Star defects on flat and spherical surfaces

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Submitted on 1 Jan 1993

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Star defects on flat and spherical surfaces

David Pettey, Tom Lubensky

To cite this version:

David Pettey, Tom Lubensky. Star defects on flat and spherical surfaces. Journal de Physique II,

EDP Sciences, 1993, 3 (10), pp.1571-1579. �10.1051/jp2:1993206�. �jpa-00247925�

J.

Phys.

II IFance 3

(1993)

1571-1579 OCTOBER1993~ PAGE 1571

Classification

Physics

Abstracts

61.30J 68.15 87.25D

Star defects

_on

flat and spherical surfaces

David

Pettey

and Tom C.

Lubensky

Department

of

Physics~ University

of

Pennsylvania~ Philadelphia

PA 19104~ U-S-A-

(Received

11

May1993, accepted

29 June

1993)

Abstract. We

investigate

the

Dierker-Pindak-Meyer

star defect observed in

freestanding

hexatic

liquid crystal

films. We derive

general

formulae for the energy of

arbitrary

vortex

configurations

both _on flat films and _on the surface of

spherical

vesicles. We find that

equi-

librium defect

configurations

in

general

have _no identifiable symmetry ^{and that}

configurations

with

nearly 6-fold~

5-fold~ and 4-fold symmetry are

possible. Configurations

with

nearly

S-fold symmetry _are~

however,

favored

by

the conditions _in the Dierker-Pindak

experiment.

1. Introduction.

The star defect in

free-standing liquid crystal

films

provides

_one of the most

striking

^confirma-

tions of the existence of the tilted hexatic

phase.

In _{a now} classic

experiment ill,

Dierker et al.

captured

_a

single

disclination of unit

strength

ⁱⁿ a Smectic-C film. Under crossed

polar- izers,

this defect exhibits two

light

and two dark _arms

emanating

from _a central _core.

Upon cooling,

Dierker et al. observed the evolution of _{a more}

complex pattern,

which

they

inter-

preted

_as shown in

figure

I in terms of six

strength 1/6

disclinations

(permitted

in hexatic

phases) arranged

in _a five-fold star whose _{arms were} domain walls _across which the relative orientation of the hexatic and tilt order

parameters change. They

calculated the _energy of the observed five-arm star and _a

competing

six-arm star and concluded that the five-arm star _was

energetically

favored.

The star defect results from the

capture

^of a

single topologically

stable

strength

_one defect in

a flat Smectic-C film. A vesicle with

spherical topology

and tilt

(Smectic-C) _tangent plane

order

necessarily

has _a total

vorticity

of

two,

^which in the lowest _energy

configuration

is

produced by

two

strength

_one defects located _at

antipodal points

on the

sphere _[2].

As the flat film is cooled into the tilted hexatic

phase,

the

single strength

_one defect breaks _up into _a star defect with

arms terminated

by strength 1/6

defects. One would

expect, therefore,

that the two

antipodal

defects of _a Smectic-C

spherical

vesicle should each

develop

into _a star defect upon

cooling

into the tilted hexatic

phase.

When hexatic order dominates _over tilt

order,

the defects should have five _arms with vertices and centers

coinciding

with the twelve vertices of _an icosahedron

[3, 4].

Fig-I-

Dierker-Pindak star. The

hexagons

show the direction of _@6 and the _arrow the direction of Hi. Both Hi and _@6

undergo

_a rotation

through

2x at the outer

boundary.

The _arms of the star

are domain walls _across which Hi _@6

changes by 2x/6.

In this

representation,

_@6 is

approximately

constant _across walls, and 91

changes by 2x/6.

'~'~'~'~

~~ ~ ^~ ~ ^~ ~ ~ ~

w ~ , ~

~ i r~

T=75.5 T=71 T=69.5 T=66

l I

[ ~ j

i

~

j

~

-

$ g

,

X x ~ $

~ fl fl

m~-~~~ ~ ~

T=63.5 T=62.382812 T=62 T=61.932586

~ ~

fl

~ ~ ~

m , ^v~

$

$ j £

X ,

~ ~ ~ ~

T=61.856901 T=61.772611 T=61.747586 T=61.702411

Fig.2.

Vortex and domain-wan

configurations

_{as a} function of temperature for Ki

#

lo~~~

erg,

K6 _"

K60exp(D/T To"),

_e

= 1.0

X10~~~ erg/pm,

and _a

= I-I _X

lo~~pm, (with

K60

" 1.55 X

10~~~

_erg, D

= 3117

K",

To ₌ 61.4 K and _{v =} 0.38 _as in reference

[ii.

The dimensions

given

correspond to those of the viewing box, recall that the actual

sample

size is taken to be _cc.

N°10 STAR DEFECTS ON FLAT AND SPHERICAL SURFACES 1573

' ' ' '

~ ,~ ~ m~ ~ _~ ^{~ ~}

m m

Q~i~i~u~~

T=72 T=71 T=68.5 T=66.884518

m

~

m m m

X X X

m m

~

~j

~

~

u~~£~$~i~ ~

T=66.884514 T=66.884513 T=64 T=63.5

[

fl fl

~l

~ ~ ⁷

o

$

o'

$

X

~ +

, ~

£~U~~Q~U~~

T=63,161499 T=62.875 T=62.5 T=62.038933

~

_fl

~ _~

~

' o

p fl _X ^'

m ~

X

~ ~~~"~

T=61.816835 T=61.752783 T=61.697351 T=61.66658

Fig.3.

