Numerical study of a charged bead-spring chain

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Numerical study of a charged bead-spring chain

J. Barrat, D. Boyer

To cite this version:

J. Barrat, D. Boyer. Numerical study of a charged bead-spring chain. Journal de Physique II, EDP

Sciences, 1993, 3 (3), pp.343-356. �10.1051/jp2:1993131�. �jpa-00247837�

Classification

Physics

Abstracts 61.40K

Numerical study ^of

_a

charged bead-spring chain

J. L. Barrat and D.

Boyer

Laboratoire de

Physique

(*), Ecole Normale

Supdrieure

^de

Lyon,

46 allde d'ltalie, 69364

Lyon

Cedex, France

(Received J2 October J992,

accepted

9 December 1992)

Rksumk. Nous dtudions par des simulations Monte-Carlo et des m£thodes variationnelles _un modble

simple

de

polydlectrolyte,

la chaine

gaussienne chargde.

Un bon accord

quantitatif

entre les deux

approches

utilisdes _est obtenu. Pour des interactions de Coulomb _non £crantdes, les rdsultats _{sont en} excellent accord _avec le modble

classique

du _« blob

dlectrostatique

_»,

Lorsque

les interactions

dlectrostatiques

sont 6crant£es, _on observe _aux dchelles de longueur

plus grandes

_que

la

longueur

d'dcran le d6but d'une transition _{vers un}

rdgime

de volume exclus

dlectrostatique.

Abstract. A

simple

model of _a

polyelectrolyte,

the

charged bead-spring

chain, is studied

using

Monte-Carlo simulations and variational methods. A

good quantitative

_agreement between both

approaches

is obtabled. For bare Coulombic interactions, excellent agreement ^{with the} classical

« electrostatic blob _» picture is found. Screening of the electrostatic interactions _{causes a crossover} to excluded volume behaviour that takes

place

at scales of the order of the

screening

length,

1. Introduction.

Our present

understanding

of

charged polymer (or polyelectrolyte)

solutions is far from

being

as

satisfactory

_as that _we have of neutral

polymer

solutions.

Theoretically,

the

difficulty

_stems

from the existence of many different

length

^{scales in} the

problem [I].

This renders the _use of

simple scaling

laws hazardous. Such

scaling

laws

[2] (later justified through

renormalization group calculations

[3])

_were crucial to the

development

of _a consistent

theory

of neutral

polymers.

The

practical

success of the

scaling approach

for neutral

polymers

is also related _to the existence of

experimental

_systems that fulfill all the

requirements

of theoretical models _:

large

^{number of} monomers, flexible

chains,

and short range interactions.

Unfortunately,

the

complexity

of

polyelectrolyte

_systems is much greater. ^Both

long

_range

(electrostatic)

and

short range interactions

(excluded volume)

are now present.

Moreover,

the short range

interactions between

polymer

backbone atoms, counterions and solvent molecules _are

likely

_to

play

_an

important

role in the _« condensation _»

phenomenon

^first described in reference

[4].

One therefore

expects

a nontrivial

coupling

between the nominal

charge

of the

polymer chain,

its local

properties (stiffness)

and its

« effective

»

charge.

(*)

URA 1325 du Centre National de la Recherche

Scientifique.

The

difficulty

in

studying polyelectrolyte

solutions is thus twofold.

Firstly,

the theoretical methods used for neutral

polymers

_are not

easily

extended _to systems ^with

long

range forces.

Secondly,

the

ability

of

simple

theoretical models to describe

experimental

situations is _not established. In this paper, we concentrate on the first aspect ^{of the}

problem.

We present a

numerical

study

of _an

extremely simple polyelectrolyte model, namely

a

charged bead-spring chain,

in very dilute solution

(Note

that in real

polyelectrolytes,

the very dilute

regime

is

difficult _to

study

because the

overlap

concentration is

small).

The Hamiltonian of _a chain made up of N beads with

positions

_rj,

, rN is of the form

~

3kBTN-i~

~~

j

~~

_~~~

~~2 ~

_~~~~~~~

~§~ _~~~~~/~

,=i i,j

with

V

(r)

= kB ^T

~~

exp

(- «r) (2)

This Hamiltonian models _a chain of beads connected

by

Hookean

springs

and

interacting

through

the screened Coulomb

potential (2).

^The

length f~

characterizes the

strength

of the screened Coulomb interaction and is called the

Bjerrum length.

