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Ion adsorption and equilibrium distribution of charges in
a cell of finite thickness
G. Barbero, G. Durand
To cite this version:
G. Barbero, G. Durand.
Ion adsorption and equilibrium distribution of charges
in a cell of finite thickness.
Journal de Physique, 1990, 51 (4), pp.281-291.
<10.1051/jphys:01990005104028100>. <jpa-00212367>
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N° 4
Tome 51
15
LE JOURNAL DE
J.
Phys.
France 51
Classification
Physics Abstracts
4l.lOD- 66.10
-
(1990)
82.45
(Reçu
281
15 FÉVRIER 1990,
61.30
-
(*, **)
Laboratoire de
PHYSIQUE
281-291
Ion adsorption and
of finite thickness
G. Barbero
FÉVRIER 1990
equilibrium distribution of charges in a cell
and G. Durand
Physique
le 2 août 1989,
des Solides, Université de Paris-Sud, Bât. 510, 91405
accepté
le 26 octobre
Orsay,
France
1989)
En utilisant une méthode self-consistente, on évalue la distribution d’équilibre de
dans
une cellule d’épaisseur finie, en présence d’adsorption par les parois. L’analyse est
charges
faite dans les deux cas limites où la densité des neutres est fixée, soit très faible (dissociation
totale), soit très grande (dissociation faible). La charge adsorbée en surface est évaluée, en
étendant le problème classique d’adsorption de Langmuir aux situations loin du régime de
saturation. On discute l’importance du problème considéré sur les propriétés interfaciales des
cristaux liquides. On montre en particulier qu’en tenant compte du champ électrique de surface
associé aux charges adsorbées, on s’attend à trouver un caractère non local à la partie anisotrope
de l’énergie d’ancrage. Ceci est en accord avec des expériences récentes montrant que des
échantillons nématiques de même traitement de surface, mais d’épaisseurs différentes, ont des
énergies de surface apparente différentes.
Résumé.
2014
Abstract.
By using a self-consistent method the equilibrium distribution of charges in a liquid
cell of finite thickness, when the adsorption phenomenon is present, is evaluated. The analysis is
performed for the two limiting cases where the neutral density is fixed : either very weak (total
dissociation), or very large (weak dissociation). The surface adsorbed charge is evaluated, by
extending the classical Langmuir problem of adsorption to the case far from the saturation
regime. The importance of the considered problem on the interfacial properties of liquid is
discussed. In particular it is shown that by taking into account the surface electric field associated
with the adsorbed charges, a non-local character of the anisotropic part of the anchoring energy of
liquid crystals is expected. This agrees with recent experimental observations showing that
nematic samples of different thicknesses, but with the same surface treatment, seem to have
2014
different anchoring
energies.
(*) Partially supported by
Ministère de l’Education Nationale, de la Recherche et des
Sports
de
France.
(**)
Also
Dipartimento
di Fisica, Politecnico, C.
so
Duca
degli
Abruzzi 24, 10129 Torino,
Italy.
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01990005104028100
282
1. Introduction.
The surface properties of liquid/solid interfaces depend on the physical chemistry of the two
media, and on long range forces like Van der Waals and double layers [1]. In this paper we
limit ourselves to consider the effect of the double layer forces, connected to selective surface
adsorption of ions dissolved in the liquid. We show that the ion adsorption can be responsible
for apparent non locality of interface properties recently observed in nematic liquid crystals
[2-5].
These ions can originate from more or less dissociated impurities (the extrinsic contribution), but also from the spontaneous dissociation of the liquid molecules themselves (the
intrinsic contribution). The positive and negative ions can have different affinities for the
boundary surfaces. Their selective adsorption creates at the liquid interface a surface field
which depends on the volume of the sample. This surface field can modify the surface
properties of the solid/liquid interface. This idea was already suggested [6], but not
quantitatively worked out.
