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Anchoring energy and easy direction of non uniform surfaces
G. Barbero, T. Beica, A. Alexe-Ionescu, R. Moldovan
To cite this version:
G. Barbero, T. Beica, A. Alexe-Ionescu, R. Moldovan. Anchoring energy and easy direction of non uniform surfaces. Journal de Physique II, EDP Sciences, 1992, 2 (11), pp.2011-2024.
�10.1051/jp2:1992248�. �jpa-00247785�
Classification Physics Abstracts 61.30
Anchoring energy and easy direction of non uniform surfaces
G. Barbero (~), T. Beica (2), A-L- Alexe-Ionescu (~) and R. Moldovan (~)
(~) Dipartimento di Fisica, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy
(~) Institute of Physics and Technology of Materials, CP MG 7 Bucharest, Rumania
(~) Politechnical Institute of Bucharest, Department of Physics,Splaiul Independentei, 313, 77216 Bucharest, Rumania
(Received 9 June 1992, accepted in final form 4 August 1992)
Abstract The erect of a periodic distribution of easy directions on the surface macroscopic properties of nematic samples are considered. Our analysis shows that the elective surface energy of the sample contains an interference term coming from the two diferent easy axis. This allows to introduce an elective uniform easy direction, depending on the two easy directions and containing an interference term too, as suggested a few years ago by some researchers.
1 Inti~oductiou.
The orientation induced by a solid substrate on a nematic liquid crystal is important both for
technological and fundamental aspects. Alany papers are devoted to this subject [1-4]. The published analyses consider the problem relevant to the nematic-substrate interaction both from pl~enomenological [5-7] and microscopic [8-10] points of view. The phenomenological description of the surface interaction is made by using the concepts of easy orientation ar and of anchoring strength w. The easy direction ar is defined as the average nematic orientation n
(the nematic director) on the surface minimizing the surface energy fs, in the absence of bulk distortion. The anchoring strength is defined as the curvature of fs around its minimum. A very popular expression of is is the one proposed long ago by Rapini and Papoular [iii, which is of the kind: fs(n ar)
= fs(1) + w(n ar)~. The phenomenological approach is widely used in 2
literature [12-16], because it gives the possibility to describe, by means of few parameters, the nematic liquid crystal-substrate interaction. The microscopic approach is, on the other hand,
very important to understand >vhich mechanism is responsible for the orientation imposed by
the substrate on the nematic. Of course the number of parameters necessary for a complete microscopic description is very large. llowever the microscopic approach also confirms that the phenomenological expressions used in literature are correct. The papers relevant to the
phenomenological approach try to connect the physical properties of the substrate with the orientation induced on a given liquid crystal. The effect of the surface geometry has also been taken into account [17-20].
In most of the published papers the surface is considered uniform and characterized by a well defined easy direction and anchoring energy strength. But, as is well known, the solid substrate
is never homogeneous over a large scale, even if the chemical composition of the surface does
not change. Hence it seems interesting to consider the effect of a spatial distribution of easy directions on the experimentally detected easy direction. This analysis could be important also in the case in which a surface is characterized by more than one easy axis, as in the case of
evaporated surfaces [21-23].
Our paper is organized as follows. In section 2 a semi-infinite periodic structure, with
strong anchoring, is considered. The analysis, which generalizes the one proposed long ago by Berreman [17] and de Gennes [18] and recently by Durand [19] et al., gives the opportunity
to introduce the equivalent surface energy of a periodic structure. In section 3 the finite
anchoring energy, in a semi-infinite sample, is taken into account. Our analysis, very near to the one performed by Faetti [20], generalizes the results obtained in section 2. We will show there that the different easy directions give rise to a kind of interference in the surface energy.
In section 4 a finite sample characterized by an imposed deformation in the bulk is considered.
In this way >ve can define an equivalent easy axis. This is done comparing our results with the
well known one, relevant to a uniform sample. It gives, furthermore, the possibility to treat a
surface with periodic structure as a homogeneous surface. In section 5 the main conclusions of
our paper are given.
