Eur.Phys.J.B1,327–331(1998)
Effect of surface electricfield on the anchoring
of nematic liquid crystals
G.Barbero1,a,L.R.Evangelista2,and N.V Madhusudana3
1Dipartimento di Fisica del Politecnico and Istituto Nazionale della Materia Corso Duca degli Abruzzi24,10129Torino,Italia 2Departamento de Fisica,Universitade Estadual de Maringa,Avenida Colombo3690,87020-900,Maringa,Parana,Brazil
3Raman Rearch Institute,C.V.Raman Avenue,Bangalore560080,India
Received:29September1997/Received infinal from:10November1997/Accepted:18November1997
Abstract.We analy the influence of adsorbed ions and the resulting surface electricfield and its gradient
on the anchoring properties of nematics with ionic conductivity.We take into account two physical mech-
anisms for the coupling of the nematic director with the surface electricfield:(i)the dielectric anisotropy
and(ii)the coupling of the quadrupolar component of theflexoelectric coefficient with thefield gradient.
It is shown that for sufficiently largefields near saturated coverage of the adsorbed ions,there can be a
spontaneous curvature distortion in the cell even when the anchoring energy is infinitely strong.We also
discuss the director distortion when the anchoring energy of the surface isfinite.
婚宴上绿新郎
PACS.61.30.-v Liquid crystals–61.30.Gd Orientational order of liquid crystals;electric and magnetic
field effects on order–61.30.Cz Theory and models of liquid crystal structure
1Introduction
As in any other sample of condend matter,the surface and interfacial properties of nematic liquid c
信贷网rystals(NLC) are rather complex.In many physical studies as well as practical devices like displays,it is necessary to anchor the orientation of the nematic director at appropriate sur-faces in specific directions.Several techniques have been invented for this purpo[1,2].In practice the anchor-ing has to be characterized by an angle dependent energy density and the simplest form propod by Rapini and Pa-poular[3]consistent with the symmetry of the NLC has the following form:
F s=−
1
a e-mail:barbero@polito.it devices the power consumption has to be reduced to the absolute minimum,and special care is taken to purify the sample.On the other hand,the effect offinite conductiv-ity has very interesting conquences,for example leading to a‘nonlocal’character of the anchoring energy itlf. Indeed there have been experimental studies[8,9]which have clearly demonstrated the necessity to take into ac-count the influence of adsorbed charges on the surfaces in understanding the anchoring properties as functions of thickness and conductivity.In thefirst theoretical models the attention was confined to the coupling of the surface electricfield produced by the adsorbed charges with the dielectric anisotropy of the medium.
All liquid crystals haveflexoelectric properties,and in particular the nonzero quadrupole density arising out of the orientational order in the medium[10]couples with electricfield gradients which can be quite large near the surfaces.In the prent paper we will discuss the gen-eral electrostatic problem near surfaces which incorpo-rates both the dielectric andflexoelectric properties of the medium.The previous treatments of the problem[11–15] were bad on the naive assumption that the dielectric and theflexoelectric torques are reduced to only surface con-tributions,ignoring the elastic torque completely.The were balanced by the torque due to the anchoring energy at the surface.In turn the problem was simply treated as a renormalization in the effective anchoring energy. This approach implies that in the ca of strong anchoring there cannot be an instability due to the surface electric field.In this paper we prent a more general analysis
328The European Physical Journal B of this problem and show that even in the ca of w→∞,
a curvature instability can indeed occur above a threshold
double layer potential.If w isfinite,the threshold poten-
tial naturally gets reduced.
2Theoretical model
We consider the specific ca of an NLC confined between
two glass plates treated for homeotropic ,
the easy axis n0is along z,the normal to the surface.θ(z)
is the polar angle made by the director with respect to the
z-axis.The problem is considered to be one-dimensional,
<,we assume that the surface has uniform properties in
the xy-plane.As we mentioned in Section1,the medium is
assumed to contain ionic impurities,and the surface lec-
tively adsorbs one type of ions(usually positively charged)
with an adsorption energy E.As is well known in the elec-
trolyte theory,such an adsorption produces a counterion
cloud over a depth L d,called the Debye screening length
[16–18].In turn,there is an electricfield which is very
strong near the surface(=E)and decays as we move
away from it.As such,there is a fairly strongfield gra-
dient near the surface.In usual liquid crystals,L d d,
where d is the thickness of the sample.Hence,it is suffi-
cient to treat the ca of a mi-infinite sample bounded
at z=0.The free energy density of the bulk NLC has the
following contributions:
(i)the elastic part which is given by
f el=1
2
K33(n×∇×n)2;(2)
(ii)dielectric coupling with the electricfield given by
f diel=−
a
2Kθ 2(z)−
a
2
sin[2θ(z)]θ (z)E(z),
(5)
whereθ =dθ/d z and e=(e1+e3)is the sum of the two flexoelectric coefficients defined in equation(4).