The mechanism we imagine is the following : the bulk liquid is neutral but contains positive
and negative ions, in fixed concentration p é
P e P e in absence of adsorption from the
is
plate. pc given by the fixed concentration of the totally dissociated impurities, or by the
dissociation constant of the liquid. To keep the calculation general, we consider Pe as an
independent parameter. We assume a selective adsorption on the solid substrate for, say,
positive ions, with an adsorption energy Ea, and for instance, a very large repulsive energy for
the negative ions, so that their surface density is always negligible. When putting in contact
the liquid and the substrate, positive ions will migrate toward the interface and create a
surface electric field Es localized inside a boundary layer of a Debye screening length
Ls. Inside this boundary layer, the coupling of Es with the dielectric anisotropy of the liquid
gives rise to an additional orientational dielectric free energy. This new contribution can be
considered as quasi-local and renormalizes simply the interfacial properties of the liquid/solid
system, although Es is, in principle, a non-local quantity. It is indeed sufficient to explain how
Es changes with the thickness of the cell d, because for instance of the saturation of
adsorption, to explain the apparent long range interaction between the two plates. Let us call
cl the adsorbed positive charge density (and the thickness integrated negative volume
screening charges). We discuss to simplify a one dimensional model. In the liquid bulk, the
new charge densities p ± (z ) depend on the distance z from the solid boundary plates. The aim
of the calculation is to estimate the surface field Es (or the absorbed charge u) vs. the initial
charge concentration p,,, for samples of various thicknesses d.
We use a self-consistent method to solve the problem : we first write that all parts of the
liquid, which exchange positive and negative ions, are in equilibrium in an arbitrary potential
V (z). This fixes a Boltzmann shape for p± [z, V (z)], which depend only on two parameters,
the densities p ± (0) at the center of the cell. We determine V (z ) from the charge densities
using Poisson equation, which can be integrated if we know the adsorbed charge density
a on the surfaces. cr is defined itself from the
equilibrium of positive charges, which are
between
and
the
the
surface
bulk, and depends also on V (z), i.e. on
exchanged
now
the
conservation
p ± (0). Writting
charge
equation (or the mass action law), we obtain
two
the two concentrations p± (0). The
to
determine
finally
self-consistency equations
one
of
contains
two
kind
the
then
problem
describing the electric properties,
equations :
in
one
section
and
the
the
2,
analyzed
charge exchange equilibrium, in the bulk
describing
and
the
and
bulk
surface
between
the
(Sect. 2)
(Sect. 3). This last equilibrium is very similar
to the classical adsorption problem of Langmuir [7]. The electric part has been already
partially analyzed in a different context for isotropic liquid [8]. Finally in section 4 we discuss
=
=
283
the influence of the surface field
local effect.
2. Basic
équations of
the electric
on
the
anchoring
energy, to
explain
the cell thickness
non-
problem.
Let us consider a liquid containing ions of density p,,, in thermodynamic equilibrium in
absence of adsorption. The equilibrium is defined by constant temperature and volume
conditions (we neglect the compressibility of the liquid). We analyze a unidimensional
problem. In our reference frame the boundary plates are placed at z ± d/2 (see Fig. 1).
Furthermore we suppose the initial electroneutrality of the liquid and that the two surfaces,
assumed identical, adsorb only positive ions. If a is the surface density of adsorbed ions,
p ± (z ) the bulk densities of positive and negative ions, the electroneutrality condition imposes
=
1.
Expected thickness dependence of the reduced potential u, and the ion charge densities
p+ and p_ . The plates are assumed to adsorb only positive ions (surface density o,). pe is the equilibrium
concentration of the neutral solution in absence of adsorption. The electric fiels is localized on a Debye
screening length close to the plate at ± d/2.
Fig.
-
In principle, the ionic dissociation obeys a chemical equilibrium, where the density of the
neutral species is important. As we have no information on the practical origin of the ions, we
have specialized our analysis for the two limiting cases where the neutral density is fixed : or
very weak (total dissociation), or very large (weak dissociation). In the case of totally
dissociated impurities, we must impose the conservation of the total number of each (positive
or negative) ions, i.e.
In the case of intrinsic
action law :
where pci is
a
conductivity
constant which
or
weakly
depends only
on
ionized
impurities,
the temperature.
we
must use the mass
284
In the
liauid
gas approximation
be written as
perfect
can
(for small Pe) the chemical potential
IL:t of the ions in the
where À th is the average thermal de Broglie wave-length [7] of the ions, ka is the Boltzmann
constant, T the absolute temperature, q the electrical charge of the ions and V (z ) the
macroscopically averaged electrical potential. Since only differences in potential are
physically significant, we put V(0) 0, in the center of the cell (where p ± (0 ) = p ô ) and
consequently E (o ) 0 because of symmetry.
