The tube diameter in polymer melts, its existence. and its relation to the quantum Hall effect

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The tube diameter in polymer melts, its existence. and its relation to the quantum Hall effect

Arkady Kholodenko, Thomas Vilgis

To cite this version:

Arkady Kholodenko, Thomas Vilgis. The tube diameter in polymer melts, its existence. and its relation to the quantum Hall effect. Journal de Physique I, EDP Sciences, 1994, 4 (6), pp.843-862.

�10.1051/jp1:1994232�. �jpa-00246950�

Classification Physics ^Abst?.acts

61.25H 46.30J 64.70D

The tube diameter in polymer melts, ^{its existence and its}

relation to the quantum ^{Hall effect}

Arkady

L. Kholodenko ('> 2) ^{and Thomas A.}

Vilgis (')

(') Max-Planck-Institut fur

Polymerforschung,

P-O- Box 3148, D-55021 Mainz, Germany (2) 375 H. L. Hunier Laboratories, Ciemson University, Ciemson, SC 29634-1905, U-S-A-

(Received 20 August 1993, ?.ei<ised 23 Feb?.uaiy 1994, accepted 2 Maich 1994)

Abstract. The self consistent single tube modei which

plays

the

major

rote in the

reptation

theory of de Gennes, Doi and Edwards is reanaiyzed ^with heip ^{of the} discovered analogy with quantum ^Haii effect (QHE). We

study

the _existence and the

stabiiity

of _a given tube caused by ils interactions with other tubes. The interaction is of _a

topoiogicai

nature due to the effects of entangiements. The formalism developed aliows _{us tu} explain the observed _crossover from the reptation tu the Rouse regime in _ternis of meiting transition of _a quasi-iattice of de Gennes tubes.

Some of the

major

resuits of the existing packing ^and scaiing theories reiating ^the monomer density

tu the tube radius _are obtained _as

iimiting

cases of _{a more} generai theory. The resuits of _Dur caiculations _are supported by the _{most recent} Monte Carlo simulations.

1.

Introduction,

basic formulation and definitions.

Recently

the

topological theory

of

reptation

has been

proposed [il.

The

reptation theory

_was

originally suggested by

de Gennes

[2]

and

subsequently

refined

by Doi,

Edwards

[3]

and

others. From the very

begmning

of the

development

of the

theory

of

reptation,

it was clear that

topology

should

play

_{an important} rote in this

problem.

In the _same time, the

existmg developments (except

that

presented

in Refs.

[1, 4])

_use

topology only indirectly,

_e-g-

by considering

the

problem

of motion of _a

polymer

chain _in the array of obstacles, _usmg the concepts ^{of tubes} and

primitive paths [2, 3]. Although

the concept ^of a tube tutus out to be decisive for the

development

and _success of the reptation

theory,

_it is introduced in this

theory axiomatically.

Indeed to

change

the lime scale of

polymer

motion from the Rouse time

r

~N~ (N being

the chain

length

or

degree

^of

polymerization)

to r

~N~

characteristic for

reptation,

a new

length

scale had to be introduced. De Gennes used _an

underlying

method with

a certain mesh _size whereas Doi and Edwards confined the chain into a

(harmonic)

tube of

diameter _a.

Analogous

to the tube model is the

primitive path

model

[3],

which is also

instructive. It fixes the

primitive path

_step

length

_a

(1.e.

the tube

diameter)

to be

forger

than the Kihn step

length ^f.

It has been difficult _to find _a

theory

which

provides

this _new

length

scale from _a

topological

point ^of view. The _aim of this paper is to

develop

a formalism and _to find _an

analogy

to relate the tube diameter _{to a more}

general physical

picture. ^In

spite

^{of the fact that}

reptation

is _a

dynamical phenomenon,

the tube is introduced in _a

traditionally thermodynamic

way

(to

be discussed below in this Sect. and in Sect.

2). Although tubes, apparently,

_can

appear and

disappear,

at time scales shorter than the terminal relaxation time

[5]

r~ N ^~.~

(where

N is the

length

of _a

polymer chain)

the melt of

fully

flexible

polymers

could be viewed _{as a} porous continuum with tubes

(pores) being randomly

distributed in it

[6].

To characterize the porous

material,

the

partition

coefficient _{X is} used

[7].

For the

fully

flexible chains it is

given by

X " exp

(- As/k~

_{m exp}

(-

_a

_(R~/Rp)~) l.1)

where As is the

configurational

entropy decrease caused

by

the confinement of the chain.

As

depends

on the characteristic size of the pore

Rp,

its geometry

(reflected

in the numerical coefficient

a)

and the character of interactions between the

polymer

and the pore watts.

R~

₌

Nf~

_{is the size of the} unconstrained macromolecule and f is of the order of _an effective

monomer size.

It is important to notice that In _{X can} also be

remterpreted

as a decrease in the entropy caused

by

thermal fluctuations. Such fluctuations take the system _away ^from

equilibrium

and,

therefore, Rp

should be considered,

rigorously speaking,

_as a random variable. From this point

of _view. the _occurrence of tubes in

polymer

melts _{is a} direct _extension of the vacancy models

of self-diffusion _m

simple liquids [8].