As in

figure

2 but with Ki

=

10~~~

erg. Notice that the size of the stars _{as a} function of temperature _are similar to

figure

2 but that the

configurations

at similar temperatures _are

quite

different. In

particular,

the transition from

a 4-arm to _a S-arm star occurs at _a lower temperature, and the 4-arm stars _are

larger

than in

figure

2 and

might,

therefore, be

experimentally

observable. Also

note the 6-arm

symmetric configuration

at T

= 72 °C in

figure

_[2]. When Ki _# 10~~~ _erg, this

shape

does not appear until T _> 100 °C.

The purpose of this _paper is to

investigate

vortex

configurations

in tilted hexatic membranes in both

planar

and

spherical geometries.

Rather than

using

conformal

mapping

to calculate the

energy of certain

highly symmetric configurations,

we obtain _an

expression

for the energy of

arbitrary configurations

of disclinations and domain walls. Our results for flat membranes differ

slightly

from those in the literature _11,

5].

^{Our results for}

spherical

vesicles _are

straightforward generalizations

^of those for flat membranes.

With the aid of _our

general

_energy

expression,

_we

numerically

calculate the minimum _en- ergy

configurations

for the star defect. We find that the lowest energy

configuration

in flat

membranes

always

has six _arms

(Figs. 2,3). However,

_{one arm may} be much shorter than

the other

five,

and the six,arm star could

easily

be confused with _a five-arm star. We also find that in _some

parameter

_ranges, two arms are shorter than the other four. There _can be

more than _one local minimum to the _energy, and discontinuous

jumps

between two minima _can

occur when their

energies

_{cross as}

parameters

are varied

(As

_occurs between

panels

5 and 6 in

figure

3 in the

vicinity temperature

^T ₌ 66.884

°C). Figure

2 shows _{a sequence} of

equilibrium

defect

configurations

_{as a} function of

decreasing

temperature for

parameters appropriate

to the Dierker-Pindak

experiment.

Note that in the range of

temperatures IT

= 66 to T

=

62)

of

Fig.4.

Three views of _a

pair

of 5,arm stars, ^and a

pair

of 6-arm stars, _on ^{the surface} of _a

sphere (Radius

20

pm)

for

a = I-I

X10~~

pm, e = 1.0 X10~~~

erg/pm,

K6

" 2.1638 X 10~~~ _erg

(left)

_:

4.7937 X 10~~~ _erg

(right),

Ki _# 10~~~ _erg

(left)

: 10~~~ _erg

(right).

The center vortex of _one star has been fixed at the North Pole.

(Top)

View from above the

sphere looking

down towards the North pole.

(Middle)

View from the side of the

sphere, displaying only

the domain walls. Notice the star near the South

pole

is not

precisely antipodal

to the star at the North

pole. (Bottom)

View from

directly

above the North Pole. Note the azimuthal correlations between the two stars.

the

experiment,

the

equilibrium configuration

is _a

nearly symmetric

5-arm star. At lower and

higher temperatures, however,

the

equilibrium configuration

is

considerably

different. Note also that the size of the defect increases

rapidly

_as the transition to the

crystal phase (at

T

= 61.4

°C)

is

approached. Figure

3 shows

equilibrium

defect

configurations

with the

splay

elastic _constant

Ki

ten times _as

large

_as that in

figure

2.

In

spherical vesicles,

_we find that there _are in

general

two six-arm stars with centers approx-

imately

at

antipodal points. However,

at low

temperature,

where hexatic order dominates _over tilt

order,

the stars become five-armed with vertices

coinciding

with those of _an icosahedron.

Typical high

and low T

configurations

_are shown in

figure

⁴

2. Flat membranes.

The tilted hexatic

phase

is characterized

by

two

complex

order

parameters:

_{~fi6 "} ~fi6e~~°6 and

~fii = ~fiie~°i When order is

well-developed,

the _energy of the tilted hexatic

phase

_{[5j can} be

N°10 STAR DEFECTS ON FLAT AND SPHERICAL SURFACES 1575

expressed

in _terms of the

angle

Hi and b61

7i =

/ _d~x ^~ _K6i7b6~

+

~Kii7bi~

₊

K16i7bi i7b6

₊

V(b6

Hi

)I, ⁽¹⁾

2 2

where

V(b)

is _a

periodic function,

such _as

-Vo cos6b,

with

period 2x/6.

The

potential

term

depends only

_on b6

9i

We _can,

therefore,

introduce the

change

of variables

9+

=

a96

+

fl91,

9-

=

b6 Hi, (2)

where

K6

+

K16

Cf "

fl (3)

"

Ki

+

K6

₊

2K16

^'

to obtain _a Hamiltonian in which

b+

and b- _are

totally decoupled:

7i =

/ _d~x ^jK+i7b+~

+

jK-i7b-~

⁺

^(9-)j

,

(4)

~~~~~

K+

₌

K6

+

Ki

+

2K16

K-

=

~~~(~ ^{~~~ (5)}

The Hamiltonian of

equation (4)

is the _sum of _an

xy-Hamiltonian

^for

9+

and _a sine-Gordon-like Hamiltonian for 9-.

In order _{to use}

equation (4)

to calculate the _energy of defect

configurations,

_we need to know what defects in

9+

^{and 9-} are

possible.

