FLUCTUATIONS IN HEXATIC MEMBRANES

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FLUCTUATIONS IN HEXATIC MEMBRANES

L. Peliti

To cite this version:

L. Peliti. FLUCTUATIONS IN HEXATIC MEMBRANES. Journal de Physique Colloques, 1990, 51

(C7), pp.C7-297-C7-307. �10.1051/jphyscol:1990730�. �jpa-00231129�

COLLOQUE DE PHYSIQUE

Colloque C 7 , supplhment au n023, Tome 51, ler dgcembre 1990

FLUCTUATIONS IN HEXATIC MEMBRANES

L. P E L I T I

Dipartimento di Scienze Fisiche and Unita INFM, Universita di Napoli, Mostra d'oltremare, Pad. 19, Associato INFN, Sezione di Napoli, I-80125 Napoli, Italy

1. I n t r o d u c t i o n

H e x a t i c membranes a r e h y p o t h e t i c a l a m p h i p h i l i c f i l m s w i t h long-range o r i e n t a t i o n a l o r d e r . They may e x h i b i t a number o f i n t e r e s t i n g phenomena due t o t h e i n t e r a c t i o n between i n n e r geometry, r e l a t e d t o o r i e n t a t i o n , and o u t e r geometry, r e l a t e d t o t h e i r embedding i n o u t e r s p a c e . However, a few problems c o n c e r n i n g t h e b e h a v i o r o f a c t u a l h e x a t i c membranes a r e n o t y e t s o l v e d . The m a s t e r c o u r s e i s o b v i o u s l y t o r e a l i z e them p h y s i c a l l y , b u t some f u r t h e r t h e o r e t i c a l work may h e l p i n c l a r i f y i n g t h e i s s u e .

I n a two-dimensional f l a t c r y s t a l , t h e r m a l f l u c t u - a t i o n s d e s t r o y , i n a s t r i c t s e n s e , long-range o r d e r . C r y s t a l - l i n e o r d e r is o n l y e x h i b i t e d by t h e slow (power l a w ) decay o f o r d e r p a r a m e t e r c o r r e l a t i o n s . T h i s b e h a v i o r a p p e a r s a t low t e m p e r a t u r e , a s long a s f r e e d i s l o c a t d o n s do n o t appear i n t h e c r y s t a l . S l n c e d i s l o c a t i o n s d i s r u p t c r y s t a l l i n e o r d e r , o r d e r p a r a m e t e r c o r r e l a t i o n s d e c a y , i n t h e p r e s e n c e o f f r e e d i s l o c a t i o n s , o v e r a d i s t a n c e

3 ^,

^where

3

-2 i s t h e d e n s i t y o f d i s l o c a t i o n .

Free d i s l o c a t i o n s c a n n o t a p p e a r a t s u f f i c i e n t l y low t e m p e r a t u r e 1

,

s i n c e t h e e l a s t i c energy o f an i s o l a t e d d i s l o - c a t i o n , i n a p l a n a r c r y s t a l o f s i d e L , i s p r o p o r t i o n a l t o I n L.

I t i s t h e n e a s y t o argue t h a t t h e appearance o f f r e e d i s l o - c a t i o n s i n a c r y s t a l would produce a d e c r e a s e i n f r e e energy o f t h e o r d e r o f kT I n L because o f e n t r o p y , b u t an i n c r e a s e o f t h e o r d e r of K I n L , where K i s some e l a s t i c c o n s t a n t ,

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1990730

COLLOQUE DE PHYSIQUE

because o f e l a s t i c e n e r g y . One e x p e c t s t h e r e f o r e t h e system t o remain c r y s t a l l i n e up t o t h e m e l t i n g t e m p e r a t u r e T f o r

m which kT m-

-

^{K .}

I f t h e two-dimensional c r y s t a l we c o n s i d e r i s allowed t o t a k e up non-planar s h a p e s , t h e s i t u a t i o n changes 2

.

