 0,6 0,8 1
x/h
1
10
100
1000
+(x)
d=1000a
L=3a
L=5a
L=10a
L=60a
L=200a
L=1000a
L=10a
L=10a
L=3a,5a
τ
τ
(x)
0
A) Co-ions
0,2 0,4 0,6 0,8 1
x /h
0,5
1
1,5
2
L=5a
L=10a
L=60a
L=3a
L=100a
L=100a
L=1000a
d=1000a
L=60a
L=10a
τ-(x)
τ (x)0
B) Counter-ions
Figure 2. Ratio of the Mean First Passage Time of ions j in a pore to the limiting
MFPT of a free diffusing particle with no interactions, τj(x)/τ0(x), as a function
of the initial position x/h, for the same system described in figure 1: A) co-ions;
B) counter-ion. Solid lines corresponds to the S-HNC theory and the dashed
line to points ions S-MGC approximation. Results are for very thick walls with
d = 1000a and pore sizes L as labeled.
For very large pores we get the one membrane slab behavior reported in pa-
per I [7], being indistinguishable from the non-interacting free diffusing particle for
distances away from the walls. As the pore becomes smaller the behavior changes
rapidly to a situation where τ+(x) is independent of the initial position except for
distances very close to the absorbing plane of the closest approach where it drops
rapidly to zero. We can clearly see from the figure that the time it takes for a co-
ion to get absorbed increases the free diffusing particle value τ 0(x) up to several
times, as the pore becomes smaller. Thus for narrow pores the co-ions get trapped
in the potential well described in figure 1A. The point ion Smoluchowski equation
with the MGC profile S-MGC, shown in dashed lines, predicts times that are always
monotonic and slightly larger than the free diffusing particle.
Figure 2b shows the results for the counter-ion ratio τ−(x)/τ 0(x). Again, for
very large pores, say for L = 100a − 1000a , we get the one membrane slab be-
havior reported in paper I [7]. That is, the MFPT ratio presents oscillations about
τ−(x)/τ 0(x) = 1, for initial distances in the neighborhood of the interface. The mean
first passage time ratio predicted by the HNC/MSA theory (continuos lines) changes
in a non-monotonic way as a function of the initial position. As the pore becomes
narrower, it first increases and then decreases. In figure 2B, the MGC (dashed lines)
only accounts for the electrostatic attraction of counter-ions to the charged wall. As
the pore becomes smaller, the time it takes to reach the surface decreases monotoni-
cally with the distance it starts from and decreases with respect to the free diffusing
particle.
In figure 3 we show the effect of the ion-ion correlations across the walls on the
Mean First Passage Time of ions j in a pore of size L = 5a. We plot the deviation
310
Transverse ion diffusion
0 0,2 0,4 0,6 0,8 1
X/h
0
10
20
30
40
τ
−
τ A) Co-ions
L = 5a
d=0
d=a
d=1000a
d=100a
d=10a
+
(x
)
(x
)
0
0 0,2 0,4 0,6 0,8 1
X/h
-0,25
-0,2
-0,15
-0,1
-0,05
0
0,05
τ
−
τ
B) Counter-ions
L = 5a
d=0
d=a
d=100,1000a
d=10a
(x
)
(x
)
0
d=1000a
-
Figure 3. The effect of the ion-ion correlations across the walls on the Mean
First Passage Time of ions j in a pore of size L = 5a. We show the deviation of
the MFPT of co-ions (A) and counter-ion (B) from the limiting MFPT of a free
diffusing particle with no interactions, τj(x) − τ0(x), as a function of the initial
position x/h, for several values of the wall thickness d as indicated. The system
and labels are the same as in figures 1 and 2.
of the MFPT of co-ions (A) and counter-ion (B) from the limiting MFPT of a free
diffusing particle with no interactions, τj(x) − τ 0(x), as a function of the initial
position x/h, for several values of the wall thickness d as indicated. The system
and labels are the same as in figures 1 and 2. For this pore size the electrostatic
repulsion dominates and co-ions take longer than a non interacting particle to reach
the absorbing boundary. For infinite walls no correlation exists among ions inside
and outside the pore. As shown in figure 3A, for finite thickness the ion correlation
gives rise to an effectively lower surface charge and the HNC/MSA MFPT decreases
as d decreases. The MGC approximation shows very little variation with d.
