Influence of chiral interactions on the winding and localization of a polymer chain on a rigid rod molecule

HAL Id: jpa-00246668

https://hal.archives-ouvertes.fr/jpa-00246668

Submitted on 1 Jan 1992

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Influence of chiral interactions on the winding and localization of a polymer chain on a rigid rod molecule

B. Houchmandzadeh, J. Lajzerowicz, M. Vallade

To cite this version:

B. Houchmandzadeh, J. Lajzerowicz, M. Vallade. Influence of chiral interactions on the winding and

localization of a polymer chain on a rigid rod molecule. Journal de Physique I, EDP Sciences, 1992,

2 (10), pp.1881-1888. �10.1051/jp1:1992250�. �jpa-00246668�

Classification

Physics

Abstracts

05.40 36.20 64.60

Influence of chiral interactions

_on

the winding and localization of

_a

polymer chain

_{on a}

rigid rod molecule

B.

Houchmandzadeh,

J.

Lajzerowicz

and M. Vallade

Laboratoire de

Spectrom£trie Physique,

Universitd

Joseph

Fourier (Grenoble I), BP 87, 38402 Saint-Martin-d'Hdres Cedex, France

(Received 9

April

1992,

accepted

in

final form

24 June J992)

R4sum4. On dtudie le

probldme

de l'enroulement d'une

longue

cha~ne de

polymdre

autour d'une borne

rigide,

_en relation _avec la transition de localisation

prdsentde

_par ce

systdme.

Le r61e des interactions chirales _et

polaires

est considdrd. Ces rdsultats _sont

compards

h des modbles

prdc£demment propos£s

_pour ddcrire

l'adsorption

d'un

polymdre

_et la transition

h£lice-pelote.

Abstract. The

problem

of winding of _a

long polymer

chain around _an attractive rigid rod is studied in relation with the localization transition exhibited

by

this system. ^{The role of chiral and}

polar

interactions is considered. The results _are

compared

with previous models of

adsorption

of _a

polymer

and of helix-coil transition.

1. Introduction.

As is well known, interactions between

long polymer

chains _can

give

rise to

cooperative phenomena

and _to

phase

transitions _: the helix-coil transition for double-stranded DNA is _one of the most celebrated

examples.

Several methods and models have been used _to describe these

phase

transitions and

they

have been reviewed

by Wiegel [1, 2].

In

particular,

this author

shows how the

problem

of

adsorption

of _a

polymer

on a surface _{or on a}

rigid

rod _can be

elegantly

solved

by

the _use of Wiener

path-integral

methods.

In _a

preceding

_paper

[3]

_we used this

technique

to

study

the

closely

related

problem

of localisation and

roughness

^of _a linear interface in _a 2-d semi-infinite

Ising

model. The results there

obtained,

were also extended to the

problem

of localisation of _a line _{near a}

straight

attractive rod in 3-d space. In the limit of _an

infinitely long line,

the system exhibits _a localisation transition _{at a} finite temperature

T~

^{in both} cases.

Thermodynamic

critical

properties

_are described

by

_{a power} law in the 2-d _case and

by

_an essential

singularity

in the 3-d

case. The

general

_case in d

=

I ₊

di (di

_m I

)

has

recently

been solved

by Lipowski [4].

The main purpose of the

present

paper is to

report

on an extension of the

preceding

model which focuses _on the

question

of

winding

of the chain around the rod and its relation _to the

JOURNAL DE PHYS>QUE -T 2. N' >0. OCTOBER >992 67

1882 JOURNAL DE PHYSIQUE I N° 10

phase

transition. It is

intuitively

clear that the

winding properties

cannot be

independent

of the localisation and

they

are

expected

_to

change drastically

at

T~.

^Edwards

[5]

and

Wiegel [I]

have shown how _to calculate the

winding

number

probability

distribution

by imposing topological

constraints _on the

path-integral.

We shall derive relations between

angular

and radial fluctuations of _a chain around _a rod in _a

slightly

different _manner. In addition, _we shall allow for

possible

^chiral and

polar interactions, having

in mind the

absorption

of _a helical and/or directed

polymer.

When such interactions _are present, ^the average

winding

number is _non-zero in the localized state and it decreases _{to a}

vanishingly

small value above

T~.

The critical

unwinding

as well _as other statistical fluctuations _are shown _to be

easily

calculated for _a

particular

model of interactions. This

problem

is

closely

related _to Brownian motions of _a

particle

in the presence of _a

magnetic

interaction, a

subject

of

continuing

interest

[6].

