PLASTICITY OF LIQUID CRYSTALS J. Friedel
To cite this version: J. Friedel. PLASTICITY OF LIQUID CRYSTALS. Journal de Physique Colloques, 1979, 40 (C3), pp.C3-45-C3-52. <10.1051/jphyscol:1979311>.
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Colloque C3, supplkment au no 4, Tome 40, Avril 1979, page C3-45
JOURNAL DE PHYSIQUE
MECHANICAL PROPER TIES, PLASTICITY AND DEFECTS. PLASTICITY OF LIQUID CRYSTALS J. FRIEDEL
Laboratoire de Physique des Solides (*) Biitiment 510, Universitt Paris Sud, 91405 Orsay, France
Rksumk. - Les comportements visqueux et plastiques des nkmatiques et des pmectiques sont comparks. Les similitudes et les diffkrences sont soulignkes. Abstract. - Possible viscous and plastic behaviours of nematics and smectics are compared. Similarities and differences are stressed.
Introduction. - Asked to talk about the plasticity of liquid crystals, I have to stress from the start that I have nothing original to report. The reason is simple : the very fluid nematics and cholesterics have been fairly fully studied in their macroscopic motions, and nothing much has to be added to what is already known ; the study of the more viscous smectics has barely started, and here very little is known and things are certainly complex. As a result, I can only compare the two situations and put a few questions.
FIG. 1 . - Dislocation of rotation
+ n (fils) (free nematic).
1. Nematics. - They are known to be liquids with a low and anisotropic viscosity. a) Thus a well ordered (( single cr.vstal )) of such a phase shears under a low shear stress o with a strain rate h proportional to o : The viscosity coefficient v is low (less than 1 poise) and anisotropic, i.e. it is a tensor. Under an increasing stress a, this simple laminar flow becomes unstable above a critical value of o, and convective cells appear. This is similar to the Benard instability of isotropic fluid ; but the anisotropy of v makes the instability more easy and more complex. Under even higher stresses, an irregular turbulent flow occurs, again more easily than in isotropic fluids [I]. b) Real nematic crystals have actually some defects : singular lines, which are rotation dislocations (or disclinations) of rotation f n (Figs. la, b), and singular points of opposite strengths  (Figs. 2a. b). These are respectively the fils )) and the (( noyaux )) of the 1920's . (*) AssociC au C.N.R.S. (LA No 2).
FIG. 2. - Singular points (noyaux) (free nematic).
These defects 'can occur in a perfect crystal as dislocation loops or as pairs of opposite points. But because these defects attract as an inverse power of their distance, very large stresses are required , even with the help of thermal agitation, which only prevails possibly in the turbulent regime. But some such defects can preexist in a real crystal, because of imperfect recovery during the preparation or because of irregularities in the boundary conditions. The dislocation lines can then possibly multiply under a low stress, by a process akin to the Frank and Read-Herring pole mechanism of perfect crystals [5, 61. Defects can also be produced under low stresses at special places of the boundaries,
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which act as stress concentrators. One can then produce imperfect crystals with many defects. c) The following questions need stressing about the processes involving the creation and motion of defects.
The production of these defects adds a small but finite friction, which will depend critically on boundary conditions ; i.e. it is not an intrinsic property of the liquid crystal. Because of their high mobility, the defects thus created in such a texture rearrange and mostly disappear fairly quickly, in times of the order of seconds. Because of this recovery, the hysteresis produced by this friction will be frequency dependent in this range of frequency. This whole field needs more quantitative measurements and it would help to understand the very similar field of production of vortex lines un suprafluid helium. - One can ask whether there is any coupling of these defects with the instabilities described above. One certainly expects defects to be coupled with the geometry of the convective cells, and it would be interesting to study the nature of this coupling. But, except in very imperfect crystals, one does not expect the critical conditions for convective instability to be very sensitive to the density of defects. Contrarywise, it is difficult to imagine a turbulent flow in a crystal which would not contain any of these defects. Indeed, one can think that turbulency comes with the cooperative creation of many such defects, indeed with a rdntinuous distribution of defects of infinitesimal strengths, in the same way as melting can formally be described as the cooperative production in a crystal of a continuous distribution of infinitesimal and fluctuating translation dislocations [7, 81. - Even the slow motion of disclinations in otherwise perfect nematic crystals poses some unanswered questions. For instance the Philips group 191 studied some time ago a simple geometry where a thin layer of nematic, of 20 to 80 A thickness d, was under a f 7112 torsion between two glass plates where molecules lied parallel to the plates (Fig. 3). Domains with 7112 torsion were separated by dislocation lines of rota71, which could be analysed in the de Gennes tion
approximation [ l o ] , because of the low value of d. The Philips group checked that the velocity of the dislocations was proportional to the applied stress, due either to the line tension and curvature or to a magnetic field applied parallel to the plates and which stabilised one type of domains with respect to the other. But they seemed to observe that the dislocations in motion had a rather broad core, of order nearly one micron ; also the mobility of the dislocations was high, and could only be explained by assuming that no energy was spent within a large core radius, of the same order of magnitude. There is certainly a problem for a rotation dislocation with a core of atomic dimensions to move without changing its form. This would clearly involve the molecules near the core to rotate by large angles, of the order of 4 2 , in times of order vll, if v is the velocity of the dislocation and I the size of the molecules. This in turn would require local stresses that would be much larger than the critical stresses for convective instability. In other words, a moving dislocation should progressively change its form with increasing velocity, and is likely to have convective instabilities within a fairly broad core. It is in particular easy to check that the solution cp
Y Arctg x - vt
which represents a dislocation moving with a velocity u along the x axis is not a solution of the equation of motion
sin 2 cp
(where the last term is the magnetic force) ; and the
acp would diverge as x - l . friction term v at Another way of stating the same problem is to remark that a disclination $2is equivalent to a parallel
FIG. 3. - Disclination between glass plates (nematic under f x/2 torsion).
