Universal shape properties of open and closed polymer chains: renormalization group analysis and Monte Carlo experiments

HAL Id: jpa-00246699

https://hal.archives-ouvertes.fr/jpa-00246699

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.

Universal shape properties of open and closed polymer chains: renormalization group analysis and Monte Carlo

experiments

O. Jagodzinski, E. Eisenriegler, K. Kremer

To cite this version:

O. Jagodzinski, E. Eisenriegler, K. Kremer. Universal shape properties of open and closed polymer

chains: renormalization group analysis and Monte Carlo experiments. Journal de Physique I, EDP

Sciences, 1992, 2 (12), pp.2243-2279. �10.1051/jp1:1992279�. �jpa-00246699�

J. Phys. I France 2

(1992)

2243-2279 DECEMBER1992, PAGE 2243

Oassification Physics Abstracts

36.20E 64.60A 82.20W

Universal shape properties of open and closed polymer chains:

renormalization _group analysis and Monte Carlo experiments

O.

Jagodzinski(~),

E.

Eisenriegler(~)

and K.

Kremer(~)

(~) Fachbereich Physik, Universitit Gesamthochschule Essen, Postfach 103764, D-4300 Essen I, Germany

(~) ^Institut fir

Festk6rperforschung, Forschungszentrum

Jfilich, Postfach 1913, D-5170 Jfilich, Germany

(Received

2 December 1991, revised 28 August 1992, accepted 2 September

1992)

Abstract. We investigate the influence of excluded-volume interaction

(EV)

_on the shape

of _a long flexible polymer chain. Both _open chains and ring polymers _are considered. We study probability distributions of shape parameters which _are typically ratios of characteristic lengths

(such

_as the principal radii of

gyration)

of _a given conformation. For _a class of shape parameters

such _as the 'asphericity' Ad of _a chain in d space-dimensions it is shown how _mean values _or higher moments of the distributions _can be evaluated by field theoretic renormalization _group

methods. The universality of these distributions is shown and the _mean asphericity

(Ad)

is calculated within

an e = 4 d expansion.

(Ad)

is found to be much _more sensitive to the EV than _a frequently used asphericity-approximant which avoids the ratic-averaging. This is the first analytical confirmation of _a result observed by other _groups in numerical simulations.

We also investigate the complete distribution of A3 and of another

(prolate

_vs,

oblate)

shape parameter ^{in d} = 3 by _means of Monte Carlo simulations. The dependence on chain length is

carefully

investigated.

This improves the _accuracy of previous estimates of universal asymptotic shape distributions. Generally the EV makes the shape _more aspherical and prolate.

Comparing

quantitatively the increase in

(A3)

due to the EV _as implied by the Monte Carlo data with that by the

(appropriately extrapolated)

first order e-expansion _one finds good

[fair]

agreement in

case of ring polymers

[open chains].

1. Introduction.

The

typical

_gross

shape

of

polymer

chain conformations is _not

spherical

_{[1, 2].} This

asphericity

seems to

play

_an

important

role for flow

properties

of

polymeric

fluids

[3]-[6].

^{It must also be}

taken into account if

polymer

solution

properties

such _as the second virial coefficient _are calculated

by

_means ^of

phenomenological

'smoothed

density'

theories [7]. ^{Other random fractal}

objects

such _as the clusters

resulting

from kinetic

aggregation

_or

percolation

_processes

show also

non-spherical shapes [8]-[10].

2244 JOURNAL DE PHYSIQUE I N°12

Consider _a

specified

conformation of N

points

in d-dimensional _space

[representing

either

the _monomer

positions

in _a

polymer chain,

_or the

occupied

sites in _a

percolation cluster, etc.]

with

corresponding position

vectors

Rj

_"

(X;,I>

X;,2>

>X;,d)

_{) "}

I,.

,

N

Ii)

The

shape

of this conformation _can be

conveniently

characterized

by

the d

eigenvalues

_qa of the radius of

gyration

tensor [2, 11, 12] ^{with elements}

N

Qop

₌

~ £ _{(x;,a x;,aj (x;,p x;,pj (2)}

;,;=1

The usual _square radius of

gyration RI

is

given by

the trace of

Q

RI

d

=

L

qo ⁺

d# (3)

«=1

and characterizes the linear overall extension _or size of the conformation. To characterize the

shape

of the conformation _one has _to consider ratios of

eigenvalues.

The

simplest

such

quantity

is the

'asphericity' ill,

_13]