It is illustrative to analyze the behavior shown in figure 3 in the context of the
mean force potential profile of figure 1. For L = 5a in figure 1, the size or volume
exclusion effect excludes counter-ions from the region around x/h = 0.5, where co-
ions form a layer. As the initial position moves away from the wall, the time shift
for counter-ions, τj(x) − τ 0(x) in figure 3B, goes through a minimum and then an
inflection point. The counter-ions tend to have MFPT that are lower than the times
for free diffusing particles since they are attracted to the wall at x = h.
In figure 3B we can see that for distances to the walls lower than one diameter
(x/h > 3/4, with h = 2a) the electrostatic attraction dominates and the counter-
ion diffuse faster than a free diffusing particle showing little dependence on d. As
the counter-ions start diffusing closer to the midplane the MFPT goes through a
minimum and then increases, showing a clear dependence on the wall thickness. The
point ion MGC approximation is monotonic and presents a small dependence on d
which goes in the opposite direction as that shown by the HNC theory.
Even though local properties, such as the local density profiles ρj(x) or the MFPT
311
B.Sulbarán, W.Olivares-Rivas, P.J.Colmenares
tMFP
j (x), are useful for understanding the physics occurring inside a micro-pore,
most experimental measurements correspond to macroscopic quantities obtained
from their space average. Moreover, in porous materials the measured quantities are
the result of averages over a given array of micro-pores. Therefore we shall end this
discussion by looking at such macroscopic quantities.
The average density of ions j inside a pore is
ρj =
1
h
∫ h
0
ρj(x)dx. (9)
So, the normalized probability of finding an ion of species j at the initial position x
inside a pore is gj(x) = ρj(x)/ρj . The position average time that an ion j takes to
diffuse from any point inside the pore to the absorbing wall is then,
tMFP
j =
1
h
∫ h
0
tMFP
j (x)gj(x)dx. (10)
1 10 100 1000
L
0,2
0,4
0,6
0,8
1
ρ d=0
d=a
-
+
_
ρ
+
d=10a d=100,1000a
/ a
d=1000a
A) Co-ion Exclusion
____ HNC
------ MGC
1 10 100 1000
L
0
2
4
6
8
10
12
14
ρ-
-
_
ρ
-
d=0
d=a
d=10a
d=100, 1000a
/ a
B) Counter-ion adsorption
Figure 4. The local particle density,averaged over the entire pore, ρ̄j , as a function
of the pore size L/a. Results are given for different wall thickness as labeled. A)
ρ̄+/ρ+ shows the co-ion exclusion; B) ρ̄−/ρ− shows the counter-ion adsorption.
For a free diffusing ideal particle confined in a pore of size 2h, ρj(x) = ρj , hence
g◦(x) = 1 and tMFP
◦
(x) = h2/(2D) · (1 − (x/h)2). One gets t0MFP = h2/(3D), or
τ 0 = 1/3. By analogy, one can define an apparent diffusion constant Da
j ,
tMFP
j =
h2
3Da
j
. (11)
In figure 4 we show the average density of co-ions and counter-ions, ρj/ρj, inside
a highly charge pore. ρ+/ρ+ is a measure of the so-called ionic exclusion or salt
rejection in porous media. As shown by the MGC dashed lines in figure 1A, purely
electrostatic effects give a monotonically increasing salt rejection, as the pore size
312
Transverse ion diffusion
decreases, with very little effect of the wall width d. That would also be the behav-
ior of the HNC theory for low salt concentrations and low electrolyte and surface
charges. A similar behavior is found when the surface or the zeta potentials are kept
fixed as the pore size h changes. However, for the high values of the fixed surface
charge density σ used here, the ion-ion correlations account for a local violation
of the electroneutrality, which can be interpreted as a net surface charge reversal
or overcharging [2]. This is reflected in the non-monotonic behavior of the co-ion
exclusion, with a strong dependance on the wall width. Since the corresponding
counter-ion concentrations are much higher this is seen in figure 4b as a small hump
in the adsorption isotherm ρ
−
/ρ− vs pore size.