These models _are also based _on the

assumption

of

surface tension-limited spontaneous vacancy formation which facilitates the diffusion of host

atoms

through

the

liquid.

In the case of

simple liquids

the

Laplace-Young equation

2 «H

=

AP

(1.2)

where _« is surface tension,

ù

_{is the}

mean curvature while AP is the pressure difference, admits

only

_one

solution,

_a

sphere.

In the _case of

polymer

solutions

(1.2)

also _can

provide

a

cylindrical (tube-like)

solution. This could be understood

simply by posing

a

question

how many surfaces of _{genus zero can}

(1.2)

describe ? A

simple analysis [9]

shows that there _are

only

two

possibilities

: a

sphere

and _an infinite

cylinder.

In

simple

fluids _an

imagmary cavity

could be created

by taking

round-like molecule out from the bulk to the surface while _m

polymer

solutions such

imaginary procedure

^would create a

cylindrical-like

tube.

As _was

already

noted

by

Casassa and

Tagami loi,

for the _case of

strongly repulsive

_pore

watts

(1.1)

is _{not very} sensitive to the actual

shape

of the pore : for pores of

spherical, cylmdrical

and slab geometry ^the differences in _{X are} very small

(e.g.

_see

Fig.

3 of Ref.

loi).

Because both in a porous medium and in the

polymer

melt

Rp

is a random variable

(restricted

_to non-negative

values), (1.1)

should be understood in some

pre-averaged

sense, i-e-

Rp

in

(1.1)

should be understood _{as some} representative

non-fluctuating (1.e. averaged) quantity.

The situation here is similar _to that which is

being developed

in the free volume

theories of the

liquid-glass

transition

[1ii.

Similar arguments are also

being

used in the theories of

liquid-liquid polymorphic

transitions

[12].

Unlike the above

theories,

in the present case

topology plays

a crucial rote _{as we} shall demonstrate below.

Topology

determmes the very

existence of the tubes and is

responsible

for the different viscoelastic behaviour of the

polymer

melts _{as a} function of the

polymer weight (e,g.

see

Figs.

7.6, 7.7 of Ref.

il 3]).

Because of the above mentioned

insensitivity

of

(1.1)

to the actual

shape

of the pore, it was

recognized

a

long

time ago

[14]

that for

R~

w

Rp

the entropy ^{decrease for} a

polymer

^{chain confined} in the tube could be calculated with the

help

of _a

simple

harmomc oscillator model.

Following

Doi and Edwards

[3],

let _us mtroduce the end-to-end distribution function

G(r,

r'; N which satisfies the equation « of _motion

»

l~ _V)

+

"~

(.<~ +

y~)

G

(r,

r' N

=

0

(1.3)

éN

supplemented

with the

boundary

condition :

G(r, r'; N)j~_~

_-

ô(r r'). (1.4)

Here

w~

_represents the

strength

of the harmonic-like

potential

which models the presence of the tube around the chain. The partition ^{function Z for such confined}

polymer

chain _can be written

as

Z

=

ldr~

^dr

^[

^{dz dz' G}

^(i~

^r

^[,

^{z z'} _, ^{N ),}

^(1.5)

where _we have taken into _account the fact that the _« horizontal _» 1 and the _« vertical _»

degrees

of freedom _are

completely decoupled. Solving (1.3)

and using

(1.5)

the free energy F _can be written _as

F

=-kBTlnZm-TAS, (1.6)

where As has the _same meaning as in

(1.1), k~

is the Boltzmann _constant and T is the

temperature. ^The

quantity

of primary mterest in the

theory

of reptation ^is not F but the average

(x~ +y~)

= =

a~ (1.7)

w

which

produces

the tube _cross section

a~.

If

a~

_{is given, then}

evidently

_{w is} fixed _as well For the purpose of this paper we introduce _an

arbitrary plane through

the three-dimensional

polymer

melt. The melt and its

density

_can be visualized

by

the number of crossmgs which _are obtained when the

polymers (tubes)

_cross the

plane.

Let _us concentrate our attention _on the distribution of

just

defined tube _cross sections _m this

imaginary plane.

Because of the natural cut-off

given by

the Kuhn

length ^Î

_{we require} that

a~

~

Î~.

_Let A be _{an area} of the surface

(plane)

and let _n~ be the number of _« tubes _»

(cross

sections) which _cross this

plane.

Introduce

now the surface

density p

_i,ia

à

=

1 (1.8)

and,

following

references

[15, 16],

define the

Wigner-Seitz

radius

i~_~

^via

arr'$-z

=

(1.9)

p

Evidently,

that 2 r~_z represents an average distance between the centres of two

neighbouring

tubes. The

filling

fraction _{v can now} be defined _as

~ 2

v = =

ara~ à (Î.ÎÙ)

'w-z

Obviously, by construction,

_{v w} 1.