~

~~ =

~~~ _~~~

did -1) j2 ~)

Note that Ad vanishes for conformations with all

eigenvalues

_{qo =} j

equal

and takes _a maximum [13] ^{value of I for} a

(completely collinear)

conformation where all

eigenvalues

vanish except

one.

Ad

characterizes the

shape's

overall deviation from

spherical

symmetry. ^Another

simple

ratio [13]

~~ ^{~~~ ~~}

~)#)~~~

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

distinguishes prolate

from oblate

shapes

in d

= 3. Positive

(negative)

values

belong

to

prolate (oblate) shape [with

_one

eigenvalue greater (smaller)

than and the other _two smaller

(greater)

than

ii. 53

is bounded [13] ^{to the interval}

[-~,2].

The value 2

(-~)

_{is taken when two}

4 4

eigenvalues

^vanish

(one eigenvalue

vanishes and the other two _are

equal).

The

probability

distribution of the conformations

(R;)

in

(I) depends,

of _{course, on} the system under consideration. Of

particular

interest _are critical systems such _as

long

flexible

polymers, percolation

clusters _near the

threshold,

etc, where correlations exist _{over macro-}

scopic lengths.

These systems show the remarkable

phenomenon

of

universality.

This _means that many of their

large

scale

properties [which

_are then called 'universal

properties']

_are inde-

pendent

of most details of the

microscopic

interactions. As _a consequence one may use

simple

models for _a

quantitative

calculation of universal

properties.

In the present context it is natural to look for

shape properties

which _are universal

[13].

We _are interested in linear

polymers,

both _open chains

(oc)

and

ring polymers (rp).

As _we shall discuss below the _mean values of

Ad

and

53 land

_even the

corresponding probability distributions]

_are universal in this

case

provided

the number N of links

(or monomers)

in the flexible chain tends to

infinity.

This _means in

particular

that for

polymer

lattice models in d

= 3, _say, the universal shape

properties

_are

independent

of the lattice structure. Here the so-called

polymer-magnet-analogy [14,

15] ^is

extremely

useful since

universality

in critical

N°12 UNIVERSAL SHAPE PROPERTIES OF POLYMER CHAINS 2245

magnetic

systems or 'field theories'has been

investigated

in great ^{detail. Similar}

analogies

_are

useful in _case of

percolation

clusters _or lattice animals [9].

Polymers

with and without excluded-volume interaction

belong

to dilTerent

universality

classes

[15], [16].

The main _concern of the present _paper ^is to

quantitatively

estimate the influence of excluded volume _on universal

shape properties

both

analytically by

_means of the renormalization group and

by

numerical simulations.

We

investigate (for

the first time

analytically)

the _mean

asphericity

(Ad) (6)

in _presence of excluded-volume interaction. Here _{we use a} method to average ratios such

as Ad _or

53,

which has been

proposed by

Diehl and

Eisenriegler [17].

^{These authors have} obtained exact results for

arbitrary

d of the

corresponding quantity (Ad)o

^{for random walk}

chains. Previous

analytical

treatments

ill, 13,

18] ^{avoided the}

ratio-averaging

and evaluated

a

quantity

j

^~~~

^~~~

_(~~

~ (~~)

which

is,

of _course, dilTerent from the _mean

asphericity (6). Contrary

to

(Ad)

the

quantity

Id

has _no direct relation to the

probability

distribution of the

asphericity

parameter Ad- It is these

shape-distributions, however,

which

provide

the clearest information. We

investigate

analytically

how

(Ad) begins

to deviate from

(Ad)

_o ^{when the} _space ^{dimension d becomes}

slightly

less than the

'upper

critical dimension' d

= 4

[above

which the excluded-volume interaction is irrelevant

[15,

16] ^and

(Ad)

becomes

equal

to

(Ad)o

as N _-

cc].

An estimate for

(A3)

is obtained

by appropriate extrapolation.

We find that

(Ad)

is much _more sensitive to the excluded-volume interaction than the

quantity Id-

_{This is in line with}

_previous

_numerical

estimates

by Bishop

and Saltiel

jig]

and

by Cannon,

Aronovitz and Goldbart [20].

We also calculate

exactly

the _average

(53)o

^{of the}

prolate /oblate shape

parameter

53

for

rp's

and oc's without excluded-volume interaction. Let _us mention that the methods of reference [17] ^{and of the} present _paper could be used to average besides Ad and 53 also other ratios of

interest in

polymer physics.

One

example is,

of _course, that of the _squares of the end to end distance and the

gyration

radius. Another _one, for _a chain

grafted

to _a

plane surface,

is that

of the _squares of

appropriate gyration

tensor components,

parallel

and

perpendicular

to the

surface.

To

investigate

universal

shape properties

we have also

performed

Monte Carlo

(MC)

sim- ulations for

polymer

_on lattice models in d

= 3. This allows to compare with _our

analytical (renormalization group)

estimates of

(A3)

^for

^rp's

^{and oc's. Besides the} mean

value,

_{one may}

extract from MC-data the

complete probability

distribution for _a

given shape

parameter

[21].

These distributions _are not

sharply peaked

for N _{- cc} and show _a non-trivial universal form which

depends strongly

_on the _space dimension

[24].

^In

higher

dimensions

id

=

4, 5)

the

most

probable asphericity (Ad)mp

^{where the}

(broad)

distribution attains its maximum

is _nonzero and

roughly equal

to

(Ad)-

^{In d} =

2, however, (A2)mp

^and

(A2)

_can be

drastically

dilTerent.

E-g-

for

random-walk-rp's (A2)mp

_" 0 while

(A2)

_" 0.28

[25].

^In this _{paper we} shall

concentrate _on the distributions of

A3

and

53.

As for _open chains these distributions have

been also addressed in reference [20] ^{and reference} [24]. ^For

ring polymers they have,

to our

knowledge,

not been considered before. We extend

previous

Monte Carlo work

by carefully investigating

the

dependence

_on chain

length

N in order to arrive at more accurate numerical

estimates for the universal N _{- cc}

properties.

We close the introduction with _a

guide

to the rest of the _paper. The

analytical approach

to evaluate

(Ad)

will

proceed

in the usual _way

by combining

the

expansion

with respect to

2246 JOURNAL DE PHYSIQUE I N°12

the excluded volume

(EV)

with the renormalization _group

[16].

^{In subsection 2.I} we discuss the random walk

(or

ideal _or Gaussian

-)

chain situation which is the

starting point

of the

expansion.

Here _we

explain

the method [17] ^of

ratio-averaging,

This leads _to somewhat

unusual Gaussian _mean values and propagators [17] ^{which will} appear

again

^later _as

building

blocks in the

perturbation expansion.

We also take here the

opportunity

to calculate

(53)o.

Details of this calculation _are

given

in

Appendix

A.

The influence of EV-interaction _on

(Ad)

^is

investigated

in subsection 2.2 which is divided into two parts. In the first part we discuss the

general

structure of the renormalization _group and show that the distributions of

shape

_parameters such _as

Ad

_are universal. Some details of this discussion have been deferred to

Appendix

B. In the second part we determine the form of the first order in EV contribution to

(Ad),

evaluate the universal result for

(Ad) /(Ad)o

to

first order in

e = 4 d

(8)

and present _an estimate for

(A3).

The necessary

explicit

numbers _are extracted from the first order

expressions

in

Appendix

C. There _we also

verify

that these expressions

obey

the

general

renormalization structure.

In section 3 _we describe _our Monte Carlo simulation and present ^{results for}

distributions,

mean values and relative variances of A3 and

53

for oc's and

rp's

with EV-interaction. For

comparison

_we present also the

A3

distribution for oc's without EV, A _summary of _our results is

given

in section 4.

2.

Analytical

treatment.

2.I IDEAL (RANDOM WALK) CHAIN. Consider _a

simple polymer

model where the N

points

R;

in

(I)

form _a harmonic chain with _nearest

neighbour interaction,

I-e- with _a Hamiltonian

d

7iRW(Rj)

"

£ _{hRW(X;,a) (9)}

a=1

