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A possible magnetic transition in smectics A
Solid State Communications,
Vol. 11, pp. 1503—1507, 1972. Pergarnon Press.
Printed in Great Britain
A POSSIBLE MAGNETIC TRANSITION IN SMECTICS A 0. Parodi Laboratoire de Physique des Solides,* Université Paris-Sud, Centre d’Orsay, 91 Orsay
(Received 13 July 1972 by P.G. de Gennes)
Possible transitions are discussed for a smectic A film placed between two plates with homeotropic boundary conditions in a magnetic field parallel to the plates. It is predicted that in a central region the molecules will be aligned and that, close to each plate, a periodic pattern of disclinations will allow for folding the layers. 4oe for a 1cm The is vary expected be approximately thickcritical sample field and to as theto inverse square rooti0of the sample thickness.
1. INTRODUCTION
in that sense that the maximum tilt angle remains very low at twice the critical magnetic field.
IN 1933, Freedericks observed, in nematic films, a magnetic transition which was later analysed by Zocher,2 then by Rapini et al.3 This transition takes place when solid boundaries and a magnetic external field have cornpeting incluences upon the orientation of the nematic liquid crystal. This is the case when the nernatic film is placed between parallel plates with horneotropic boundary conditions (the director n is normal to the plates) and when the magnetic field is parallel to the plates (Fig. 1). The critical field is then given by 7T =
(K33
Another kind of magnetic transition was proposed by Heifrich for planar cholesteric structures.6 It consists in an ondulation of the cholesteric planes with a wave vector paraUel to the magnetic field. A complete theory of this transition was given by Hurault,7 and can be extended to the case of smectics A. The author has shown that, in that case, the critical magnetic fields varies like d”2, and is as high as 20koe for a 1cm thick sample. Moreover, this transition appears to be also a ‘ghost transition’.
1/2
—
2d
The physical reason for the ‘ghost’ character of both these transitions is the existence of nearly
where d is the film half-width, K
33 the ‘bend’ ~ elastic constant and the anisotropic part of the magnetic susceptibility. Above this critical field, one observes a distorted configuration that results from a ‘bend’ deformation.
incompressible layers. As a result, the only allowed deformation is a splay deformation and a true magnetic transition should lead to a structure very different from those considered in both Freedericks and Hurault—Helfrich transitions.
The question arises as to whether such a transition is possible in a smectic A. Freedericks’ 5 who transition discussed Rapini has shown has thatbeen it was in fact aby‘ghost transition’,
One can easily see that an orientation of the smectic by thelayers, magnetic fieldforinvolves a foldingmolecules of the smectic which, geometrical reasons that are illustrated in Fig. 2,
*
cannot occur in a perfect film. However, this last restriction can be removed by the creation of de-
Laboratoire associé au C.N.R.S. 1503
1504
A POSSIBLE MAGNETIC TRANSITION IN SMECTICS ~
Vol. 11, No. 11
for the semi-sandwich structure. It follows from / / I
2d
, ..
valueequation this of H, which that means f~is negative that the for semi-sandwich any finite structure is the stable one, and that the critical field H~= 0. The penetration depth A is obtained from the minimization of f~and depends on the
H
//‘“ ,
/
FIG. 1. Configuration of a nematic film for H > H~.in a Freedericks transition.
exact configuration of the transition.
~!1.
(ii) Consider now the case of a film with half-width d. The free energy per unit-surface of the plates is now: = a~d (2) This expression becomes negative, and the sand-
wich structure stable, for H> H,. where H,., the
bi
critical field, is given by: H~ = d 2(2fp/~a)~ (C)
(3)
As will be shown later, the magnetic energy in the transition region is much smaller than the
f I I I IA i~iiHi~Hi1W[HLI f~ I~ I
FIG. 2. A Freedericks-like configuration (a) or a folding of the layers (b) are impossible without defects and provide only a local perturbation of the planar structure. In (c) the central region is field aligned; in two transition regions a periodic pattern of defects allows for folding the layers. fects. One then expects, as a high magnetic field configuration, a sandwich structure where a central aligned region, with the layers normal to the field, lies between two transition regions, extending over a penetration depth A, where a periodic pattern of disclinations allows for the folding of the layers [Fig. 2(c)].