The order

parameters

~fii and _{~fi6 are} invariant under the

respective replacements

91 - 91 +

2xki

and

96

-

96

+

2xk6/6, (6)

where

ki

and

k6

_are

integers. Thus,

there _can be defect

singularities

in which 91

changes by 2arki

and

96 changes by 2ark6/6

in _one circuit around _{a core.}

Alternatively, 9+ changes by 2arq+

^{and 9-}

changes by 2arq-

in _one circuit around the _core, where

q+ =

°

k6

+

Ii a)ki,

6

~~ ~~ ~~~

are the

"charges"

for

9+

and 9-. Note that the

charge

_q+ is in

general

irrational because

a is determined

by

the

arbitrary potentials Ki, K6,

and

K16.

^The

charge

_{q-, on} the other

hand,

is _an

integral multiple of1/6.

Each defect is characterized

by

two

numbers,

which _can

conveniently

be

represented

_as

2-component charges [

=

(ki, k6)

_or

equivalently #

₌

(q+,q-).

The _energy of _an

arbitrary configuration

of N defects characterized

by charges ii Ii

=

i,..,N)

_can be calculated with the aid of

equation (4).

The _energy associated with the _q+

charges

is

simply

that of _a Coulomb _gas. The _energy associated with _q- is somewhat _more

complicated.

The

potential V(9-)

favors 9-

= 0 mod

2ar/6

Because 9- must

change by 2arq-

in _one circuit around _a

defect,

it is

impossible

to

satisfy

9-

= 0

everywhere

around the defect. The lowest energy

configurations

of 9-

satisfying

the defect constraint _are those in which 9-

changes rapidly by 2ar/6

_across domain walls

emanating

^from the defect _as shown in

figure

i.

Thus,

there will be

6q-

domain walls

emanating

^{from each} defect. The

equilibrium

configuration

for _a

given position of defects

is the _one in which the total

length

L of domain walls is _a minimum. The energy associated with variations of 9- is then EL where

e is the

energy per unit

length

of _a domain wall. If _{Vo is} the

depth

of the

potential V,

then the domain wall width _w is of order

Ii

and _e is of order

l§.

We will _assume that _w is small

enough

that _{we can} treat the domain walls _as lines. The total _energy of _a

configuration

of defects in

a

sample

of size R

is, therefore,

E =

-arK+ ~j

_q+iq+j

_{In(r~ la)}

₊

_{arK+Q~ In(Rla)}

₊

_{EL, (8)}

iii

where

Q

=

£~

_{q+i, a} is the _core

radius,

and r~ = ri rj where _ri is the

position

of the ith defect.

In the _case of _a

single strength

_one disclination in the Smectic-C

phase,

the hexatic and tilt order

parameters

^{follow each other in the far field} as shown in

figure

i and

undergo

_one

complete

2ar revolution. Therefore

£ kit

"

I,

and

£ k6i

"

6;

_or

£

_q-1

= 0 and

£

_q+i

=

Q

₌ I. The

most

general

^defect structure

satsifying

these constraints with _no

negative

defects will have six defects with

[

=

(0,1)

_or

=

(o/6,1/6)

and _one defect with

[

=

(1,0)

_or

#

=

(I

_a,

-I).

Thus,

there will be six defects with _a

single

domain wall and _one defect with six domain walls.

Figures

2 and 3 illustrate the

typical

_sequence ^of

configurations

^{that should} be _seen in materials similar to those used

by

Dierker and Pindak

ill.

In _our calculations _we have taken the

cross-coupling K16

"

0,

the line tension _e

=

10~~ erg/cm,

and the _core radius _a

=

10~~cm.

We also took

K6

to vary with

temperature

as

K60 exp(D /T-To"

_as it would _near the transition to the

crystal phase [6, 7, Ii. Figure

2

corresponds

to the values of

Ki

and

K6 given

in

ill

(Ki

"

10~~~ erg).

Notice that at T

= 63.5 oC _we indeed _{see a} five-arm _{star in}

agreement

with Dierker and Pindak's observation.

Figure

3

provides

_a simlar collection of _stars with

Ki

increased

by

a factor of _ten

(I.e., Ki

_"

10~~~ erg).

In this _{case we see a}

symmetric

six-arm

configuration

at

high temperature.

While this

configuration

will exist for _some

sufficiently high temperature

when

Ki

_"

10~~~

_erg

(since

the

configurations only depend

_on the

temperature through

the variable

K6 ),

it exists in _an

experimentally

_more accessible _range when

Ki

=

10~~~

erg.

Although

^{the sixth} arm is not

always

visible in the

pictures

it does _{appear to} be

real,

in

the _sense that it is

always

much

larger

than the _core

radius,

and

thus,

in

principle,

should

be

distiguishable

from the _core

(to

convince the

reader,

we ask that the

varying

scales _on the

pictures

be noted _{as a} function of

temperature). Finally

_we remark that in

sufficiently large samples (such

that the

approximation,

maximum _arm

length

<

sample size,

is

valid) larger

stars without 5-fold

symmetry

^should appear as the

temperature approaches

the

crystal

transition

(Fig. 2,

^T = 61.7

°C).

3. Stars _on the

sphere.

The calculations of the

preceding

section _can

easily

be

generalized

to describe vortices _{on a}

sphere.

Positions _{on a} twc-dimensional surface in

R~

are

specified by

_a vector

R(fi)

_as function of _a twc-dimensional coordinate fi ₌ (~1~, ~1~). For the

spherical surface,

_{we can} choose fi to be

the

polar

coordinates

(9, #).