We require at equilibrium that the chemical potentials are uniform across the liquid
sample ; from equation (3) we then obtain that the steady-state distributions of mobile
charges obey the Boltzmann distribution
=
=
In all considered cases, pô , différent from pe,
account (4) Poisson équation reads
where - is
an
average dielectric constant, and the
and taking into account that, for symmetry
hypothesis U(z ) U (- z ) and U’(0) 0,
=
from which
Boundary
we
=
obtain
condition
US
=
on
It is easy to
write :
verify
=
By taking into
d/dz. By putting :
considerations coming from the electroneutrality
we can rewrite equation (5) as :
the electric field at
we
case
(2)
of
we
z
=
-
d/2 gives
have
U(- d/2). By equation (7)
We consider first the
into account equation
symbol ’
détermine.
easily :
By using equations (7) and (8)
where
are two constants to
totally
we
obtain
dissociated
impurities. Using equations (4)
and
taking
deduce
that the total system remains neutral since,
using equation (9),
we can
285
The integrand in equations (10) and (11) present an apparent divergence at z
0;
furthermore they are written in absolute units. It is better to rewrite them in a dimensionaless
form. By putting
=
equations (9), (10)
and
(11)
rewrite
as :
where :
«
D is measured in terms of the equilibrium Debye length Le, whereas U e
p e Le is a natural
surface charge density unit.
In the case of intrinsic conductivity (or weakly ionized impurities) equation (16) is changed
in
=
coming
For
a
from mass action law.
fixed cr, the z-dependence of p±
3. Surface
(z)
and
U(z)
is shown
qualitatively
in
figure
1.
charge and the adsorption phenomenon.
In order to determine the surface charge density adsorbed by the solid surface let us
consider the classical Langmuir problem of adsorption [7]. Let us call 03C3M/q the maximum
number of adsorption site per cm2 on the surface (q is the proton charge). The simplest model
to explain the selective attraction Ea is to imagine that positive ions are very small compared
to negative ions, and that they are attracted by a highly polarizable solid medium.
UM can then be estimated as q/m 2, where m is a molecular size. Writing that IL+ is the same
on the surface and in the bulk, we obtain the covering ratio 9 = U / UM in the form :
where
Ea
is
positive
in the considered
case
of
adsorption. Using equation (13)
we can
write
ka T ln (p c À3th) is the chemical potential in the absence of adsorption. By putting
equation (20) into equation (19) we obtain
where IL c
where
=
equations (13)
have been taken into account, and furthermore
we
have put
286
We are interested in
A &#x3E; 1. This leads to
a
non-saturating regime, i.e.,
as
follows from
equation (21),
UM/ A. Note that our hypothesis of « infinite » repulsion for negative ions from
would not be valid any more if 0-L were comparable to UM, because of the
electrostatic attraction between positive and negative ions on the surface, i.e. in the case of a
too small A coefficient.
Equations (14)-(16) or (16’) and (21) are valid for any d, but are quite complicated in
general. We consider first the fully ionized case, i.e. the one where each number of positive
and negative charges is separately conserved. We are interested in two limits : 1) the small
thickness cells are expected to produce a small u, then a small U ; 2) the large thickness cells
are expected to give a saturation of o-, with a large U.
In the limit of small U(i.e. q V IkB T « 1 ), which implies ts « 1, we can expand in the usual
way equations (14)-(16) to obtain
where 0-L
the plate
From
=
(15’)
and
(16")
we
deduce
Consequently
and
Equations (23)-(25) have been deduced by supposing ts 1; hence they are valid for
(u / u c) D 8. By considering that R1 &#x3E; 0, from equation (25) we also obtain 0’/U,, D/2.
Since D is small in the considered limit, from the above conditions we deduce that
into equation (21’) and by
equations (23)-(25) hold for D 4. By substituting equation (25)
solving with respect to u we obtain :
from which
Equation (27)
shows that in the limit of small thickness, cr is proportional to d. In this regime
(Us : 1), the potential drop between the plate and the bulk is small compared to
kB T/q. Most of the positive ions are trapped on the surfaces, which are each covered by half
of the initial charge number p e d. We point out that equation (26) is not valid in the
287
limit since it has been deduced
equation (26) holds only for
D --&#x3E;
00
by supposing
that ts
1. From
(24)
we
obtain that
since usually 0-,,/O’L - 1. Previous estimation agrees with the one reported above.