of a rotation dislocation from f2 to
PLASTICITY O F LIQUID CRYSTALS
disclination S2' (SZ' = SZ) shifted by d, plus a continuous distribution of edge dislocations (Fig. 4). When a disclination moves from to a', it must [ll]. - Bring along with it the cloud of translation dislocations which relaxes its long range stresses other than splay, torsion and bend. Disperse the continuous distribution of edge dislocations along its trail. The first effect leads to the long range friction computed by the Philips group. The second could lead to convective instability of the core. There has been little detailed study of the core of moving disclinations so far. The observations on nematic polymers  are however qualitatively coherent with the preceeding remarks.
FIG. 6. - Shearing of layers in a smectic with intralayer ordering) : a) by dislocation glide ; b) by vacancy diffusion.
In the second one, the layers glide past each other (Fig. 7).
2. A smectics. - Smectics present, a priori, a much more complex behaviour which we shall try and discuss only in the simpler case of A smectics. 2.1 LIQUIDVISCOSITY. - A perfect single crystal of A smectic is expected to have two modes of easy shear with liquid viscosity [I]. - In the first one, each laver is sheared with a constant amount (Fig. 5).
FIG. 7. - Shearing layer on layer (A smectic).
NB : In other smectics with some short or long range molecular order between layers, one expects again a solid friction, due to the formation and glide of translation dislocations gliding between layers (Fig. 8). But again by analogy with what is known of glide in lamellar structures, and especially in crystalline polymers, this type of glide is expected to be easy, i.e. to occur at very low (solid) friction.
of layers (A smectic).
NB : It can be remarked that in other smectics with an intralayer order, such a shear would require either the production and glide of translation dislocations (Fig. 6a), or- the production and diffusion of molecular vacancies (Fig. 6b). The first type of process would correspond to a solid friction, the second to a viscous friction by Nabarro creep. By analogy with plastic molecular crystals, it is expected that Nabarro creep  is dominant and easy, leading to a low viscosity even for these smectics with intraplanar order. No clear cut experimental study of this mode of shear has been done so far.
FIG. 8. - Shearing by dislocation glide in the presence of interlayer'ordering.
The only simple experiment which refers to this type of shear is the work of Kleman and Horn  reported at this conference. They checked that a slow shear of this kind of a
good smectic crystal does not destroy it. Also in a shear between cone and plate at small angle, which is near to such a shear, they observed the progressive development, from irregularities such as specks of dust, of a regular array of cofocal domains (Fig. 9a) which might possibly be akin to that described by Meyer et al.  (Fig. 9b). The stress a necessary to produce a strain rate E: can then be written as
possibly be due to the anchoring of the focal conics on the boundaries, which might require their conformation and energy to vary strongly during shear. However more experiments are required in this range. There is an obvious similarity between the occurrence of these cofocal domains and the Benard convective instability of nematics and isotropic fluids. Both are essentially periodic perturbations of the structure. The similarity is reinforced by the fact that straining at high rates (i: > 80 s-') produces irregular cofocal domains reminiscent of turbulent flow. However the similarity must not be stressed too far : - The regular and irregular cofocal domains are static or quasistatic perturbations. The regular cofocal domains appear by heterogeneous nucleation, and not as a uniform mode of instability throughout the volume of the crystal. ,-
FIG. 9. - a) Area of cofocal domains (seen from above) ; b) proposed coupling of two parabolic focal conics (seen from the side).