~~~~~

hRW

(~j ^~2 ~ _~'

~~ ^~

^~ _~~~~

j"2(J)

The summation _over

j

starts at

j

₌ 2 if the chain is _open, and at

j

₌ I if it is closed. In this latter

case one must

interpret

^the variable Xo

occuring

in the

j

₌ I term _as XN. The Gaussian model

(9), (10)

is _one of the

simplest

ⁱⁿ the

universality

class of ideal

(or

random

walk, RW)

chains where true many

body interactions,

such _as the

EV-interaction,

_are

disregarded.

The ratios

(4)

and

(5)

_can be written _as [13]

and

53 ₌

27~lll)~ ⁽¹²⁾

respectively.

Here ~j means the traceless tensor

ij

"

Q ~~~

ld

(13)

N°12 UNIVERSAL SHAPE PROPERTIES OF POLYMER CHAINS 2247

and the definition

(3) off

has been used. Note that numerator and denominator in

(11)

and

(12)

have _a

simple (polynomial)

form in terms of the basic variables R _or X. This would be

more

complicated

for other

interesting shape

parameters such _as the ratio

qm;n/qmax

of the minimal and maximal

eigenvalue

of

Q.

In order to average the ratios

II), (12)

it is

advantageous

to

exponentiate

the denominators

[17]

by

_means of the

identity

(~~Q)~~

_"

~~

_~~j

/~dV V~~~ ~~~~~~ (~~)

and to rewrite the

ensuing

_average as

1' _(Qi

_~~

~~~io

^-

i' _IQ)

io,~ l~~

^~~~

_{lo (15)}

where

F(~j)

is

tr(~j2)

_or

det~j

_in

case of

(11)

_or

(12).

Here

)o

^{denotes the}

original

_average

over

(R)

conformations with respect to the Hamiltonian

~Rw

in

(9)

while )o,~ ^is an average with respect to _a formal Hamiltonian

~Rw,~

_" ~Rw + vtr

Q (16)

From the form

(2)

of _one concludes that

~Rw,~

has

again

the form

(9)

with hRw

replaced by

N ~

hRw

~

(X; )

₌ hRw

(X; )

₊ ^~

~

£ _{(Xi X;) (17)}

' 2N

1,j=J

The

subscript

0 _on the above averages indicates that _we still

disregard

the excluded-volume interaction. Since both ~Rw and

~Rw,y

_are

quadratic

forms in the basic variables R _or

X,

the above _{averages can} be

easily performed,

_e-g-

by introducing

normal coordinates. In the

continuum limit t

-

0,

N

- cc with

L ₌

Nt~ (18)

~~~~ _~~~ ~~~~ ~~~

(e~~ _~~~)$~~ sin~~~/2)~

^~

~~~~

~~

(e~

^~~

~)i~~

_"

sint(fl)1

^~~~

~~~~

for

ring polymers

and _open

chains, respectively,

where

fl

₌

2/j. ₍₂₁₎

Note that the

quantities (19), (20)

_are

closely

related to the distribution functions for the square radius of

gyration

of random walk chains

[26, 27].

^The other _average

(F(4))o,~

_can be

expressed

in terms of the propagator

G>,>, =

lx> xii lxj, xii _)~ (22)

by applying

Wick's theorem. The continuum limit

G(t,t')

= lim

Gj=i/ia,j,=t,/t2 (23)

2248 JOURNAL DE PHYSIQUE 1°12

of the propagator ^reads [17]

G~»~(tt~)

=

p

inl~p/~~ C°Sh fl II

^'~

_i~")I

C°Sh

fl II I)I

cosh

fl ~

~)j

⁺

^{cosh(fl/2) (24)}

2 L

and

cosh

fl l

)j

^cosh

^{fl l} _~')

+

osh(fl))

L L

(25)

Of _course, t and t' both _vary in the interval [0,

L].

This formalism has been used [17] to evaluate

(Ad)o,

the _mean

asphericity

of random walk chains. Here _we calculate the

corresponding

_mean value

(53)o

of the

prolate/oblate

_parameter

(5).