2.

Winding

of _a chain _{on a} rod in presence of chiral and

polar

interactions.

Let _us consider _a

long

chain made up of N identical segments of

equal length f.

The segments

are assumed _to be

freely orientable,

but this

hypothesis

is _not

essential,

since it _can be shown that any stiffness of the chain _can be taken into _account

by

_a redefinition of the

length

^f

(the

persistence length

is substituted for the segment

length) [7].

Self interactions of the chain with itself will be

neglected.

We shall _{use a} continuous

approximation,

_so that the index _n which labels the segments ^{will be considered} as the curvilinear abscissa

along

the chain. As usual

[8]

the statistical

properties

of the chain _are calculated

by asserting

that each segment at

position

R~ ^has a

potential

energy

U(R~)

« T

(T

is the temperature ⁱⁿ energy

units).

We consider _a

potential

_energy of the

following

form _:

U(R~)

= V

(R~) A~(R~)(&~

~,

&~) A~(R~)(Z~

~ j

Z~)

=

V(R~)-A~(R~)~~-A~(R~)~ _(l)

Where

R~,

&~, Z~ are

polar

coordinates of

R~. V(R~)

is _a scalar

potential

with

cylindrical

symmetry ; it is

infinitely repulsive

^for

R~

_- 0

(hard core)

and attractive at short distance. It is assumed to vanish

beyond

_a characteristic radius d. This

corresponds

to the presence of _an attractive

rigid

rod

along

the Oz axis.

A~(R~)

and

A~(R~)

describe the chiral and

polar

character of the

interaction, respectively.

One may

imagine,

for

example,

intermolecular Van der Waals interactions which

depend

on

the relative orientations of the molecules

[9].

In real

situations,

these interactions have _a

complicated dependence

_on all the coordinates. We shall make the drastic

assumption

of

neglecting

_any

dependence

_{on &~ and}

Z~,

I.e. _we shall _assume that chiral and

polar properties

are uniform and

isotropic along

the rod. One may have in mind, ^for instance, the _case of _an

helix-shaped

^rod.

(We remark, however,

that in this _case the interaction

depends only

on the relative

sign

of

(&~~,

_&~) and

(Z~~, -Z~)

and _{not on} the

particular sign

of

either.) Conversely,

one may consider the _case where the rod has full

cylindrical

_symmetry but the segments of the chain _are chiral and

point preferentially

ⁱⁿ a

given

Z direction

(like climbing plants growing along

the bars of _a

garden gate I).

^The

potentials A~

and A~ ^{have the} same short

range character _as

V,

since

they

arise from the intermolecular interaction. The components A~ and A _~, however,

might

also involve _a uniform part

corresponding

to an extemal field. For _a

sufficiently large

_A~, the

probability

for _a segment to

point

in _{a «} wrong » direction with respect

to A~ is very small and the chain becomes very

elongated along

the Oz axis. Its conformation

can then be described

using only

^the two coordinates

R(z)

and

&(z)

_: this

limiting

_{case was}

considered in reference

[3].

The

potential

_energy then involves _a term

A~(R)(d&/dz)

which favours _a helicoidal conformation. It is

interesting

to note that this term

plays exactly

the _same role _{as a} Lifshitz invariant in _{a two} component ^order

parameter system.

^{This invariant is known}

[10]

_to induce _an incommensurate modulation in

crystals.

In the present case it leads to _a helix with _a

pitch

incommensurate with the segment

length ^f.

In the

following

_we do not consider this uniform field component ⁱⁿ A~.

The

probability P~(R~, Ro)

of

finding

a chain with its ends at

Ro

^and

R~

at a

temperature

T

=

I/p,

_can be

expressed

_{as a}

path integral ill

N 3 jR ²

P~(R~, Ro)

=

fl~[R~]

_exp +

pU(R~))

^dn

⁽²⁾

o 2

f~

3n

P~(R~, Ro)

is the Green function of the

partial

derivative

equation

_:

a~i _R

~~

R

~

~&

~ ~~~

^~ ~

Z ~~~~~~j ~

~

~~ ~~~

~

x

P~(R~, Ro)

=

(R~ Ro) (N ) (3)

This

equation

is similar _{to a}

Schr6dinger equation (with imaginary time)

for _a

charged particle

in presence of scalar and _vector

potentials.