elasticandcoreenergies.Hence,inequation(3), can be taken as field independent, and, for any structure of the transition region, one gets d1’2. It is clear from equation (3) that, in order to see if there is any change to observe such a transition for reasonable values of the film width and of the magnetic field, one must look at the different possible configurations of the transition region and estimate the corresponding free energies. Unfortunately, these numerical estimates rely on numerical values of some physical constants for which no experimental data are presently available. By heuristic arguments, one can obtain an order of magnitude estimate for these physical constants, which permits one to decide
2. GENERAL DISCUSSION The penetration depth A and the critical field
whether or not the transition is observable although it does not permit one to choose among the possible configurations of the perturbed region.
H~can be estimated using the following arguments. (i) Consider first a semi-infinite sample. Let f, be the free-energy per unit surface of the transition region In f~are included the elastic and magnetic energies as well as the core-energies of the disclinations. In the central region the smectic is aligned by the magnetic field and the configuration energy vanishes. Then the total free-energy per unit-surface of the plate is:
f~= f~-.~2 IH2dr 2 C
(1)
These physical constants are (i) the elasticity modulus of the layers E, (ii) the surface tension of the smectic material A, and the smectic—nematic interface tension, A’, and (iii) the core-energies per unit-length of a disclination, f~.and of a dislocation f~. A first evaluation for E is k 3, where a, 3T/a a typical molecular distance, is of the order of 107cm. This gives F 10~cgs, but this value seems too low: one can expect F to be of the
Vol. 11, No.11
A POSSIBLE MAGNETIC TRANSITION IN SMECTICS .4
order of a typical Young’s modulus for an organic crystal, i.e. F S x iO~°_i0~ cgs. A typical value for A, for organic liquids, is 2ocgs. A’ is probably smaller and can be estimated to k 2, 8T37,~/a where 1’s,., is the smectic—nernatic transition temperature. This gives .4’ ~ 4 cgs. An upper limit for f~,is given by the melting energy of the core, i.e. f~ lO5cgs. A more reasonable value is f~= ~K, where K, the splay Frank constant4 is of the order of kBT/a, i.e. K 4 x 10~cgs, f~is the core-energy per unit-length for an edge dislocation, and is of the order of K. As in nematics, ~ lO7cgs.
2x (1.9 x lO3oe x cm 1/2 ). But field is H d~” this value 0 is= probably unrealistic, and, for E = 1010 cgs, f~,, has no minimum, which means that such a transition configuration is not stable. (b) Cylindrical structure + nematic (Fig. 4) The transition region is made up of nondistorted cylindrical structures. These structures do not fill entirely the transition region; the remaining space is filled with nematic. One gets: 1
=
(1 \
K A ——ln—+ 4 A a
+ 37T
—
4/ ..A 2
Where Q
=
p CV(TSI.
f~
—
A
1 +_XaH2A~ 4
7T,
3. POSSIBLE CONFIGURATIONS FOR THE TRANSITION REGION
1505
—
T)
+
10 cal/cm3.
Five different configurations for the transition region will be now examined.
FIG. 4. Th
FIG. 3. A spatial period of the transition region in the case of distorted structures.
e layers are folded in a periodic pattern of cylindrical structures; in the remaining space, the material is nematic.
(a) Distorted configuration (Fig. 3) The transition region is made up of distorted cylindrical structures. The elastic free-energy per unit-volume g is obtained from a generalization of de Gennes8 free-energy FIG. 5. Cylindrical structures again, but here g
=
~i(Vc~2
—
1)2
-4-
(div
the remaining space is unfilled.