2.2 PERMEATION . - This mode of deformation assumes that molecules can jump from layer to layer by a thermally activated process reminiscent of vacancy or interstitial diffusion in crystals (Fig. 11). This process can alter the area of the corresponding layers. And the velocity of liquid diffusion due to permeation is obviously related to the local pressure gradient normal to the layers :
The viscosity v is low, of order of 1 poise and varies but little with the proportion of region with cofocal domains ; but the solid friction A increases linearly with the proportion of area of cofocal domains (Fig. lo), with a maximum value A,,, K/12 if K is the elastic bend coefficient and I the size of the cofocal domains. This law shows that the perfect crystal regions have a fairly low viscosity, as expected. It suggests that the regions of cofocal domains have a solid friction A,,, of the order of the elastic energy stored in the production of the domains. This might
process involved in permeation.
The permeation rate could in principle be measured in an experiment where a smectic single crystal fills a cylindrical tube so that its layers and anchored normal to the sides of the tube and a pressure gradient is established along the tube (Fig. 12). Then eq. (3) should hold, except for a small region near to the sides of the tube of extent 
Variation of solid friction A and viscous coefficient with percentage of area of cofocal domains.
where v is the coefficient of viscosity defined in (2) for the flow of matter parallel to the layers (K-' is of molecular dimensions).
PLASTICITY OF LIQUID CRYSTALS
FIG.13. - Shearing of a smectic crystal with permeation. FIG.12.- Geometry and velocity u in an experiment to measure permeation.
Experiments of this kind have failed to materialise up ti1 now. However there is ample proof of the existence and ease of permeation from experiments where the size and number of layers are altered. The simplest of such experiments might be thought that where a smectic crystal is sheared in the way pictured figure 13. It is however obvious that such a shear or its opposite will involve the creation or destruction of layers. This must in turn involve the creation and motion of defects which will act as sources and (or) sinks for permeating molecules (Fig. 14) : - steps on surfaces, - elementary or multiple dislocations with an edge character, - focal conics in their parts which are not parallel to the layers. Now except perhaps at the surface under suitable boundary conditions, the creation of such sources and sinks requires finite and fairly large stresses . It is then expected that at least under low stresses, percolation to be effective must involve the motion and possibly the development of preexisting defects. The only such motion which has been studied from a theoretical point of view is that of an edge dislocation . A study of the flow of matter can be made, during the climb of such a dislocation normal to its glide plane (defined by its geometrical position D and its Burgers vector b) and thus parallel
FIG.14. - Sources or sinks of permeating molecules : a) surface step ; b) elementary translation dislocation with an edge cornponent ; c) multiple translation dislocation with an edge component ; d) focal conic which is not parallel to layers.
FIG. 15. - Range of permeation behind a climbing dislocation.
to the smectic layers. It shows that only a thin parabolic zone is affected by permeation in the wake of the climbing dislocation (Fig. 15).
G ( K - I x)'l2
and that the force per unit length necessary to produce a velocity u is multiplied  by a factor Kb compared with a classical viscous force vv : F = Kbvv.
The only experiments relating to the set up of figure 13,are those of Durand et al. . By applying a sudden strain and measuring the elastic stress and its relaxation (Fig. 16), they showed that, at low stresses, the fairly fast relaxation time could be explained by a percolation process by the climb of a small density of edge dislocations. The amount of
FIG. 17. - Instabilities under large stresses ; a) undulation of layers under extension , b) molecular tilt under compression.
where 6/d is the extension of the layers and 8 their angular rotation. Hence 
2 a d
where d is the thickness of the sample. Here again a similarity with the convective instability of fluids is obvious. But it seems clear also that defects - translation dislocations and focal conics strongly interfer with the oscillations of the layers, and stabilise the buckled state, in a way that has not been completely elucidated. Again, at larger stresses, irregular cofocal domains appear, reminiscent of the turbulent state of fluids . 2.3 DISLOCATION GLIDE. - The equivalent of low temperature deformation in crystals would be the glide of translation dislocations parallel to their Burgers vector . The following predictions can be made : - Glide of screws, which extends or contracts a surface step, should be easy, because it does not FIG. 16. - Application of strain ~ ( t and ) stress production and relaxation u(t) in Durand el al's experiment. 'changes the nature of the core of the dislocation (Fig. 18). relaxation seemed to show a minimum value compatible with the climb of an elementary dislocation of bminequal to a layer thickness. It would be of interest to check where these climbing dislocations come from, and whether they nucleate under the applied stress or more likely move and eventually multiply from ,some initial (screw ?) dislocations or from regions of stress concentrations at the boundaries. It is also not clear that a relaxation of less than b, could not be produced by an elementary dislocation climbing only part of the way through the region where the thickness is measured. Of course at higher fast stresses, buckling instabilities are observed , by bending of the layers in FIG. 18. - Screw dislocation. extension (Fig. 17a) and by molecular tilt under compression (Fig. 17b). In extension for instance, the wave length of buckling A, is obtained by mini- Glide of elementary edge dislocations involves mizing the total free energy the periodic variation of the molecular structure of the core. For a straight dislocation perpendicular to the plane of figure, the configurations of figures 19a, c are equivalent, but with different energy from those
PLASTICITY O F LIQUID CRYSTALS
exchange between them of a continuous distribution of translation dislocations which would strongly distort the core and increase its energy [I I]. Very little is known experimentally about these motions; and the method developed by Meyer et al. [211 to underline their cores by producing locally a A -+ C phase transformation should also block significantly their glide (but not their climb !). The only type of experiments which has invoked such a glide is in the early studies by Mrs Williams  of the shear of a smectic crystal in the geometry of figure 20a. Above a critical elasti&deformation pictured figure 20b, Mrs Williams invoked the compression 1 - cos 8 of the layers, due to their inclination 8, to assume the periodic rupture of some layers, as shown figure 20c. If by permeation the dislocations thus formed in the volume of the sample come near
FIG. 19. - Glide of elementary edge dislocation : a, b, c, three successive stages.