Associated with

R(fi)

is the metric tensor gab "

baR(fi) bbR(fi),

its inverse

g~~,

and determinant _g. The

angles

91 and 96 now

specify

directions relative to local orthonormal coordinates _on the

sphere,

which _{we can} take to be

along

the

longitudinal

and azimuthal directions

specified by

the unit vectors ei " eo and _e2

" e~. Since the

sphere

has

nonzero curvature, _~fii and _~fi6 will have

spatial

variations

arising

from

spatial

variation of the

vectors ei and _e2. This leads to a modification

[8,

9] ^{of the}

flat-space

_energy of

equation (I)

in

N°10 STAR DEFECTS ON FLAT AND SPHERICAL SURFACES 1577

which derivatives _are

replaced by

covariant derivatives: i7191 -

ba9j-Aa,

where

Aa

=

eibae2.

The

resulting

Harniltonian is

7i ₌

/ _d~~lfi ^jK6(ba96 _{Aa)(b~96 A~)}

+

jKi(ba91 ^{Aa)(b~91 A~)}

+K16(b~96 Aa)(b~91 A~)

+

V(96 91))

,

(9)

where b~9 A~

=

g~~(bb9 Ab).

The transformation to

9+

and 9-

(Eq. 2)

^{then leads} to 7i =

/ _d~~tfi ^jK+(bag+ _{Aa)(b~9+ A~)}

+

jK-bag-b~9-

⁺

^{(9-)j (10)}

Thus,

_{we see} that the 9- contributions to 7i _are unaffected

by

the curved

geometry

^{of the}

sphere.

The

b+ part

is modified

by

the connection

Aa

and is

simply

the Hamiltonian for fixed

amplitude complex

order

parameter

^{in the}

tangent plane.

The _energy of _a

configuration

of defects _{on a}

sphere

is thus the _sum of the vortex energy for

interacting

vortices of

strength

_q+ and the _energy of domain walls which lie _on the surface of the

sphere.

^The vortex energy was calculated in reference

[4].

^{The domain wall} energy is

proportional

to the _arc

length

L of the walls. The

resulting

_energy is

E =

-arK+ ~j

_q+iq+j

_log ^~"

₊

_{2arK+ (2 log}

^~~

_i)

₊

_{EL, (ii)}

iii ^{~ ~}

where

d~

is the

length

of the chord

connecting

vortices I and

j (I.e.,

distance in

R~)

The

geometry

^of the

sphere

forces

[10]

a total

vorticity

of +2 in both Hi and

b6, I.e.,

it

requires

that

£ kit

_" 2 and

£k6i

= 12. There

will, therefore,

in

general

be 12 vortices of

strength #

₌

(a/6,1/6)

and 2 two vortices of

strength #

₌

(i

_a,

i).

We have

numerically

minimized

equation (ii)

with

respect

to the

positions

of these 14 vortices. For the lowest energy

configuration

we find two

approximately antipodal

six-arm stars

(just

_as in the

planar

case all stars are six-arm

stars,

which

occasionally

^look like five-arm

stars).

When the _{arms are} all _very

short,

numerical minimization shows _{no, or}

little, energetic favoring

of _a correlation in azimuthal

angles

for the two stars.

However,

_as the _{arms grow} and

we find two

essentially

5-arm

symmetric

stars, their azimuthal orientations lock

in,

^such

that,

as

K6/Ki

_{- cc}

(hexatic

order

dominates)

_we obtain the

placement

of vortices at the vertices of _an icosahedron.

Figure

4 shows _some

typical configurations

_on the

sphere.

Acknowledgments.

This work _was

supported

in part

by

the National Science Foundation under grants ^DMR-91- 20668 and DMR 9i-22645. The authors _are

grateful

to Suzanne Amador for

helpful

discussions.

Appendix

A.

Symmetric

stars.

In section

2,

we derived _an

expression (Eq. (8))

for the _energy of _any number and _ar-

rangement

^of vortices and used it to calculate minimum energy

configurations

when the total

vorticity,

for both Hi and

96,

is i. In this

section,

we will consider the

symmetric

N-arm _con-

figurations

^studied in references _11,

5].

^{If there} are N _arms of

length

R

emerging

from _a central

point,

then each _arm is terminated

by

_a vortex of

strength #

₌

(a/6,1/6)