Because both

a~

_and

à ultimately depend

on the

polymer (monomer)

concentration

p m the melt _{we rewrite}

(1.10)

_as

v t

ara2(p)à(p) _(i.ii)

JOURNAL DE PHYSIQUE T 4 N' 6 JU~E lQ94 ~I

where both

a~

_and

#

_{are m}

general

unknown functions of _p. We

expect, however,

that

à (p

cc p in view of

(1.8). Suppose

_now that there is _some interaction between the _«

Wigner-

Seitz discs _»

(cross sections)

in the

plane

_so that the gas of discs cannot

overlap

and may form a

lattice _{or a} disordered structure. In the latter _case the _«

liquid

_» of uniform

density might

occur so that the

initially

well defined tube radius

a~

_{may loose its}

meaning.

Like _m the

theory

of

liquid-gas

transitions

(based

on the

Ising model),

we

anticipate

that, if the

solid-liquid

transition indeed _occurs, it will be characterized

by

_v ^* which is

just

_some

particular

model-

dependent

number, 0 _w v* w1. If this number is known, then _{we can} write v*

=

ara~(p *) à(p*) (1.12)

which

implicitly

determines the critical

polymer (monomer) density p* According

to

experiments [13],

the viscoelastic behaviour of

polymer

solutions

dramatically changes

from

Rouse-like to

reptation-like beyond

_some «critical»

polymer

molecular

weight.

Such _a

change,

in view of the

picture just described,

could be associated with the

melting

transition _: when the tubes

(cross sections)

form _a quasi solid lattice, we expect to have the reptation

behaviour,

while when

they

form _a

liquid,

we expect to hâve Rouse-like behaviour

(no

tubes

are

required

to exist for Rouse~hke

regime).

To make the above statements more

precise,

^the «

magnetic

_»

analogy presented

in section 2 is

helpful. By noticing

that the harmonic oscillator

model,

_given

by (1.3), formally

coincides

with that for _an electron in _a constant

magnetic field,

we translate _our tube

problem

into

magnetic language

used in the

theory

^of quantum Hall effect

(QHE).

This then allows _{us to}

generalize

one tube

(one electron)

model to the _case of many interactmg ^{tubes in section 3} while _m section 4 _we reduce the

problem

of

phase

transitions in the

assembly

of

interacting

tubes _to that of

phase

transitions _m one-component two-dimensional Coulomb

plasma (CCP).

This then enables _us to

study phase

transitions in such system ^based on available Monte Carlo and

analytical

results. Dur results

strongly _support

the conclusions of

packing [17]

and

scaling [18] reptation

models and _{are m} agreement ^with recent Monte Carlo simulations

[19].

In section 5 _we discuss the obtained results and

provide

_some additional arguments m favour of the

topological

_origin of the

viscosity _exponent

3.4 observed in the reptation regime. In this section _we also consider _some extensions of _our results _to

polymer

networks. In the _case of

networks,

the presence of crosslinks

plays

a similar role in _our model _as Coulombic

interelectron interactions _in the _case of

QHE.

To make the paper accessible _{to a} broad

readership

_we introduce the basic concept ^{in both}

languages,

_{i e, usmg} the

polymer

notations and that of the

QHE.

Therefore similar formulae will appear several times for the two cases.

We nevertheless think, that this

parallel

introduction of the concepts is useful and will make the paper more readable.

2. Landau

diamagnetism

and

polymers

^confined into tubes.

We would like _to remmd the reader about _some facts from the

theory

of Landau

diamagnetism

w order _to make connections with the

existing

hterature.

The Hamiltonian

Ù

_for

an electron in a

magnetic

field H

=

Vx A _can be written as

(h

= c = e ₌

1)

Ù

=

V)

+ A V~ +

~~

,

(2.1)

where A is vector

potential

of the field H. In the _case of constant

magnetic

field

H _we may choose

[20]

~~

Î~~~ ~~'~~

or

A

=

(- Hy, Hx, 0)

,

(2.3)

2 2

1-e- the _constant field H is

being perpendicular

to the

xy-plane.

The Bloch

equation

for the

density

matrix p

(r,

r' _;

p )

^for the electron in the

magnetic

field H _can be written _{m a} standard way ^as

~ p =

Ùp _(2.4)

éP

The above

equation

is

supplemented by

the initial condition

p (r, ^r'

p

_- 0+

)

= à

(r r'), (2.5)

where

p

=

(k~ T)~'.

The combined _use of

(2,1)-(2.4) produces

the

following equation

for _p

[20]

_:

lÎ_ _{v2_}

^{H é é}

^j~2

~

éô ² m ^~ 2 mi ^~

éy

^Y

£

^~

$ ^(~

^{+ Y~}

))

^{P = 0}

^(2.6)

The

corresponding pjlymej ^problem

^{is obtained}

^{by making}

^the

^following replacements

_; p # N,

=

~~

,

~

# ^'~ and

considering only

states with the total

angular

monomentum

2 _m 6 8 _m 6

zero. After these

replacements,

and in _view of

(2.5),

we obtain

equation (1.3)

as

required.