Let _us introduce the

generating

^function :

Z~(Ro, R~ r,

F

)

₌

le~ ~~

~~~~~° ~~~~°~

P~ (R~, Ro) d&~ dZ~ (4)

Where

&~o

₌

&~ &o, Z~o

₌

Z~ Zo (rotational

^and Z-translational invariance of

P~ (R~, Ro)

has been

used).

The transformation

(4)

^is a

Legendre

transform which introduces the

thermodynamic

variables r

(torque)

and F

(force along

the _z

axis) conjugate

to Band Z

N

respectively.

Let _us

emphasize

that

&~o= (d&/dn)dn

takes values from _-aJ to

o

+aJ

(&~o

=

2 arv +

4~o

where _v is the number of _tums and

4~o

the

angle

modulo

2

ar). Equation (4)

is _a bilateral

Laplace transformation,

the convergence of which

requires

a

sufficiently rapid

decrease of

P~

when

&~o

and

[Z~o

go to

infinity.

The

independence

of the

potential

_energy with respect to & and Z

actually

ensures this convergence, since these

variables behave

essentially

_as normal random variables. The transformation

(4)

avoids

using

the usual trick

[1, 5]

which consists in

imposing

a constraint _on the number of _tums

v in

calculating

the

path integral (Eq. (2)).

This method leads to _a calculation of the Fourier transformation of

P~

^with respect to _v instead of the

Laplace

transformation with respect to

&~o. ^{Both methods} are

equivalent

_{as soon} as the results can be

analytically

continued from the

imaginary

to the real axis in the

complex

r

plane,

which is

actually possible

in our case. The

Laplace

transformation will be

preferred

both for its

physical clarity (as Legendre thermodyna-

mic

transformation)

and for mathematical

simplicity (the

Hamiltonian-like

operator appearing

in

equation (3)

after the transformation

(4)

is

Hermitian,

_even in the presence of _{a « vector-}

potential »). Z~

_can be

expanded

with the

help

of the

eigenvalues

_A~ and

eigenfunctions

~~(R)

of the

equation

_:

~- j(

(R (

+

p2(

^~

^{°~~~ ~}

⁺

^p2(F _-A;(R))2j

⁺

pv(R)) ^~~(R)

⁼

= A~

~~(R)

_x

x

°~ ~z(R) ~~(R)R

dR

= &~~

(5)

o

zN (RN, R01 r,

F

"

£ ~l(R0) ~n(RN)

^{e~ ~~~}

(6)

1884 JOURNAL DE PHYSIQUE I N° 10

The _moments of the random variables

&~o

^and

Z~o

^for fixed

Ro

and

arbitrary R~

_can then be deduced from the characteristic function _:

f~(r,

F

)

=

lm ^{Z~ (R~, Ro r,}

^F

^{R~ dR~}

o

~ m

Z~ (R~, Ro

_;

0,

0

R~ dR~

o

= p

(r ( &~o)

+ F

(Z~o)

₊

P ~(r~( _&(o)

₊

^F~ (Z(o)

₊ 2 Fr

( _&~o Z~o)

₊

(7)

2

When

equation (5)

_{possesses a}

non-degenerate ground

state

eigenvalue

A

o,

Z~

is dominated

by

the

corre8ponding

contribution in

equation (7) (in

the

thermodynamic

limit N

-

aJ).

Then _:

Log f~ (r,

F

= N

(Ao(r,

F A

o

(o,

o

)) (8)

The average

winding

^number

(&~o)

and the average extension of the chain

along

the _z axis

(Z~o)

^take on the

simple

forms _:

(8N0) ~~ ^{0 ~~ ~~} _~~~~ ~~

~~

jz~~)

=

) )

)~

^"

^~~

₃

^{~~°~~~~~~~ ~~}

°

lm

where

( ~o

O

~o)

~ means

~? (R

O

~o (R

R dR with r

=

F

=

0.

o

The fluctuations _can be calculated in the _same way :

iAf+<o) i Iii ^~

f2 A~(R) a~o

~~

3

°~

R~ ~'°

o

~~~

~°

R~

31~

~

~~~~~

~

~~~~

~~ll° ~~~°> jl)j~ij~~~~~

no ^~

₊

^~oi ~ii~ ^ii° _ioi ~~°~~

We _note that all these

quantities

_are

proportional

to N.