and the center is _a

vortex of

strength #

₌

(i oN/6, -N/6). Using

these values in

equation (8),

_we obtain

EN

₌

~°~~~

(12

a

Na) log

^~

~~~(~~ ~j log(2

sin

))

_{+ EL}

_(Al

~~ _~

=~

where L

= RN. Note the term

involving sin(arn/N).

This arises from interactions among

terminating

vortices and _was not included in references _11,

5].

^This term will not affect the minimum energy arm

length

R for _a

given

N. It

does, however,

affect the value of N with the lowest energy and thus lead to a

slightly

^different

region

in the _space of

parameters Ki, K6,

and

K16

where the 5-arm _star is stable.

To make contact with

previous work,

_we will _now rederive

equation (Al) using

conformal

mapping

for _o

= i. The

generalization

to _a

#

i is

straightforward.

We wish to find the solution to

Laplace's equation

in the domain

(-ar/N<#<ar/N,0<r<cc)

with the Dirichlet

boundary

conditions:

l~(m-j) for0<r<R, #"~§

8(~,

_T)

(A2)

"

~f

for _r >

R, ~

_"

~(

Where R is the

length

of the _arms

(recall

that there is _a vortex at the end of each

arm),

and for later reference _we will denote

by

_a the _core size of each vortex.

We take _our

problem

to be in the

complex z-plane (as

usual _{z = x} +

iv). Now,

_suppose that _an

analytic

function _w

=

f(z)

=

~1(x,y)

+

iu(x,y)

_{maps a} domain

Dz

in the

complex z-plane

into _a domain

Dw

in the

w-plane.

If

0(~1,u)

is _an harmonic function defined _on

Dw

then

8(x,y)

=

0(~1(x,y),u(x,y))

is harmonic _on

Dz. Thus, by taking

the _{map w}

=

zi (w

₌

exp( )Logz)

for

definiteness,

where

Logz

refers _{to a}

specific

choice of the branch

cut,

namely #

=

0),

_{we can} reduce _our

problem

to

finding

the solution of

Laplace's equation

in the half

plane

~1 > 0 with Dirichlet

boundary

conditions:

The solution _to this

problem

is known:

O(~1,

~)

=

l Ill

~~

i(l~l

~z~

dt, (A4)

where

fi

for _{t >}

Ri

~~~~ ~fi [

~~ ~~ (~ ^{~~% ~~~~}

-§

^{for RT} > t Thus _we

have, b(~1,u)

=

Im[2() _))Logw

+

)Log(w~

⁺

^R~/)],

^and

8(z)

= Im

(2((

₆

^)Logz%

⁺

(Log(z~

⁺

R~)j ^(A6)

N°io STAR DEFECTS ON FLAT AND SPHERICAL SURFACES 1579

(The

reader _may

verify

that this

equation

does indeed

satisfy

the

boundary

conditions

). Finally

because

(z~ +R~/)

=

(z -Rwo)(z-Rwi) (z-AWN-i)

where

(wi)

=

((-1)~/~/),

there exists

some set of branch cuts

(log # Log, I.e., log

does _not

necessarily

have the branch _cut

#

=

0)

such

that,

8(z)

= Im

2(( )Logz+

₊

f

log(z j)j

,

where _{zi =}

Rwi. (A7)

~

i=o

Now let _us calculate the _energy difference between the _star

configuration, equation

A7 and the

configuration

with _a

single

+i vortex at the center

(8nostar(z)

=

Im[log(z)]), remembering

to take into account the wall _energy. We let the

energy/length

of each wall be _E, then:

~ ~~

~~ ~

~~~~~ _~~

_{? ~}

~~°~~~~ ?

~n°Star)

ids ₊ eRN

~~~~

In

carrying

out the

integration

_we will

neglect

the circular _arcs of radius _a about each vortex.

Then

taking

the

sample

size to be

S,

and

using

_{a <} R < S _{we can}

obtain,

I

=

-I ~j _log

_{2 cos} ^~~~ ₊ ^~~

_{(N ii) log}

^~ ₊

eR~, _(A9)

§~N

^~~

=~

N 36 _{a lG}

or

AF

=

-~(

_II

_{N) log}

^~

~ ^flog

(2

^sin

))

_{+ eRN}

_(A10)

~

n=1

in

agreement

^with

equation

Al with _a

= 1.

References

[Ii

^Dierker

S.,

Pindak R. and

Meyer R.B., Phys.

Rev. Lent. 56

(1986)

1819.

(2] MacKintosh

F., Lubensky T., Phys.

Rev. Lent. 67

(1991)

l169.

[3] ^Park

J., LubenskyT.,

MacKintosh F.,

Europhys.

Lent. 20

(1992)

279.

[4]

Lubensky T.,

Prost J., J.

Phys.

II IFance 2

(1992)

371.

[5]

Selinger

J., Nelson D.,

Phys.

Rev. A 39

(1989)

3135.

[6] Nelson D.

R., Halperin

B.,

Phys.

Rev. B19

(1979)

2475.

[7]

Young A., Phys.

Rev. B 19

(1979)

1855.

[8] Nelson

D.,

Peliti L., J.

Phys.

II IFance 48

(1987)

1085.

[9] David F., ^Guitter

E.,

Peliti L., J.

Phys.

II IFance 49

(1987)

2059.

[10]

Spivak

M., A

Comprehensive

Introduction to Differential

Geometry (Publish

_or

Perish, Berkeley,

1979).

Classification

Physic-s

Abstiacts

61.40K 66.10 47.00

Dynamics of telechelic ionomers. Can polymers ^diffuse large

distances without relaxing stress ?

L. Leibler

('),

M. Rubinstein

(2)

and R. H.

Colby (2)

(') Groupe

de Physico-Chimie

Thdorique,

E-S-P-C-1-, 10 rue

Vauquelin,

75231 Paris Cedex 05, France

(2)

Corporate

Research Laboratories, Eastman Kodak

Company,

Rochester, New York 14650- 2ll0, U-S-A-

(Received 8 April 1993, accepted 18 June 1993)

Abstract. We consider

dynamics

of

entangled

telechelic ionomers in the limit of strong

association, where there _{are no} free chain ends. Stress relaxation _occurs in such _a system by an

exchange between pairs of chain ends in the associated _state. For complete ^{relaxation of} stress, _an

exchange

event must take place on every entanglement strand. However, diffusion _{can occur on an}

arbitrarily

shorter time scale,

leading

_to the

interesting

result that chains can diffuse distances many times their coil size without

relaxing

stress.

