Ion adsorption and equilibrium distribution of charges in a cell of finite thickness

HAL Id: jpa-00212367

https://hal.archives-ouvertes.fr/jpa-00212367

Submitted on 1 Jan 1990

HAL is a multi-disciplinary open access

archive for the deposit and dissemination of

sci-entific research documents, whether they are

pub-lished or not. The documents may come from

teaching and research institutions in France or

abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est

destinée au dépôt et à la diffusion de documents

scientifiques de niveau recherche, publiés ou non,

émanant des établissements d’enseignement et de

recherche français ou étrangers, des laboratoires

publics ou privés.

Ion adsorption and equilibrium distribution of charges in

a cell of finite thickness

G. Barbero, Geoffroy Durand

To cite this version:

281

Ion

_adsorption

and

_equilibrium

distribution of

_charges

in

a

cell

of finite thickness

G. Barbero

_{(*, **)}

and G. Durand

Laboratoire de

_Physique

des Solides, Université de Paris-Sud, Bât.

_510,

91405

_Orsay,

France

(Reçu

le 2 août _1989,

_accepté

le 26 octobre

₁₉₈₉₎

Résumé. 2014 En utilisant _une méthode

self-consistente, on évalue la distribution

_{d’équilibre}

de

charges

dans une cellule

_{d’épaisseur}

_finie, en

_{présence d’adsorption}

_{par les}

_{parois. L’analyse}

est

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. Abstract. 2014

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

different

_{anchoring energies.}

J.

_Phys.

France 51

₍₁₉₉₀₎

281-291 15 FÉVRIER 1990,

Classification

Physics

Abstracts 4l.lOD- 66.10 - 82.45 - 61.30

LE

JOURNAL

DE

_PHYSIQUE

(*) 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.

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

contri-bution),

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

plate.

pc is

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

exchanged

between the surface and the

bulk,

and

_depends

also on

_{V (z),}

i.e. on p ±

(0). Writting

now the

_charge

conservation

_{equation (or}

the mass action

_law),

we obtain

finally

two

_{self-consistency equations}

to determine the two concentrations _p±

_(0).

The

problem

contains then two kind of

_{equations :}

the one

describing

the electric

_properties,

analyzed

in section

_2,

and the one

_describing

the

_{charge exchange}

equilibrium,

in the bulk

(Sect. 2)

and between the bulk and the surface

_{(Sect. 3).}

This last

_equilibrium

is very similar

the influence of the surface field on the

_anchoring

_energy, to

_explain

the cell thickness

non-local effect.

2. Basic

équations

of the electric

_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

Fig.

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.

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

conductivity

or

_weakly

ionized

_impurities,

we must use the mass

action law :

In the

_perfect

gas

approximation

(for

small

Pe)

the chemical

potential

IL:t of the ions in the

liauid can be written as

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, are two constants to détermine.

_{By taking}

into

account

₍₄₎

Poisson

_équation

reads

where - is an _{average dielectric} constant, and the

_{symbol ’}

=

d/dz.

By

putting :

and

_taking

into account

_that,

for

_symmetry

considerations

_coming

from the

_{electroneutrality}

hypothesis

U(z )

=

U (- z )

and

U’(0)

=

0,

we can rewrite

_{equation (5)}

as :

from which we obtain

_{easily :}

Boundary

condition on the electric field at z = -

d/2

gives

By using equations

(7)

and

₍₈₎

we have

where

_US

=

U(-

d/2).

By

equation

(7)

we obtain

We consider first the case of

_totally

dissociated

_impurities.

_Using

_{equations (4)}

and

_taking

into account

_equation

₍₂₎

we deduce

It is easy to

_verify

that the total

_system

remains neutral

since,

using equation

(9),

we can

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

from mass action law.

For a fixed cr, the

z-dependence

of p±

(z)

and

U(z)

is shown

qualitatively

in

_figure

1.

3. Surface

_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

where IL c =

ka

T ln

(p c À3th)

is the chemical

potential

in the absence

of adsorption.

_By

putting

equation

(20)

into

_equation

₍₁₉₎

we obtain

We are interested in a

_{non-saturating regime,}

_i.e.,

as follows from

equation (21),

A > 1. This leads to

where 0-L

=

UM/ A.

Note that _our

hypothesis

of _« infinite _»

repulsion

for

_negative

ions from the

_plate

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

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

>

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

equations

(23)-(25)

hold for D 4.