In

Appendix

A _we show how

(det ij)o,~

_can be

expressed

in terms of the propagator

G(t,t'). Using (14)

with _m

= 3 and

(19), (20)

one finds

lS3lo

=

/~°dflz(fl) ⁽²⁶⁾

with

~~~~~(fl) =

~°~

p2

~°~~(~fl)

6

cosh~)p)

~ j

~~°Sh(fl)

lo

~fl Cosh(3fl/2)

~~~~~~ ~ ~ ~

+2fl~ cosh(fl/2)

+ 6fl~

sinh(fl/2) 3fl cosh(fl/2)

+ 48

sinh(fl/2)j ⁽²⁷⁾

and

/

~~~~~~~~

~~ _cosh(4fl) -~~~(2fl)

+ 3

(~~

^{~°~~~~~~ ~~~~~~~~~ ~}

~~

_{~°~~~~~ ~}

+12fl~ sinh(fl) 3fl cosh(fl)

+ 24

sinh(fl)j ⁽²⁸⁾

Numerical

integration yields

the final results

j~~jjoc)

= 0.475,

which show that the _average form of open as well _as closed Gaussian chains is

prolate.

2.2 CHAIN WITH EXCLUDED-VOLUME INTERACTION. To describe _a

polymer

chain in _a

good

solvent _we have to add _a

repulsive

interaction V between _monomers to the Hamiltonian ~Rw.

N°12 UNIVERSAL SHAPE PROPERTIES OF POLYAIER CHAINS 2249

Due to

universality

there is _some freedom _{as to} the form of V. For

analytical

calculations it is most convenient to _{use a} continuum model with the form

v

jRj

₌

j _f~ dial'&d (R(i) R(i')j (30)

of V. The

perturbation theory

of the continuum model is well defined if the

occuring

_unper-

turbed chain

partition

functions contain

only

'monomer

numbers'larger

than _some

microscopic

cutolT

[16]. Actually

_we shall

adopt

dimensional

regularization

_[16] instead which is _{more con-} venient.

In the

following

_we shall

study

how the _mean value

(Ad)

^{of the}

asphericity ill)

is influenced

by

the excluded-volume interaction.

Using

the

identity (14)

^with m = 2 _one finds

(Ad

_" ^~

/ _dy

y

(tr _(ij~)) (e~~

^~~Q

(31)

d I

~

~

Here the second _mean value _on the rhs has to be calculated with the usual Hamiltonian

~ ₌ ~Rw + V

(see Eqs. (9), (10), (30))

while the first _mean value involves

~~

=

~Rw,~

+ V

(see Eqs. (16), (17)).

2.2,I Renormalization and

universality. Straightforward perturbation theory

in _u is useless for

long

chains. The _reason is that in

spatial

dimensions d < 4

increasing

_{u powers} get

multiplied by increasing positive

_powers of the chain

length

L. One _may circumvent this

problem by

_means of the renormalization group

(RG),

which _{maps a}

long

onto _a short chain.

How to set up the RG for the above-mentioned continuum model is well known

[16].

^Since

the statistics of

polymer

conformations is

closely

related [14] to a

(Landau-Ginzburg type)

field

theory,

one may use field theoretic renormalization

[28, 29].

Here _one locates the ultraviolet

(UV) divergencies

of the

theory (which

_appear, within dimensional

regularization,

as

pole

terms in _e

= 4

d)

^{and absorbs them} into _a

reparametrization.

We discuss in

Appendix

B how the

reparametrization

for the _{two mean} values _on the rhs of

equation (31)

can be obtained. The result is

(tr _(~j~))

₌

~~~

Fj

L~,u~,

(32)