Due _to the relation between _stress and orientational correlations in

polymers [I],

there is _a

general

belief that diffusion and _stress relaxation _are

coupled

in

polymer

systems. ^There are

examples

where _stress relaxation _occurs much faster than

diffusion,

such _as in _a melt of star

polymers [2],

but the

opposite

_case, where chains diffuse many times their size without

relaxing

stress, is

quite

_rare. One

exception

is semidilute solutions of

disordered,

rod-like

polymers

I

],

for which translational diffusion is fast

compared

to rotational

diffusion,

which

determines the time scale for _stress relaxation. Due to the

large

aspect ^{ratio of}

long

rods, rotation

through

a small

angle requires large

translation. Another

exception

^is a

polydisperse

system of flexible

chains,

where the measured diffusion coefficient reflects _some average dominated

by fast-moving (small) species,

while the relaxation time is dictated

by

the slowest

(largest) species.

The

question

arises whether

monodisperse

flexible

polymer

chains _{can ever} diffuse

arbitrarily large

^distances without

relaxing

orientational correlations, and hence _stress.

The time scale for diffusion in

polymers

_{r~,~~ is} defined _as the time it takes for _a chain _to diffuse

a distance of order of its coil

size,

which _we take _to be its end-to-end distance

R m

bN'/~,

_N

_being

the number of _monomers

(Kuhn segments)

and b

being

the _monomer

length

~2

~~'~~ ~

fi

' ~~

1582 JOURNAL DE PHYSIQUE II N° 10

where D is the

(three-dimensional)

self-diffusion coefficient. Note that _we

drop

numerical

prefactors

of order

unity throughout

the paper.

The relaxation time is

typically

associated with the

longest

mode encountered in _{a stress} relaxation

experiment.

In other words, for times

longer

than the relaxation time, the _stress

«

decays

as a

single exponential

in time t

« exp

(-

_t/r~~~~,

(2)

Experimentally,

for

monodisperse

linear

polymers

_r~~~~~ is

slightly

shorter

[3, 4]

than

r~,~~, and for branched

polymers

_{r~~j~~ can} be

considerably

smaller

[2]

than _r~,~~. However,

examples

of

monodisperse

flexible

polymer

systems ^with _{r~~~~~ ~ r~,~~} are not found in the

experimental

literature, Here _we

develop

_a

theory

for _stress relaxation and diffusion in concentrated solutions of

monodisperse

flexible linear telechelic

polymers

with _no free chain ends. This situation could be realized in chains

having opposite charges

_on each of their two ends

[5] (with

_no unattached

counter-ions)

or in chains with

singly-charged

ionic groups of the

same

sign

attached _to each of their ends, in the presence of divalent counter-ions

[6].

The ends of such chains exist

solely

in

pairwise

association _states, due to strong ^ionic interaction. Pairs of chain ends _are

capable

of

forming

reversible

junctions through polar

interactions. We show that in certain circumstances the chains

comprising

these reversible networks _can indeed diffuse distances that _are

large compared

to their size _on time scales that _are short

compared

to

the _stress relaxation time.

Consider _a concentrated solution _or melt of linear chains

(with

N

monomers)

with

singly charged

ionic groups ^at each of their ends. If the

charges

are of

opposite sign,

_we have the telechelic ionomer shown

schematically

in

figure

la. In

hydrocarbon

media

(low

dielectric

constant)

the ends will be

paired

to minimize free energy, as shown in

figure

16. If the

charges

are of the _same

sign,

_we have the more conventional telechelic

ionomer, schematically

shown in

figure

2a. For

charge neutrality

there _must be

oppositely charged

counter-ions present. ^We focus _on the _case of divalent counter-ions. In low dielectric _constant media, each divalent

counter-ion will be

strongly

bound to two chain ends, shown in

figure

2b.

In either _case, the chain end

pairs (hereafter

called

stickers)

can lower their free energy further

by associating

_to form

larger

_groups of ions

(so-called multiplets) through polar

a)

₊

b)

^~

~

fi

C)

~

+

~

Fig.

I. Schematic

representation

of _a telechelic chain with

charges

of

opposite sign

at its two ends a) telechelic ionomer b) ionic association of two chain ends c)

quadrapolar

association state.

a)

b)

e

C)

Fig.

2. Schematic representation of _a telechelic chain with like charges at its _two ends _: a) telechelic ionomer b) ionic association of _two chain ends with _a divalent counter-ion _; c) polar association _state.

interactions. For

simplicity

_we focus _on the strongest ^{of these} interactions, shown schemati-

cally

in

figures

lc and 2c. These

polar

interaction

energies

are

considerably

weaker than the ionic _ones of

figures

16 and 2b

(of

order _a few kT at room

temperature),

and thus these

polar

associations _are reversible

junctions.

We define p ^to be the

probability

of _a sticker _to be in _a

multiplet,

and thus the

probability

of _a sticker to be free from

multiplets (I.e.

the

simple

chain-

end

pair

states of

Figs.

lb,

2b)

is p. We define the lifetime of the sticker in the associated

state

(multiplet)

to be _r. The situation is very similar to that discussed in reference

[7].