The surface energy which is generally assumed to be of the Rapini-Papoular form given by equation(1)can have another contribution if the molecules are polar and the anchoring is homeotropic[19].It is now experimentally established that there is a surface polarization P s in such a ca[20–22].The angle dependent part of the total surface energy density now becomes[23,24]
f s=−
1
什么水果降糖
2
w cos2θ0−P s E0cosθ0,
(6) whereθ0and E0are the values of the polar tilt angle and the electricfield at the surface(z=0).Equation(6)was ud to describe planar to homeotropic transitions at the nematic surface[25].
The equilibrium configuration in the bulk medium is given as usual by the Euler-Lagrange equation which yields
Kθ (z)−
商场规划a
2
E (z)sin[2θ(z)]=0,
(7) which has to be solved with the boundary conditions
−Kθ +1
4π
E2(z)+eE (z)
θ=0,(10)
for the Euler-Lagrange equation,and
−Kθ +(eE0+w+P s E0)θ0=0,(11) for the boundary condition at z=0.
We recall that the electricfield E(z)is generated in the prent problem becau of the adsorbed ions
on the surfaces and the counterion cloud forming the diffu dou-ble layer in the liquid crystal.Thefield distribution in this ca is well known and it can be written to a very good approximation as[16]
短整型E(z)=E0exp(−z/L d).(12) The total energy for unit surface area is given by
F=
∞
f d z+f s
=
∞
1
8π
E2(z)θ2(z)−eE(z)θ(z)θ (z)
d z
+
1
G.Barbero et al.:Surfacefield and anchoring energy329 3Analysis
Thefield which acts as the source term for the instability
(e Eq.(10))is localized clo to the limiting surface at
z=0.Thus it is appropriate to consider an approximate
solution of the form
θ(z)=θ0+∆θ[1−exp(−z/L d)]=θb−∆θexp(−z/L d),
(14)
whereθb is the value ofθin the at z L d).
The bulk energy density now takes the simple form
f=
1
2L d +
a在我的生活中
4π
E20L dθb∆θ+eE0(∆θ)2,(17) and C=
a
4L d +
eE0
32π
E20L d+
w+E0P s
2
+
a
16π
E20L d+
w+P s E0
@(∆θ)
=2α∆θ−βθb=0,(23)
and
∂F
∂(∆θ)2
=2α>0,(25) and the Hessian determinant
H=∂2F
∂θ2
b
−
∂2F
16π
E20L d+
2e
2L d
+w
>0,(27)
and
K
3
+
a
4π
E20L d
+
K
3
+
a
2
+
a
2L d
−4π
a L d
(29)
and thefields corresponding to zero crossings ofµare
given by[26]胆小如鼠的意思是什么
E0=
8πe
θ
8πe
a L2
d
.(30)
We can now discuss different possibilities.
4.1Nematic with negative dielectric anisotropy, a<0
In this ca the term in the square root of equation(30)
is always positive and larger in magnitude than thefirst
term.Hence in general there are two values of the surface
field,one negative and another positive,corresponding to
different species of adsorbed charges,between which the
homeotropic anchoring is stable,and beyond which it gets
destabilized.If theflexoelectric coefficient e is positive,the
negative thresholdfield is much larger than the positive
330The European Physical Journal B field,and vice versa for a negative e.The physical meaning
of the results is obvious:while a negative a leads to an
instability of the director if the electricfield is large enough
and has either sign,theflexoelectric term stabilizes the
homeotropic alignment for one of the signs of thefield
gradient depending on its own sign.
4.2Nematic with positive dielectric anisotropy, a>0
In this ca two possibilities have to be considered accord-
ing to the value of a.