The

partition

function Z is

given by

z

=

dr _p

(r

t r' _;

p (2.7)

Evidently,

both

(1.5)

and

(2.7) (after replacements)

will

produce

the _same results for As _w the limit N

- m.

Equations (1.5)

and

(2.7)

are m fact somewhat different _: while the first is in accord with the

general

definitions

given

in Doi and Edward's book, the second is in

accord with reference

[3],

_page

1816, equation (A.6).

This could be

easily

understood if _we recall that the

density

matrix _{p can} be

expressed

in the form

P

(r, r'Î P )

=

z

_e~ ^~~~~P

iii(r) ji~(r'), (2.8)

where _w (n

)

and #i~ are

respectively

the

eigenvalues

and the

eigenfunctions

of the Hamiltonian

Ù _(2.1).

_For

_{large p (or N)}

_the

_ground

_{state dominance}

_[14]

_leaves

_only

_{the lowest}

n-terni

contributing

to

(2.8).

In the present

study,

m view of

(1.7),

we are interested in the _{mean cross} sectional _size of _a tube in the

plane.

To

generalize

our results for many

interacting

tubes, the _use of

complex

variables _{is most convement.}

By introducing complex

variables _z

= x + ty, and f

= x iy, ^we rewrite the Hamiltonian

Ù

_given

_{by (2.1)}

_{in the form}

_{[21] (for}

_{the constant magnetic} field

case)

Ù= ~àé+£[z[~-~(zé-Ià) _(2.9)

where à

=

),

_à

=

~

,

[z

^~ = zf, etc.

z ai

The

angular

momentum

operator

^J = zé

fà

_con be

ignored

if _{we are} interested

only

_in the

states with _zero

angular

momentum. For such _states the

ground

state

eigenfunction

of

Ù

_is

given by

4ro(z, ¹⁾

= N exP

(

lz1~ (2.10)

or,

taking

into _{account our}

replacements (after Eq. (2.6)),

_we

obtam, 4io(z, i)

=

N exp

) _jz2

,

(2.

ii i where N is the normalization factor.

Equation (1.7)

_{can now} be rewritten _as

a~

=

d~z[#fo(z,

f)]~ _(z(~

m

d~zà(z, I)[z[~ _(2.12)

Thus,

the _existence of

a~ depends

_on the convergence of the

mtegral

_w the r-h-s- of

(2.12).

In the limit of

non-interacting

tubes this

integral

is well defined, while if tubes

(disks)

_are

interactwg,

^{the conclusion about the} convergence of such _an

integral

_is much _more difficult _to make _{as we} will

explicitly

demonstrate below in section 4. For the

beginning,

however, _we have to find the mechanism causwg the tubes _{to wteract.}

3. Statistical mechanics of

polymers

_m the presence of

topological

constraints.

The

study

of statistical mechanics of

polymers

_w the presence of

topological

constraints was

mitiated _a

long

time ago

by

Edwards

[22].

To

develop

his ideas further, it is

helpful

_to recall

briefly

_some of his

findings.

The

winding

number _w for _a fine _{w a}

plane encirclmg

an

infinitesimally

small puncture in

this

plane

_is

given by

l

l~

^~~

^xi,

^yx

^(3.1)

~ 2

ar o x~ ₊ y~

where

j

=

~Y

,

.i

=

~

and L is the

length

of the lwe

lywg

_in the

plane.

In writing ^{the above}

dr dr

equation we

had,

without loss of

generality,

^{chosen location of the} puncture as the origin of coordinate system. ^{The result}

(3.1)

is

conveniently

rewritten w the

followmg

_way

[23]

w=

j~dr)8(r(r)). (3.2)

" o ^~

where

&ir(r)j

= tan~ ^'

fi _(3.3)

and

r(r

=

(x(r ), y(r )),

The unconstrawed propagator ^{for the}

polymer lying

in the

plane

is given

by

r(L)=r~

lj

^L

G

(rj,

_{r21 L}

)

=

D

[r(r

_exp

/ ^dré~

,

(3.4)

r(0) _r 0

while _w the presence of _a hole _{we can write} instead,

G

(ri,

_r~

,

L, _w

=

r(L) r~ j ^L ~ ^L

r(0>

~

r

~ ~~~~~~ ~

l~ _"

^~~

_~ ^~~~~~~l

^~~~

¹⁰

^~~~

^~~~~

so that

G

(ri,

_{r~ ;} L

=

ldwG ^(ri,

^{r~ ;}

^{L, w) (3.6)}

Equation (3.5)

can be

equivalently

rewritten _as

G(rj,

_{r~ ;}

L)

=

7r Î~

^~~

^{~"~ ~~~_}

^{~~ ~ ~~~~~~ ~~~}

Î)

^{~~ ~~ ~}

^~ÎÎ /r _~~~

_{~~~~ ~~} ~~

At the classical level, the action S in the exponent ^{of the functional}

wtegral (3.7)

_given

by

S

=

j~

^dr

^é~

⁺

£ ) _{8(r(r))) (3.8)}

o ^{" ~}

could be

simplified

because the second _{term is} the total

« lime _» derivative

and, whence,

_can be

dropped.