The system may t h e n t r a d e p a r t o f t h e s t r e t c h i n g e n e r g y needed t o accommodate t h e d i s l o c a t i o n f o r bending e n e r g y , by a c q u i r i n g warped c o n f i g u r a t i o n s . Some e x p e r i m e n t a t i o n by c u t t i n g o u t a d i s l o c a t i o n i n a p a p e r s h e e t , f o r which s t r e t c h i n g energy i s e s s e n t i a l l y i n f i n i t e , may h e l p i n c l a r i f y i n g t h i s p o i n t . Q u a l i t a t i v e arguments2 and s i m u l a t i o n s 3 may convince one t h a t t h e e l a s t i c e n e r g y o f a d i s l o c a t i o n i n a warped membrane i s f i n i t e . A s a consequence, f r e e d i s l o c a t i o n s a p p e a r a t any nonzero t e m p e r a t u r e , u n l e s s t h e i r c o r e e n e r g y i s i n f i n i t e ( a s i n t h e c a s e o f polymerized membranes).

D i s l o c a t i o n s i n a t r i a n g u l a r l a t t i c e a r e i n f a c t a bound p a i r o f 5- and 7 - d i s c l i n a t i o n s . While f r e e d i s l o c a t i o n s d i s r u p t c r y s t a l l i n e p o s i t i o n a l o r d e r , t h e y do n o t h i n d e r bond o r i e n t a t i o n a l o r d e r . Hence, f l u i d s w i t h a f i n i t e d e n s i t y o f f r e e d i s l o c a t i o n s o f t e n d i s p l a y t h e r e s i d u a l bond o r i e n t a - t i o n a l o r d e r o f t h e h e x a t i c phase1. The e l a s t i c bending energy o f an i s o l a t e d d i s c l i n a t i o n may be s e e n t o be p r o p o r t i o n a l t o I n L. I t seems q u i t e p r o b a b l e , i n p r i n c i p l e , t h a t many membranes which a r e c r y s t a l l i n e when c o n f i n e d t o a p l a n e w i l l become h e x a t i c a t s u f f i c i e n t l y l o n g l e n g t h s c a l e s when allowed t o f l u c t u a t e i n t h e t h i n d dimension.

I s h a l l t r y i n t h e f o l l o w i n g t o d e s c r i b e t h e e f f e c t s o f t h e r m a l f l u c t u a t i o n s i n such a membrane.

2 . Model

I n a p l a n a r h e x a t i c , t h e d i r e c t i o n s o f t h e bonds c o n n e c t i n g e a c h molecule t o i t s s i x n e i g h b o r s e x h i b i t q u a s i - l o n g range o r d e r , i . e . , slow c o r r e l a t i o n decay. Let @ be t h e a n g l e between one o f t h e s e bonds and t h e X a x i s . One t h e n h a s

where

9

^{i s} a temperaure-dependent exponent. The f a c t o r 6 i n t h e e x p o n e n t s reminds u s t h a t t h e r e a r e s i x e q u i v a l e n t d i r e c - t i o n s around e a c h molecule, and t h a t t h e p h y s i c a l s i t u a t i o n i s t h e r e f o r e l e f t unchanged by t h e r o t a t i o n of a l l t h e bonds by 6 0 ° , a t r a n s f o r m a t i o n c o r r e s p o n d i n g t o

E q u a t i o n ( 1 ) may be d e r i v e d by t h e c o n s i d e r a t i o n of t h e f o l l o w i n g o r i e n t a t i o n a l f r e e energy:

where i s t h e two-dimensional n a b l a o p e r a t o r and K i s t h e A

h e x a t i c s t i f f n e s s c o n s t a n t .