~2
n) where the function çf(r) characterizes the layers (a layer is defined by çf~(r)= mb where in is an integer and b the lattice parameter). Here, we take for ç6(r) the following expression: g~(r)= (x2
.~
y2 —?C2x2y2)”2
0.036 EaA
+
K (—1.19
+
1.18 in aj
+
+
the latent heat of the smectic—nematic transition. For f,, = 7TK, f~has no minimum. A significant value of A is obtained only for f,, = 20 K, which is probably much higher than the real value. One then gets A 105cm and H, d~’2x (3.1 x lO4oe x cm1’2).
One then gets:
f~,=
)~xQ~, is
0.25 ~aH2A .
A For E = 4 x 10~cgs, f,~= 7TK, I is minimized for A = 8.5 x iO~cm. The corresponding critical
(c) Cylindrical structure + vacuum (Fig.5) In the preceeding configuration, most of the energy comes from the nernatic region. It is therefore natural to ask if one could lower the energy by allowing the remaining space between the smectic cylinders and the plates to be unfilled. One then gets:
1506
A POSSIBLE MAGNETIC TRANSITION IN SMECTICS 4
1
‘
= (
+
7T 71 K A 1 f 1—.- pA+——ln————0 4) 4 A a 2 A
7T
A
+
A”
Vol. 11, No. 11
(e) Grain boundary (Fig. 7) A grain boundary can be analysed in terms of a continuous density 1/b (b is the lattice parameter) of dislocations with core~nergies
~~aH2X
Here A” is the surface tension of the glass, and can be taken of the same 6cm order and than A. One gets, for f,, = 77K, A = 2.5 x 10 /2
2 x (3.3 x lO4oe x cm
~
A different kind of dislocation with an extra coreenergy f~(f,~ K) takes place at each edge of the periodic pattern. One then gets
f~=L~+L~+...)(~ 0HA. b A
) . H0 value = d”depends strongly on the value of This last the surface tensions.
In all other configurations, the magnetic energy of the transition could be neglected for fields
(d) Multicylindrical structure (Fig. 6)
smaller than lO7oe. Here, minimizing
Another possibility is that the transition region could be entirely filled with tangent
A2
f~,one
gets:
=
~a112
cerr~0~ region
and, neglecting terms in (b/d)1’2, H2=~~~ bd~
Note that, at the critical field, A 0 depends only
FIG. 6. Multicylindrical structures: the remaining space is filled with tangent cylindrical structures.
on the sample thickness:
cylindrical structures. The minimum radius of these cylinders should be of the order of amolecular distance, a. Then ~ ln R + Nj’ 2A fp=—271/i’ ~g+— KTr a~R
A~= 2db (1~ fCI 7). For a 1 cm thick7cgs, sample, taking one finds bA f~= K = 4 x lO H 3oe. 4. CONCLUSION 0 = 9 x lO
where N is the number of cylinders and R their radius. Using a computer, one finds that the three terms of f~,vary as powers of ~ = A/a:
Except for a specially low value of E(E 10~) which would lead to the distorted structure, the grain boundary configuration seems to have the lowest critical field. It is probably the easiest to identify, since the periodicity of the transition region could be detected by optical methods.
~ R
=
0.53X~°32 ~ in
N
=
a
=
0.25~1
31
20 A, f~ 4.5k and=
0.83~132.
One then finds that f~has no minimum, i.e. that there is no such stable configuration.
_______
= =
_______
___________________
FIG. 7. Here the transition region is made up of
periodic boundary grains,
The most efficient methods could be the detection of coherent scattering of a laser beam at Bragg angles. It seems possible to perform such an experiment since (1) one can make 1cm thick monociomain samples, 10 (ii) coherent scattering should not be very much perturbed by the presence of disclinat ions and (iii) the intensity of the signal should not depend on the penetration depth A, provided that A is not much smaller than the laser wavelength. An interesting result is that the experimental values of A 0 and H0 would lead to the determination of the core-energies f~,and f~’.