At the quantum level _we cannot

simply drop

the second term but some important

simplifications

still _can be made _{as we} shall demonstrate below.

The

generalization

to no punctures is

straightforward (see

also

[23])

j ^L ~ ^"o

w ₌

m

^dr

p jj

&

(r(r

_r,

(3.9)

o

, =1

To make the

important

connection with

topology,

consider

again

^the

problem

of _a

«

particle

_»

(polymer)

_«

moving

_» in the

plane

in the presence of infinitesimal hole

(puncture).

The

setting

of the

problem

will _remain the _same if instead of _a hole _we would have another

polymer

_piercing

through

the

plane.

Consider _now these _two

polymers

m 3-dimensional space.

Following

the

argument by

Delbrück

[24] (see

also Ref.

[l]),

_we consider the ratio

R~/N

N~ ^~'~ For N

- m, i e. for

sufficiently long polymer chains,

the distance between the

polymer

ends _is much smaller than the total

polymer length

so that the chains could be considered _as

effectively

closed. For _limes less than _r~, defined in section 1, the two

entangled polymers

could be considered _as

forming

_a link

(1,e,

two mterlocked

rings)

e,g, see

figure

1,

The field

theory

which generates ^the

lmking

numbers is known _to be as the Abelian Chem-

Simons

theory

which we have

briefly

discussed and used earlier _w reference

[1].

It is

relatively

easy ^to

consider,

mstead of

just

two mterlocked _« rings », say, n interconnected rings.

Indeed, following

references

[21, 23],

_we consider the functional

integral

of the form

G

=

fi

_D

_{jr (ri )i}

_D

_jA~

eXp

j- 50 ir (T )i Sl~ iA i

Sini

iA, ri j (3. ÎÙ)