I f t h e membrane i s warped, one h a s t o c o n s i d e r f i r s t i t s bending f r e e e n e r g y , r e p r e s e n t e d by H e l f r i c h ' S hamiltonian!

where i s t h e bending r i g i d i t y modulus, dS t h e a r e a e l e m e n t , and H t h e mean c u r v a t u r e . We s e t t h e spontaneous c u r v a t u r e t o z e r o (symmetric membrane) and d i s r e g a r d t h e Gaussian c u r v a t u r e

CoLLoQUE DE PHYSIQUE

term. I n o r d e r t o c o n s i d e r t h e e f f e c t s o f h e x a t i c o r d e r , we remark t h a t c u r v a t u r e i n t r o d u c e s a f r u s t r a t i o n i n o r i e n t a t i o n a l o r d e r . One may e x p e c t t h a t t h e s t a t e o f l o w e s t o r i e n t a t i o n a l e n e r g y would be reached i f bond d i r e c t i o n s a t d i f f e r e n t l o c a - t i o n s c o u l d be mapped o n t o one a n o t h e r by p a r a l l e l t r a n s p o r t . However, whenever t h e s c a l a r c u r v a t u r e R i s n o n z e r o , i t i s n o t p o s s i b l e t o d e f i n e a d i r e c t i o n f i e l d on t h e s u r f a c e , which i s mapped o n t o i t s e l f by p a r a l l e l t r a n s p o r t a l o n g a c l o s e d

c u r v e . I f we d e f i n e a s e t o f c u r v i l i n e a r c o o r d i n a t e s CC: 6 2 ) on t h e s u r f a c e , i t may t h e n be shown5 t h a t t h e h e x a t i c energy s h o u l d r e a d

where g . . i s t h e induced m e t r i c t e n s o r ,

g

i s t h e bond a n g l e

l J l

w i t h t h e

c

d i r e c t i o n , and t h e v e c t o r p o t e n t i a l % i s r e l a t e d t o t h e s c a l a r c u r v a t u r e R by

. .

Here g = d e t ( g . . ) and

6 ' )

i s t h e asymmetric symbol.

1 J

I t i s remarkable t h a t e q u a t i o n ( 5 ) r e p r e s e n t s t h e o n l y l o w e s t - o r d e r c o u p l i n g between o r i e n t a t i o n a l and geometric d e g r e e s o f freedom a l l o w e d by t h e symmetry w i t h r e s p e c t t o r o t a t i o n s by 6 0 ° . I n f a c t , i f one c o n s i d e r s systems o f lower symmetry, l i k e nematic o r S membranes.-, o t h e r terms appear i n 6

C

g e n e r a l , i n v o l v i n g t h e c u r v a t u r e t e n s o r

4 -3

where r =

rC5)

i s a p a r a m e t r i c r e p r e s e n t a t i o n o f t h e membrane embedding, and D i s t h e c o v a r i a n t d e r i v a t i v e . However,the hex-

i

a t i c i n t e r a c t i o n ( 5 ) i n d u c e s a long-range c o u p l i n g between t h e g e o m e t r i c d e g r e e s o f freedom, which l e a d s t o t h e i n t e r e s t i n g b e h a v i o r d i s c u s s e d below. On t h e c o n t r a r y , t h e e x t r a terms

-z.

i n v o l v i n g

Kij ,

which a p p e a r i n systems w i t h r e d u c e d symmetry, o n l y l e a d t o l o c a l i n t e r a c t i o n s between g e o m e t r i c a l d e g r e e s o f freedom. T h e i r o n l y e f f e c t i s t h u s t o r e n o r m a l i z e t h e bending r i g i d i t y modulus a p p e a r i n g i n H e l f r i c h f s h a m i l t o n i a n .

3. H e x a t i c membranes w i t h o u t d i s c l i n a t i o n s

The b e h a v i o r o f t h e model d e f i n e d by e q u a t i o n s ( 4 ) - ( 6 ) , i n t h e absence o f d i s c l i n a t i o n s , h a s been a n a l y z e d by r e n o r m a l i z a t i o n group methods5 and i s r e p o r t e d i n r e f . 7 . These r e s u l t s a r e o b t a i n e d i n t h e l i m i t of l a r g e moduli K and K They show t h a t

A '