~ p

and

(e~~

_{~~Q) =} F2

L~,

u~, ₍₃₃₎

~ Here

L =

~~~ (Zt) ^{L~ (34)}

and

u =

16x~f(e)~~Zu ^u~ (35)

with _{~ an}

arbitrary

inverse

length

scale. We will not

specify

the

e-dependence

of

f(e)

= I +

Eli

+ e~

f2

+

(36)

beyond f(0)

= 1, ^{since it} must

drop

out in

(a systematic e-expansion of)

universal

quantities.

The Fi in

equations (32)

and

(33)

_are dimensionless functions and _are

finite

for _e

-

0,

if their

arguments

are

kept

fixed. All

e-poles

_are contained in the factors

Zg = I +

e~~Zj~) (u~)

+

e~~Zj~) (u~)

+

(37)

g # t, U

2250 JOURNAL DE PHYSIQUE I N°12

ofthe

reparametrization.

These _are indentical with the

corresponding

factors ofa n-component

#~ ^field

theory

_{as n -} 0. For later _{use we} note the

explicit expressions

Zt ₌ I +

(

+ O

(u~)

^~

Zu ₌ +

~

+ O

(u~)

^~

(38)

The arbitrariness _{of p} leads to renormalization flow

properties

of the

Fj,2.

^This is shown in

Appendix

B and reads

Fi

L~,u~, Fill~(I),fi~(A),

~

~

e~~'~ (39)

~

(e-Ap)

with _a free

parameter

^I. Here

Ej = -4

,

E2 _" 0

(40)

and fi~

,

£~ are determined

by

(~iR(>)

=

-fl(a~(>))

,

fi~(o)

₌

^uR (41)

and

£~(l)

~

=

L~

exp

/

_dA'

1l(fi~(1')) (42)

o

The functions

fl

and 1l _are related to the factors

Zg

ⁱⁿ

equation (37),

_see

Appendix

B. For

e > 0 the function

fl

=

fl(u~)

has _a

non-vanishing

_zero

u~

=

(~

= e + O

(e~)

,

(43)

which is _an attractive fixed

point

for

positive

fi~

as I _{- cc.}

Equation (39)

_can be used to _{map a}

long

chain

L~

- cc onto _a short chain

£~

= l. This

requires

_{- cc}

since1l((R)

is

positive. Quantitatively,

increases with

LR (in

the

asymptotic

limit

L~

-

cc)

_as

e~ -

[D (u~) L~)~

,

(44)

where

v = =

(I

⁺

)j

^{+ O}

(e~) (45)

"( "~)

and D is

given

in

equation (I II).

Thus in this limit

Fi

L~,u~, _-~bi

e~~

e~~'~ (46)

P ~

with I from

equation (44)

and

~b;(Y)

= l§

(1,(~, _Y) (47)

We _{may now} obtain the renormalization behaviour of the _mean

asphericity (Ad).

From

equations (31),(32),(33)

it follows that

(Ad)

_"

) /

_dz

z

Fi (L~, u~, z)

F2

(L~, u~, z)

_{=: a}

(L~, u~) (48)

~

N°12 UNIVERSAL SHAPE PROPERTIES OF POLYMER CHAINS 2251

is _a function of

L~, u~,

which remains finite _{as e}

- 0.

Equations (40), (44), (46)

and

(47)

tell

us that in the

asymptotic

limit L~

- cc

lAd)

= a

I, uR

,

u~

_{> 0}

(49)

~

Equation (49)

shows the

universality

^of

(Ad).

Its value for

long enough

chains is

independent

of the

strength

^u~ > 0

(I.e.

of _{u >}

0)

^{of the} excluded-volume interaction. It does of _course

depend

on whether _we consider _{an open}

polymer

chain _{or a}

ring polymer (For given

arguments ^the functions

Fi,~bi,

_a have dilTerent values in the _two

cases).

We show below that these universal

(Ad)-values

_are

dijfeTent

from the

corresponding

values

(Ad)o

#

ail, ₀₎ (50)

for random walk type ^{chains which have been}

given by

Diehl and

Eisenriegler [17].

^In

Appendix

B it is also discussed how the two mean values in the

integrand

of

(31)

can in the

asymptotic

limit be

parametrized

in _a universal way.

(53)

and the _mean values

((Ad)~), ((53)~)

of

positive integer

_powers of Ad and

Sd

_are universal in the _{same sense as}

(Ad)-

Each of these _mean values has _a

representation

with

essentially

_[30] the form of the rhs of

(31). By repeating

the

previous arguments

one realizes that the

ensuing

flow factors

e~,

_see

equation (44), always drop

out from the

y-integral-

As _a

consequence the distributions such _as

(6(A _Ad)

₊

P(A) (51)

are universal functions in the limit of

long

chains.

2.2.2 Perturbation

theory

and

e-expansion

results. In this section _we present

explicit

_es-

timates for the universal _mean

asphericity (49)

of _a

ring polymer

and _an open

chain,

in the swollen state. We shall

investigate

how the

a(I, (Rl's

_{start to deviate from their random walk}

counterparts

a(1,0)

_as the dimension d

= 4 _e of the

embedding

_space becomes

slightly

less than

4,

and try to

extrapolate~towards

^{d = 3.}

^Actually

^we

^perform

^a

^systematic e-expansion

to first order of the ratio

~~~'"~~

a(1, ₀₎

According

to

equations (32)-(38)

this

requires expressions

for the two _mean values _on the rhs of

(31)

to order

u°

and

vi,

which _we denote

by subscripts

0 and

I, respectively.

Zero order

expressions

for

(e~~

~~Q) have

already

^been

given

in

equations (19), (20).

For the second _mean value _one finds from

(2)

and

(22)

ltr(ij~))

_" ^{~~ ~}

((~

^~~

_/~dti ^dt[dt2dt[ _{(2j (ii,} ^t[,

t2,

t[)

^~

₍₅₂₎

o,~ _o

Here

~ (il>i~>12>i~) (~1J42)01 ~(il>12)

₊

G(i~>i~) ~(il>i~) ~(i~>12) (53)

and _we have introduced the notation

A; ₊

x~ji;) x~(ij)

,

i _m 1,2

(54)

To arrive at

(53)

_we have used that for the Gaussian

(0,y)-statistics

Wick's theorem holds and the dilTerent Cartesian components Xa _are

independent

and

equivalent. Inserting

the

propagators

(24)

and

(25)

one

finds, respectively,

for

ring polymers

(tr ^Q~)))[~

=

L~

)[))(j)) ²

₊

(

+

( _sinh(fl)

2

Cosh(fl)j (55)

2252 JOURNAL DE PIIYSIQUE I N°12

and for _open chains

with

fl

from

(21).

Of _course, these

expressions appeared already

in the calculation of

a(1, 0) J71.

To evaluate the u~-contributions

(e~~~~Q)~

=

(e~~~~Q)~ ((V) (V) ⁽⁵⁷⁾

o o,~

an~

jtr _(Q2)j

~~

=

ltr ^(°~)

vlo,~

⁺

l~~ ~°~~10~ 1~1°Y

^~~~~

we rewrite

(30)

_as

v

jRj

=

"

/~didi, / ~ik jR~t)-R(i,)j

6 ~~g~

o k ^'

with

~x~d

~

~~~~

The

remaining

_averages with the Gaussian Hamiltonian

~Rw,~

all follow from Wick's theorem and

simple

combinatorics. The average

(V)o,~,

^which we

require

for

(57),

is

particularly

_easy

u

~~~~~, / ~-k2sy(t,t')

=

? f~didi' (s~ji,i')j

^~~~~~

⁽⁶¹⁾

~~

o,~ 6

o k

(2vfi0~~

_°

Here

S~(t, t')

=

j(A~)

=

jG(t, ^{t) G(t, t')}

⁺

jG(t', ^{t') (62)}

o,~

with _a notation

A ₊

Xd(t) Xd(t') (63)

similar _to

(54).