where

~ ,,

Li

~°

~

2f~~~

~

~~~~~~~~~~

'

~~'~~~

Fig,

l,

Fig.

2.

Fig. l. -Link made of _two rings dissected by _an

arbitrary plane.

Fig.

^2. Braid-type representation of the lmk

depicted

in

figure

1.

and

Sj~iAj

= i

1 d~x e"~PA~ (x) ô~4~(x), (3.12)

*

n L~

S~~~IA, r

j

=

£ dT~

_lj

(T

~) ^/Î~

(r(T~ ))

,

(3.13)

1=1 0

and summation _over the

repeated

Greek indices _is assumed from to 3.

The functional

integral

_over A-fields _can be

easily

calculated

(by completing

the

square)

because it is Gaussian. A

straightforward

calculation

produces [1, 23]

:

G

=

fl Djr(r~)j exp(- ^soir(r)j

^-14S

^{£ ijr(r~),} _r(r~)j) ^(3.14)

where

]

~ ~ (r(T, ) r(Tj ))~

I

[r(r~), r(r~)]

₌

dr~ dr~ e~~~1." ^(r~

~

iP(r~), (3.15)

4 _"

c, ,~

(r(r~) r(r~)(

and _c~(c~

)

denotes the

contour(s)

of1 th ~j

th) chain(s) respectively. Equation (3.15)

_represents the _sum of

linking

(1 #

j

and self

linking

₍₁

=

j

numbers while ~tl in equations

(3.12), (3.14)

is to be determmed below. As _we have

briefly

demonstrated in reference

[Ii (see

also Refs.

[25, 26]

and Sect.

5),

the self

linking

numbers may

effectively

contribute to the

rigidity

of the individual

polymer

chains while the

linking

numbers could be rewritten after _some

algebra (e.g,

_{see p,}

198,

199 of Ref.

[23]

and Sect. 7.2 of Ref.

[55])

_as follows

j

L

~

I

jr(r~ ), r(r~ )]

=

dr

8(r~ (r

_r~

(r ))

₊

I~. ^(3.16)

2 _"

0

~~

It was

argued

in reference

[23]

that, for closed

paths, I~

^{vanishes, while for} open

paths

it does not contribute _to

dynamics appreciably.

This _can be

easily

understood if _we consider _one of the

rings

(or quasi-rings, if

they

are not

chemically closed)

_as

lying

_{m some}

plane

while the other

one as piercmg the

plane just

_{m two points.}

Then, locally,

the picture will look

exactly

the

same as described

by equations (3.1), (3.3)

in accord with reference

[22].

Obviously,

the Gauss

linking

number

(3,15) (for

#

j)

is

only

the lowest order

topological

invariant and other

topological

mvariants

describing

links and knots should

be,

in

principle,

considered. Because of the skein relations between the different knots

(links) [27],

_{we can}

always

unknot the knot

[link]

thus

reducing

more

complicated

knot

(link)

_to less

complicated.

It _was shown in references

[28, 29]

that the

quantum

^{Hall effect}

picture

remains in _{most cases} almost the _same if

higher

order

topological

invariants are

being

used instead of the

linking

numbers. Therefore _we restrict ourselves _to the

simplest linking

number

picture.

In order to utilize the results

given by

equation

(3.16),

consider the situation

depicted

in

figure

1. In

figure

we have

depicted

an

arbitrary plane

which _« dissects _{» our} link while in

figure

2 _we have made _a

braid-type projection

of the link _on the

arbitrary plane

so

that, according

_to

figure

2, our «

particles

_» start «

moving

_{» at} the _{same «} time _»

(1.e.

their mtemal

« times _{» are}

synchronized). Examples

of such

synchronized polymers

are known in

polymer

physics

as directed

polymers [30].

The directed

polymers

differ from the usual flexible

polymers [31] only by

the choice of _{a «} gauge »

[32].

For the flexible

polymers

the contour

length

is chosen _{as a} natural

parametrization

for the

spatial position

of segments

along

the

polymer

chain while for the directed

polymers

one of the

spatial

coordinates, _e.g. z-direction,

is

subject

to the additional constiaint _:

~j~~

^{= l for 0 w r w N. Such a choice of constraint is}

r

motivated

by

the

physics

of the

problems

which involve the _use of directed

polymers.

The choice of the additional gauge condition introduced above

usually

leads to the

synchromzation

of _« lime _» for different

polymer

chains _as discussed in _some detail _m the

Appendix.

We consider the

polymer

chains _as

asymptotically

closed, and the

synchronization

_is

always possible

if _we

mitially require

that the individual action

(1.e,

that for _a

single chain)

to be

reparametrization-invariant [32].

The addition of

magnetic

field

only

^{thickens the fines} thus

converting

them _mto tubes. From this

point

of

view,

the first term w the r-h-s- of

(3,16)

represents a mutual

winding

number for such tubes. For such

synchronized

_« motion

» of tubes

we can

write,

instead of

(3,14),

the

following

final result which takes into _account

only

the

cross sectional _{« motion »}

(in

the

plane)

:

-

i,i

D

ir (T)i xP1 Il _dT1,f i _ri

+ i

i,i i _&(rJ

_(T

_))11(3.17)

where r,~

(

_r

)

_{= r~ r r~ r}

).

The form of

equation (3.17)

is

just

that used in the

theory

of

QHE [21, 23],

_except that _{we use}

here the Euclidean _« time _» mstead of Minkowski which amounts of

considering

the

QHE

at

finite temperatures. ^{In view of the discussion}

presented

in reference

[23],

this

replacement by

the Euclidean time

produces practically

no

changes

_m the

subsequent development

of the

formalism. The constant «

magnetic

field _{» causes our} tubes _to have _a fimte thickness. This _is also

being

used in the

theory

of the

QHE. Although

the distribution function given

by (3.17) (with

added _constant magnetic

field),

_{represents a quantum} mechanical

many-body problem,

its solution _is rather

simple

if and

only

if _we shall _use the _«

synchromzed

time _»

(directed) polymer

formalism

(e.g.

see

Appendix).

To illustrate the

idea, followmg

reference

[33],

let _us

consider _a system ^described

by

the collection of

smgle partiale

Hamiltonians

Ù

=

£ ^Ù (r,, ^Î

^{m H}

_(r,,

^~

^(3.18)

,

ér, ér,

Topological