In order to have some information at large D let us consider now the case where
ts &#x3E; 1. Since d &#x3E; Le, the liquid remains practically neutral in the center of the sample, and
hence R1 ~ R2 ~ 1. From (21) we obtain for the saturation charge density
and for ts the
We
can now
expression
define
a
critical thickness d * for which the two
condition 1/22 Pc d * = U L’
from which,
by taking
régimes merge. It is given by the
equation (28), we obtain
into account
where we use the fact that at room temperature q2/ eka T is an order of magnitude larger than
a molecular size m. The maximum surface covering ratio uL/uM’" 1 /A can be taken
0.1, i.e. A - 10. A small ion, of atomic size, could have a dielectric attraction energy with
the boundary Ea ’" 1 eV. This results in a practical concentration Pe ’" q .1013 cm- 3, easy to
achieve [9]. In this case, we find a maximum value for d* - 50 ilm comparable with typical
nematic liquid crystal cell thickness. The fact that d* is macroscopic is obviously related to the
amplification factor 40 Le/mA.
To conclude, for small cell thickness, (d d* ~ 50 ktm for typical Pe), most positive ions
are adsorbed on the two plates with a surface density proportional to d. This is a very
reminiscent of the surface purification process already demonstrated [10]. For large thickness,
the surface density a is saturated classically by the now fixed p, e of the solution. This
thickness dependence of U (d) is sketched in figure 2.
=
Reduced surface charge density versus reduced thickness for a fully dissociated impurity.
Fig. 2.
Below D*, most positive ions are adsorbed on the plate and the surface field increases with
D. Above, D*, the surface density saturate, and also the surface field.
-
288
discuss the case of intrinsic conductivity (or weakly ionized impurities).
Equations (14)-(15) and (16’) must be solved with equation (21’). Using equations (16’) and
(21), from equation (14) we obtain
Let
us
now
in the limit of
we have
uL/2 o-c &#x3E; 1, usually verified. By substituting equation (24’) into equation (15)
giving D = D (R1 ) and plotted in figure 3. This
approximated graph of the previous case (Fig. 2).
Fig.
3.
-
Same
as
figure 2,
As previously, let
equations (14’), (25’)
us
we
but for
a
weakly
exact
plot
must be
compared
with the
dissociated system.
consider first the limit of small ts, which
obtain :
implies
small D. From
and
Equations (24"), (25") hold for D 4 (2 U e/ UL)I/3, usually very small. Hence these
approximate equations are not very useful.
In the opposite limit of large D we have 7?i - R2 - 1. Consequently, as in the previous case,
tS - U L/2 u c and cr = 0- L, as expected. One can practically define a d * - 5 Le for which the
saturation is obtained. For samples thicker than d *, one is in the classical Langmuir limit : the
surface charge a is fixed by the ion concentration and not by the volume of the sample. Below
d *, U decreases with the thickness d.
4. Influence of the adsorbed
substrate surface energy.
charge
on
the
anisotropic part of
the nematic
liquid crystal,
In sections 2 and 3 it is shown that the ionic charge that is stuck at the solid-liquid surface
a diffuse layer of ions of the opposite sign in the liquid. The adsorbed charge and the
attracts
289
layer of oppositely charged ions that it attracts constitute an intrinsic double layer
depending only on the solid and liquid. The thickness of this double layer is of the order of the
Debye screening length.
It follows that the selective surface adsorption of ions creates an electric surface field
Es which extends in the bulk over the Debye screening length Ls. If an isotropic liquid is
considered this surface electric field can change the effective surface tension, which is an
isotropic quantity. In the opposite case where an anisotropic liquid, as a nematic liquid
crystal, is considered, the surface field can also change the anisotropic part of the surface
energy. In this section we wish to analyse the influence of the selective adsorption
phenomenon on the so-called nematic /substrate anchoring energy.
As known a nematic material is characterized by anisotropic physical properties [1] as
dielectric, or magnetic, permittivity. The physical properties of a nematic depend on the
average molecular orientation, usually indicated by fi. û is known as nematic director. When
a nematic is put over a solid substrate, û alignes parallel to a well defined direction
à called easy direction [1]. If now an extemal field is applied, whose effect is such to change
the surface orientation of n, the surface tends to maintain the surface director ns parallel to
Tf, by means of a restoring torque. This restoring torque takes origin by the anisotropic part of
diffuse
the surface energy which controls the surface orientation of the nematic. The surface energy
results from short range forces, so that it is expected to be a quasi-local property of the
nematic/substrate interface. Recently some experimental determination of surface energy
have shown an apparent long range dependence of this parameter vs. the macroscopic size of
the cell d (d 3 --. 100 ktm ). Since a direct influence between the two surfaces a macroscopic d
apart is difficult to imagine we think this strange d-dependence could be explained by
considering the influence of the surface electric field E,, coming by the selective adsorption,
on the surface energy. Since the nematic presents anisotropic physical properties, the simplest
coupling of Es with the director is the one related to the anisotropy of the dielectric constant
8. Calling, as usual, Ea
El - E 1. the dielectric anisotropy (where Il and 1 refer to
the
dielectric
of
n),
coupling Es with n gives a surface free energy density
=
in the
simplest exponential approximation for E (z ). In (29), ûs is the surface director and
LS penetration length given by : Ls Le[(7?i + R2)/21+ 1/2, not very far from Le.
