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Influence of surface tension on the stability of edge dislocations in smectic A liquid crystals
L. Lejcek, P. Oswald
To cite this version:
L. Lejcek, P. Oswald. Influence of surface tension on the stability of edge dislocations in smectic A liquid crystals. Journal de Physique II, EDP Sciences, 1991, 1 (8), pp.931-937. �10.1051/jp2:1991117�.
�jpa-00247565�
Classification
Physics Abstracts
61 30J 61.70G
Influence of surface tension on the stability of edge dislocations in smectic A liquid crystals
L. Lejcek (*) and P Oswald
Ecole Normale Supdneure de Lyon, Laboratoire de Physique, 46 Alike d'ltalie, 69364 Lyon Cedex 07, France
(Received13 March 1991, accepted15 May 1991)
Rksumk. La solution gdnkrale au probldme d'une dislocation coin dans un film smectique A est
pr6sent6e Nous calculons son knergie d'mteraction avec les surfaces en tenant compte ngoureusement de la tension de surface Cette formule est utfliske pour discuter la stabihtk de la dislocation vis-i~vis d'un mouvement de ghssement vers la surface Si la tension de surface domlne l'dlasticitd smectique (caractknsde par les constantes K et B), c'est-I-dire
si y ~ (KB)"~,
alors la dislocation se stabihse I l'mtbneur du film Si, par contre, la tension de surface est petite (y ~ (KB)"~), la dislocation est attirhe vers la surface
Abstract. The general solution describing an edge dislocation inside a smectic A film is
presented. Its elastic interaction energy with the limiting surfaces is calculated by taking into account ngorously the surface tension y This formula is used to discuss the dislocation stability with respect to a slip motion towards the surface If surface tension prevails over smectic A elasticity, charactenzed by elastic constants K and B, i e if y
~ (KB)"~, then the dislocation is stabilized within the film If, by contrast, the surface tension is small (y ~ (KB)"~), the
dislocation is attracted to the surface of the film
1. Introduction.
The surface tension is usually neglected in problems connected v/1th dislocations near a free surface. In solids, this constitutes a good approximation because the surface tension y is small compared to pb, Where p is a shear modulus and b the Burgers vector of the dislocation In
smectic A liquid crystals, this cnterion is different and can be expressed as y « fi [1-4] K
is the curvature constant and B the elastic modulus of the layers This condition is not always fullfilled expenmentally, so that surface tension must be taken into account exphcitely in the
calculations To our knowledge, this has never been done ngorously
In this article, We give the solution to the problem of an edge dislocation lying in a smectic A film limited by two surfaces of different surface energies The total energy of the edge
dislocation and its position of stable equilibnum (When it exists ') will be also determined
(*) Permanent address Institute of Physics, Czechoslovak Academy of Sciences, Na Slovance2, 18040 Prague 8, Czechoslovakia
932 JOURNAL DE PHYSIQUE II bt 8
Let us assume that the sample surfaces are situated at z=D and z=0 and let
yi and y~ be their surface energies, respectively The layers, parallel to the sample surfaces,
are perpendicular to the z~axis and parallel to the (x, y)~plane. The edge dislocation is situated at z = z' and lies along the y-axis (Fig. I).
z
(A~Az)2A~b z ~ ~z'+ SD
z - SD
(A~Az)2b z - z'+ 4D
z - 4D
(A~Az)A~b z ~ -z' + 4D
z m 3D
(A~Az)b z = z' + 2D
z - 2D
A~b z - -z' + 2D
z - D
b z
- z'
z - 0 x
Azb . z - -z'
z - -D
(A~A~)b . z - z' 2D
z - -2D
(A,A~)A~b z » -z' 2D
z -
-3D
(A~A~)2b . z - z' 4D
z-~D
(Ai~2)~A2b . z - -t 4D
Fig. I. Burgers vectors and posiuons of the first five image dislocations. The satnple of thJckness D is between z
= 0 and z
= D and contains an edge dislocation of Burgers vector b at z = z' Coordinates x,
z are also displayed
The total energy W of the sample ~per unit~length of dislocation) is the sum of the smectic A elastic energy W~ [1]
W~ =
dx dz(K
~~ ~
+ B ~ j
(I)
ax z
and the surface energies Wi and W~ [1, 2]
~ ~ ~j~ ~
)~=D ~' ~ j~j~ ~~ )~=o ~~ ~~~
In these equations, u = u(x, z) is the layer displacement in the z~direction. Minimrang W v/1th respect to u gives the equilibnum equation
K°~=B°~~ ~ (3)
in the bulk of the sample, v/1th the two boundary conditions
at the sample surfaces. One recognizes above the well-known Gibbs~Thomson equation, a~ulax~ being equal to the curvature of the surface if it is slightly distorted, while B aulaz is the elastic stress normal to the layers.