Vol. 11, No. 11
A POSSIBLE MAGNETIC TRANSITION IN SMECTICS A
Another important result is that, for any of the stable configurations, the critical field is not too large for a sample of reasonable size. It must also be noted that this transition can be much less perfect than is assumed here. There
is a possibility for the existence of a ‘Shubnikov phase’, with many defects, for fields smaller than
H0, and above H0 the central part of the sample will probably be far from perfect. However the change occuring at 1-Ia should be detectable.
Acknowledgements
— The author is greatly indebted to Drs. G. Durand and M. Kléman and to Professors F.C. Frank, J. Friedel, P.G. de Gennes and R.B. Meyer for many fruitful discussions.
REFERENCES 1.
FREEDERICKS V.K. and ZOLINA V., Trans. Faraday Soc. 29, 219 (1933).
2.
ZOCHER H., Trans. Faraday Soc. 29, 245 (1933).
3.
RAPINI A. and PAPOULAR M., J. Phys. 30, C4-54 (1969).
4.
FRANK F.C., Disc. Faraday Soc. 25, 1 (1968).
5.
RAPINI A., J. Phys. 33, 237 (1972).
6.
HELFRICH W., J. Chem. Phys. 55, 839 (1971).
7.
HURAULT J.P., J. Chem. Phys., to be published.
8.
DE GENNES P.G., J. Phys. 30, C4-65 (1969).
9.
See e.g. READ W.T., Dislocations in Crystal, McGraw-Hill, N.Y. (1953).
10.
1507
DURAND G., Private communication.
On étudie la possibilité de transitions pour un film de smectique A place entre deux plaques avec des conditions aux limites homéotropes et soumis a un champ magnétique. On prévoit que, dans une region centrale, les molecules sont alignées par le champ, tandis qu’au voisinage de chaque paroi un réseau périodique de disclinations permet aux couches smectiques de se plier; le champ critique est inversement proportionnel a la racine carrée de l’épaisseur et atteint environ io~oe pour une épaisseur de 1 cm.
Vol. 11, pp. 1503—1507, 1972. Pergarnon Press.
Printed in Great Britain
A POSSIBLE MAGNETIC TRANSITION IN SMECTICS A 0. Parodi Laboratoire de Physique des Solides,* Université Paris-Sud, Centre d’Orsay, 91 Orsay
(Received 13 July 1972 by P.G. de Gennes)
Possible transitions are discussed for a smectic A film placed between two plates with homeotropic boundary conditions in a magnetic field parallel to the plates. It is predicted that in a central region the molecules will be aligned and that, close to each plate, a periodic pattern of disclinations will allow for folding the layers. 4oe for a 1cm The is vary expected be approximately thickcritical sample field and to as theto inverse square rooti0of the sample thickness.
1. INTRODUCTION
in that sense that the maximum tilt angle remains very low at twice the critical magnetic field.
IN 1933, Freedericks observed, in nematic films, a magnetic transition which was later analysed by Zocher,2 then by Rapini et al.3 This transition takes place when solid boundaries and a magnetic external field have cornpeting incluences upon the orientation of the nematic liquid crystal. This is the case when the nernatic film is placed between parallel plates with horneotropic boundary conditions (the director n is normal to the plates) and when the magnetic field is parallel to the plates (Fig. 1). The critical field is then given by 7T =
(K33
Another kind of magnetic transition was proposed by Heifrich for planar cholesteric structures.6 It consists in an ondulation of the cholesteric planes with a wave vector paraUel to the magnetic field. A complete theory of this transition was given by Hurault,7 and can be extended to the case of smectics A. The author has shown that, in that case, the critical magnetic fields varies like d”2, and is as high as 20koe for a 1cm thick sample. Moreover, this transition appears to be also a ‘ghost transition’.