The average

IV)

_o ⁱⁿ

(57)

follows from

(61) by putting

_{y =} 0. For the

quantity (58)

_one finds

l~~

~~~~i,y

~~

4~~~~ /~~ _/~

~ ~~~~~~~~~~

~~~~

~~~~~~~~~0& ~~2~~0a~~~~2~0w

(k~)~ (lAiAlo,vj

~

(IA~A)o,yj

~

l

(d I)(d

₊

₂₎

_u

(2~6od

4L4 6 ^~

x

/dS (Sy(t,t')j ~~~~~2j(tj,t[;t,t')2j(t2,t[;t,t')2j(tj,t[;t2,t[)-

js~(i, i,)j

^~~+~/~

in (ii, ii

i,

i,)j

^~

in (i~, ii

_i,

i,)j

^~

¹⁶⁴⁾

with the notation

/dS

⁺

/

~ _dt

_{dt'dti dt[ dt2 dt[ (65)}

o

N°12 UNIVERSAL SHAPE PROPERTIES OF POLYMER CIiAINS 2253

These

expressions

allow to calculate the

uR-expansions

Fi

L~,u~, §,e

=

£ (u~)~F;jlL~, §,e ₍₆₆₎

~ ;-oi ~

(i

₌ 1,

2)

^of the rhs's of

(32)

and

(33)

_up to and

including j

₌ I. Here _we have

explicitly

noted the

e-dependence

of the Fi's which has been

suppressed

in subsection 2.2.I. The first order in

u

expressions (e~V~~Q)i

^and

(tr (~j~))i,y

contain

([or

_u, L

fixed) e-poles

at _{e =} 0. The _reason for these

poles

is the

vanishing

Sy(t,t')

- It

t'[

,

t _- t'

(67)

of

Sy

as t

approaches

t'. This leads _to

singularities

in the

integrands

of

(61)

and

(64)

which

are

non-integrable

for _e

= 0. In

Appendix

C _we calculate the

leading

_{e -} 0 contributions

c~ ue~~ _and ue° _to

(e~V~~Q)i, (tr (~j~))j,y.

^We

verify

that the

poles

_are indeed absorbed

by

the

reparametrizations,

I-e- that the functions

Fi;i(L~,z,e)

contain _no

e-poles,

and present

explicit expressions

for

Fi;i(L~,

_z,

_0).

These enter the ratio

~j)j,(jj~

^{= i +}

^uR

^{b + o}

(uRe, (uR)~j ⁽⁶⁸⁾

with

~

b =

~

/

_dz

_z(Fj.j(1,

~,

0)F2 o(I,

_z,

0)

+

Fio(I,z,0)F2.i(I,

_z,

0)) ⁽⁶⁹⁾

a(1>°>°

3

o ^{' ' ' '}

We show in

Appendix

C that is indeed

independent

of the form of the non-universal factor

f

in

equation (36)

and present in

equations (166), (167)

and

(169)

numerical results for the

two contributions to the

integral

in

(69).

For the final evaluation of b _one needs the random walk results

~~~'~'~~

"

~~~~° .~~~

)$(

~~~~

for

ring polymers (rp)

and _open chains

(oc) [17].

This leads to

Surprisingly

_we find within the _accuracy shown

nearly

the _same result for both

ring polymers

and _open chains.

To estimate

(A3)

+ a

(I, (R, I)

we use

equation (68)

with uR

=

(R

_and

approximate

its rhs

by

the first order

e-expansion.

Then

equations (43)

and

(71)

lead to

To arrive at the final numbers _we have

again

^{used random walk} results [17]

~^~ o

1°.246

0.394

^(rP) (oc) ^(~3)

Equation (72)

represents _our renormalization _group estimate for the

asymptotic

_{mean as-}

phericities

for

ring polymers (rp)

and open chains

(oc),

in the swollen state and in three

2254 JOURNAL DE PHYSIQUE I N°12

dimensions. The

prefactor

I +

b)implies,

for both types ^of

chains,

_an increase of 5

~it

_in

8 2

the _mean

asphericity

due to the excluded-volume interaction.

We note that the

quantity ld,

where _numerator and denominator in

(4)

_are

averaged

^inde-

pendently,

is much less sensitive to the excluded-volume interaction. Within the

e-expansion

Aronovitz and Nelson [13] ^estimated [31] an increase of

l~~

_of

_only

_I

^~it

_{Monte Carlo simu-}

2 lations in d

= 3 which will be discussed in the next section arrive _at the _same

qualitative

conclusions.

3. Monte Carlo calculations.

In the

analytical

_{part we} used

a continuum

model,

both in the _monomer index _space and in the

embedding

_space. For Monte Carlo simulations of _{course we} have _to confine ourselves to discrete

models,

at least

concerning

the _monomer index. In this work _we will also discretize the

embedding

_space, I-e- _{we are}

going

to

investigate

the

shape properties

of lattice

polymers

[32]

3.I OPEN _CHAINS.

3.I.I Model and method. As _{we are}

only

interested in the _case d

=

3,

the

polymer

_con-

formations _are modelled _as N-step walks _{on a}

simple

cubic lattice

(lattice spacing

₌

I).

We denote _one end of the _monomer I

by

the lattice _vector Ri ^with I < I < N. The

only

constraint for RWS is that all successive vectors in the set

(Ro,

,

RN)

denote

neighbouring

lattice

points.

The excluded-volume condition is

easily implemented by

the

simple requirement

that

no lattice vector R may appear twice in _one conformation

(Ro,

,

RN

).

To go from _one

single configuration

to proper

ensembles,

_we need _an efficient

algorithm

which

scans the _space of all

possible

conformations with respect to

ergodicity.