K i s n o t r e n o r m a l i z e , i . e . , t h a t o r i e n t a t i o n a l o r d e r w i t h i n A

t h e membrane i s u n a f f e c t e d by shape f l u c t u a t i o n s . Moreover, G r e n o r m a l i z e s t o some f i n i t e v a l u e K I ( K ) , g i v i n g r i s e t o

A

a rinkl led*^ phase: i n which t h e c o r r e l a t i o n s o f t h e normals t o t h e membrane decay l i k e a power law, c o r r e s p o n d i n g t o a s e l f - s i m i l a r embedding i n ambient s p a c e . The e x i s t e n c e o f t h i s p e c u l i a r c r i n k l e d phase makes h e x a t i c membranes a p a r t i c u l a r l y i n t e r e s t i n g system.

G u i t t e r and ~ a r d a r ~ have r e c e n t l y r e c o n s i d e r e d t h e r e s u l t s q u o t e d above. They p o i n t o u t t h a t , i f

K

i s s m a l l e r

t h a n a c e r t a i n v a l u e ^K* (K,), h e x a t i c membranes s h o u l d undergo a crumpling t r a n s i t i o n analogous t o t h a t e x p e c t e d i n polymerized, o r t e t h e r e d , membranes ^{2 r} ( F i g . 1 ) . A t low ^K, i n t h e crumpled phase*, K r e n o r m a l i z e s t o z e r o . I t i s l i k e l y t h a t t h e l i n e o f f i x e d

*The main d i f f e r e n c e between crumpled and c r i n k l e d membranes l i e s i n t h e c o r r e l a t i o n f u n c t i o n f o r t h e normals t o t h e membrane which i s :

< n ( o ) . n ( r ) > -. e-''6 f o r t h e crumpled phase I r l - q f o r t h e c r i n k l e d phase

COLLOQUE DE PHYSIQUE

p o i n t s K*(K ) j o i n s t h e crumpling t r a n s i t i o n l i n e &*(K ) a t

A

*

A

some p o i n t

( c

,K;). T h i s p o i n t i s a f i x e d p o i n t o f t h e renormal- i z a t i o n group f l o w , r e p r e s e n t i n g a " m e l t i n g t t t r a n s i t i o n ,

c h a r a c t e r i z e d by u n i v e r s a l jumps i n K and KA. For KA(K;;, t h e membrane i s always i n a crumpled s t a t e . However, t h e n a t u r e o f i t s m e t r i c f l u c t u a t i o n s depends on t h e v a l u e o f K*. I n

f l u i d membranes, e n t r o p y f a v o r s t h e a p p e a r a n c e o f s p i k e s 12

,

1 3 making them t o resemble branched polymers a t l o n g d i s t a n c e s

.

Such a phenomenon i s a b s e n t i n t e t h e r e d membranes, whose i n d u c e d m e t r i c i s e u c l i d e a n by d e f i n i t i o n .

G u i t t e r and ~ a r d a r ~ a n a l y z e m e t r i c f l u c t u a t i o n s i n t h e crumpled phase by means of t h e conformal gauge ( i s o t h e r m a l

-

c o o r d i n a t e s ) , f o r which g . . = e q

5

^.

^,

^and

R; = L

r 5 : -

2;$.

The

1 J J 2 J

hexatic free energy takes up t h e form

showing t h a t t h e f l u c t u a t i o n s o f

d

d e c o u p l e from t h e f l u c - t u a t i o n s of

g ^.

On t h e o t h e r h a n d , i n t h e crumpled p h a s e , t h e f l u c t u a t i o n s o f

g

and t h o s e o f t h e embedding 3 decouple a s i n

1 4

P o l y a k o v ' s model of t h e bosonic s t r i n g

,

which i s an e f f e c t i v e .model f o r t h e crumpled membrane15. One.-may i n t e g r a t e o v e r

and f

+ ,

y i e l d i n g an e f f e c t i v e f r e e energy8 f o r t h e m e t r i c f i e l d

- 9

:

where d i s t h e dimension o f embedding s p a c e . For K = l / l 2 ' % t h e A

r e s u l t c o i n c i d e s w i t h t h a t o f f l u i d membranes. By a n a l y z i n g t h e s t a b i l i t y o f t h i s model w i t h r e s p e c t t o s p i k e f o r m a t i o n 12

