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Comptes Rendus Physique
Volume 17, n° 3-4
pages 242-263 (mars 2016)
Doi : 10.1016/j.crhy.2015.12.002
Dislocations and other topological oddities
Dislocations et autres bizzareries topologiques
 

Fig. 1




Fig. 1 : 

Charles Frank, friend of Jacques Friedel since Bristol years: a) smoking a pipe and walking with John Bechhoefer during the Jacques Friedel's Jubilée in Orsay, b) figure from the generic paper of Burton, Cabrera and Frank [2].


Fig. 2




Fig. 2 : 

Coexistence of fcc and bcc crystallites detected by means of the microspectrometry: a) setup, b) λ 110 =527 nm, bright Bragg reflection from the bcc crystallite in the center, c) λ 111 =542 nm, bright Bragg reflection from fcc crystallites surrounding the bcc crystallites, d) spectra of Bragg reflections. Blue and yellow arrows in b point respectively to large-angle grain boundaries (lagb) and small-angle grain boundaries made of dislocations (sagb).


Fig. 3




Fig. 3 : 

System of edge dislocation in a colloidal crystal confined between a glass sphere and a glass cover slide: a) scheme of the setup, b) image taken with an inverted reflecting microscope equipped with an immersion objective. Variations of colors unveil strains in the vicinity of dislocations. Arrows point large angle grain boundaries (lagb) which are melted and small angle grain boundaries (sagb) made of dislocations visible as black dots.


Fig. 4




Fig. 4 : 

Jacques Friedel (a) and the beginning of his hand-written calculation of the energy of a system of dislocations (b). One can read: On a une collection de dislocations coin. Les conditions aux limites rigides sont équivalentes à l'adjonction de dislocations mirages. (Annotations in red have been added with the aim to improve the readability of a pencil-made drawing.)


Fig. 5




Fig. 5 : 

Series of structures in a suspension of hard spheres confined in a wedge formed by two glass walls: a) geometry of the setup, b) view in transmitted light at low magnification, c) view of the 2Δ⇒3□ frontier at high magnification. (PhD work of B. Pansu.)


Fig. 6




Fig. 6 : 

Explanation of the “patchy” aspect of stacks of triangular layers: a) definition of a, b and c positions in stacks of triangular layers, b and b') two stacks of three optically equivalent layers, c and c') two other stacks of three optically equivalent layers, d, e and f) three stacks of four optically different layers.


Fig. 7




Fig. 7 : 

Theory of structural transitions: a) high-pressure limit, stacks of triangular and square layers [15]; the inset shows alternating “up” and “down” motions splitting one square layer 1□ into the stack 2□ of two square layers, b) high-pressure limit, dashed lines: stacks of triangular and square layers [15]; plain lines: rhombic deformation of square stacks, c) result of the Monte Carlo simulation performed by Schmidt and Löwen [16]. (Domains labeled “forbidden” are not accessible without overlapping of particles.)


Fig. 8




Fig. 8 : 

Buckling transition in a monolayer of N-isopropyl acrylamide microgel spheres dispersed in water and confined between two glass plates: a–b) schematic representations of the sample, top and side views, c–d) optical images taken at T =27.1°C and 24.7°C. (Adapted from reference [17] with permission of The Royal Society of Chemistry.)


Fig. 9




Fig. 9 : 

Observation of the growth of a BPI crystal from the isotropic phase (Iso.): a) experimental set-up. Temperatures T 1 and T 2 are regulated with an accuracy of 0.01°C, T 2 <T 1 , b) growth of the BPI crystal by the spiral motion of the step connected to the dislocation emerging on the (110) facet at isotropic/BPI interface, c) growth of the BPI crystal by nucleation of steps on the (110) facet. Photographs taken with a reflecting microscope in monochromatic illumination with the wavelength λ =558 nm tuned to the Bragg reflection from (110) crystal planes. Knowing that the average refractive index of BPI is n 1.5, one obtains from the Bragg formula the interplanar distance h 110 =λ /(2n )=186 nm. Composition of the sample: 58.5% of CB15 in ZLI1140. (Collaboration with P.E. Cladis and R. Barbet-Massin.)


Fig. 10




Fig. 10 : 

Model of the BPI phase of symmetry I 41 32 built around the periodic minimal surface G (gyroid) of symmetry Ia 3d . a) Two adjacent chiral molecules, like two screws matching their threads, have the tendency to adopt a twisted configuration. b) Double-twist configuration around the molecule located in P. c) An elementary piece of the minimal surface G. The whole surface G can be built from such pieces. At F, the flat point, principal curvatures are zero. Thick lines represent a chiral director field n that has a singularity at F: a −1/2 disclination. d) two labyrinths separated by the surface G .


Fig. 11




Fig. 11 : 

Three dislocations piercing the BPI crystal. a) Side view of the setup, dislocations are drawn with dotted lines. b) Focus on the isotropic/BPI interface. c) Focus on the BPI/glass interface. Photographs taken with a reflecting microscope in monochromatic illumination. (Collaboration with P.E. Cladis and R. Barbet-Massin.)