If a<8π
K
(31)
the term in the square root is still positive,but smaller in magnitude than thefirst term in equation(30).The insta-bility occurs for some value of thefield,but the homeotro-pic alignment gets restabilized at a cond higher thresh-oldfield in view of the quadratic dependence of the stabi-lizing
dielectric torque on thefield.The sign of e decides the sign of thefield for which the destabilization occurs: for positive e,E0also should be hefield gradient should be negative,and vice versa.
If a>8π
K
(32)
the term under the square root becomes negative and there cannot be any destabilization of the homeotropic alignment.
The ca when a=0will be discusd parately.
5Threshold values forfinite anchoring energy When the anchoring energy isfinite,the zero crossings in relations(27,28)have to be numerically evaluated for given material parameters and the value of w.This has been done,and as can be expected,as w gets smaller,the thresholdfield needed for instability becomes lower.For example,if w=10−2erg/cm2,which corresponds to an extrapolation length of∼0.5µm,which can be attained in the laboratory[8],and a=−1,P s=−10−3esu[23,24], L d=0.1
µm[17,18],K=10−7dyn[27],the threshold double layer potential is about22mV for e=+5×10−4 esu and∼30mV for e=−5×10−4esu[28].The values are easily attained in conducting nematic liquid crystals [17,18].We consider now two simple limiting cas.
5.1Threshold for a dielectrically isotropic medium
For simplicity we assume that P s=0in further analysis. In this ca,when a=0,the stability conditions equa-tions(27,28)read as
2e
2L d +w
>0,(33)
and
K
3
w−
eE0
2|e|
K
3e
−
3e
2
+
2Kw
3e
+
3e
2
+
2Kw
eL d
.(38)
E3is of cour positive for positive e.This means that the
threshold occurs for a double layer potential[27,28]
V th∼
K
5×10−4
∼10−3stat V∼0.3V.(39)
Indeed such voltages are possible across double layers[29].
5.2Threshold for a nonflexoelectric medium
In this ca e=0and as before we assume P s=0.Now
the instability threshold,which can occur only if a<0,
is given by
| a|
2L d
+w
<0,(40)
and
−
2
a
4π
K
12
E20−
Kw
L d
| a|,(42)
who form is reminiscent of the condition for Freedericksz
transition.Again we get a double layer threshold voltage
for a=−4to be V th∼1V;of the same order as in the
previous ca.
The above analysis shows that even when the anchor-
ing energy w is considered to be infinite,a sufficiently
G.Barbero et al.:Surfacefield and anchoring energy331
strong surface electricfield generated by adsorbed ions can lead to a destabilization of the homeotropic alignment.If the dielectric anisotropy is negative and theflexoelectric coefficient e is positive,and the anchoring energy is mod-erate,the destabilizing double layer voltage can be quite low,of the order of0.1V,which can be easily attained in practical cas.
Similar considerations are valid for planar alignment. In this ca it is easier to destabilize nematic liquid crystals with positive a and for a positive surfacefield,materials with negative e.
6Conclusions
We have reexamined the influence of adsorbed ions on the orientation of nematic liquid crystals doped with ionic im-purities.The earlier approaches treated the problem only for the ca of a weak surface anchoring,and treated the effect of the double layer potential as a purely surface ef-fect,ignoring the elastic distortion in the bulk.In this approximation,the surface electricfield just renormalizes thefinite anchoring energy at the surface and hence it does not influence the director profile if the anchoring is strong.We have removed this limitation in our analysis and shown that the surface electricfield can affect the bulk orientation of the director by distorting the director pro-file near the surface.Indeed such a distortion is found even when w→∞.This destabilization has two origins,due to both the dielectric anisotropy coupling with E20,and to theflexoelectric coefficient e=e1+e3coupling with the strongfield gradient near the surface.For homeotropic anchoring,a positive sign of e leads to a destabilization of the director for a negativefield gradient and hence a positive surfacefield,and is the origin of destabilization in materials with positive a.In this ca at a cond larger threshold the director profile gets restabilized in view of the E2-dependence of the dielectric coupling.In the ca of negative dielectric anisotropy materials,such a restabi-lization is not possible.In fact it is often found that it is rather difficult to get a homeotropic alignment of materi-als w
ith negative dielectric anisotropy[29].We feel that the phenomena discusd in this paper can account for the experimental results.
Many thanks are due to A.K.Zvezdin for uful discussions. This work has been partially supported by Istituto Nazionale della Materia and by Dipartimento di Fisica del Politecnico di Torino in the framework of the collaboration between Politec-nico di Torino and the Raman Rearch Institute.References
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