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Trapping of edge dislocations by a moving smectic-A smectic-B interface
P. Oswald, L. Lejcek
To cite this version:
P. Oswald, L. Lejcek. Trapping of edge dislocations by a moving smectic-A smectic-B interface. Jour- nal de Physique II, EDP Sciences, 1991, 1 (9), pp.1067-1076. �10.1051/jp2:1991205�. �jpa-00247575�
J. Phys. II France 1 (1991) 1067-1076 SEPTEMBRE 1991, PAGE 1067
Classificatlon Physics Abstracts
61 30J 61 50C
Trapping of edge dislocations by a moving smectic-A smectic-B interface
P. Oswald and L. Lejcek (*)
Ilcole Norraale Supdneure de Lyon, Laboratoire de Physique, 46 Alike d'Itahe, 69364 Lyon Cedex 07, France
(Received 6 May 1991, accepted 14 June 1991)
Rksumk. Nous analysons comment le mouvement des dislocations coin du cristal hquide
smectique A perrnet de relaxer plastiquement les contraintes induites par la croissance du cnstal
plastique smectique B. Ces contralntes sont engendrkes I la fois par la dif§Erence de densitk qul
exlste entre les deux phases et par la vanatlon d'Epaisseur des couches h la transition Nous
calculons en particuher dans quelles conditions une dislocation coin pout dtre pikgke par le
smectique B Enfin, nous suggdrons que ce pidgeage dynamlque est I l'or1glne de la trds forte densitd de fautes d'empilement qui est couramment observke aux rayons X dans la phase B
Abstract. We analyze how the motion of the edge dislocations of the smectic-A liquid crystal allows the system to relax plastically the stresses that are generated dunng the growth of the smectic-B plastic crystal. These stresses are both due, to the density difference between the two
phases, and to the layer thickness vanation at the phase transition. In particular, we calculate under which conditions a dislocation can be trapped by the smectic-B phase Finally, we suggest that this dynamical trapping is responsible for the very large amount of stacking faults observed by X,ray d1f§raction
1. In«Oducfion.
Recently, we have studied the destabilization in the directional growth of the faceted smectic, A smectic-B interface of the liquid crystal 40 8 (butyloxybenzilidene octylanibne) ill. We
recall that a smectic A liquid crystal is a lamellar phase whose molecular layers are fluid whereas, in the smectic-B phase, the layers have a hexagonal ordenng and strong inter-layer correlations [2] In tills experiment, the temperature gradient was perpendicular to the layers
so that the macroscopic front was parallel to them, an onentation that is faceted [3] We showed that, above a cntical pulling velocity, the front destabilizes by making macrosteps. At the same time, we observed that certain topological defects, such as focal comics [4], are set in
motion and repulsed by the moving front inside the smectic-A phase The same effect applies
(*) Permanent address Instltute of Physics, Czechoslovak Academy of Sciences, Na Slovance2, 18040 Prague 8, Czechoslovakia.
iub8 JOURNAL DE PHYSIQUE II bt 9
to edge dislocations Although they are not directly visible through the microscope, their motion can be studied via the motion of the small dust particles that they have trapped [5]. So far, there is no systematic study of their motion m the vicinity of the growth front, only a few observations showing that they start moving m the smectic-A phase when they approach the front within a few tens of ~m [Ii.
In this article we study how the movement of edge dislocations allows the system to relax its elastic stresses dunng growth. The stresses anse from the density change and the layer thickness difference between the two phases in contact. For the sake of simplicity, we have divided this study in two parts. First, we calculate the viscoelastic stress field and the
permeation flow throughout te smectic-A layers as a function of the front velocity V, by neglecting the elastic repulsion between the front and the dislocations (Sect. 2). This allows us to estimate the climb velocity of edge dislocations parallel to the layers as a function of their distance to the front, a quantity that is measurable expenmentafly. Second, we analyze m
detail how the dislocations interact with the front itself (Sect. 3). We show that they are repulsed by the front and that they can glide m a direction perpendicular to the layers if their
core is wide enough We also discuss under which conditions the dislocations are trapped by
the growing smectic-B plastic crystal.