of figure 19b. The dislocation is then expected to tend to lie along the most stable of these two extreme configurations (probably that of figures 19a, c) ; and a finite Peierls stress is necessary to induce this straight dislocation to glide past less stable configurations such as that of figure 19b. At finite temperatures, a thermally activated process involving the kinking of parts of the dislocation between the equivalent configurations of figures 19a, c, ... should lower the frictional Peierls stress . There is no numerical estimate of these friction ; however by analogy with what is known of molecular crystals, they should not be unduly large at the temperatures of experiments. - ~ l i d eof multiple edge dislocations should on the contrary be practically impossible. The reason is that the glide of the two rotation dislocations in which they split (Fig. 14c) would involve the
FIG. 20. - Shearing a smectic crystal in Mrs Williams' geometry : a) initial state ; b) elastic distortion ; c) hypothetical elementary dislocations ; d) multiple dislocation ; e) Granjean wall of cofocal domains.
. I . FRIEDEL
to the surfaces, the average distance of layers could go back near to its equilibrium value in the volume of the sample if the average distance between elementary dislocations was not much bigger than
What was observed by Mrs Williams was actually not these elementary dislocations, but multiple ones. of Burgers vectors of 50 to 100 b, regularly spaced on the surfaces of the sample and which she assumed were produced by a regrouping by glide of the elementary dislocations (Fig. 20d). It is easy to see that this indeed reduces the total strain energy and that the distance between layers is preserved in the volume if the distance between dislocations is still given by (8). where b is now the Burgers vector of the multiple dislocations. Because of the inclination 8 of the layers, the cores of the dislocations, figure 20d. are asymmetrical, and produce a sharper angle of the layers on one side of the core. It is then easy to understand qualitatively that, on increasing the shear, these parallel multiple dislocations are progressively replaced by a Grandjean ~lall of focal conics, with ellipses on or near the surfaces and hyperbolae plunging into the volume of the sample. It is indeed well known that multiple dislocations have a tendency to split into rows of cofocal domains , and that the Grandjean wall provides the arrangement of lowest energy of these domains to produce a general bending of the smectic structure along a grain boundarv which is here the surface . It must be stressed that if the later stages pictured figures 20d and e are well established experimentally, the details of this analysis leave room for some controversy : - The stage pictured figure 20d of elementary
dislocations has not been observed. If such elementary dislocations were formed and were able to glide parallel to the surface to regroup, they would also be able to glide so as to relax the bending of the layers. Such a relaxation processes has not been observed in this geometry. This clearly indicates that the elementary dislocations are not formed or not able to glide. - One can imagine that the layers of the multiple dislocations of figure 20e are directly nucleated and formed from the surfaces by a percolation process. - More recent studies by Parodi et al.  under alternating shear stresses have shown that focal conics were periodically produced in the volume of the sample, with their ellipses parallel to the surface ; and under high enough amplitudes of shear, these ellipses could separate and go to the surfaces of the sample, where they would form a configuration similar to that of figure 20e. This observation seems to indicate that focal conics can move fairly easily by percolation. 3. Conclusion. - This discussion shows that nearly perfect smectic shears easily by glide of layer over layer, with a low viscosity. However under large enough stresses, the structure becomes unstable and becomes dislocated or produces cofocal domains which are more difficult to deform. The deformation of these less perfect structures involves the motion by permeation of dislocations or of focal conics. The higher viscosity or even solid friction observed in this range does not arise from the permeation process, which seems easy, but from the small number of defects involved or from their long range elastic interactions respectively. No clear observation of dislocation glide has been made. There is obviously more studies to be made, especially of the motion of isolated defects and their interactions.
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