2. Semi-infinite sample with sti~ong alichoi~ing energy,
Let us consider a solid surface characterized by a periodic structure of wavelength ~ along the z-direction- Les a and fl be the easy axes for 0 < z < a and a < z < ~ (see Fig. I):
~~
T
a
li
O ~ ~ x
Fig. i. Semi-infinite liquid crystal sample ~vith periodic distribution of easy directions.
aforo<z<a,
8(z) = (I)
fl for
a < z < ~.
The total elastic energy of one period, of the nematic liquid crystal (unbounded along y) is given by:
F = lk it°' ~iv~)~d~dY 12)
in the
one constant approximation, and by supposing that
n is everywhere parallel to the
(z,y)-plane. Equation (2) holds in the strong anchoring case, I-e- in the event in which R, the angle made by the director n with y, at y = 0 is imposed by the surface treatment. In our
case:
~(z,0) " 8(z) (3)
where 8(z) is given by (I). By minimizing (2) we obtain for R(z, y) the equation
fi2@ fi2@
4 ~ fiy2 ~ ~~~
i-e- the function minimizing (2) is a harmonic function. This function has to satisfy the
boundary condition (I). This is the well known problem, called Dirichlet's problem, in which
we have to determine
a harmonic function knowing its value
on the boundary. Equation (4)
can be solved directly, as shown in Appendix I, but for further generalization it is better to expand R(z, y) in Fourier series. By putting
~> "
R(z, y) = ° + £ [D[cos(nqz) + E(~sin(nqz)] e~~~Y (5)
~
n=1
and imposing boundary condition (3), the expansion coefficients are found to be
~l ~ ~" + l~ ~)fl
0 ~
D[ "
) ~
~ ~si~l'~q~)
~~ /q ~ ~ ~~~ ~°~~~~~~~' ~~~
where q =
~'~
is the wave vector of the periodic structure. By substituting (5) and (6) into
~
(2) one obtains
~ ~
~
~ ~
j~ ~~~~~~~ ~~~
n=
This is the total energy per unit length along the z-direction- It can be considered as the
equivalent surface energy of the sample [17, 18]. Note that for a
= 0 or a = this energy
is zero. Furthermore it is zero for a = fl. These results are obvious. Keeping in mind the
definition of q, it is easy to show that F/~ is proportional to (a fl)2/~, i-e- to the elastic energy due to the periodic distortion imposed by the surface treatment. Note that:
D[ au + (~ a)fl
I ~ ~~
is the average of the easy axis, where
lim R(z, y) = ~" ~ ( ~~~
= Be, (8)
y - c<
is the tilt angle imposed by the surface at y - c<, I-e- very far from the surface. It follows that Be can be considered as the easy axis due to the periodic structure under consideration (see
Appendix 2). The results obtained in this section generalise the analysis presented in reference [24] for the particular case of anchoring on vicinal silicon surfaces.
3~ Semi-infinite sample with weak anchoi~ing energy~
Let us consider now a more general situation in which the easy axes are still the ones of equation (I), but the anchoring energies are finite on 0 < z < a and a < z < ~, and they have the same
value w. In this event the total elastic energy of the nematic sample is given by
F = )k/~~(VR)~dzdy
+ jw/~[S(z,0) 8(z)]~dz. (9)
o o o
As in the previous case the sample is considered unbounded along the y-direction and undis- torded for y - c<. In equation (9) the first term represents the bulk contribution, and the second term, the surface contribution. The surface term is written in the parabolic form, valid for small deviation from the easy axis. By minimizing (9) trivial calculations show that for 0 < z < ~ and 0 < y < c<, R(z, y) is still given by equation (4), but now it has to satisfy the
boundary condition
()) + )lsl~>0) 81~)1" °> (lo)
y=o
at y = 0. In (10) L = k/w is the extrapolation length. The mathematical problem is now called Neuman-Dirichlet's problem (mixed), in which we have to determine a harmonic function
knowing a relation betwen its normal derivative and its value on the border. As shown in
Appendix I this problem can be solved easily only by means of a Fourier expansion. By putting expansion (5) into equation (10) we obtain now for the expansion coefficents
,, aa + fl(~ a)
Do " 2
~
~~
l +~qL ~n ~° ~~~~~~~~~~
~~
l
~iqL (ii
~° ~~~~ ~°~~'~~~~~ ~~~~
instead of (6). Note that also iii this case equation (8) holds, I-e- very far from the solid substrate S(z, y) tends to Be (See Appendix 2). By assuming 8~ as the easy direction imposed
by the non uniform surface treatment, 8~ can be deduced by minimizing a surface energy of a z-uniform sample defined as:
Is = wo(S a)~ + ~wp(R fl)~
where wa
= w(a/~) and wp = w(~ a)/~ and is assumed z-independent. By putting fifs/fiR = 0 one obtains
= 8~
= [aa + fl(~ a)]/~, as previously reported. Note that if for 0 < z < a and a < z < ~ the anchoring energies are different, previous results have to be modified. However in our case in which the chemical nature of the surface is everywhere the same, to assume w(0 < z < a) = w(a < z < ~) seems to be justified.