_By

_{substituting equation (25)}

into

_equation

_(21’)

and

_by

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

D --> 00 limit since it has been deduced

_{by supposing}

that

_ts

1. From

₍₂₄₎

we obtain that

equation (26)

holds

_only

for

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

> 1. Since d >

_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

_expression

We can now define a critical thickness d * for which the two

_régimes

_{merge. It is}

_{given by}

the

condition 1/2 Pc d * = U L’

from

which,

by taking

into account

_equation

_(28),

we obtain

2

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.

Fig.

2. -

Let us now 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

in the limit of

_uL/2

_{o-c >}

_1,

_usually

verified.

_By

_substituting

_{equation (24’)}

into

_equation

₍₁₅₎

we have

giving

D = D (R1 )

and

_plotted

in

_figure

3. This exact

_plot

must be

_compared

with the

approximated graph

of the

_previous

case

(Fig. 2).

Fig.

3. -

Same as

_figure

_2, but for a

_weakly

dissociated _system.

As

_previously,

let us consider first the limit of small

_ts,

which

_implies

small D. From

equations

(14’), (25’)

we obtain :

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

_charge

on the

_anisotropic

_part

of the nematic

_{liquid crystal,}

substrate surface energy.

In sections 2 and 3 it is shown that the ionic

_charge

that is stuck at the

_solid-liquid

surface

diffuse

_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 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

n),

the dielectric

_coupling

of

_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

a

_{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:> 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 :

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

(1/B/2)

_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}

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 > 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

Fig.

4. - Predicted

dependence

of the

_anchoring

_energy versus

thickness,

for the case of nematic

_liquid

crystal

5 CB. One assumes a

_simple

dielectric

_coupling

between the nematic orientation and the surface field. The

_points

are

_experimental

data from reference

_[4].

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> ~ (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

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.

References

[1]

YOKOYAMA H., Mol.

_Cryst.

_{Liq. Cryst.}

165

₍₁₉₈₈₎

_{265 ;}

YOKOYAMA H., KOBAYASHI S., KAMEI H., J.

_{Appl. Phys.}

61

₍₁₉₈₇₎

4051. BLINOV L. M., KATS E. I., SONIN A. A.,

Usp.

Fiz. Nauk. 152

₍₁₉₈₇₎

449.

[2]

CHUVIROV A. N., Sov.

_Phys.

_Crystallogr.

25

₍₁₉₈₀₎

188.

[3]

BLINOV L. M., SONIN A. A., Sov.

_Phys.

JETP 60

₍₁₉₈₄₎

272.

[4]

BLINOV L. M., KABAENKOV A. Yu., Sov.

_Phys.

JETP 66

(1988)

1002.

[5]

BLINOV L. M., KABAENKOV A. Yu., SONIN A. A., Presented at XII Int. L.C. Conference,

Freiburg

15-19

_{August (1988)}

_p. 397 ;

Liq. Cryst.

5

₍₁₉₈₉₎

645.

[6]

BARBERO G., DURAND G.,

Liq. Cryst.

2

₍₁₉₈₇₎

401.

[7]

KUBO

_Ryogo,

Statistical Mechanics

_(North.

Holland, Publ. Co.

_Amsterdam)

1967, p. 92.

[8]

THURSTON T. N., J.

_{Appl. Phys.}

55

₍₁₉₈₄₎

4154.

[9]

THURSTON R. N., CHENG J., MEYER R. B., BOYD G. D., J.

_{Appl. Phys.}

56

₍₁₉₈₄₎

263.

[10]

YOKOYAMA H., KOBAYASHI S., KAMEI _H., J.

_{Appl. Phys.}

56

₍₁₉₈₄₎

2645.

[11]

RAPINI A., PAPOULAR M., J.

_Phys.

_Colloq.

France 30

₍₁₉₆₉₎

C4-54.

[12]

KARAT P. P., MADHUSUDANA N. V., Mol.

_{Cryst. Liq. Cryst.}

36

₍₁₉₇₆₎

51.

[13]

YOKOYAMA H., VAN SPRANG H. _A., J.

_{Appl. Phys.}

57

₍₁₉₈₅₎

4520.

[14]

ZHANG FULIANG, DURAND _G., J.

_Phys.

France 50

_(1989).

[15]

BARBERO G., DOZOV I., PALIERNE J. F., DURAND _G.,

_Phys.

Rev. Lett. 56

₍₁₉₈₆₎

2056.