It _we take _as units of energy and

length respectively

the thermal energy

k~

^{T and the} r.m.s. distance _a between two

monomers of _a neutral chain

(v(r)=0),

the model is

fully specified by

the three

dimensionless parameters u ₌

iBla _(coupling parameter),

_«a

(screening parameter)

and N

(number

of beads per

chain).

Several

assumptions

underlie the _use of such _a model _: short range excluded volume

interactions _are

neglected,

the solvent

being

thus _{a «} 0 solvent

» for the neutral

polymer.

Linear

screening

of electrostatic interactions

by

the counterions is assumed. This results in the form

(2)

for monomer-monomer

interactions,

the inverse

screening length

_«

being

related to the

ionic

strength

I of the solution

through

the usual

Debye-Hiickel relation, «~=

₄

griBI.

Finally,

the Hookean

spring

that connects two

neighbouring

_monomers

actually

models the

elastic behaviour of _a whole subchain of atoms. Our model therefore represents a

weakly

charged polyelectrolyte,

in which _two consecutive

charged

_monomers

along

the chain would be

separated by

a very flexible subchain. This subchain must thus be much

longer

than the

« bare _»

persistence length

^{of the actual}

polymer

backbone. All these

assumptions

would

obviously

be

highly questionable

if _{we were}

aiming

at a realistic

description

of

polyelectrolyte

systems. ^Our purpose here is however very different. The model described

by (I)

is the

counterpart

^for

charged

chains of the well known _« two parameters ^model » of neutral chains

[5].

As such it is taken _{as a}

starting point

in many theoretical studies

[2, 6-10].

A better

understanding

of this model,

independent

of any consideration of its

experimental relevance,

is therefore desirable.

The

properties

of

(I)

in the absence of

screening («

₌

0)

_are indeed well known from

previous

theoretical studies

[2].

The chain is made up of _a linear

string

of _« electrostatic blobs _» of size

f. f

is defined _as the

length

scale below which correlations between _monomers

are unaffected

by

the electrostatic

interactions,

and _are therefore those of _an ideal

bead-spring

chain.

Specifically, f

veRfies the two

equalities

~~ i fi

=

(3)

which expresses the fact that the electrostatic energy of the blob

(made

_{of g}

monomers)

is of the order of the thermal energy and

f

=

9~'~

_a

(4)

which expresses that its statistics _are those of _an ideal chain. f ^{scales thus} as u~ ^~/~ a. The characteristic size of the chain is R

(N/g ) f Nau~'~

_The

swelling exponent

^is _{v =} ^{I. These}

results,

and indeed the model

itself,

_are sensible

only

for

coupling

_{parameters u} smaller than

unity,

since

larger

values would _cause the Hookean

springs

_to reach

unphysically large

extensions.

When _« is _nonzero, which

physically corresponds

to

having

a finite concentration of salt added to the

solution,

_{a new}

length

scale «~ appears in the

problem.

This

length

scale is assumed to be

larger

than the blob size

f. Otherwise,

the

potential (2)

is short range ^and can be treated _{as a} usual excluded volume interaction. The behaviour of the chain _at short and

large length

scales is

easily

understood in this _{case :} at veTy

large length scales,

the range of the interaction

potential

is irrelevant and _one

expects ordinary

^{excluded volume}

behaviour,

with _a

swelling

_{exponent v}

=

0.6. At short

length

scales _on the other

hand,

the

screening

^is irrelevant

and the _monomers will form _a linear

string

of blobs. Two different

theories, however,

have

been

proposed

to describe the _crossover between these two

limiting regimes. Pfeuty [7]

_argues that the

screening

becomes relevant _{as soon as} the scale

« is

reached,

and that the _crossover

to excluded volume behaviour should take

place

around this

length

scale. Khokhlov and

Khachaturian

(KK) [8],

_on the other

hand,

extend to this weak

polyelectrolyte

model the concept ^of an « electrostatic

persistence length

_» first formulated

by Odjik [I I]

and SkoInick and Fixman

[12]

in their

study

of

charged

semi-flexible chains. These authors considered _a

semi-flexible

(Kratky-Porod)

chain with «bare»

persistence length i~o, bearing charged

monomers

interacting through

the

potential (2)

and

separated by

the distance A

along

the chain

contour.

