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Counterions Released from
Oppositely Charged Surfaces: a
Liquid Theory Study
Cite as: AIP Conference Proceedings 885, 155 (2007); https://
Published Online: 15 February 2007
A. Flores-Amado, and M. Hernández-Contreras
Time Continuity and the Positivity Problem of the Floquet Hamiltonian
AIP Conference Proceedings 885, 26 (2007); https://
Supertubes and Superconducting Membranes
AIP Conference Proceedings 885, 63 (2007); https://
AIP Conference Proceedings 885, 155 (2007);
© 2007 American Institute of Physics.
885, 155
Counterions Released from Oppositely Charged
Surfaces: a Liquid Theory Study
A. Flores-Amado and M. Hernández-Contreras
Departamento de Física, Centro de Investigación y de Estudios Avanzados del IPN, A.P. 14-740,
México D.F., Mexico
Abstract. The strength of the attractive interaction of two oppositely charged surfaces in an electrolyte solution, and the counterion release from the slit between them was determined within the
hypernetted chain approximation and compared with the Poisson-Boltzmann equation. Liquid theory predicts small deviations, for the attraction between the plates and release of ions, than mean
field theory at moderate and low added salt. At higher salt concentration both theories agree on their
predictions for these thermodynamic properties.
The release of counterions upon association of macromolecules of opposed electrical
sign is a mechanism of self-assembly formation [1]. Here we present a liquid theory
study of this phenomenon and its effects on the interaction between two charged flat
plates of contrary electrical sign and same strength embedded in an aqueous electrolyte
solution. The counterion release has been studied on this system by Fleck et al [2] and
Safran [3] by means of the Poisson-Boltzmann (PB) mean field theory. On the other
hand, our investigation relies on the use of the anisotropic hypernetted chain approximation (AHNC) [4] which has the capability of taking into account, the charge-charge
density and ion’s size correlations which are neglected in PB theory. We considered
the cases of smeared out and discrete distribution of surface charges on the plates. It is
found that in all cases PB theory always predicts a more enhanced, pressure between the
plates, and number of counterions released, than liquid theory does. We found that systems with the least concentration of added salt display a stronger attractive interaction
than those cases where the electrolyte concentration was higher. Similar effects are also
noted on the excess number of monovalent counterions. Thus, the strength of released
counterions is bigger at low salt concentrations than at larger values of bulk electrolyte
concentration. These findings will be described below in the next paragraph.
The model system consists of two identical surfaces with density of electric surface
charge σs = 0.267 C/m2 , each one of opposite sign separated the distance L by an 1:1
NaCl electrolyte solution where the two species of ions are formed by spheres of the
same size of hydrated diameter d = 4.25Å dispersed in an aqueous solvent of dielectric
constant ε = 78.5. The ions direct interaction is represented by the long range Coulomb
electrostatic potential plus a hard ATTACHMENT
core to avoid overlapping between the spheres. We
shall not
image TO
The theoretical
we useOF
is based
ON THEapproach
Ornstein-Zernike (OZ) integral equation of inhomogeneous electrolytes for the: total,
and direct correlation functions and the profile distribution of both species of ions n∓ (z)
in the space between the plates [5].
Such integral equation is solved numerically for the total correlation function and
CP885, Advanced Summer School in Physics 2006, Frontiers in Contemporary Physics—EAV06,
edited by O. Miranda, M. Carbajal, L. M. Montaño, O. Rosas-Ortiz, and S. A. Tomás Velázquez
© 2007 American Institute of Physics 978-0-7354-0385-7/07/$23.00
FIGURE 1. Profile distribution of positive and negative ions n∓ (z) between two asymmetrically
charged surfaces separated the distance L = 9Å. Figure (a) for three cases of σs = 0.267 C/m2 : discrete
(black continuous line, results of AHNC theory), smeared out (dashed line, and AHNC theory), and
smeared out (dotted line, PB theory results using pointlike ions), respectively, with salt concentration
of n0 = 0.01M NaCl. Figure (b) same as Figure (a) but salt concentration n0 = 1.0M NaCl.
n∓ (z) by imposing the AHNC closure relation between the total and direct correlation
functions, and the direct pair potential of two particles. The converged solution of the
OZ equation corresponds to the thermodynamic equilibrium of the whole system and
therefore leads to the profile distribution of positive and negative ions in the slit between
the plates n∓ (z) as a function of the perpendicular distance z from the origin of coordinate located on the left surface. These structural properties are plotted in Fig. 1 for
two bulk electrolyte concentrations n0 = (0.1, 0.01)M of NaCl, wall to wall separation
L = 9Å, and absolute temperature T = 300K. In those figures, black continuous line
corresponds to discrete σs on the surfaces and use was made of the OZ+AHNC equations, dashed line are results of OZ theory for smeared out σs , whereas dotted line is
the PB result of pointlike ions with σs being a smooth distribution of surface charge.