1 10 100 1000
L
0,001
0,01
0,1
1
_
/a
D
D
a
d=0
d=a
d=10a
d=100,1000a
+
+
A) Co-ion
_____ S-HNC
-------- S-MGC
1 10 100
L
5
10
15
20
25
30
35
40
45
_
/a
D
D
d=0
d=a
d=10a
d=100,1000a
-
B) Counter-ions
a _____ S-HNC
------- S-MGC-
Figure 5. Ratio of the apparent transverse ion diffusion constant of confined ions
to that of ions in the bulk solution, Da
j /Dj , as a function of the pore size L/a, for
different wall width d as labeled. As before, solid lines correspond to the S-HNC
approach, while the dotted lines correspond to the point ion limiting approach
S-MGC.
Figure 5 shows the ratio of the apparent transverse diffusion constant Da
j of ions
confined in porous media to the regular bulk solution ion diffusion constant Dj. As
expected the ions diffuse to the absorbing charged walls as if counter-ions had a larger
apparent diffusion constant D−, and co-ions had a lower D+. However, the trans-
verse diffusion constants go through maxima and minima as the pore size becomes
of molecular dimensions. The change in the drift of the ions in a porous media is
much larger in the Smoluchowski-HNC description than in the Smoluchowski-MGC,
showing the importance of the entropy size effects on the diffusion in confined sys-
tems. Figure 5B clearly shows the importance of ion-ion correlation across finite wall
in reducing the enhancement of the diffusivity of counter-ions. For co-ions the thin-
ner are the pore walls the higher is the apparent diffusivity, as shown in figure 5A.
313
B.Sulbarán, W.Olivares-Rivas, P.J.Colmenares
4. Conclusions
We have studied the transverse diffusion process of ions confined in a fixed surface
charge planar micro-pore of several ionic diameters wide. We solved the Fokker-
Plank-Smoluchowski equation for the survival probability in terms of the mean Mean
First Passage time for ions to reach the pore walls. The assumption of the IRA
approximation allowed us the use of the MGC and HNC/MSA equilibrium potential
profiles. The observed oscillations in the HNC particle density and the potential of
mean force profiles in the vicinity of the pore walls gives rise to a similar behavior
for the Smoluchoski-HNC (S-HNC) mean first passage time, τj(x), which oscillates
about the value for a free diffusing ideal particle.
We also showed that the ion diffusion towards the interface is modulated by
the ion-ion correlation across a thin pore wall. The results depend not only on the
magnitude of the surface charge density at the inner and outer faces but also on
the difference between them. For the conditions chosen, namely large inner surface
charge and zero outer surface charge density, the effect of decreasing the pore wall
width is to decrease the net charge density seen by the ions inside the pore. As
a result,a compensation of the typical differences between the S-MGC and S-HNC
theories for thin walls was observed for low values of d.
An apparent diffusion constant Da
j for a porous media was obtained and shown
to depend strongly on size and ion-ion correlations within the pore and across the
pore walls. The presented results should be useful in the interpretation of diffusion
controlled processes occurring at highly charged interfaces. The classical MGC the-
ory fails to show these features. In fact, it is useful only for very low surface charges
and low concentrations of 1:1 electrolytes.
5. Acknowledgement
This work was supported by grant G97000741 from the Consejo Nacional de
Investigaciones Cient́ıficas y Tecnológicas (CONICIT) of Venezuela and grant ADG-
CDCHT. We thank the Computer Center CeCalCULA.
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B.Sulbarán, W.Olivares-Rivas, P.J.Colmenares
Поперечна іонна дифузія всередині вузької пори з
фіксованим зарядом
Б.Сулбаран 1,2 , У.Оліварез-Ріваз 1 , П.Дж.Колменарез 1
1 Природничий факультет університету де Лос Андес, Венесуела
2 Технологічний університет, Ексідо, Венесуела
Отримано 9 листопада 2004 р.
Дифузійна динаміка іонів всередині плоскої щілини з великою пост-
ійною густиною поверхневого заряду досліджувалась шляхом розв’-
язку супер-ланцюжкового рівняння Смолуховського для середніх час-
ів перших переходів. Поведінка локальних властивостей, таких як
потенціал середньої сили, корелює з часом, необхідним для іону
бути адсорбованим на поверхні, а також з макроскопічними влас-
тивостями, такими як коефіцієнт несприйняття солі та константою
уявної дифузії в пористих середовищах як функція розміру пори.
Ми показуємо, що ширина стінок має суттєвий вплив на поперечний
іонний дифузійний процес.
Ключові слова: іонна дифузія, заряджені мікропори, середній час
перших переходів, константа уявної трансверсійної дифузії
PACS: 05.20.Jj, 82.65.+r
316
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