2. S«ess relaxation and dislocation niovenient in front of the moving intedace.
The geometry is described m figure I : the front, parallel to the layers, moves at a constant
velocity V. The smectic-B phase, assumed at rest, grows into the smectic-A liquid crystal. The
layers are perpendicular to the z-axis and infinite m the (x, y) plane, an assumption that amounts to the neglect of the anchonng of the molecules to the glass plates that bruit the
samples We assume that, at time t, the front is at z
= 0 As usual, we call u the layer displacement in the smectic,A phase, v the barycentric velocity of the molecules, b the layer tllickness and p the density. As the phase transition is first order, there is a density change Ap = p (Sm. A ) p (Sm. B) and a layer thickness variation Ab
=
b (Sm. A b (Sm. B ),
between the two phases. These parameter differences generate stresses normal to the layers which accumulate in the vicinity of the interface and produce some plastic deformation. In order to tackle this problem, we hypothesize that there exists a uniform density of mobile edge dislocations p~ m the smectic-A phase. These dislocations expenence two forces. The
first one is a repulsion force due to the elastic interaction with the interface. This force can
displace the dislocations perpendicular to the smectic layers (glide) if it is greater than the Peierls-Nabarro force [6] We shall calculate this force m the next section. For the moment,
we hypothesize that the Peierls-Nabarro force is large enough to prevent any glide movement
SM A liquid crystal
~
~ A
~ ~ ~
~ ~
fi ~
~ ~
'
y
fiA
-
1
edge parallel to the interface. A is
the average distance between dislocations.
M 9 TRAPPING OF DISLOCATIONS BY A SmA-SmB INTERFACE 1069
of the dislocations, an assumption that we shall discuss in great detail m the next section The second force is associated with the normal stress «~~ = «. This force is perpendicular to the
glide plane and displaces the dislocations parallel to the layers (climb) This movement allows the system to relax its growth stresses when the front propagates.
In order to calculate these effects, we must first wnte the equations of conservation of both the density and the layer number Because v
= u =
0 in the smectic-B phase, we have [7] :
Vzo "
) V (I)
uo =
~ V. (2)
The subscnpt 0 indicates that these quantities are taken at z
= 0, 1-e- on the interface, on the side of the smectic-A phase
In the bulk, the mass conservation implies divv =0, assuming the smectic-A phase incompressible. The translation invanance in the (x,y) plane imposes d/dx =d/dy =0 whence
dv~
= 0 and v,
= v~o Vz ~ 0. (3)
dz
As v~o and uo are not identical on the interface, there is a permeation flow across the layers.
Tins flow is described by a phenomenologlcal equation which is identical to a Darcy law in a porous medium, that is [8]
Q-v~= A~G (4)
A~ is the bulk perrneation coefficient and G the elastic force acting on the layers :
G =
~"
= B
~~ (5)
dz az
where B is the elastic modulus of the layers
In order to calculate the stress field «(z), we must take into account the movement of the dislocations. Let v~(z) be the climb velocity of the dislocations at distance z. According to the
geometrical Orowan relation [9], we have
lz
U(Z)
= U0 + pm bvc(Z) dZ (6)
0
In smectic A phases, the climb velocity v~(z) of an edge dislocation is proportional to the
applied stress «(z)
v~(z)
= mm (z) (7)
where m is the mobility of the dislocation. It has been measured expenmentally and is close to
10-6 crn2/poise [10].
From equations (1)-(7), we obtain an equation for «(z) :
lj ~
~'
~~" ~~~ ~~ ~ ~ ) ~ ~
P
~ ~~~
iuJu JOURNAL DE PHYSIQUE II bt 9
This equation shows that the stress field results from a competition between the permeation flow due to the density change in the layers (which is given by the difference Ap/p Ab/b)
and the relaxation by climb of edge dislocations. Equation (8) is readily integrated to give
~r (z)
= ~ro exP I (9)
A = (p~)- ~/2 is the ean
distance between islocations. The distance f, over which
essentially can be writtenifferently if we
that the mobility of an edge
at
jl~
~ ~
~
b
~~~~
where l~ = fi is a permeation length of the order of the
comparable with the mean distance between dislocations and is of the front
velocity V By contrast, the stress increases
linearlywith V and ismaximum on the interface
On the other and, its sign
depends
on the sign of V
~positive the hase grows
and negative when it melts) and on the sign of the difference
Ap/p - Ab/b = 2
is the average distance between molecules in the
layers)that we expect to be negative.
case, the smectic-A
layers are
nder compression (« ~ 0) when the phase grows
and nder dilation (« ~ 0) when it melts. The stress sign is very mportant because exists a buckling instability of the layers dilation8] Thus we
expect
that the layers
break m melting above a ntical
elocity [ V* [. In order to this velocity, we
hat a sample of
hickness d is unstable when it is submitted to a stress greater than
2 ~ /~d, where
«0~2"fit. (ii)
Using (9) and f
= A, tills condition can be rewntten m the form
~
~aA~/~
V~V
= ~
~0. (12)
AaA
The existence of this cntical melting velocity, below which the layers undulate, is observed
experimentally, m particular because this instability induces, almost immediately, focal comics which destroy the molecular alignment of the sample [4].
