polarizer analyzer rubbed plates 4 N EMATICS IN T WO D IMENSIONS : - PowerPoint PPT Presentation

F UNDAMENTAL G ROUP : N OT THE WHOLE STORY Theorem (Po´ enaru) Let n be a field of directors [a line field] in R 2 with 128 V. Poenaru an isolated singularity at 0, defining a measured foliation. Then I ( n ) ≤ 1. In particular, a vector field ξ on R 2 , with an isolated singularity at 0, such that ∇ × ξ = 0, has the property that I ( ξ ) ≤ 1. V. Poenaru 128 Measured: (no singularity) Fig. 1. Measured foliations (2 dimensional "smectics") singularity) ( n o Fig. 1. Measured foliations (2 dimensional "smectics") Not: Fig. 2. Non measured foliation N. D. Mermin, Rev. Mod. Phys. 51 , 591-648 (1979); V. Poénaru, Commun. Math. Phys. 80 , 127-136 (1981) Fig. 2. Non measured foliation be putting to use is the "theory of foliations". We mathematical tool which we w i l l 14 now a very sketchy idea of what this is all about more mathematical w i l l g i v e details are to be found in the appendix at the end of this paper. Assume, for simplicity, that the physical space M is an open region of the n and that V is the set of all /c dimensional linear subvarieties of mathematical tool which we will be putting to use is the "theory of foliations". We euclidean � space R n � ). Assume, also, that the order n (this is called the Grassman manifold G R will give now a very sketchy idea of what this is all about more mathematical parameter � associates to every peM — � such a /c dimensional linear subvariety say that � defines a foliation if M — � can be w i l l � (p) passing through p. One details are to be found in the appendix at the end of this paper. completely covered by two by two disjoint, /c dimensional smooth, connected Assume, for simplicity, that the physical space M is an open region of the is the tangent space of the (unique) layer passing through the layers such that � (p) point p. n and that V is the set of all /c dimensional linear subvarieties of euclidean � space R If k= 1, such layers always exist, because one just has to integrate an ordinary differential equation in order to get them. But if k ^ 2, a field of /c dimensional n (this is called the Grassman manifold G R n � ). Assume, also, that the order planes � very seldom defines a foliation. The condition for this to be the case is a parameter � associates to every peM — � such a /c dimensional linear subvariety non linear "integrability condition" involving the first order derivations of the map � . If the condition is satisfied, we say that � is "integrable" (and if this is so, � (p) passing through p. One will say that � defines a foliation if M — � can be then a foliation # " is defined by � ). completely covered by two by two disjoint, /c dimensional smooth, connected A very important class of foliations are the so called "measured foliations", for which the layers (or rather "leaves" as they are usually called) are all equidistant. layers such that � (p) is the tangent space of the (unique) layer passing through the One can think of a measured foliation as being a very rough mathematical model of a smectic liquid crystal. Figure 1 below shows some examples of such measured point p. foliations, with singularities, in dimension 2 (n = 2, k—i). If k= 1, such layers always exist, because one just has to integrate an ordinary By contrast, the foliation in Fig. 2 is not measured. Now, with respect to the standard homotopy theory, here comes a new fact. If differential equation in order to get them. But if k ^ 2, a field of /c dimensional our ordered medium is modeled by a (measured) foliation with V— G n >k , although planes � very seldom defines a foliation. The condition for this to be the case is a � • V is not every individual value � (p)eV is acceptable, a global map M — necessarily acceptable. All this is very much in line with Mermin's critique. non linear "integrability condition" involving the first order derivations of the The first two paragraphs of this paper will give instances of the following two map � . If the condition is satisfied, we say that � is "integrable" (and if this is so, basic facts (in this framework of ordered media defined by foliations): (i) N ot every (homotopy class of) defect(s) predicted by pure homotopy theory then a foliation # " is defined by � ). is necessarily realized. In particular, we show that for a punctual defect of a two dimensional smectic the index of the corresponding plane field takes only the A very important class of foliations are the so called "measured foliations", for which the layers (or rather "leaves" as they are usually called) are all equidistant. One can think of a measured foliation as being a very rough mathematical model of a smectic liquid crystal. Figure 1 below shows some examples of such measured foliations, with singularities, in dimension 2 (n = 2, k—i). By contrast, the foliation in Fig. 2 is not measured. Now, with respect to the standard homotopy theory, here comes a new fact. If our ordered medium is modeled by a (measured) foliation with V— G n >k , although every individual value � (p)eV is acceptable, a global map M — � • V is not necessarily acceptable. All this is very much in line with Mermin's critique. The first two paragraphs of this paper will give instances of the following two basic facts (in this framework of ordered media defined by foliations): (i) N ot every (homotopy class of) defect(s) predicted by pure homotopy theory is necessarily realized. In particular, we show that for a punctual defect of a two dimensional smectic the index of the corresponding plane field takes only the

S MECTIC P HASE F IELD AS A H EIGHT F UNCTION Chen, Alexander, Kamien, PNAS 106 , 15577-15582 (2009) 15

S MECTIC P HASE F IELD AS A H EIGHT F UNCTION φ ( x, y ) y x Chen, Alexander, Kamien, PNAS 106 , 15577-15582 (2009) 15