The

polar

associations act _as temporary ^{crosslinks and the} system as a whole is _a reversible network. The strong

pairwise

association of chain ends _causes formation of trains of chains between

multiplets,

hereafter called

superchains,

with effective

degree

of

polymerization

that

can be much

larger

than N. Thus the system ^is

usually highly entangled

_even when N _~

N~,

the number of _monomers in _an

entanglement

strand. The effect of

surrounding

chains is modeled

by

a tube of diameter _a

=

bN(/~.

The details of the

multiplet

state are not

important

for

dynamics

_any sticker in the associated _state is confined _{to a} volume of

roughly a~,

from which it cannot move until it leaves the associated _state.

We _assume there is _a thermal

equilibrium

between the associated and free _states of stickers.

As discussed in reference

[7],

the _two parameters p and _{r are} sufficient to

fully

describe the kinetics of association and dissociation of the stickers. For

example,

^the _average ^{duration of} a

single

sticker free from the

multiplets

is rj =

r(I p)/p.

We consider the

dynamics

of _a labeled telechelic chain in this reversible network. The situation is

analogous

to reversible networks made up of

long

chains with stickers

regularly spaced along

the chain

[7],

with the main difference _now

being

that the effective chain

length

is

infinite

(no

free

ends).

At short times the curvilinear motion of the labeled chain

along

its tube is subdiffusive. In contrast to the standard situation of

entangled polymers,

^{the effective}

friction for times

longer

^than _r ^{is controlled}

by

the

opening

and

closing

^{of reversible}

junctions.

The time

dependence

^{of the} mean-square curvilinear

displacement

of _{a monomer}

along

the tube is still

A~(t)

=

a~(t/r~~~)'~~, (3)

1584 JOURNAL DE

PHYSIQUE

II N° 10

but with r~j

being

the effective Rouse time of _an

entanglement

strand, modified

by

the presence of the stickers. The detailed mechanism of motion of _a chain of S stickers and finite

number of _monomers M

=

N

(S

₊ I _was discussed in reference

[7].

Since _{r~~~ measures} local chain

dynamics,

it does _not

depend

_on the overall chain

length

in the multisticker

problem.

On short time scales chain motion is

Rouse-like,

with

A~(t

=

) a~(t/T~~~~~)'/~

,

(4)

where the effective Rouse time of the chain is

proportional

to the square of the number of stickers in the limit of

large M, S,

and p m 1.

~Rouse " ~~

(5)

Hence,

the effective Rouse time of _an

entanglement

strand scales like

r~~~ m r

(N ~/N

)2

j6)

This

simple

result is

only

valid when the

majority

of

junctions

_are closed ~p _m

).

The _more

general

situation has _a friction that

depends

on p, the average fraction of closed

junctions

r~~~ m r

(N~/N

)~/

f ~p,

k~~~

(7)

The function

f~p, k~~~)

was derived in reference

[7].

(~

^~ ~~

~ '

f~P'

~~~~~

~i

~~ ~~~ ^~

(kmax + ' )~

~i~

^~~~

~~~

~

where _we have summed the contributions from the various k-strands

(successions

of

k open

stickers) weighted by

their

probabilities

of _occurence, whereas

only

the k

= process is active when _pm I

(cf. Eq. (6)).

^The sum is

split

into two parts : for k

~

k~~,

the strand of k open stickers _can

equilibrate by

Rouse motions

during

the lifetime of the

k-strand,

while for k ~

k~~,

the

equilibration

is

incomplete.

^The parameter

k~~,

was thus evaluated

by matching

the Rouse time of the k-strand with the lifetime of the k-strand

[7], resulting

in

l~

^{/f 2 Tj}

~~~

~

lie

^~~ _~max ^~

where rj is the lifetime of _a

single

_open sticker and _r~ is the Rouse time of _a chain of

N~

_monomers.

The _monomer

displacement

remains subdiffusive

(with

A~

t'/~)

_as

long

_as the chains _are

confined to their

original

tubes. Since _we consider the limit with _no free ends, the

only

_way for

a chain _to get out of its

original superchain

tube is

by

a

multiplet exchange [8].

^This

exchange

process is shown

schematically

in

figure

3 for the

simplest

association of chain-end

pairs.

We demonstrate the

exchange

_process for the _case of _a conventional telechelic ionomer with divalent

counter-ions,

but the _case of telechelic ionomers with

opposite charges

at their two ends

(Fig. I)

is

perfectly analogous. Initially (Fig. 3a)

the _two chains _are confined _to their

original

tubes.

Through

subdiffusive

motion,

their sticker

pairs

meet and form _a

polar

association

(Fig. 3b).

After time _r the

polar

association breaks, and the chains can either recombine in their

previous pairs (Fig. 3a)

_or

they

_can

exchange

in the associated state and

Fig. 3. The quadrapolar

exchange

_{process :} a) initial _state showing two chains trapped ^inside their tubes b) the

quadrapolar

association _; c) final state

showing

chains

moving

into _new tubes after the

exchange.

emerge in _new

pairs (Fig. 3c).

Notice that after the

exchange

_occurs, the chains _can diffuse into

new tubes.

The

multiplet exchange

_process is described

by

the characteristic time

r*,

which is the average time between

exchange

events for _a

given

chain end. The

probability

of _a

given

sticker to be

unexchanged

after _a time _t is

Q (t)

= exp

(-

t/r ^*

) (9)

We need _to calculate the

(time-dependent)

_average contour

length

between

exchanged

^chain

end

pairs, L(t)

=

(X(t)) aN/N~.

The average number of

primary

chains between

exchanged

chain end

pairs

is

given by

one-dimensional

percolation [9]

(x~~~l

~ i

) $)tl'

~~~~

1586 JOURNAL DE PHYSIQUE II N° 10

which leads _to the tube

length

of the average sequence of

unexchanged

stickers

The subdiffusive _monomer motion described

by equation (3)

ends _at time scale T~, ^{when the}

typical

monomer's motion

begins

to be influenced

by exchange

events

(when

the Rouse time

of the chain between

exchange

events is

reached).