1/2
—
2d
The physical reason for the ‘ghost’ character of both these transitions is the existence of nearly
where d is the film half-width, K
33 the ‘bend’ ~ elastic constant and the anisotropic part of the magnetic susceptibility. Above this critical field, one observes a distorted configuration that results from a ‘bend’ deformation.
incompressible layers. As a result, the only allowed deformation is a splay deformation and a true magnetic transition should lead to a structure very different from those considered in both Freedericks and Hurault—Helfrich transitions.
The question arises as to whether such a transition is possible in a smectic A. Freedericks’ 5 who transition discussed Rapini has shown has thatbeen it was in fact aby‘ghost transition’,
One can easily see that an orientation of the smectic by thelayers, magnetic fieldforinvolves a foldingmolecules of the smectic which, geometrical reasons that are illustrated in Fig. 2,
*
cannot occur in a perfect film. However, this last restriction can be removed by the creation of de-
Laboratoire associé au C.N.R.S. 1503
1504
A POSSIBLE MAGNETIC TRANSITION IN SMECTICS ~
Vol. 11, No. 11
for the semi-sandwich structure. It follows from / / I
2d
, ..
valueequation this of H, which that means f~is negative that the for semi-sandwich any finite structure is the stable one, and that the critical field H~= 0. The penetration depth A is obtained from the minimization of f~and depends on the
H
//‘“ ,
/
FIG. 1. Configuration of a nematic film for H > H~.in a Freedericks transition.
exact configuration of the transition.
~!1.
(ii) Consider now the case of a film with half-width d. The free energy per unit-surface of the plates is now: = a~d (2) This expression becomes negative, and the sand-
wich structure stable, for H> H,. where H,., the
bi
critical field, is given by: H~ = d 2(2fp/~a)~ (C)
(3)
As will be shown later, the magnetic energy in the transition region is much smaller than the
f I I I IA i~iiHi~Hi1W[HLI f~ I~ I
FIG. 2. A Freedericks-like configuration (a) or a folding of the layers (b) are impossible without defects and provide only a local perturbation of the planar structure. In (c) the central region is field aligned; in two transition regions a periodic pattern of defects allows for folding the layers. fects. One then expects, as a high magnetic field configuration, a sandwich structure where a central aligned region, with the layers normal to the field, lies between two transition regions, extending over a penetration depth A, where a periodic pattern of disclinations allows for the folding of the layers [Fig. 2(c)].
elasticandcoreenergies.Hence,inequation(3), can be taken as field independent, and, for any structure of the transition region, one gets d1’2. It is clear from equation (3) that, in order to see if there is any change to observe such a transition for reasonable values of the film width and of the magnetic field, one must look at the different possible configurations of the transition region and estimate the corresponding free energies. Unfortunately, these numerical estimates rely on numerical values of some physical constants for which no experimental data are presently available. By heuristic arguments, one can obtain an order of magnitude estimate for these physical constants, which permits one to decide
2. GENERAL DISCUSSION The penetration depth A and the critical field
whether or not the transition is observable although it does not permit one to choose among the possible configurations of the perturbed region.
H~can be estimated using the following arguments. (i) Consider first a semi-infinite sample. Let f, be the free-energy per unit surface of the transition region In f~are included the elastic and magnetic energies as well as the core-energies of the disclinations. In the central region the smectic is aligned by the magnetic field and the configuration energy vanishes. Then the total free-energy per unit-surface of the plate is:
f~= f~-.~2 IH2dr 2 C
(1)
These physical constants are (i) the elasticity modulus of the layers E, (ii) the surface tension of the smectic material A, and the smectic—nematic interface tension, A’, and (iii) the core-energies per unit-length of a disclination, f~.and of a dislocation f~. A first evaluation for E is k 3, where a, 3T/a a typical molecular distance, is of the order of 107cm. This gives F 10~cgs, but this value seems too low: one can expect F to be of the
Vol. 11, No.11
A POSSIBLE MAGNETIC TRANSITION IN SMECTICS .4
order of a typical Young’s modulus for an organic crystal, i.e. F S x iO~°_i0~ cgs. A typical value for A, for organic liquids, is 2ocgs. A’ is probably smaller and can be estimated to k 2, 8T37,~/a where 1’s,., is the smectic—nernatic transition temperature. This gives .4’ ~ 4 cgs. An upper limit for f~,is given by the melting energy of the core, i.e. f~ lO5cgs. A more reasonable value is f~= ~K, where K, the splay Frank constant4 is of the order of kBT/a, i.e. K 4 x 10~cgs, f~is the core-energy per unit-length for an edge dislocation, and is of the order of K. As in nematics, ~ lO7cgs.