Using the boundary condition (10), equation (9) becomes:
(
= jk1/~ ~(VR)~dzdy + L/~ ())~ dz). (12)
o o o Y y=o
By substituting into (12) equations (5) and II) we obtain
~ ~ ~ (°~fl)~f ~"~~ ~~f~ 1(~~)
~ 2 L ~
n=1 n2q2 (1+ ~)
Equation (13) shows no>v that the total energy in this case is proportional to (a fl)~/L, I-e- the elastic energy is localised over L. As shown in Appendix 1.
f ~~~~~ (~f~
~
l~(~ ~) (~~)
)2 8
~=i ~~
Consequently
~ ~~~~ ~~~
i (nq)2 (1~ ~) ~ 8~~~ ~~~ ~~~~
and (13) can be rewritten as
~' ~~ l~~ ~ ~
~ ~~L ~~l' ~~~~
Equation (16) works well if qL » I, I-e- for L
-~ ~. Equation (16) can be interpreted as an interference between the two surfaces characterized by different easy directions.
4~ Sample of finite thickness and weak anchoi~ing energy,
Finally let us consider a real problem of
a sample of thickness d
= 2e limited by two solid surfaces of the kind described above. Let us suppose first that the surfaces are arranged in an
y
e
-a -fl
0 a ~ x
~
fl
Fig. 2.- Finite liquid crystal sample of thickness 2e limited by two solid surfaces with periodic distribution of easy directions arranged in an antisymmetric way.
antisymmetric way, and hence the tilt angle is an odd function with respect to the middle of the sample, as shown in figure 2.
By using the reference frame having the origin of the y-axis in the middle of the sample (see Fig. 2), the total energy of one period and half thickness is given by
F = 2k/~ ~(VR)~dzdy + w /~[S(z, -e) 8(z)]~dz (17)
_e 0 2
0
where 8(z) is still given by (I). Simple considerations show that R(z,y) is now a harmonic function odd w-r-t- y : R(z, y) = -R(z, -y). Its Fourier expansion is written
S(z,y)
=
~°
y + £ [Dncos(nqz) + Ensin(nqz)]sh(nqy). (18)
2
The expansion coefficients are now determined by the boundary condition (10) rewritten for y = -e. Simple calculations give
2 aa + fl(~ a)
~°
e + L ~ '
a fl sin(nqa)
~" ~
~ flq (flqLcn + Sn
~ ~a fl i cos(flqa)
~jg~
" ~ flq (flqLcn + Sn)
where Cn = ch(nqe) and Sn
= sh(iiqe).
By operating as in the previous case, relevant to weak anchoring, the total energy for unit
length along the z-axis is found to be
I
= lk Ii + I) l~ +1 1°1~)~ ~~[l~)lT'~j
(2°)
nq
where Tn = Sn/Cn and
os = °~ + (~ ~~
~
j
~ = (Six, -e)1, 121)
is the average surface tilt angle.
y a fl
0
a ~ x
a fl
e
Fig. 3. Finite liquid crystal sample of thickness 2e limited by two solid surfaces with periodic distribution of easy directions arranged in a symmetric way.
