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Attachement kinetics at the smectic-A smectic-B interface
P. Oswald, F. Melo
To cite this version:
P. Oswald, F. Melo. Attachement kinetics at the smectic-A smectic-B interface. Journal de Physique II, EDP Sciences, 1992, 2 (6), pp.1345-1351. �10.1051/jp2:1992204�. �jpa-00247732�
Classification Physics Abstracts
61.30 81.30F
Attachment kinetics at the smectic.A smectic.B interface
P. Oswald and F. Melo
Ecole Normale Sup6rieure de Lyon, Laboratoire de Physique, 46 Al16e d'Italie, 69364 Lyon
Cedex 07, France
(Received 23 December 1991, revised 17 February 1992, accepted 20 February 1992)
R4sum4. Des donndes exp6rimentales ant6rieures [Ii sur la cin6tique d'attachement moldculaire h l'interface smectique-A smectique-B du 408 (butyloxybenzilidene-octylaniline) sont
analys6es th60riquement. Deux modbles sont discut6s en d6tail, selon lesquels la croissance est dominde suivant la vitesse, soit par les dislocations vis, soit par la nucldation bidimensionnelle. La mobilit6 d'une marche est 6galement estimde ; sa valeur est environ loo h 000 fois plus grande
que celle d'une dislocation. Un modble rh6010gique permet d'expliquer cette diff6rence.
Abstract. Previous experimental data [I] on the molecular attachment kinetics at the smectic-A smectic-B interface of 408 (butyloxybenzilidene-octylaniline) are analyzed theoretically. Two models are discussed in detail, in which growth is either dominated by screw dislocations, or by
two-dimensional nucleation, depending on the velocity. The step mobility is estimated too ; its value is about loo to 000 times larger than that of a dislocation. A rheological model allows us to
explain this difference.
1. Introduction.
Recently [Ii, we described the directional solidification of the faceted smectic-A smectic-B
interface of 408 (butyloxybenzilidene octylaniline). The main result was that moving
macrosteps nucleate above a critical velocity that is given by the classical constitutional-
undercooling criterion. We never observe the stationary crenellated solutions proposed by the theorists [2] and we have shown [3] that these solutions are not observed because of kinetic effects. The explanation is that the macrostep height is always smaller than the kinetic length,
defined to be
8T~ 2
~~~
2 1AT ~~~
ID IT
where 8T~
=
T- T~~jj~t~~ is the kinetic undercooling, I~
=
2D/V the diffusion length and
I~
= AT/G the thermal length. V is the pulling velocity, G the temperature gradient and
AT
= Tj~~~~~t~~ T~~ji~tu~ the freezing range.
1346 JOURNAL DE PHYSIQUE II N° 6
Such an explanation supposes that we were able to measure accurately the front
undercooling as a function of its velocity. This was done by using the double-sample method (for further details, see Ref. [I]). This technique allows us to measure the front temperature to about 10 mK. We found that 8T~ varies roughly as the square root of V and we concluded (without calculation) that the system can be described by a screw dislocation growth
mechanism.
In this article, we resume the discussion about kinetics in great detail. In section 2, we recall the experimental results. In section 3, we discuss the possible growth mechanisms of the facet and we estimate the step mobility. In the last section, we show that this mobility is related to that of an edge dislocation of the smectic-A phase. The latter explains the apparently large mobility of the steps.
2. Experimental data.
Figure I shows the front undercooling 8T~ as a function of the front velocity V. These
measurements are new but quite comparable with those we obtained previously [I]. At
velocities smaller than 5~m/s, the best two-parameter fit, 8T~=AV~, would give
m = 0.6 and A
= 14 in CGS units. For reasons discussed in the next section, we prefer using a
fi law. The best fit with
a square root gives then
8T~ (°C)
= 5.I fi (cm/s) (0
~ V
~ 5 ~m/s). (2)
This curve is plotted in figure I and passes through the experimental points within the
experimental error bars. In contrast, one sees clearly a saturation of the kinetic undercooling
at velocities ranging between 5 and 15 ~m/s.
0.14
______.--
O.1 '~~'~'
,- ,-
--'
0.1 ,-"
,' ,
l' ',
F-il cO
o-o 0.2 0.4 0.6 0.8 1-o 1.2x10'~
V(cm/s)
Fig. I. Kinetic undercooling versus the velocity. The different symbols correspond to different
samples. The solid line is the best fit below V = 5 ~m/s by a $ law corresponding to a screw
dislocation growth mechanism, whereas the dashed lined is the theoretical curve calculated from the two-dimensional nucleation law (Eq. (lo)) which fits at best the experimental data for V
~ 5 ~Lm/s.
Measurements at larger velocity are impossible because of the nucleation of smectic-B germs allead of the interface.