2. Displacement field around an isolated edge dislocation.
The displacement u
= u (x, y) around the dislocation must satisfy the equilibrium equations (3) and (4) It is given by the general method of image dislocations [4, 6]. The solution is thus
obtained by adding the displacement corresponding to an isolated dislocation in an infinite medium [5, 7j to all the displacements associated vlith the images The position of the image
dislocations can be found using the method described m reference [6] Their positions and their Burgers vectors are given by the recurrent relations obtained from equation (4) :
z=z'+2Dm b~=b(AIA~)~ mm0
z=z'-2Dm b~=b(AIA~)~ mml
(Ai A~)~
z = z' + 2 Dm b~
=
b m m I m integer
A~
z=-z'-2Dm b~=b(AIA~)~A~ mm0
Burgers vectors and positions of a few dislocations are depicted in figure I. The resu1tlng displacement is therefore
u(x, z)
=
~
sg (z z') I + erf ~
~~
4~ 2(A(z-z'() +
+A~sg(z+z')(I+err ~ ~~)
2(A(z+z'()
~
~~l ~~~~~~~ ~~~~~~'~~~'ll)(l+err 2(Ajz-zi~~
~ ~~)
+(AIA~)~sg(z-z'+2Dm) 1+err ~
2(A(z-z'+2Dm ()'~~
(Ai A~)~
x
~ A2 ~~ ~~ ~ ~~ ~ ~'~~
~
~ ~~
2(A z + z'- 2 Dm )~'~
+ (Aj A~)~ A~ sg (z + z' + 2 Dm ) I + err ~
~~
(5) 2(A(z+z'+2Dm()
~~
In this expression, z' describes the dislocation position in the sample (0
~ z' ~ D ), b is the
Burgers vector, A
= (K/B)~'~ is a charactenstic length of the order of the layer thickness and
erf is the error function :
~ ix
err (x)
= ~ exp (- t~) dt
w o
The parameters Ai and A~ determine the values of the Burgers vectors of the image dislocations and have the fornl
ai I yi a~ I y~
~~~ai+1 ~~~ ~~~BA' ~~~fi ~~'~~ ~~~BA°
934 JOURNAL DE PHYSIQUE II bt 8
The values and the signs of At and A~ depend on the ratios at and a~ which characterize the influence of surface tension on the dislocation behaviour. Many cases can be considered
depending on the values of ai and a~.
For free surfaces, one usually takes at
= a ~ = 0 As we shall see below, this constitutes a very crude approximation which gives wrong results in general. In this limit At
= A~
= I
and the image dislocations have Burgers vector equal to b and b alternatively [4~6].
In order to descnbe a sample sandwiched between two plates, it is sufficient to take at and a~ - + cx~. In this limit, At
= A
~ = l and all the image dislocations have their Burgers
vector equal to b [4, 6].