FIGURE 2. Figure (a) provides the comparison of Pslit calculated with AHNC and PB theories as a
function of surfaces separation L at two bulk electrolyte concentrations n0 = (0.1, 0.01)M. Figure (b)
depicts the comparison of the excess number of monovalent counterions N/Å2 as a function of surfaces
separation at the same electrolyte concentrations of Figure (a). In both cases same symbols for lines as in
Fig. 1 and σs = 0.267 C/m2 .
According to Fig. 1, the OZ equation always predicts for the case of discrete density
of surface charge σs a higher ion’s concentration close and in contact with the surfaces,
while for a continuous density of charge σs both OZ and PB theories lead to profiles
n∓ (z) roughly of the same order of magnitude when the salt concentration is low, thus,
for instance at n0 = 0.01M (see Fig. 1(a)). However, OZ y PB approaches lead to different numerical values for n∓ (z) at contact when the concentration of salt is raised up to
n0 = 1.0M (see Fig. 1(b)). In previous work [6] we reported also of an observed increase
in the ions contact densities n∓ (zcont ) for the case of equally charged symmetric surfaces
due to the positional correlations among the bulk ions densities with those charges located at the surfaces when there are present discrete surface charge distributions. This
high contact values of n∓ (zcont ) has a direct consequence in another structural property,
namely, the pressure between the two walls, which is given by the contact theorem as
Pslit = kB T [n+ (zcont ) + n−(zcont )] − 2εs , where zcont is the distance of closest approach of
the center of an ion to one of the charged surfaces.
Figure 2(a) depicts the pressure Pslit between the walls given in Molar units as a
function of wall’s separation L for two concentrations of NaCl, n0 = (0.1, 0.01)M. In
such a figure, black continuous and dashed lines correspond to results of OZ theory
for discrete and smeared out charge σs on the surfaces, respectively, while dotted line
is the PB theory result. We note that the value of the slit pressure between the walls
is found to have higher values at the PB level of approximation than those of OZ
theory with σs being a smooth distribution of charge, and also than the OZ equation
prediction for σs discrete, consecutively. And the strength of the slit pressure is in
any case larger for n0 = 0.01M of NaCl than the case at hight salt concentration of
n0 = 0.1M. Yet the maximun number of counterions released N/Å2 takes place at the
lowest concentration of salt and it occurs in the same order of their relative magnitudes
as for the pressure curves shown in Fig. 2(a) obtained according to the different theories
quoted above. Both Fleck et al [2] and Safran [3] found that the Pslit pressure value
is maximun at the largest counterion release with low salt concentrations in solution.
However, we found quantitative differences among the predictions made by the different
theories, namely, PB and OZ equation, on the structural properties such as Pslit and
N/Å2 . Particularly, those differences are larger for wall to wall separations on the order
of two or more ion’s diameters (see Fig. 2).
This work was supported by CONACyT Grant No. 48794-F, México.
K. Wagner, D. Harries, S. May, V. Kahl, J.O. Rädler, A. Ben-Shaul, Langmuir 26, 303-306 (2000).
A.A. Meier-Koll, C.C. Fleck, and H.H. von Grünberg, J. Phys.:Condens. Matter 16, 6041-6052 (2004).
S.A. Safran, Europhys. Lett. 69, 826-831 (2005).
R. Kjellander and S. Marcelja, J. Chem. Phys. 82, 2122-2135 (1985).
R. Kjellander, T. Åkesson, B. Jönson, and S. Marcelja J. Chem. Phys. 97, 1424-1431 (1992).
O. González-Amezcua, M. Hernández-Contreras, and P. Pincus Phys. Rev. E 64, 041603(1-3) (2001).
A. Flores-Amado and M. Hernández-Contreras, submitted to Phys. Rev. E, August (2006).
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