3. Two possible growth processes.
The variation of 8T~ with fi
at small velocity suggests straightaway a screw-dislocation
growth process [4]. As quoted in reference [I], this mechanism is quite plausible inasmuch as
screw dislocations are numerous in our samples, their role being, among other things, to
release the slight twist of the layers, that always occurs during the sample preparation. Let us
now recall briefly the model. Let I/d be the step density and u~ the velocity, the growth rate of the facet is
2 bu~
V
= ~ (3)
2 b is the step height (in fact twice the layer thickness if the dislocation is perfect) and d the average distance between steps which has been calculated by Burton, Cabrera and Frank [5].
These authors have shown that each step anchored at the point of emergence of a dislocation develops into a spiral whose pitch has a well-defined value and is a function of the
undercooling only
~ 20pT
~L2b8T~ (4)
p is the step free energy that we measured previously [6] and L is the latent heat per unit volume.
In the next section, we shall show that the step velocity is proportional to the difference in chemical potential between the two phases :
mL 8T~
u~ = m Ap
= (5)
m is called the step mobility.
After regrouping (3), (4) and (5) one yields
Thus 8T~ is proportional to fi
as expected. In this expression everything is known except the step mobility. One has typically b
= 3 x 10~~ cm, L
=
5 x 10~erg/cm~, T
= 323 K and
p
=
7 x 10~~ erg/cm (by taking twice the value that we have found for an elementary step [6]). Comparison between the experimental law (2) and the theoretical expression (6) gives the mobility of the step : m = 0.6 x10~~ cm~sg~~. This mobility is about two orders of
magnitude larger than that of a dislocation. This point is explained in the next section.
Let us now discuss whether a nucleation-mediated growth process could be efficient and
explain the saturation in kinetic undercooling observed at velocity larger than 5 ~m/s. This mechanism is govemed by three quantities the rate of two-dimensional nucleation I (per
unit surface area and per unit time), the spreading velocity of monomolecular steps
v~ and the crystal face area S. Recently, Obretonov et al. [7] have proposed an unified expression for the facet growth rate, combining the known equations for mononuclear and
1348 JOURNAL DE PHYSIQUE II N° 6
polynudear growth mechanisms. It reads :
~
i 2/3 ~~~
l +
~~
~
The denominator in this formula represents the number of germs N occurring on the facet per unit time. If it is large enough (N » I ), the facet surface area does not influence its velocity (polynudear regime). By contrast, V is proportional to S if N is near to I (mononuclear regime). In this case, the spreading time of the nucleus over all the facet is shorter than the
nucleation time of a new nucleus.
The nucleation rate is given by
~ ~ ~~~
kB Tb Ap ~~~
where the preexponential term C is a frequency per unit area which depends, among other
things, on the step mobility m. This term is difficult to calculate exactly. An estimate is given
in reference [8] and reads C
= (kB TIP ) (m Ap la~). Experimentally S is very large (especially
below the onset of instability) so one expects a polynudear process. In this case, (7) can be rewritten in the form (by using Eqs. (5) and (8)) :
kB T 1/3
~ p2 sT~
~ ~~~
pa~ ~~~ 3 kB bL 8T~ T ~~~
We have plotted this theoretical curve in figure I (dashed line) by adjusting p in order to fit at best the experimental points obtained at large velocities (V
~ 5 ~m/s ). This procedure gives fl
=
4.3 x10~~erg/cm, a value that is in good agreement with previous measurements
(p = 3.6 x 10~~ erg/cm from [II) and
V (cm/s)
=
15.3 8T~ exp ~'~~~ (°C) (10)
8Tc by taking b
= 3 x 10~~
cm, a = 5 x10~~
cm, L
= 5 x 10~erg/cm~ and the preceding value for the mobility I-e- m
= 0.6 x 10~~cm~ sg~ (the mobility is independent of the step height
as we shall see in the next section). Let us also emphasize that the kinetic undercooling 8T~ given by this formula is much larger than the experimental value as long as the front
velocity is smaller than about 5 ~Lm/s. One thus concludes that at small velocity, kinetics are probably dominated by screw dislocations. In contrast, homogeneous nucleation should play
an important role at velocities larger than 5 ~m/s and should even become dominant.
One must note however that all these conclusions are very sensitive to the values of the parameters chosen, and notably to that of p, as it enters as p~ in the exponential term. For instance, increasing p by 10 fG shifts the theoretical curve (Eq. (10)) up by about 0.02 °C.
Another difficulty is the estimation of the preexponential factor C. The corresponding error is difficult to estimate but is minimized by the fact that C enters in equation (9) with the power 1/3. Nevertheless and in spite of all these uncertainties, it is clear that a two-dimensional
nucleation process cannot explain the very small values of 8T~ observed at small velocities, where we believe that a screw-dislocation mechanism is responsible.
In the next section, we discuss the step mobility using an hydrodynamic approximation,
which we know to be valid for dislocations.