,

one f i n d s t h a t t h e y a r e e n t r o p i c a l l y f a v o r e d f o r K n 5 i ( t = d / d 2 n . T h e r e f o r e , f o r K (KA t t h e membrane behaves l i k e a f l u i d o n e ,

A

and s p i k e s a r e p r e s e n t , w h i l e t h e y a r e s u p p r e s s e d f o r l a r g e r v a l u e s o f K T h i s i n t r o d u c e s t h e n o t i o n o f a " t e t h e r i n g "

A '

t r a n s i t i o n f o r h e x a t i c membranes.

4. D i s c l i n a t i o n s and s e l f - a v o i d a n c e

So f a r , we have r u l e d o u t d i s c l i n a t i o n s by t h e argument t h a t an i s o l a t e d d i s c l i n a t i o n c o s t s a l o g a r i t h m i c a l l y d i v e r g e n t energy on a h e x a t i c membrane. One s h o u l d however c a r e f u l l y c o n s i d e r t h e e f f e c t s of d i s c l i n a t i o n s on e a c h o f t h e d i f f e r e n t p h a s e s d e s c r i b e d i n t h e p r e v i o u s s e c t i o n . We s h a l l now t a c k l e t h i s problem, f o l l o w i n g a g a i n G u i t t e r and Kardar 8

.

I n t h e crumpled p h a s e , one may c a l c u l a t e t h e e f f e c t i v e i n t e r a c t i o n among d i s c l i n a t i o n s by i n t e g r a t i n g o v e r

0

^and

@

i n a s i t u a t i o n i n which t h e p o s i t i o n and c h a r g e s o f t h e d i s c l i - n a t i o n s a r e f i x e d , and by u s i n g P o l y a k o v ' s h a m i l t o n i a n a s an e f f e c t i v e f r e e e n e r g y f o r t h e embedding. I t t u r n s o u t t h a t d i s c l i n a t i o n s unbind a t any nonzero t e m p e r a t u r e i n a crumpled h e x a t i c membrane. T h e r e f o r e , t h e t e t h e r i n g t r a n s i t i o n i s wiped o f f by d i s c l i n a t i o n s , and crumpled h e x a t i c membranes a r e i n d e e d f l u i d , u n l e s s t h e c o r e e n e r g y o f d i s c l . % n a t i o n s i s i n f i n i t e .

I n t h e c r i n k l e d p h a s e , P o l y a k o v ' s model does n o t

a p p l y . One s h o u l d t h e n go back t o t h e o r i g i n a l h a m i l t o n i a n ( 4 ) - ( 6 ) . A c a r e f u l a n a l y s i s o f t h e e n e r g e t i c c o s t t o produce a s i n g l e d i s - c l i n a t i o n a t z e r o t e m p e r a t u r e a l l o w s t o draw t h e f o l l o w i n g c o n c l u s i o n s . ( F i g . 2 ) . There i s a l i n e K, (KA) i n t h e ( K , K A ) p l a n e such

COLLOQUE DE PHYSIQUE

t h a t , i f K < & ( K A ) , t h e system i s u n s t a b l e a g a i n s t d i s c l i - n a t i o n s . T h i s l i n e s t a r t s from i - C . = G O / l l i t f o r K A = W and r e a c h e s

K = &

f o r K = 7 2 / % . Let u s compare t h i s l i n e w i t h t h e

A

boundary o f t h e c r i n k l e d phase i n t h e ( c , K A ) p l a n e . There w i l l be i n g e n e r a l a nonempty s u p e r p o s i t i o n o f t h e c r i n k l e d phase r e g i o n w i t h t h a t o f s t a b i l i t y a g a i n s d i s c l i n a t i o n s , a t s u f - f i c i e n t l y l a r g e v a l u e s o f t h e e l a s t i c moduli. However, t h e m e l t i n g l i n e ( s t a r t i n g a t K =K*) may l i e i n t h e s t a b l e o r

A A

u n s t a b l e r e g i o n . I n t h e f i r s t c a s e , m e l t i n g o c c u r s due t o i n s t a b i l i t y towards crumpling, w h i l e i n t h e second i t o c c u r s due t o i n s t a b i l i t y towards d i s c l i n a t i o n s . I n t h e second c a s e , t h e f r a c t a l dimension o f t h e m e l t i n g membrane w i l l be i n g e n e r a l d i f f e r e n t from t h a t a t t h e crumpling t r a n s i t i o n .