In the event in which the surfaces are arranged in symmetric way, and hence the tilt angle
is an even function with respect to the middle of the sample, as shown in figure 3, R(z,y) is
given by
~~ «
R(z,y)
= + ~ lDlcos(nqz) + Elsin(nqz)I ch(flqy), (22)
n=i
where the expansion coefficients are now given by
~* ~"~ + fl(~ ~)
0 ~ '
~
a fl sin(iiqa)
~"~ ~
~ flq(flqLsn+Cn)~
~~ ~~
~
~ q(nq~~~~~~~nl'
~~~~
By substituting expansion (22) into expression (17) and taking into account equations (23),
the total energy of half sample per unit length along the z-axis is
In this section, it is supposed that the two regions with easy axes
a and fl have to be in exact
register as in figures 2 and 3. Iii a practical sample, one can expect a random arrangement of the two regions on the two plates. In this case the above reported calculations can be easily generalised. Of course the results depend on the actual arrangement of the considered surfaces.
This means that equations (19-21) cliange for an arrangement different from the one considered
above. However the aim of our analysis is to show the expected results for two very simple,
and completely different, surface arrangements.
In order to deduce the "equivalent" easy direction of a uniform surface we choose as criterion the equality of measurable quantities. From the experimental point of view we can only measure birefringence or other physical anisotropies of the sample. These quantities are proportional, in the limit of small angles to the average square tilt angle, (R~). Consequently it is possible,
according to our point of view, to define two samples as equivalent if they are characterized bj
the same (R~). By defining, for the periodic structure under consideration, (R~) by means of
(R~) =
~ /~/~ S~(z, y)dzdy, (25)
e 0 -e
and for the z-uniform structure characterized by a R(y) tilt angle the same quantity by means of
(Sll ~
=
) / Sl(y)dy> (26)
-e
the "equivalent" easy axis is obtained by putting
(S() = (S~). (27)
In the event in which the sample is in the antisymmetric arrangement, by substituting expan-
sion (18) and equations (19) into (25) one obtains, in the limit of e » ~,
where
m ~~~~2 j@)
j~(~ ~ ~) 2 (~g)
' ' (iiq)3(1 + nqL)2'
To the contrary for an uniform antisymmetric sample, for which Su(y) = -(yle)Ssu, [25]
equation (26) gives
~~
=
~~~ (~~)
u) 3 su.
It follows that in this case the condition (27) gives:
Consequently the "equivalent" easy axis is found to be [25]
R) =
°~ ~ (~ ~~ ~ + ~
(~
~ ~ @)~R(q,
a, L). (32)
e e
If the non homogeneous sample is in the symmetric arrangement, and hence R(z, y) is given by expansion (22), similar calculations give
(R~) = (°~
~ (~ ~j ~ + ~ (~
~ ~)~R(q, a, L). (33)
e
Since in this case no distortion is present in the homogeneous sample, (RI = R]. Consequently
the equivalent easy axis is now given by
S( = (°~
~ (~ ~j ~ + ~ (°
~
)~
R(q, a, L). (34)
e
In both cases the equivalent easy axis contains a term proportional to
1° ~ ~) R(q, a, L) (35)
~
vanishing for a = fl or a
= 0 or a
= ~. Note that in the case of strong anchoring I-e- for L - 0, the interference term reduces to
m 2 @)
Rlq, a,0) = L ~)~~~] > Rlq,
a, L).
n=1
This means that the interference term is always present. It takes origin from a geometrical
effect, and it vanishes only for very weak anchoring (I.e. for L - c<), as expected. An interference term in the effective anchoring energy was proposed a few years ago by Yokoyama
et al. [23] to interprete the temperature variation of the surface tilt angle experimentally
observed in nematic samples. It is important to underline that the analysis presented in section 3 and in the present one holds only for a fl « I, I-e- a
-~ fl. In fact the parabolic expression
for the surface energy used in the text is valid only for small deviations of the surface tilt angle
from the easy axis. This implies that ii ai « I and ii pi « I, and consequently a
-~ fl. In
the general case of ii ai
-~ I and ii pi
-~ I the analysis has to be performed by using for the surface anchoring energy the Rapini-Papoular or more general expressions. In this frame the analysis is more complicated, due to the non linear character of the boundary condition.
However an interference term is still expected in the surface energy.
5 Conclusions.
We have considered the effect on the macroscopic surface properties of a nematic liquid crystal
of a periodic distribution with two different easy axes. More precisely we have shown that the effective easy axis is
a weigthed average (with respect to the geometrical extension) of
the two easy axes plus an interference term. This interference term is usually negligible if
optical measurements are involved. In fact in this case
a sample is considered uniform if the