2x (1.9 x lO3oe x cm 1/2 ). But field is H d~” this value 0 is= probably unrealistic, and, for E = 1010 cgs, f~,, has no minimum, which means that such a transition configuration is not stable. (b) Cylindrical structure + nematic (Fig. 4) The transition region is made up of nondistorted cylindrical structures. These structures do not fill entirely the transition region; the remaining space is filled with nematic. One gets: 1
=
(1 \
K A ——ln—+ 4 A a
+ 37T
—
4/ ..A 2
Where Q
=
p CV(TSI.
f~
—
A
1 +_XaH2A~ 4
7T,
3. POSSIBLE CONFIGURATIONS FOR THE TRANSITION REGION
1505
—
T)
+
10 cal/cm3.
Five different configurations for the transition region will be now examined.
FIG. 4. Th
FIG. 3. A spatial period of the transition region in the case of distorted structures.
e layers are folded in a periodic pattern of cylindrical structures; in the remaining space, the material is nematic.
(a) Distorted configuration (Fig. 3) The transition region is made up of distorted cylindrical structures. The elastic free-energy per unit-volume g is obtained from a generalization of de Gennes8 free-energy FIG. 5. Cylindrical structures again, but here g
=
~i(Vc~2
—
1)2
-4-
(div
the remaining space is unfilled.
~2
n) where the function çf(r) characterizes the layers (a layer is defined by çf~(r)= mb where in is an integer and b the lattice parameter). Here, we take for ç6(r) the following expression: g~(r)= (x2
.~
y2 —?C2x2y2)”2
0.036 EaA
+
K (—1.19
+
1.18 in aj
+
+
the latent heat of the smectic—nematic transition. For f,, = 7TK, f~has no minimum. A significant value of A is obtained only for f,, = 20 K, which is probably much higher than the real value. One then gets A 105cm and H, d~’2x (3.1 x lO4oe x cm1’2).
One then gets:
f~,=
)~xQ~, is
0.25 ~aH2A .
A For E = 4 x 10~cgs, f,~= 7TK, I is minimized for A = 8.5 x iO~cm. The corresponding critical
(c) Cylindrical structure + vacuum (Fig.5) In the preceeding configuration, most of the energy comes from the nernatic region. It is therefore natural to ask if one could lower the energy by allowing the remaining space between the smectic cylinders and the plates to be unfilled. One then gets:
1506
A POSSIBLE MAGNETIC TRANSITION IN SMECTICS 4
1
‘
= (
+
7T 71 K A 1 f 1—.- pA+——ln————0 4) 4 A a 2 A
7T
A
+
A”
Vol. 11, No. 11
(e) Grain boundary (Fig. 7) A grain boundary can be analysed in terms of a continuous density 1/b (b is the lattice parameter) of dislocations with core~nergies
~~aH2X
Here A” is the surface tension of the glass, and can be taken of the same 6cm order and than A. One gets, for f,, = 77K, A = 2.5 x 10 /2
2 x (3.3 x lO4oe x cm
~
A different kind of dislocation with an extra coreenergy f~(f,~ K) takes place at each edge of the periodic pattern. One then gets
f~=L~+L~+...)(~ 0HA. b A
) . H0 value = d”depends strongly on the value of This last the surface tensions.