According to the sign of Ea, FE is minimum for ns parallel (Ea:&#x3E; 0) or perpendicular
(Ba : 0) to Es, which is obviously normal to the limiting surface. The dielectric energy
FE must be added to the intrinsic surface energy form, which tends to align û along the easy
axis Tr (fr2 =1 ), normal or parallel to the plates. We assume that the Rapini-Papoular form
[11] is a good approximation for the intrinsic anchoring energy. The total surface free energy
density is then :
a
=
where Ws is the anchoring strength and Es is given by equation (8). Since Es depends now on
d, a non-local « size effect » is expected. Note that Ls appearing in equations (25-31) also
depends on d, but in a weak way. In fact this parameter changes from Le to
Le for d
ranging from infinite to zero. Equation (30) shows that the anisotropic part of the
nematic/substrate surface energy contains two terms : the first connected to short range
interaction forces and hence independent of the sample thickness ; the second coming from
the dielectric interaction between the nematic and the surface electric field Es. Since
(1/B/2)
290
Es depends on the adsorbed charge, which is thickness dependent, we conclude that also the
effective anisotropic surface anchoring energy is expected to be thickness dependent. This
agrees with recent measurement performed by a Russian group [2-5], showing a long range
dependence of the anchoring energy vs. the macroscopic size of the cell d (3 : 100 um).
We can now compare these predictions with the experimental data reported in reference
[4]. Note first that the various samples at various d are differents, which allows each sample to
represent an equilibrium situation at constant p e, probably the same for each sample. The
starting geometry is planar and the used nematic liquid crystal is 5 CB, with a positive
Ea =13 [12]. This combination is the one of the two which can give rise to a decrease of
Ws and even to an instability if Ea &#x3E; 2 WS E2/ (Ls ul) as follows from equation (30) and
equation (8). In reference [4], only a decrease and a saturation is observed, which means that
the above mentioned condition is probably not fulfilled. To fit the data of reference [4], we
should know the conductivity of used sample, to estimate Pe. The only reported data
connected with the conductivity is the 1 kHz AC voltage frequency which suppresses the
electrohydrodynamical effects. The dielectric relaxation frequency of the sample O’IE is then
lower than 103 Hz. Using for the ions mobility the value of reference [9], we estimate
pc~ 10-4 C/cm3, and hence L,, - 0.1 itm. We can now try to fit the reference [4] data to
determine Ws and 0-L. Note first that the model of weak dissociation is not very good to
explain the observed saturation around d - 20 = 30 lim, since it predicts a
d* - 5 Le - 0.5 um. We have tried a fit with the fully dissociated impurities model. The result
is shown in figure 4. The best values are Ws - 1 erglcm2, and 0 L - 4.3 x 10 - 7 C/CM2
Predicted dependence of the anchoring energy versus thickness, for the case of nematic liquid
5 CB. One assumes a simple dielectric coupling between the nematic orientation and the surface
field. The points are experimental data from reference [4].
Fig. 4.
crystal
-
The value for Ws is a bit larger than the one usually measured [13], but must be accepted
since it is a reasonable extrapolation of the data of reference [4]. From OL’ we can estimate
the mean distance between ions on the surface is l&#x3E; ~ (UL/q )-1/2 ’" 2 x 10- b cm, and hence
A - 102. These values are at the limit of validity of our model. They just indicate that the
general trend is correct. To check the model more carefully, one should know better the
experimental parameters, and describe in more details the interaction between neighbouring
sites, which has been neglected in our model.
291
5. Conclusion.
In this paper we have extended the classical Langmuir problem of adsorption in order to take
into account the electrical properties of a sample selectively adsorbing ions contained in a
given liquid. The analysis shows that the surface electric field built by trapped ions depends on
the sample thickness, in a macroscopic range. This implies that in ordinary liquids the
interface properties strongly depend on the adsorbed charge. In particular we have focused
our attention on the anisotropic part of the anchoring energy nematic/solid substrate, showing
that apparent long range dependence of this quantity can be easily explained with our model.
The solutions derived here for the electrical problem can be useful to test different mechanics
proposed in order to study the surface properties of liquids, when a selective adsorption is
present.
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