Thus

Tim (L(Tj)la)~

_r~~~ and

using

equation ill),

_we obtain the

equation

for time scale

Tj

j~-~/r~~~)>/2 » ~~~

~

Coth

lT'/~~

^*

This time criterion

corresponds

to a mean-square curvilinear

displacement A~(Tj

_m aL

(Tj ).

In

the limit where subdiffusive motion is fast

compared

with

exchange

events

(Tj

MT

*),

coth

(Tj/2r

^*

m 2r

*/Tj,

and _we get

Ti

₌ 2 ~

r ^*

~~~

~ i/3

N ^off

~

(l

3)

On time scales

beyond Tj

the curvilinear

displacement

of _{a monomer}

along

the tube is

diffusive,

with _a curvilinear diffusion coefficient

Dj

_m

aL(Tj )/Tj

_m

a~/(T,

_r~~~)~/~

(14)

The three-dimensional _monomer motion remains subdiffusive until the curvilinear _monomer fluctuations reach the coherence

length

between

exchange

events. This time scale is T~, determined

by A~(T~

_m

D,

T~ _m L~

(T~),

which leads _to

1/f

²

coth~ (T~/2r *)

^T2

= ~.

(15)

~e (Ti ~effl'~

In the slow

exchange

limit

(T~

«

2r*),

this reduces to

~2

_~ 2 ~

y *

~~

~ 2/9

N ^en

~

(16)

The

(three-dimensional)

self-diffusion coefficient of the

primary

chains is

D

m

~ Dj

_m

^~~~~~~

_m

~

~

coth

(T~/2r *) (17)

L

(T2)

T2 T2

Ne

In the slow

exchange

limit

(T~

«

2r*)

the self-diffusion coefficient is

/f ~* /f ^-5/9

Dm2a~-qma~

_2-r*

_{rj~~/~, (18)}

Ne _T2 Ne

and the time

required

for _a chain to diffuse _a distance of order of its size is

fit 14/9

r~;~~ m

(2r

*)5/~

r((/ ₍₁₉₎

fife

The three-dimensional mean-square monomer

displacement ~fi(t)

is calculated from the

mean-square curvilinear

displacement A~(t) _along

the tube in the usual _manner

ill,

remembering

that the tube itself is _a random walk, ~fi

(t )

and

A~(t

_are

plotted

in

figure

4 and the relevant time scales _are summarized in table I, This

plot

is very similar _to that for

simple reptation

of flexible linear

polymers II ], Up

to the Rouse time of the average chain strand

between closed

stickers,

_{T~ the}

dynamics

_are identical _to those of _a linear chain _on time scales

less than its Rouse time, Between T~ and the lifetime of _a sticker _{r, no net monomer}

displacement

_{occurs, as} the Rouse modes of _a strand between closed stickers have

completed,

and further

displacement

must await the

opening

of the associations

(on

time scale

r).

The curvilinear _monomer

displacement

at _r is

A~(r)maz

where

zma(T~/r~)'/~m

a(r/r~~~)'/~

and the three-dimensional

displacement

is

~fi(r)ma~/~z'/~

_Between

r and

T, (the

time where

exchange

events start to influence _monomer

displacement)

the curvilinear

displacement_is

subdiffusive

(A~~ t'/~

_{and thus ~fi}

t'").

As discussed

before,

_r~~~ is the

effective Rouse time of _an

entanglement strand,

determined

by extrapolation

of the

subdiffusive motion between _r and

Tj

back to the tube diameter

(see Fig. 4). Tj

is the Rouse time of the

largest portion

of chain _or

string

of chains that _moves without

knowledge

of

exchange

events. The curvilinear _monomer

displacement

at

Tj

is

A~(Tj )m aL(Tj)

and the three-dimensional

displacement

is

~fi(T,)m a~/~[L(T,)]~/~

T~ ^{is the time where}

exchange

events allow the _monomer motion _{to no}

longer

be confined _to the

original

tube. Between

Tj

^and T~ ^the curvilinear

displacement

is

diffusive,

but the three-dimensional

displacement

is not (~fi t

'/~),

in _exact

analogy

with normal linear

polymers

between the Rouse and

reptation

times of the chain.

Beyond

T~ ^the three-dimensional _monomer

displacement

is diffusive. The diffusion time _r~,~~ is determined

by extrapolation

of the three-dimensional mean-square

displacement

in the diffusive

regime (t

_~

T~)

back _to the end-to-end distance of _a

single

chain

R~,

_as shown in

figure

4.

1'

,

1'

,1'

/

~fi

-,

A~ 1'

' 4l

~/

_;

/%

~e ~efl _T~ ~ T ~ ~ ~

t

^{I d'fl 2 relax}

Fig. 4. Three-dimensional mean-square displacement of _{a monomer 4~} (solid curve) and mean-square

curvilinear

displacement

of _{a monomer} along ^the tube A~ (dashed curve) as functions of time.

Logarithmic scales. See table I for _an

explanation

of time scales.