In all other configurations, the magnetic energy of the transition could be neglected for fields
(d) Multicylindrical structure (Fig. 6)
smaller than lO7oe. Here, minimizing
Another possibility is that the transition region could be entirely filled with tangent
A2
f~,one
gets:
=
~a112
cerr~0~ region
and, neglecting terms in (b/d)1’2, H2=~~~ bd~
Note that, at the critical field, A 0 depends only
FIG. 6. Multicylindrical structures: the remaining space is filled with tangent cylindrical structures.
on the sample thickness:
cylindrical structures. The minimum radius of these cylinders should be of the order of amolecular distance, a. Then ~ ln R + Nj’ 2A fp=—271/i’ ~g+— KTr a~R
A~= 2db (1~ fCI 7). For a 1 cm thick7cgs, sample, taking one finds bA f~= K = 4 x lO H 3oe. 4. CONCLUSION 0 = 9 x lO
where N is the number of cylinders and R their radius. Using a computer, one finds that the three terms of f~,vary as powers of ~ = A/a:
Except for a specially low value of E(E 10~) which would lead to the distorted structure, the grain boundary configuration seems to have the lowest critical field. It is probably the easiest to identify, since the periodicity of the transition region could be detected by optical methods.
~ R
=
0.53X~°32 ~ in
N
=
a
=
0.25~1
31
20 A, f~ 4.5k and=
0.83~132.
One then finds that f~has no minimum, i.e. that there is no such stable configuration.
_______
= =
_______
___________________
FIG. 7. Here the transition region is made up of
periodic boundary grains,
The most efficient methods could be the detection of coherent scattering of a laser beam at Bragg angles. It seems possible to perform such an experiment since (1) one can make 1cm thick monociomain samples, 10 (ii) coherent scattering should not be very much perturbed by the presence of disclinat ions and (iii) the intensity of the signal should not depend on the penetration depth A, provided that A is not much smaller than the laser wavelength. An interesting result is that the experimental values of A 0 and H0 would lead to the determination of the core-energies f~,and f~’.
Vol. 11, No. 11
A POSSIBLE MAGNETIC TRANSITION IN SMECTICS A
Another important result is that, for any of the stable configurations, the critical field is not too large for a sample of reasonable size. It must also be noted that this transition can be much less perfect than is assumed here. There
is a possibility for the existence of a ‘Shubnikov phase’, with many defects, for fields smaller than
H0, and above H0 the central part of the sample will probably be far from perfect. However the change occuring at 1-Ia should be detectable.
Acknowledgements
— The author is greatly indebted to Drs. G. Durand and M. Kléman and to Professors F.C. Frank, J. Friedel, P.G. de Gennes and R.B. Meyer for many fruitful discussions.
REFERENCES 1.
FREEDERICKS V.K. and ZOLINA V., Trans. Faraday Soc. 29, 219 (1933).
2.
ZOCHER H., Trans. Faraday Soc. 29, 245 (1933).
3.
RAPINI A. and PAPOULAR M., J. Phys. 30, C4-54 (1969).
4.
FRANK F.C., Disc. Faraday Soc. 25, 1 (1968).
5.
RAPINI A., J. Phys. 33, 237 (1972).
6.
HELFRICH W., J. Chem. Phys. 55, 839 (1971).
7.
HURAULT J.P., J. Chem. Phys., to be published.
8.
DE GENNES P.G., J. Phys. 30, C4-65 (1969).
9.
See e.g. READ W.T., Dislocations in Crystal, McGraw-Hill, N.Y. (1953).
10.
1507
DURAND G., Private communication.
On étudie la possibilité de transitions pour un film de smectique A place entre deux plaques avec des conditions aux limites homéotropes et soumis a un champ magnétique. On prévoit que, dans une region centrale, les molecules sont alignées par le champ, tandis qu’au voisinage de chaque paroi un réseau périodique de disclinations permet aux couches smectiques de se plier; le champ critique est inversement proportionnel a la racine carrée de l’épaisseur et atteint environ io~oe pour une épaisseur de 1 cm.