4. Step mobility.
The main result is that the step mobility is about one thousand times larger than that of an
edge dislocation in the smectic-A phase. In order to explain this result qualitatively, we first
discuss the origin of dissipation. This is due to both the layer thickness variation
Ab/b and the density change 2 Aala in each layer during crystallization. From a rheological point of view, these two processes can be treated separately.
Let us first consider that only the layer thickness changes (Ab/b # 0 while 2 Aala
=
0). In this case, the step can be considered as an edge dislocation of Burgers vector Ah propagating
at velocity u~ in the smectic-A phase (Fig. 2a). In a first approximation, this process does not
require permeation (I,e, flow across the layer) because the density in the layer does not vary.
In contrast, the layers are slightly deformed in the smectic-A phase. In the frame of the step,
- '
v~
-- sm.B
a)
z
-
V~(l ~
x
fi V~
5m.B ~
b)
Fig. 2. Step at the smectic-A smectic-B interface when a) ha
= 0 and Ab # 0. The layers are distorted
within the smectic-A phase, which generates a velocity gradient avjax# 0; b) ha do and
Ab
= 0. The velocity being different far from the step and close to it, a permeation flow is induced.
1350 JOURNAL DE PHYSIQUE II N° 6
advancing with velocity v~ with respect to the smectic-B phase, the layer displacement u reads 191
~~ x )j (II)
~~ 2 ~~~~~
~2/~
in the smectic-A side whereas u
= 0 in the smectic-B crystal. Coordinates x, z are pictured in
figure 2 and A
=
/~ is
a characteristic length of the smectic-A phase, of the order of b [10]. The barycentric velocity of the molecules in the smectic-A liquid crystal is thus
V~ = V~ (12)
V~ = U~
~~
aX
while u~ = u~ and u~ =
0 in the smectic-B phase. In the absence of permeation, the dissipation
is given by [10]
4l =
'~
1?
dx dz (13)
2
~~~
~
where ~ is a viscosity (of the order of a few poises). We have neglected the velocity gradient component 8ulaz « aujax. Using equations (I I) and (12), we calculate the first contribution to the total dissipation :
*~~ ~
32 fi i ~~' ~~~~
Let us now consider the second case. We assume that the layer thickness does not vary with the in-layer density (Ab/b
= 0 and Aala # 0, Fig. 2b). In the frame of the step, the molecules
move at velocity v~ in the whole smectic-B phase and at the same velocity far from the step in the smectic-A liquid crystal. In contrast, the velocity is slightly different at the vertical side of the step and is given by the equation of conservation of the mass density :
v~=v~(1+2~~) at x=0and0~z~bintheSm.A (15)
a
where ha
= as~_~ as~_B. This problem is thus quite equivalent, from a rheological point of view, to that of an obstacle propagating into the smectic-A phase in a direction parallel to the
layer with the velocity 2 v~ Aala. This problem has been treated theoretically by Clark [I II (from a calculation of de Gennes [12]) who has shown that a permeation boundary layer develops. From the formula given by Clark for the velocity components, it is possible to
calculate the dissipation exactly. A straightforward calculation gives
4l~ = 8 '~ ~~ ~bu( (16)
A~ a
where A~ is the permeation coefficient [10].
In practice, these two mechanisms act together, however, it is easy to see that the second
one is largely dominant. Indeed, one knows that the permeation length, defined to be
fi, is of the order of b [13]
so that 4l~/4lj = 256 $
» by assuming Aala
= Ab/b. The
mobility m of the step can be calculated by equilibrating the viscous force F~t (4l
= F~t u~) with
the supersaturation force F~
= b Ap :
j A~ ~ 2
~ 8 ~ ha
~~~~
This mobility is thus about 100 to 1000 times larger than the climb mobility of an edge
dislocation (of the order of /fi). One
notes that it does not depend on the step height.
With reasonable values of the parameters, I,e, fi=101, ~=2 poise and Aala=
2 x 10~ ~, one calculates m
= 10~~ cm~ sf
~, a value that is in agreement with the experimental
one. Nevertheless, the validity of equation (17) is unclear because of the lack of precision with
which we know the physical constants, and particularly the permeation coefficient
A~, and because of uncertainties in the theory. Indeed, the hydrodynamic approximation used in this calculation is questionable inasmuch as the scales involved in the dissipation are
microscopic whereas the concept of permeation is essentially macroscopic.
5. Conclusion.
We have measured the surface dissipation at the smectic-A smectic-B interface of 408. The
experimental data suggest that the molecular attachment kinetics are dominated by screw
dislocations at small velocity (below 10~m/s) and by nucleation and growth of two-
dimensional nuclei beyond this limit. Finally, we suggest that the step mobility is related to a
permeation flow of the molecules in the smectic-A phase, itself due to a density change
between the two phases.
Acknowledgments.
We ,hank Pr. D. Temkin for very fruitful discussions. This work was supported by the Centre National de la Recherche Scientifique and the Centre National d'Etudes Spatiales.
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[4] WOODRUFF D. P., The Solid-Liquid Interface, Cambridge (1973).
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