As _{we are} concerned

with

strongly fluctuating

matter, we have to generate at first

large samples

of chains of fixed

length,

in _a second step the chain

length

itself has to be varied in order to

extrapolate

to the

asymptotic

limit. Whereas the

procedure

_{to generate} RW lattice chains of

length

N is

obviously

clear

(make

N steps at random

on the above

lattice),

the choice of the

algorithm

for SAWS needs _{more care.} For _our purpose a static method like dimerization [33] seems to be best suited. This method, which respects all above

requirements,

enables _us to generate

large samples

of

statistically independent

chains

(100000 / length)

for fixed

lengths

N with 30 < N < 220. This _set of data allows _{a proper}

scaling analysis

for all

properties

in

question.

The

method,

which is _a modification of the

orginal

dimerization method of Alexandrowics [33],

proceeds

_as follows [34]. ^For a SAW chain of

length

N _we have to generate at first _a SAW of

length N/2 by simple sampling,

^I-e- generate _a non-reversal random walk

(NRRW)

of

length

N/2,

start at the

beginning

when the chain intersects itself. In _a second step another NRRW

is started at the end of the first _one. If the second half intersects itself

(and perhaps

also the first

one),

_one has _to start

again

at the end of the first

half,

the first part itself

remaining unchanged. Only

in the _case that there _are

merely

intersections between the first and the second

half, being

SAWS of

length N/2

for

themselves,

both parts are cancelled and _{anew run} has to be started. This

method,

which is

completely equivalent

to the

original

dimerization

method,

has _some technical

advantages

to the latter _one,

sharing

all their other

advantages,

e-g-

high efficiency compared

to

simple sampling.

For

larger

chains this method _can be used

recursively

_on several levels

[34].

^{As the} excluded-volume condition omits all

intersecting

N°12 UNIVERSAL SHAPE PROPERTIES OF POLYMER CHAINS 2255

1.50 '

P(A3) 1.00

0.50

0.00 0.20 0.40 0.60 0.80 1.00

A3

Fig.

I. Probability distribution for the aspheficity A3 for _open RWS

(loo

coo chains with N

=

1024);

(A3)f~~, (d3)f~~

denote exact _mean values.

configurations

from the

sampling,

all

possible

conformations have

equal probability leading

to

simple

arithmetic _averages for all

properties.

In the first part ^of _our calculations _we will have _a short look _on the

shape properties

of _open RW chains. The main part,

however,

is devoted to the

properties

of open SAW

chains,

where the size _as well _as the

shape

will be

investigated.

3.1.2 Results and

scaling analysis.

Besides _mean

values,

which will _now be denoted

by

_((A3))

or

((53)),

we also

generated

_a

histogram

for the

asphericity

of RW chains

(see Fig. I).

The main _purpose for this calculation

(100

000 RWS with N

=

1024)

^{is the}

possibility

to

investigate

the influence of the excluded-volume interaction _on the

shape properties

not

only

via average parameters, but also via distribution functions.

Using

this _{as a} basis of

comparison

_we treated the

problem

of SAW chains

by

_means of the above dimerization method. All _average

quantities

were calculated for chain

lengths

between N = 30 and N ₌ 220. For each N 100 000

statistically independent

chains _were

sampled (see

Tab.

I) giving

_{an error} bar for

((R~))

^{of about 0.3it.} In addition _we calculated

histograms

^for

the two

shape

parameters A3 and

53

from the data for fixed

length

N ₌ 140.

Mainly

_{as a} check for _our computer _program we first

investigated

the

scaling

behaviour of the end to end distance

((R~))

^and the radius of

gyration ((RI)).

A double

logarithmic plot

of these

quantities

_versus chain

length (see Fig. 2)

^leads to well-known results for the correlation

length

exponent

vR2 = 0.590 + 0.005

vRj

= 0.592 + 0.008

(74)

As detailed corrections _to

scaling

have _not been taken into _account

[34],

^{the results} are consis- tent both with other numerical and

analytical

work.

We also

analyzed

the behaviour of the universal ratio

((R~))/((Rl)),

which

is,

due to

large

fluctuations in the

samples,

a rather delicate

quantity.

A

scaling plot

of

((R~))/((R[))

_versus

the

leading

finite size correction N~~

IA

= 0.47 in d

= 3

[35],

see also

Fig. 3)

shows that

2256 JOURNAL DE PHYSIQUE I N°12

jj~~~j~«)

iiRiii~"~

20 40 80 160 300

N

Fig.2.

Double logarithmic plot of ((R~))1°~> ^and ((R[))1°~> _vs. ^N.

6.

o o

o

jjR2jj<oc> ^°

iiRiii~"~ ^~

0.00 o-lo o.20

11-A

Fig.3.

scaling plot of

((R~))1°~)/((R[))(°~)

_vs. ^N~~

universal

scaling

behaviour starts at chain

length

about 80. Below N

= 80 lattice

dependent

effects dominate. The

extrapolation

to infinite chain

length yields, assuming

the behaviour _as indicated in the

figure,

jj

j~2))(oc)

jj j~2

jj(oc)

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

(~~)

G

which should

only

be

compared

with other simulation data. Shanes and Nickel

[36],

who combined Monte Carlo data and _exact enumeration

results, got

^the

asymptotic

ratio

((R~))(°~)/((R[))(°~)

₌ ^6.2487 +

0.0013,

which is in excellent agreement to _our data.

Based _on

this,

_we think that _our data _are also

reliable,

when

non-diagonal

elements of

Q

_are considered. We calculated the _mean values of A3 and 53 _as well _as of

Al

and

S(

for _a

variety

of chain

lengths.

These averages also contain information about the width of the distribution

N°12 UNIVERSAL SHAPE PROPERTIES OF POLYMER CHAINS 2257

Table I. Simulation data

for

_open SAW chains in d

= 3

(100

000 chains

/ length).