interactions

change Ù

_into

Ù~ =

H

(r,, ) £ _£

V~,

8(r,

_r~

~

(3.19)

~, "

, #j

so that the

Schrôdinger-like equation

_can be written _{now as}

) _#i((r,),

_t)

= Ù~

#i((r~), t) (3.20)

where, as in quantum

mechanics,

the _wave function #i is

required

to be

single

valued. This requirement ^makes calculation of #i nontrivial.

Fortunately,

there is another way of

solving

the above

many-body problem. If1( _{jr, ), r)}

_is multivalued wa~e function such that

1((r,), _r)

=

exp(-) £ 8(r, -r~)j ^{#i((r,), r) (3.21)}

"1<j

and, _m addition, _{we require}

that,

1

=

Ù#i

,

(3.22)

then the

single

valued function #i in

(3.21)

evolves

according

to

(3.20). Thus,

_use of the

multivalued function

1

_{allows us to solve} the above quantum

many-body problem exactly.

With the

help

^of

complex

variables _z and f, _we can rewrite

1

as follows

[23, 33]

~

i

=

fl (z, _z~)~

^"

j(z,, _{2, ).} (3.23)

Taking

_equations

(2.8), (2.10)

and

(2.11)

into account we obtain the

following

result for the

ground

state wave function

#io.

4ro(z~, f,

= N

fl

_{(z~ z~}

⁾²

^« _exp

_1' jj

_(z~(~

(3.24)

, ~j

~

,

In the

theory

of

QHE

the _wave function m the form of

(3.24)

_was first

proposed by Laughlin [34]. Taking (2.12)

into account the calculation of

a~

_is

given

_{now as}

a~

=

N

d~z

z ~

#,~~

(z,

2)

,

(3.25)

where

ni

àint(Z, 1)

~

=

1, fl

_d~Zi 4~0(Z,,

1,)j (3.26)

=?

Thus, the

completely

solved quantum

many-body problem

leads to the classical

many-body problem

which admits _{an exact} solution for _à~~~

only

m

exceptional

_{cases as we} shall

explaw

below.

Moreover,

_as we shall demonstrate, the

exponent Polyakov spin

factor ~

which _so 2 _7r

far _is left undetermmed will be

specified

based _on the

analogy

with two-dimensional _one- component

plasma.

4. Classical statistical mechanics of one~component ^two dimensional Coulomb

plasma.

The

equation

of state for classical _one and two-component

plasmas

_are derived in refer-

ence

[35] by

^standard

scaling arguments.

^{It is remarkable that bath} one and two-component

plasmas

have the _same equation ^of state

given by

P

= p (kB ^T

g~/4

,

(4.1)

where _{p, as}

before,

is

given by

_p

=

NIA

(e.g.

see

(1.8))

while _{g is} the

magnitude

^{of the}

« electric _»

charge.

_{Let T~ =}

g~/4 k~,

then for T

~ T~ ^the system is m a « gas »

phase

while for T _~ T~ ^{it is in} a «

liquid

_»

phase.

In the _case of _a two-component

plasma,

_a

liquid

state consists of clusters of

dipoles, quadrupoles,

etc. formed

by charges

^of opposite

signs.

The transition _to

such _a state is known _as Kosterlitz-Thouless

(K-T) type

^of transition. In the _case of CCP there

are no

explicit charges

of opposite sign, except ^{those which contribute} to the uniform

background (to

insure the overall electrical

neutrality).

^When

T~T~

the pressure in

equation (4.1)

becomes

formally negative

and the system

undergoes

_a

phase

transition which is not of the K-T type so that the standard RG

analysis [36]

_cannot be

applied

to the CCP. Recall that the Hamiltonian for CCP is given

by [15, 21]

H

=

g~ ~j

^In

[z,

_z~[ +

"~~~

~j ^[z,

[~

(4.2)

,<j

2

,

The combined _use of

equations (3.24)-(3.26) produces

_as well

a~

=

l,~j ^d~z,

^{1zi ~ exP}

^{1- ùl}

_,

^(4.3)

where

ù

_{is given}

by

ù

=

Î ₁

_in

_lz,

z/1

+

~/ _{i lzi}

l~

(4.4)

,<j ,

and Z is the

partition

function.

Comparison

between

equations (4.2)

and

(4.4) produces,

in view of

(1.8),

f

^'

~~'~~

and

g2

=

~

(4.6)

Combining

these two equations ^and

taking (1.7)

into account we obtam

~

=

7ra~ à

,

(4.7) g~

which

produces,

in view of

(1.10),

v=

~=~" _(4.8)

g ~°

Whence,

the requirement ^{of the overall}

electroneutrality permits

us to fix the value of

~tl with

help

^of

(4.8).

This

produces

the final form of the Hamiltoman which _{we are} going to use

ù

=

Z

in

lzi

_zj1 +

~/ _{i lzi}

l~

(4.9)

, <j

Notice

that, by

construction, Mm1 _so that the coefficient in front of the first _{term is}

~ 4. Denote _m

=

~

and consider the

partition

function Z for

arbitrary

non-negative ^{values of}

v m. We obtain

z

=

là _d~z

exP

(+

^m

1

in 1zi

zj1 ~/ _i zi1~~ _{(4. io)}

, i <j ,

As a side remark we mention that the

partition

function in the form given

by (4.10)

was

considered in the

theory

of random matrices

[37],

where the exact solution _was found

[38, 39]

for the

special

value of _m

= 2, and in the

theory

of

strings,

formulated _as random matrix models

[40],

for

arbitrary

_m. In the latter _{case no exact} solution _was found but the

Schwinger- Dyson equations

^which connect correlation functions of various

complexities

_were obtained and

anlysed. Alternatively,

m the

theory

of

QHE

the

partition

function Z _was considered from the point of view of the

theory

of

integral equations

for

liquids [41],

^while m reference

[15]

the extensive Monte Carlo calculations combined with _various

analytical approximations

_were

discussed. In _view of such

already existing approaches leading

towards approximate ^solution for Z, we have _{not to} go mto much of the formal details _m this paper thus

restricting

ourselves to several illustrative but

important examples.

Consider first the

exactly

^soluble case m = 2. Even

though

it _cannot be

directly

used for _our case, smce m is

arbitrary,

it is

physically

_very

illummating

and will

provide

_an interesting ^limit later _on.

Let

l~ni(Zl,

22, _, ZJ,~, Zl, 22,

., Z,,~) "

Î~J,i( (Zi)

,

(Z,)

)

~ COnSt eXp

(- H) (4,

ii

)

where H is given

by (4.9)

with ~

= m. The

n-point

correlation function _is defined _as

v

R~(z~,

,

z,,

,

Ii,

..,

f~)

=