The derivatives of

~o appearing

in

equation ( lo)

can be evaluated

by perturbation theory

_:

~~~ ~~j ^2A~(R) ~j

i)o~~

A~-~o

^~

3~o

#~n)

l~n _l~ ~z(~)l _#~0)0

W 0~~ _~n~~0

The fluctuations _are thus found to contain

negative

contributions that _are

quadratic

in the

vector-potential

components. ^The cross correlation

(Eq. (10c))

is

directly

related to a rotation induced

by

_an

applied

force

along

_z

(or

to an

elongation

induced

by

_{a torque} :«

piezogyration

_»

effect).

In order _to derive

explicit expressions

for these

quantities,

let _us consider _a

potential

_energy

defined _as follows _:

~2 (2

V

(R)

= j

Vo

6R

A~(R)=G

for R~d

(11)

A,(R)

₌ A

and

U(R)

= 0 for R ~d

(see Fig. I).

v

o d R

Fig.

I. Model

potential

used to describe the rod-chain interaction.

One notes that in this _case, the

only

role of the _« vector

potential

_» is _to

change

r into

r G and F into F

A,

for R

~ d.

Equation (5)

_can then be

easily

solved. A localised state

corresponding

to a

non-degenerate ground-state

with

negative eigenvalue _Ao

exists

only

for

temperatures

^{less than} a critical

temperature T~ [3]. T~

is found _to be the solution of the

equation Po(T~)

=

2

B(T~)

with

Po

₌

Pvo j

+

(PAd)~

and a

=

(PG)~ (12)

For r

=

F

=

0,

_Ao is

given by

:

Ao= -~(~exp(-~ ⁽¹³⁾

d t

with t ₌

° B _oz

(T~ T)/T~

1886 JOURNAL DE

PHYSIQUE

I N° lo

By expanding Ao(r,

F up to 2nd order in r and

F,

_one

has,

for t «1 _:

(&~o) =Ng

^~ 2/i

t~

~- ^2/i

(Z~o)

=

Nfa

t~

~- ^2/i

(A&(o)

₌ ^N

(1

6

g~)

j

t

j ~z2

^{fi~~~ ~}

~~

~~_~ ^~~'

N0 ~

(2

_~3

(h8No _AZNO)

~

Niga

^~

_(~

t

(14)

with g =

P~

and _a

=

p j.

c

For T _~

T~,

_no

negative eigenvalue

of

(6) exists,

and the chain is delocalised. To calculate its statistical

properties,

_one has to sum all the contributions from the continuous

spectrum

ⁱⁿ

equations (6), (7).

To

leading

order in the limit N

- aJ, one

finds,

for

Ro

_~ ^d :

~~N0) "~~°~ )

_~~~

_{~~N0) ~~~~°~} )

For

Ro

_~

d, &~o)

^and

(Z~o)

_vary as a power of

d/Ro,

but

they

_are also

independent

of N and

so of order I/N when

compared

with their values for

T~TR.

The

angular

fluctuation

(A &(o)

is found to be

=

Log

~

,

also much smaller than for T _~

T~.

This last result

B

Ro

is in agreement ^{with that found}

by

Desbois et al.

[I I]

in their calculation of _a 2-d Brownian

particle

with _a

repulsive

_core

potential

=

C~/R~. _[As

_noted

by

several authors

[I, I1, 12],

the

presence of such _{a core}

potential plays

_an essential

role,

since without

it,

& would be _a Lorentzian variable

(Spitzer's law) [11],

and all its moments would be infinite. This is in agreement with our results which show

divergences

for F _-

o-J

The delocalisation of the chain _at

T~, although being

a true

phase

^transition in the limit N

- aJ

[13],

is in fact very smooth,

owing

to the very strong decrease of _e~ ^~~~ when t goes to

zero. This _means that the delocalisation process

begins

well below

T~

and finite size effects _are

expected

to be

important

near

T~.

the

approximation

of

ground-state

dominance for

T ~

T~

_{ceases to} be _correct when N e~ ^~~~ ₌

l,

I-e- in _a

temperature

_range t

=

2/Log

N. In this

region

there is _a continuous _cross-over from _a localised state to a delocalised state.

Owing

to the

logarithmic

law, finite size effects _are

expected

to be

important

in _a

relatively large

temperature range

(in comparison

with the usual

power-law

critical

behaviour).

3. Discussion and conclusion.

The model discussed above shows how chiral and

polar

interactions _can be

incorporated

into the mechanism of

adsorption

of _a chain _{on a}

rigid

rod and it allows _{one to} deduce the role

they play

in this

phase

transformation.

First of

all,

_{one can see} that

they

favour

localisation,

since

they

increase the transition temperature

T~ (the

effective

repulsive

_constant

C~ PG~

is less than

C~

_{and the effective}

depth

of the attractive

potential _Vo

₊

PA~ i16

_is

larger

than

Vo).