The case a1 # 0 and
a~ - + cx~ is also of interest. It corresponds to a free dop deposited on
a solid, a situation that has been studied in detail in recent expenments [2, 8]
3. Interaction energy.
We shall first consider the general case of two parallel dislocations. Then, we shall restnct ourselves to the special case of an isolated dislocation interacting with the limiting surfaces
3.I INTERACTION ENERGY BETWEEN TWO PARALLEL EDGE DISLOCATIONS. Let us
consider two edge dislocations with Burgers vectors hi and b~ situated at points
[xi, zi] and [x~,z~], respectively. Their interaction energy is equal to the work that is necessary to create the first dislocation in the stress field «=~ = B au ~az of the second one [6], itself charactenzed by the displacement field u~. It can be expressed in the form
% = Bb
i l~ " °"2 dx. (6)
x~
az
z z~
Using equation (5) yields
~-
i ~~ ~l ~2 e
~~> x2)~/4> z~ =~
~ ~~ ~ ~
e~ ~~i x2)~/4> =~ +.
zi z
2
'21
2 ~~ ~~
+
+ jj (A
A )~ ~
~~ ~~~~~~ ~~ ~~ ~ ~~ ~~ (x, x~)2/4> ;j z~ +2 D~
"
~ ~/ ~
+ (A
i A
2)~
~ ~2 ~'~~ ~/ zi z~ + 2 Dm
~ ~ )m (xj x~)~/4> zj + =2 2 Dm (xj x~)~/4> zj + ;2 +2 Dm
~ ~
i
~ ~
~/jzi
+ z~ 2 Dm
~ ~~ ~~~~ ~~ ~
~/ zi + z~ + 2 Dm
~~~
The first term of this expression corresponds to the interaction energy between two
dislocations in an infinite medium [5]. All the other terms come from the image dislocations which model the sample surfaces. Their presence modifies quantitatively the interaction
between the two dislocations. If at and a~ are both greater than I, then Ai and
A~ are positive and the term in brakets in equation (7) is always positive In consequence, two dislocations of the same sign repulse each other whereas dislocations of opposite signs attract each other, as expected In the other cases, when one at least of the A,'s is negative, the
behaviour is more difficult to analyze but it is qualitatively similar (except when dislocations
are very close together in the same climb plane [10]) In the next subsection, we analyze in
more detail the case of a single dislocation. In particular, we shall discuss its stability with respect to a slip motion perpendicular to the layers
3.2 INTERACTION oF A SINGLE DISLOCATION WITH SURFACES. The interaction energy
between a single dislocation and the surfaces can be easily found by putting hi
=
b~ = b, xi
= x~ and zj = z~ = zo in equation (7) If the 3-function corresponding to the dislocation self-energy in an infinite medium is omitted, Wj can be rewntten as
~ BA b2 A~ m 2(Ai A~)~ (A
i A~)~ j (A1 A2)" A2
8 fi / ~ ~i /~ ~
A2 jzo Dm
~
zo + Dm
(8)
The equihbnum position zo of the dislocation can be found from the condition of lazo " 0 After putting t
= zo/D, this condition reads
equationnviteseveral comments .
- first, it is
satisfied whatever zo when at = a~ = I
case, the islocation has no preferred
exactly for
surface effects,
- when at = a~ = a # I (i e yi = y~ = y #
t = 1/2. In quihbnum, the islocation lies in the middle of the sample, which is bvious
view of mmetry of
the oblem. On the
other
hand, this position is not always stable. If
a = y/ KB ~ l, the is repulsed by
both so that
the equihbnum is
stable By contrast, the islocation is
attracted to the surfaces is unstable and the islocation slips to the surface ;
- hird,
equation
(9) allows us to find the quihbnum position in the general
yj
~ y~.
j Aj /3
As before, this equihbnurn is unstable from the moment that one of the a~'s is smaller than I.
One case that is particularly interesting is the one of a smectic droplet deposited on a solid surface. As a~
- cx~ and A~
- I on the solid support, the solution (10) takes the form
z~ =
j
~~~
with A
=
Y fl
(i i)
+ y +
where y is the surface tension at the free surface.