Before discussing how edge dislocations interact with the front, let us conclude this section
by giving a few orders of magnitude. Expenmentally v~(z=10 ~m)
= lo ~m/s at V
=
I ~m/s [I] By taking l~
= b e A~ =
10~'~ CGS one finds «(z = 10 ~m )
= 3 000 dyn/crn~
whereas equation (9) gives «o
= 10 000 dyne/crn~ and « (z
= 10 ~Lm = 3 000 dyn/cm~
with Aala
=
-10~~ and f =10
~m. the accord is therefore quite satisfactory. One also calculates V*
= 3 ~m/s with B
= 10~erg/cm~ and K
=
10~~ dyn. Melo has recently
measured the cntical velocity for which focal comics occur and has found a value of the same order of magnitude as our theoretical estimate (about 10 ~m/s). Thus, one must melt the samples slowly so as not to break their molecular alignment.
bt 9 TRAPPING OF DISLOCATIONS BY A SmA-SmB INTERFACE 1071
3. Glide of edge dislocations and «apping by the mo~ing interface.
So far, we have assumed that each dislocation stays in its Peierls valley and does not glide.
Th1s assumption is perhaps incorrect expenmentally. Indeed, we have observed that the moving dust particles suddenly stop when they are very close to the growth front. This means
that the dislocation comes free from the dust particle There are two possibilities to explain
this phenomenon :
either the climb velocity becomes too large and the dislocation separates from the
particle by leaving around it a dislocation loop. This happens when its climb velocity is greater
than [5]
v~ =
~ (13)
4 ~Rr
where T is the line tension of the edge dislocation, close to , SK b [12] and r the radius of the particle, supposed sphencal For r= I ~m, R
= I pulse. h = Ii) ~jn ~m~ B =
10~erg/cm~, one finds v~ = 30 ~m/s, a depmmng velocity which is of the same order of
magnitude as that which is observed. Further, one can calculate the growth rate V~ below which the dust particle remains trapped until its impnsonment by the smectic-B
phase It is given by writtmg the equality m«o
= v~. From equations (9) and (13) one gets
~~8~~a~/~'
~~~~
Numencally V~=05~Lm/s with the previous values of the physical constants and
r = I ~m. So far, this prediction has not been checked experimentally
or the dislocation glides perpendicular to the layers when it is sufficiently close to the interface. Because the particle cannot move perpendicular to the layers [13], the dislocation escapes from it and the particle stops The calculation of the elastic force acting on the
dislocation m its sup plane is given in Appendix I For the sake of simplicity, we have assumed in this appendix that the smectic-B phase behaves elastically as a very viscous
smectic-A phase whose layer elastic modulus B' is larger than the modulus B of the smectic-A
phase [14]. Such an assumption is very realistic and has been proven both expenmentally and
theoretically Indeed, the smectic-B phase cannot sustain a shear stress parallel to the layers
without creeping viscously [14]. This absence of a yield stress parallel to the layers is due to the very large elastic amsotropy of the smectic-B phase (Cm is much smaller than the other
elastic modulus by 3 or 4 orders of magnitude [15]) and to the unusual properties of the stress
field around the basal dislocations [14] Another consequence is the existence of an
undulation instability of the layers under dilation [16] which can be descnbed, as m smectic-A
liquid crystals, by assuming that the smectic-B layers have an elastic modulus B' and a
curvature constant K' [5]. In this framework, the force F ~per unit length) that the interface
exerts on an elementary edge dislocation, situated at the distance zo, can be calculated
exactly :
~ 16
z~~
~+ ~~ ~~~~
where A
=
fi and A'= fi
are two charactenstic lengths of the two phases. In smectic-A, A =b, whereas m smectic-B, A' is rather the average distance between basal dislocations, so one expects A' » A. Th1s force is parallel to the z-axis and tends to glide the
dislocation perpendicular to the layers. This formula calls for a few comments :
lull JUURNAL DE PHYSIQUE II bt 9
firstly, F
= 0 when B'
=
B and K
= K'. This case corresponds to an isolated dislocation
m an infinite medium
secondly, F ~ 0 when B'~ B and A'
~ A, which is the case in our expenment Thus, the dislocation is repulsed by the growth front within the smectic-A phase with a force
approxhnately equal to
finally, tills force is maximum when the edge dislocation is on the interface. At tills position, one must introduce a cut-off length of the order of zo
= A
= b in the preceding
expression. One then arrives at
~~~~ 16~ ~~~~
Tl1is repulsion force must be compared to the Peierls-Nabarro force Fp~ that lfinders the
movement of the dislocation in its slip-plane This force has already been calculated and
crucially depends on the core width 2 ( of the dislocation ii?]