S MECTIC P HASE F IELD AS A H EIGHT F UNCTION φ = 0 Chen, Alexander, Kamien, PNAS 106 , 15577-15582 (2009) 16

S MECTIC P HASE F IELD AS A H EIGHT F UNCTION φ = 1 Chen, Alexander, Kamien, PNAS 106 , 15577-15582 (2009) 17

S MECTIC P HASE F IELD AS A H EIGHT F UNCTION φ = 2 Chen, Alexander, Kamien, PNAS 106 , 15577-15582 (2009) 18

S MECTIC P HASE F IELD AS A H EIGHT F UNCTION φ = 3 Chen, Alexander, Kamien, PNAS 106 , 15577-15582 (2009) 19

S MECTIC P HASE F IELD AS A H EIGHT F UNCTION φ = 4 Chen, Alexander, Kamien, PNAS 106 , 15577-15582 (2009) 20

C ONTOUR M APS : S MECTIC D ISCLINATIONS 21

C ONTOUR M APS : S MECTIC D ISCLINATIONS Peaks or Basins (+1) 21

C ONTOUR M APS : S MECTIC D ISCLINATIONS Passes (-1) Peaks or Basins (+1) 21

E DGE D ISLOCATIONS IN T WO D IMENSIONS Maps from R 2 \{ 0 } → S 1 22

+2 D ISLOCATION Dislocation is a helicoid! 23

+2 D ISLOCATION Dislocation is a helicoid! 24

+2 D ISLOCATION Dislocation is a helicoid! 25

+2 D ISLOCATION Dislocation is a helicoid! 26

+2 D ISLOCATION Dislocation is a helicoid! 27

+2 D ISLOCATION Dislocation is a helicoid! 28

+2 D ISLOCATION Dislocation is a helicoid! 29

+2 D ISLOCATION Dislocation is a helicoid! 30

+2 D ISLOCATION Dislocation is a helicoid! 31

+2 D ISLOCATION Dislocation is a helicoid! 32

+2 D ISLOCATION Dislocation is a helicoid! 33

+2 D ISLOCATION Dislocation is a helicoid! 34

+2 D ISLOCATION Dislocation is a helicoid! 35

+2 D ISLOCATION Dislocation is a helicoid! 36

+2 D ISLOCATION Dislocation is a helicoid! 37

+2 D ISLOCATION Dislocation is a helicoid! 38

+2 D ISLOCATION Dislocation is a helicoid! 39

+2 D ISLOCATION Dislocation is a helicoid! 40

S MECTIC S YMMETRIES : L AYER OR L AYERS ? density wave: Phase is periodic ... ... and unoriented 41

S MECTIC S YMMETRIES : L AYER OR L AYERS ? density wave: Phase is periodic ... ... and unoriented 41

S MECTIC S YMMETRIES : L AYER OR L AYERS ? density wave: Phase is periodic ... ... and unoriented ‣ sheets cross at the fixed points of these point symmetries ‣ only slices at these heights yield consistent smectics ‣ critical points are constrained to these heights 41

+1/2 D ISCLINATION 42

+1/2 D ISCLINATION 42

+1/2 D ISCLINATION 42

+1/2 D ISCLINATION 42

-1/2 D ISCLINATION 43

-1/2 D ISCLINATION 43

-1/2 D ISCLINATION 43

-1/2 D ISCLINATION 43

-1/2 D ISCLINATION 43

P INCH 44

P INCH 44

P INCH 44

P INCH 44

P INCH 44

P INCH 44

T HE D ISLOCATION 45

T HE D ISLOCATION 45

T HE D ISLOCATION 45

T HE D ISLOCATION 45

T HE D ISLOCATION 45

T HE D ISLOCATION 45

T HE D ISLOCATION 45

T HE D ISLOCATION 45

T HE D ISLOCATION 45

D ISCLINATION D IPOLE : +1 D ISLOCATION 46

D ISCLINATION D IPOLE : +1 D ISLOCATION 47

D ISCLINATION D IPOLE : +1 D ISLOCATION 48

D ISCLINATION D IPOLE : +1 D ISLOCATION 49

D ISCLINATION D IPOLE : +1 D ISLOCATION 50

D ISCLINATION D IPOLE : +1 D ISLOCATION 51

D ISCLINATION D IPOLE : +1 D ISLOCATION 52

D ISCLINATION D IPOLE : +1 D ISLOCATION 53

D ISCLINATION D IPOLE : +1 D ISLOCATION 54

D ISCLINATION D IPOLE : +1 D ISLOCATION 55

D ISCLINATION D IPOLE : +1 D ISLOCATION 56

D ISCLINATION D IPOLE : +1 D ISLOCATION 57

D ISCLINATION D IPOLE : +1 D ISLOCATION 58

F REE E NERGY AND R OTATIONAL I NVARIANCE density wave: Linear elasticity: 59

F REE E NERGY AND R OTATIONAL I NVARIANCE density wave: Linear elasticity: Nonlinear elasticity: 59

S URFACE E NERGETICS Viewing φ as a graph: Equal spacing of curves: 60

S URFACE E NERGETICS Viewing φ as a graph: Equal spacing of curves: Candidate: “Willmore in a field” 60

E QUAL S PACING 61

E QUAL S PACING K = 0 isometric to the plane 61

E QUAL S PACING K = 0 isometric to the plane 61