~~~

ij

d~z~

P,,,(

_(z~ ;

(2~ ). (4,12

)

~~t '~~'

, =,I +

It is convenient to rescale z' _{s i z}

- z and 2

- l. This

rescaling

will contribute to

~2

w

Î2

w

the _{constant term m} equation

(4.11).

It _is important to realize that

[37]

fl Îz,

_zjÎ~

"

fl (zi

z~)

(f,

_2~

(4.131

1<j , ~j

and

_d~z l~

f' zJ e~

= 7r à

~~

j (4. 14)

It _can be shown

[37]

that

,,1-1 ~t

à,m(z, 1)

oz

Ri (z, 1)

= ar

' e- ^~

£ ^Î~ (4,15)

(-o

where #~~~ was defined in

(3.26).

In the

thermodynamic

hmit _n~

- m and

( ÎzÎ~~

_ ~+

lzl~

(4.16)

~~o

~!

The _use of

equations (3.25), (3.26), (4,15)

and

(4.16)

indicates

that,

for _m

=

2,

_we would have

a~

- m.

Hence,

for _m

=

2 the CCP model _is in _«

liquid

_» state. As m the _case of normal

liquids,

the

density Ri

_is

uniform,

_i-e- constant in the

thermodynamic

limit. The

similarity

with

the usual

liquids

ends if _we realize

(based

_on Monte Carlo and the approximate

anlytical

data

[15])

that for small

(1) degrees

of

filling (1.e,

for _v

-

0)

the CCP could form _a

crystal.

At

higher filling

fractions it _{is m a}

«liquid»

state. In

reference[15]

it _was shown that

crystallisation happens

for ~

~140

(or

_v ^~

=

0.0286).

v 140

To _see what

happens

to the

original

tube in the presence of

interactions,

it is instructive to

consider _a

simple

variational calculation

using

the

thermodynamic inequality [42]

_:

F _w

Fo

+

(ù _Ùo) (4.17)

Here the free energy F is defined _{as m}

(1.6)

while for the trial

Fo (or ùo)

the

non-mteracting

case is chosen.

Ho=(1-A)£[z~[~,

Am0.

(4.18)

It is _convement to rescale z-variables both in

(4,10)

^and

(4.18) by mtroducing

_w

=

~

/

Equation (4.10)

will

acquire

the

following

form

nj

Z

= const

fl ^d~w

_exp

(-

_n~

^à [w

,

(4.19)

i

where

à[w]

is given

by ù _lwl

=

1

_1W, ^~

) _z

_in

1W, WI1~ _,

(4.20)

, 1~

~~

and, accordingly,

_in

Ùo

we

replace [z,[~ by _[w~[~

Using equation (4.14),

_we obtain _:

~~

^{~~ ~~ ~ ~~ ~ ~~}

~ ^{~ ~ ~°~~~} l À

~°~~~

,

(4.21)

which leads _to the

following

result for

Fo.

n~

Fo

= n~ In (1 A + const'

(4.22)

For

(Ù Ùo)

,

we obtain _as well

o

(Ù _Ùo)

₌ ^{~~ ~} +

'~~~ In

(1

À

)

₊ const.

(4.23)

o À 4

~

Minimizmg,

_we obtain _:

1 [Fo

⁺

(Ù Ùo)

= 0

(4.24)

which is

equivalent

to the

equation

À mn~ 4 n~ n~

À 4 _v

'4

v

~~'~~~

This

produces

_:

nt(1

À n~

=

(4.26)

nt

~ v

For _n~ w we obtain _:

n~(1

À

w v

~4.27)

This

produces

for the average size of the tube result

~2 n

~~'~ ~ ~~~~

l À

~~

'

~~'~~~

i-e- for fixed

v and _n~

- cc we would obtain

again a(~~

- m in agreement with the result

(4.16).

The above calculations served

only

to illustrate how the interaction between tubes may

destroy

their existence. At the _same time, for _n~

- m the

partition

function

(4.19)

_can be

calculated with the

help

of the saddle

point

method. In _view of

(4.20), by

_minimizing

Ù

we obtain

respectively

1<~ ⁼

~

~j

,

(4.29)

n~ w~ w~

Ju #,>

and

w~ =

~

~j ^(4.30)

lli

j~ ~,

fil,

lij

To solve these

equations, multiply

equation

(4.29) by

_w~ while

(4.30) by

_fiJ~ ^and suppose ^that w~ =

Rexp(2 7rik/n~),

_{le- we assume} that solution represents

«particles

_»

sitting

_on the

ring [43]

of radius R in the

complex plane.

For this _{case we} obtain R~

=

~

~j

~t

J

e~"~'~'

~ J~~ m 2

(4.31)

"

~

₂ 2 _v ^'

where m the second fine we have used the sum rule given m reference

[43] (e,

_g. see

Appendix

of this reference and

Eq. (5)). Thus,

the ring

configuration always provides

_an extremum for

Ù[w,].

This does _not

imply,

that the solution which _we

just

^found is the

only

one. There _are other stable solutions which _are

briefly

described in references

[43, 44].

All other solutions _are obtamed

numerically. Using

equation

(4.31)

we can obtain _an average distance _r between the

particles

(1)

=

~ "~

=

~ _"

j~ (4.32)

nt nt "