^As is obvious from the

path

integral

formulation this effect is

quite analogous

to the

localising

effect of _a

magnetic

_vector

potential

on a

charged particle.

Furthermore these interactions which _are

responsible

for _{a non-zero}

winding angle

and _an extension

along

the _z

axis,

in the localised state,

play

_a

negligible

role above

T~

where

entropic

effects dominate _:

unwinding

and

depolarisation happen

as the delocalisation process takes

place.

Conceming

the essential

singularity

that characterises the critical behaviour of the system, ^it is

symptomatic

of _«

marginality

_». The _same kind of behaviour _was

already

^found

by

Rubin

[14] (although

not obvious in his

complicated

mathematical

expressions,

the presence of _an

essential

singularity

is clear

through

the behaviour of the number of adsorbed _monomers, which exhibits

vanishing

derivatives of any order _at T

=

T~).

It is also

interesting

_{to compare}

our results with the Poland and

Sheraga

model of the helix-coil transition

[13].

First of

all,

_one

must remark that chiral interactions

play

no role in their

model,

in

spite

of the helical

shape

of

DNA molecules

they

are

describing. They only

^consider «

sticking

» energy between the

chains which is

equivalent

to our scalar

potential V(R).

Their model leads to a

power-law

critical behaviour with _an exponent

P

=

~ ~~~

for the number of ordered segments. ^For d/2

d

=

3,

this is _at variance with _our essential

singularity.

This difference reflects _a real difference between the two

models,

and it _can be understood in the

following

_way. In Poland's model the entropy ^of a disordered part of the double chain is calculated _as

being

that of _a random walk _on

a closed

circuit,

whereas in _our

problem,

where _one chain is _a

rigid-rod,

the entropy ^of a

disordered part ^{of the chain is that of} a random walk with both ends

anywhere

_on the _z axis.

One _can

conjecture

that _our model

corresponds

to an effective dimension d

= 2

(I.e.

_to

P

=

aJ)

in the Poland's model.

As noted

by

Fisher

[15],

self

avoiding

walks, ^which have been

neglected throughout

this paper, are relevant in the delocalised _state. This constraint makes the delocalization _more

difficult,

and it lowers the effective

dimensionality

in the

expression

of P. One can expect that in _our model it will lead to a

change

from _an essential

singularity

to a

power-law,

but with _a

rather

high

value of

p.

A

possible application

of the present ^model concerns the

problem

of

linking

of several

parallel

rods

by

a chiral

chain,

which winds around them. It would also be relevant in

studying polymer crystallisation.

Acknowledgments.

The authors would like _to

acknowledge

A. Comtet and J. Desbois for fruitful discussions.

References

[1] WIEGEL F. W., Introduction _to path

Integral

Methods in Physics and

polymer

Science (World Scientific, 1986).

[2] WIEGEL F. W., Conformal Phase Transition in _a macromolecule, Phase Transition and critical phenomena, ^{vol. 7} (Domb et Green Eds, Academic Press, 1983).

[3] VALLADE M. and LAJzEROWICz J., J. Phys. France 42 (1981) 1505.

[4] LIPOWSKY R., Europhys. Lett. 15 (1991) 703.

[5] EDWARDS S. F., Proc. Phys. Soc. 91(1967) 513.

[6] ANTOINE M., COMTET A., DESBOIS J, and OUVRY S., J.

Phys.

A Math. Gen. 24 (1991) 2581.

[7] KLEINERT H., Path

Integrals

in Quantum mechanics and

polymer physics

(World Scientific, 1990).

[8] _DE GENNES P. G.,

Scaling Concepts

in

Polymer physics

(Comell

University

Press, 1979).

1888 JOURNAL DE

PHYSIQUE

I N° 10

19] KIHARA T., Intermolecular Forces (John Wiley & Sons, 1976).

[10] R. Blinc and A. P.

Levanyuk

Eds., Incommensurate

phases

in dielectrics, Modem

problems

ⁱⁿ

condensed matter sciences series, Vol. 14 (North Holland, Amsterdam, 1986).

ill] DESBOIS J., J.

Phys.

A Math. Gen. 23 (1990) 3099.

[12] RUDNICK J. and YUMING H., J.

Phys.

A 20 (1987) 4421.

l13] POLAND D. and SHERAGA H. A., J. Chem. Phys. 45 (1966) 1464.

[14] RUBIN R. J., J. Chem.

Phys.

44 (1966) 2130.

lls] FISHER M. E.. J. Chem. Phys. 45 (1966) 1469.