4. Concluding remarks~
We have shown that surface tension plays an important role in problems related to
dislocations in smectic A liquid crystals and that it can be taken into account ngorously. Our
main conclusion is that edge dislocations a@ulsed from the free surface as soon as its
surface tension y is greater than BA = KB. This inequality is satisfied very often
@mentally. For instance, at the contact mth air we have y=25erg/cm~ while KB =10erg/cm~ if
one takes K=10~~dyn and B=10~erg/cm~. Thus, the common
JOURNAL DE pHysiouE ii T i v 8. AooT iwi 43
936 JOURNAL DE PHYSIQUE II bt 8
assumption of free surface with zero surface tension turns out to be completely wrong and leads to erroneous conclusions conceming the stability of bulk dislocations. Our theoretical
results are moreover in accordance with certain expenmental observations of Williams [2]
who found that dislocations are not attracted by the free surface rather they are found at
large distances from it. Williams note also that bulk dislocations influence the free surface of the sample, leading to fine fines that are vJsible through the microscope in reflected light [2]
It is possible to calculate, within the framework of this model, the surface distortion For the sake of simplicity, we shall only consider the simplest case of an edge dislocation sandwiched
between two free surfaces of surface tension y
~
fi. We shall
assume the dislocation in
equilibrium at z'
= D/2. The surface deformation is charactenzed by the inclination 0 of the
superficial layers with respect to the (x, y) plane and by the width f over which 8 varies significantly
au and f
" Xi X2'
8 " Q
~ D,z' D/2
where xi and x~ are the solutions of the equation
°~~ = 0
0X z=D,z'=D/2
By taking into account the first image dislocations only, it is possible to calculate
approximately f and the maximum tilt angle 8M
= 8 (x
= 0).
In order to evaluate these geometnca( factors, let us take a =5 [2], A =20A and
D
= 10 ~cm. We obtlidf
= 0.28 ~cm (hi 8M )
= 27 b (~cm) For « giant dislocations, that
are very numerous in smectics [2], the angle 8M can be fairly important and clearly visible
through the microscope It increases as the film thickness D decreases and it can even reach large values in the thinnest films Thus, for b=10 layers
= 3x10~2~cm we get 8M = 0.8° with D
= 10 ~cm whereas 8M reaches 2.5° for D
= I ~cm
Experimentally, the condition a ~l is rarely observed at the contact with air in
therrnotropic liquid crystals. Conversely, it should be fulfilled at the interface between the smectic A and its isotropic liquid because the surface tension is expected to be much smaller in this case Thus, the smectic A~isotropic interface should attract the bulk edge dislocations.
Th1s phenomenon is perhaps important in explaining certain anomalies observed in recent
experiments on the shape of smectic A droplets [9] Analogous situations may be encountered
in lyotropic liquid crystals where surface tensions are very small too Nevertheless, very little
is known about dislocations in these systems.
Finally, situations with two interfaces of different surface tensions could be investigated expenmentally, for example by spreading a smectic A on a suitable liquid.
Acknowledgments.
This work has been supported by the Centre NaUonal de la Recherche Scientifique and the Centre National d'Etudes Spatiales. L. L, acknowledges support from the Chaire Louis Nkel.
References
ii] DE GENNES P G, The Physics of Liquid Crystals (Oxford, Clarendon Press, 1975)
[2] WILLIAMS C, Thkse d'Etat, Orsay (1976).
[3] DE GENNES P G, Europhys Lett 13 (1990) 709
[4] PERSHAN P S, J. Appl Phys 4s (1974) 1590
[5] KLtMAN M
,
Points, Lines and Walls in Liquid Crystals, Magnetic Systems and Vanous Ordered Media (J Wiley, New York, 1983)
[6] KLtMAN M., J Phys. France 35 (1974) 595
[7J DE GENNES P G, CR Acad Sci Paris 27SB (1972) 549
[8] BECHHOEFER J, OSWALD P, Europhys Lett, in pre8s
[9] BECHHOEFER J, LEJCEK L., OSWALD P, in preparation
[10] When the two dislocations are in the 8ame climb plane and when one at least of the
A,'s is negative, the interaction energy and the force between the two dislocations can change
sign as a function of their distance Thus, two dislocations of the same sign can attract when they are very close together (typically for xi x~ ~ (AD )~i whereas they repulse if they are
of opposite sign. Nevertheless, this peculiar phenomenon is not of great interest because both dislocations are unstable with respect to a slip motion towards the surface (see Sect 3.2)