FP N =
~ Bb fiexp (- ~ ~t
~ t b (18)
In tills formula, ( is defined as the distance along the slip plane in the glide direction over
which the displacement u of the layers vanes from 0 to b/4. One thus expects
( m b/2. Two cases must be considered :
(i) either, FM~z ~ Fp~, a condition that is satisfied if the core width 2 f is smaller than 1.34 b. In tills case the dislocation does not glide easily because this motion requires the thermally activated nucleation of small bulges limited by two kinks which can then spread sideways. Under tills condition one expects that all the dislocations are trapped by the moving
interface (except perhaps at very small pulling velocity). One must also emphasize that, m this
case, there is no piling up of the dislocations in front of the interface, so that
p~ = constant, as assumed m the previous section.
(u) or, FM~~ ~ Fp~, a condition that is satisfied as soon as the core is dissociated, I e. for 2 ( m 1.34 b. In tills case the dislocation begins to glide easily when it arrives at a certain
distance zt from the front, for which F(zf)
= Fp~ when its distance z from the front is smaller than zt, the dislocation expenences an effective force that can be taken equal to
F(z) FpN and acquires a velocity
vg(z) = ) (F(z) Fp
~ ) (19)
M is the mobility of the dislocation during a motion of easy glide. M is estimated in
Appendlx II, and is given by
~ 32 ( $~
(20)
~~ ~
In a stationary regune (if one exists), v~(z) equals the front velocity V, which fixes the distance z separating the dislocation from the front In any case, this distance must be greater than b, otherwise the dislocation is trapped by the growing smectic-B phase. The smallest
bt 9 TRAPPING OF DISLOCATIONS BY A SmA,SmB INTERFACE 1073
value of V, let us say V+, above which the edge dislocations are trapped by the moving interface is thus given by the following equation
V+
=
) (FMaX Fp
N ) (21)
By takJng A
= b and after putting 2 f
= ab, one obtains explicitly, from equations (17), (18)
and (20) :
v+
= i~
ji
jjj
i1- 12~ ~~Pi"~ (22)
2
It is easy to see that the exponential term rapidly becomes negligible as soon as
a ~ l.34 (for this value V+
= 0) 1.e. as soon as the dislocation is dissociated in its glide plane.
In tills case, V+ is close to Bb/R, witllin a numencal factor of order I. It is instructive to calculate this velocity: with b
=
3x10-~cm, B =10~erg/cm~ and R
= I poise one gets V+
= 30 crn/s1 Th1s velocity is very large and unaccessible expenmentally. In this case, the smectic B grows by repelling the dislocations into the smectic A phase where they pile up and
eventually recombine. The calculation of section 2 is then questionable because the density
p~ is no longer constant, but time dependent
4. Concluding remarks.
The main conclusion of the previous section is that the trapping of edge dislocations crucially depends on the width of the core of the edge dislocations in the smectic A liquid crystal.
If the core is narrow (a = I), the edge dislocations are easily trapped by the growth front
into the smectic B phase. In this case, a topological remark concerning the Burgers vector is necessary. Indeed, tills vector is generally equal to the thickness of a layer b m the smectic A
phase whereas the penodicity along the director m the smectic-B phase is equal to 2b Thus, each trapped elementary dislocation becomes an imperfect dislocation in the B phase, to which is necessarily coupled a stacking fault We thus predict the existence of a very large
amount of stacking faults inside the smectic B plastic crystal.
By contrast, the dislocations are not trapped when their core is wide because they are then
repulsed within the smectic A phase In this case, we predict a very small number of stacking
faults
It has been observed expenmentally by X-ray diffraction [18] that smectics B contain a very
large amount of stacking faults The results of this paper suggest that edge dislocations are
narrow m smectics A and are trapped by the moving interface Thls assumption is reasonable
inasmuch as the molecules are soft m smectics A and can accommodate the large strain that exists m the core region. Such a conclusion is also compatible with the few measurements of stress and climb velocity of the edge dislocations m front of the moving interface. Finally,
one can argue that, if the dislocations glide easily, they should not drag dust particles over
very large distances, as is observed expenmentally
Acknowledgements.
We thank F. Melo for fruitful discussions. This work was supported by the Rkglon Rlibne-
AJpes, the Centre National de la Recherche Scientifique and by the Centre National d'Etudes Spatiales. One of us (L. L.) benefited from a grant from the Rbgion Rlibne-AJpes.