N llR~ll~°~~

llRlll~°~~

llA3ll~°~~

llAlll~°~~

llS3ll~°~~

llslll~°~~

9.86 0.4404 0.2305 0.5570 0.4905

40 88.29 13.92 6.341 0A410 0.2311 0.5586 0.4928

50 lls.34 18,19 6.339 0.4395 0.2297 0.5556 0.4887

60 143.65 22.64 6.345 0A401 0.2301 0.5569 0.4893

70 173.09 27.31 6.339 0.4398 0.2300 0.5566 0.4893

80 202.91 32.00 6.342 0.4387 0.2291 0.5543 0.4868

90 232.50 36.78 6.322 0.4377 0.2282 0.5526 0.4847

100 265.15 41.87 6.333 0.4388 0.2294 0.5550 0.4887

l10 295.98 46.83 6.321 0.4374 0.2282 0.5520 0A851

120 327.56 51.85 6.317 0.4372 0.2278 0.5518 0A831

130 360.95 57.16 6.315 0.4373 0.2279 0.5519 0.4842

140 393.51 62.44 6.302 0.4368 0.2276 0.5508 0A831

160 462.97 73.36 6.311 0.4372 0.2277 0.5515 0A831

180 533.58 84.48 6.316 0.4373 0.2280 0.5523 0.4843

200 602.37 95.62 6.300 0.4356 0.2264 0.5484 0.4790

220 674.57 106.99 6.305 0.4363 0.2269 0.5506 0.4805

function ofA3 and

53, given by

the relative variances

jj

~2))(oc)

jj^~~))(oc)j^~ fi~~~~

~

~

(~~)

~~

llA3ll(°C)

~~~~

~/jjs())(OC)

(jjS3))(OC)j~

~S3

ii s~))(OC) ^~~~~

Scaling plots

similar to that in

figure

3

yield following asymptotic

limits

(see Figs. 4-7)

((A3))~°~~ = 0.431 + 0.002

(78)

A

~)~

= 0.442 + 0.004

(79)

((53))~°~~ = 0.541 + 0.004

(80)

At~

= 0.773 ~ 0.005

(81)

The value of ((A3))~°~~ ^fits very well to the result of

Bishop

and Saltiel

[19],

^{who used} a

pivot algorithm

for two chain

lengths (N

₌

201,401). They

find 0.431+ 0.002 for both N which

seems not to follow

properly

the correction to

scaling

_curve

(of Fig. 4).

Other simulations

by

Cannon et al. [20]

give

0.447 + 0.011 and 0.572 + 0.025 for the _mean values of A3 and

53, respectively.

These calculations _were confined to _one

single

chain

length

N ₌ 100 and _no finite chain

length analysis

_w,as

performed.

Their

(rather large)

_error bar is due to statistical scatter of the data and cannot account for

systematic

elTects. As these authors used the _same lattice

type _{as we}

did,

their result _can

directly

be

compared

_to the data of

figures

4 and 6. As _can

2258 JOURNAL DE PHYSIQUE I N°12

*

0.

o

~

iiA311~"~ ^°

o o

0.

o.00 0.10 o.20

/f-~

Fig.4.

Scaling plot of ((A3))1°~~ _vs. N~~

(statistical

_error is of the order of the scatter of the

data);

(o) from reference [19],

(*)

from reference [20].

~(oc) o

o o

o.oo o.10 o.20

/f-A

Fig.5.

Scaling plot of

a(]~

=

fin

_{((A(11 ((A3))~}

vs. N~~

be _seen from _our

scaling plots (see Figs. 4, 6),

this

extrapolation brings

_up corrections which cannot be

neglected

in this _case.

The value of

A~)~

^{indicates the}

^large

^{width of the} distribution

function,

_as the variance is about half the value of the first _moment itself.

From the

positive

_mean value of

53

_{we see} that _open SAWS _{are on average}

prolate objects.

The relative variance of

53

of about 75~ indicates

again

_{a very} broad distribution

function,

which will _now be examined.

We calculated

histograms

for A3 and

53

from _a

sample

of100000 chains of fixed

length

N ₌ 140

(see Figs.

8,

9).

The distribution function of A3 for SAWS is _even broader

compared

to the

non-interacting

_case

(see Fig. I).

In addition the distribution is _now

quite symmetric,

most

probable

^value and _mean value

being nearly equal.

N°12 UNIVERSAL SHAPE PROPERTIES OF POLYMER CHAINS 2259

o

~ o

(lS3))~°~~

~

~

o.oo o-lo o.20

j/-A

Fig.6.

Scaling plot of ((53))~°~~ vs. N~~

o.780

0.770 _o

o ~

~(°C) _~

o o

°

0.760 O

o.oo o-lo 0.20

j/-~

Fig.7.

Scaling plot of

a(]~

=

@fi ((S())

_{((53))~ vs.} N~~

The distribution function for

53, however,

is rather

unsymmetric

with _a most

probable

value

of _zero. The _gross

shape

of the

histogram signals

that the vast

majority

of

configurations

has

prolate

asymmetry. It should be noted that the maximum of the distribution function at _zero does not

imply

that these

configurations

have

spherical

symmetry.

3.2 RING _POLYMERS.

3.2.I Model and method. In

analogy

to open chains _we model

ring polymers

as

N-step

walks _{on a}

simple

cubic lattice

adding

the obvious

requirement

that Ro

equals

RN-

By

that

means the ensemble of all

possible ring polymers

is _a

special

subset of all

possible

_open chains.

Using

the _same notation _as in section 3,I _{we are now} in need of _a

dynamical procedure,