Moreover, t h e g e o m e t r i c a l p r o p e r t i e s o f d i s c l i n a t i o n s change a c r o s s a l i n e Kc(K ) . I f %(KA), d i s c l i n a t i o n s s t a y

A

p l a n a r ; o t h e r w i s e t h e y buckle i n t o a cone. The f i x e d l i n e iC = i < ( K ) l i e s i n t h e p l a n a r r e g i o n f o r d < 5 4 / 1 1 ( a t l e a s t

A

f o r l a r g e enough K ~ ) . The c o n c l u s i o n i s t h a t t h e c r i n k l e d phase p e r s i s t s a l s o when d i s c l i n a t i o n s a r e t a k e n i n t o a c c o u n t .

T h i s does n o t mean t h a t t h e c r i n k l e d phase would be o b s e r v e d i n r e a l h e x a t i c membranes i n t h r e e d i m e n s i o n s , i f one were t o produce them. A l l t h e arguments above have n e g l e c t e d

s e l f - a v o i d a n c e e f f e c t s , which a r e h i g h l y r e l e v a n t i n t h r e e d i m e n s i o n s . They indeed modify s o s t r o n g l y t h e b e h a v i o r o f t e t h e r e d memebranes t h a t t h e y a p p e a r t o wipe o u t t h e crumpling t r a n s i t i o n 1 6 . The membranes a p p e a r t o 3 e f l a t and r i g i d a t any t e m p e r a t u r e . F l o r y - l i k e arguments a p p e a r t o s u g g e s t t h a t t h e same t a k e s p l a c e f o r h e x a t i c membranes i n t h e c r i n k l e d phase 8

.

T h e r e f o r e , a l t h o u g h t h e b e h a v i o r we have d i s c u s s e d above may

apply when the membrane is embedded in a space of high enough dimension (a simple estimate yields d>4), it might turn out that the behavior of hexatic membranes in the real world is indistinguishable from that of their tethered counterparts.

5. Conclusion

One should not feel too disappointed by the sad note ending the previous section. The interplay of inner and outer geometry may turn out to be highly relevand for more complex systems, like stacks of hexatic membranes. The investigations I? have reported here also encourage to reach a better understanding of other potentially interesting systems of reduced symmetry,

1 7 like chiral (S*) membranes _C

.

Acknowledgments

I thank E. Guitter and M. Kardar for having communicated their work before publication, and M. Cates, M. Feigel'man and S. Leibler for clarifying discussions. I am grateful to I. Mazzini for

much-needed encouragement and to E. Dubois-Violette and B. Pansu for having invited me to such a pleasant and stimulating

meeting.

COLLOQUE DE PHYSIQUE

FLAT TETHERED

1

^/K

4

Fig. 1 : Suggested phase diagram and renormalization group flows for hexatic men in high dimensions. The "tethering" transition (dashed line) disappear disclinations are included.

Fig. 2 : The locus of instability of a nonfluctuating (classical) membrane, to for of a single disclination of charge l / m (m = 6 for hexatics). Above the line K/K* = (2m - 1)2m2, the disclination buckles out to form a cone.

I , I ,

I/K*

1,

I

I\

A

I

0 I

11K.i' 12.n Id f-

1 /KA

TETHERED-LIKE CRUMPLED

,

I FLUID-LIKE CRUMPLED CRINKLED

.

I I

I I

,, ^A

^I

I I I I ) I

. .

j(

I ,

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