Fig. 12




Fig. 12 : 

Relationship between Burgers vectors and steps connected to dislocations. a) Dislocation emerging on a facet. The height h of the step connected to the dislocation depends on the length of the component perpendicular to the facet. b) Decomposition of the Burgers vector into components respectively parallel and perpendicular to the facet. c) Orientation of Burgers vectors with respect to the (110) facet.


Fig. 13




Fig. 13 : 

Observation of growth spirals on the (110) facet of a BPI-in-Isotropic crystal: a and f) photographs of steps taken with a reflecting microscope, b) schematic representation of the initial position of two steps connected to three dislocations labeled as +, + and −, c) the arrow points two steps coming into a contact during their motion (former position of steps is drawn with a dashed line), d) result of the recombination of the two steps, f) further motion of steps resulting from recombination. (Collaboration with P.E. Cladis and R. Barbet-Massin.)


Fig. 14




Fig. 14 : 

Splitting of steps: a) a perspective view of a splitted step, b) three successive planes containing disclinations.


Fig. 15




Fig. 15 : 

Structure of the bicontinuous lyotropic phase Ia 3d : a) bilayer made of surfactant molecules oriented with their hydrophobic tails toward the minimal surface called gyroid (G), b) the two labyrinths separated by the bilayer are filled with water. Remark: The two sides of the gyroid and the two labyrinths are drawn with different colors for a better visibility but, as the d -glide plane symmetry operation of the space group Ia 3d (see Fig. 18b) exchanges the two labyrinths, they should have the same colors.


Fig. 16




Fig. 16 : 

Inverted bicontinuous lyotropic phases. Upper row: spherical domains cut out from periodic minimal surfaces. Lower row: experimentally observed faceted crystal habits. (Collaboration with W. Gozdz and L. Latypova [26,27].)


Fig. 17




Fig. 17 : 

Observation of steps on facets of the Ia 3d crystals: a) scheme of the setup, b) phase diagram of the monoolein/water system. (Collaboration with S. Leroy [28].)


Fig. 18




Fig. 18 : 

Steps on the (110) facet of a Ia 3d crystal: a–d) nucleation of successive steps, e) glide plane d, f) successive (110) planes related by the glide plane symmetry. (Collaboration with S. Leroy [28].)


Fig. 19




Fig. 19 : 

Splitting of the step connected to the dislocation emerging on the (110) facet of the Ia 3d crystal: a) equivalent (110) planes related by the d -glide plane symmetry, b) spiraling steps connected to a dislocation with the Burgers vector  , c) splitting of the step into two “partial” steps, red and blue sections of the two partial steps are respectively fast and slow, d) result of the anisotropic mobility of partial steps. (Collaboration with S. Leroy [28].)


Fig. 20




Fig. 20 : 

Three types of dislocations emerging on the (112) facet: a) Burgers vectors 1(111) and 2(1-1-1) are oblique with respect to the (112) plane while 3(11-1) is parallel to it, b-e) motion and recombination of double and simple steps connected to dislocations of types 1 and 2 [28].


Fig. 21




Fig. 21 : 

Structure of the inverted Im 3m lyotropic phase: a) surfactant bilayer inside one cubic unit cell. b) Assembling of he whole crystal from unit cells. One distinguish two labyrinths separated by the bilayer (nodal approximation).


Fig. 22




Fig. 22 : 

Trivial and Möbius dislocations in Pm 3n crystals: a) definitions of Burgers vectors, b) spherical domain of the perfect Im 3m crystal (the dotted red line indicates the cut surface orthogonal to the page surface), c–f) generation of a “Möbius” dislocation by the Volterra process using vector bM , c–i) generation of a “trivial” dislocation by the Voterra process using vector bT .


Fig. 23




Fig. 23 : 

Trivial and Möbius dislocations in Im 3m crystals: a) definitions of the shortest Burgers vectors belonging to the simple cubic (bT ) and to the bcc (bM ) Bravais lattices, b) spherical domain of the perfect Im 3m crystal, c) the “Möbius” dislocation generated by the Voterra process using vector bM , d) the “trivial” dislocation generated by the Voterra process using vector bT .


Fig. 24




Fig. 24 : 

Free edge of the bilayer forming a closed loop associated with the Möbius dislocation. It follows a complicated trajectory from N to S on the crystal surface and pierces the crystal from S to N.


Fig. 25




Fig. 25 : 

Melting of the Im 3m crystal containing a Möbius dislocations into the sponge phase.


Fig. 26




Fig. 26 : 

Twist Grain Boundary made of screw dislocations in the TGBA smectic phase.


Fig. 27




Fig. 27 : 

a) −2π /6 disclination in the hexatic phase, b) −2π /6 disclination (rotation dislocation) in the triangular crystal.


Fig. 28




Fig. 28 : 

Conformal crystal generated by equation (12): a) on a rotating disc [43], the system of arrows in proves that topologically it is equivalent to a −2π disclination, b) soap bubbles confined between a sphere and a plan [44].


Fig. 29




Fig. 29 : 

Locally Favored Structures [45]: a) in the antiferromagnetic Ising model, b) so-called (24) structure, c) one of ground states of the (24) LFS, d) crystal–liquid transition obtained by the Monte Carlo method.


Fig. 30




Fig. 30 : 

A system of 25 umbilics generated by magnetic and electric fields in a nematic layer.

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