They

_were then able _to show that when the chain is in _a rodlike

configuration,

the increase in electrostatic energy ^{that results from}

bending

the chain _can be rewritten in the _same

form _as the usual curvature energy. The monomer-monomer interactions could then be

described _as

causing

an increase in the

persistence length,

the total

persistence length

iosf _being

_{the sum of the}

« bare _{» one} and of _{an «} electrostatic

persistence length

_» :

i~

~°sF

_~

_~P°

~

(5)

4 _«

~A~

KK assumed that this formula still holds for flexible

polyelectrolytes, provided

both the bare

persistence length

and the distance between

charges

are taken _to be

equal

to the blob size

f,

and the _monomer

charge

is

replaced by

the

charge

of the blob. This

yields

_{a «}

persistence

length

_» for the chain of blobs _:

g~ i~

i~~

=

f

+

4

«~ f2 i~ f2

~

~

~

~2 ~4'

^~~~

Reference

[8]

thus concludes that the _crossover between rod-like and excluded volume

behaviour takes

place

at a

length

scale of order

i~~.

_{For weak}

_screening,

_this

_length

_scale

varies _{as «}

~2,

which is very different from

Pfeuty's prediction

«~ ~. Our main

goal

in this paper will be _to find _out which of these _two theoretical

predictions

best accounts for the

structure of the

charged bead-spring

chain.

Our

investigations mainly rely

on Monte-Carlo simulations

(Sect. 2)

which in

principle provide

exact results _on the

system.

The simulation results _were

compared

to the variational

approaches

described in section

3,

thus

allowing

_a

quantitative

assessment of the accuracy of such methods. The results obtained for _«

=

0,

which

largely

confirm the classical

picture [2]

summarized

above,

_are

presented

in section 4.2. Section 4.3 discusses the influence of

screening

_: simulation results _seem to indicate that

Pfeuty's picture

is the _{correct one} for _our

system.

The paper ends with _a brief discussion.

We shall close this introduction

by considering

how _our work relates _to other simulations of

polyelectrolyte

systems. ^{The main difference between} our

study

and several other realized in the _same

spirit [13-15]

is the fact that _{we use an}

oversimplified Hamiltonian,

of which short-

range excluded volume effects are absent. This

simplification

allows direct

comparisons

with theoretical

calculations,

which

permit

_e.g. to

quantitatively

define the blob size

(Sect. 4.I).

Our

analysis

and most but not all _our conclusions _are nevertheless similar to those

presented by Higgs

and Orland

[15].

2. Monte-Carlo simulations.

The Monte-Carlo

(MC)

method used in this work is _an off-lattice

implementation

^{of the}

standard

reptation algorithm [16].

Each MC _move consists in

deleting

_one of the two extreme bonds of the

polymer

_to create a new bond at the

opposite

end. The

only original

_aspect of _our

algorithm

consists in the fact that _we

impose

_a Gaussian bias _on the choice of this _new

bond,

I.e. the bond _{vector u} is chosen

according

to the

probability

distribution

3 3J2 3

~2

p(U)

₌

~ eXp

(7)

2 _aTa 2

a~

The

Metropolis

_acceptance

probability [17]

is then corrected for this

bias,

and involves

only

the variation in electrostatic

energies

caused

by

the

attempted

_move. The choice of the bias

(7)

is motivated

by physical

considerations _i for _an ideal hookean chain

(v(r)

=

0),

this choice would

yield

a loo fb acceptance rate in the

Metropolis algorithm

and therefore be

extremely

efficient. The _« electrostatic blob _»

picture

discussed in the introduction suggests ^{that short} range correlations which _are

important

in the choice of the _new bonds remain dominated

by

the elastic

part

^{of the} Hamiltonian

(I).

Therefore the bias

(7)

is

expected

to

yield

a

high acceptance

rate. This is bome out

by

the fact that _even for the strongest

coupling

_we

considered,

u =

0.5,

the acceptance rate is about 60

fb, higher

rates

being

obtained for weaker

couplings.

Chain

lengths varying

^from N

= 50 to N

=

400 beads _were studied

using

this

algorithm,

The number of _moves

required

for _a renewal of the chain

configuration

in the

reptation

algorithm

is

typically

of order

N~.

_{Since the} interactions _are

long-range

the

computational

_cost

of _a simulation varies like

N~.

For the

longer

chains

(N

₌

400),

the simulations _{were run} for

typically

2 _x

10~

_to

10~

MC _{moves, a} duration that is

only slightly larger

than the renewal time of the chain. The results _were nevertheless found _to be

reproducible

within _a few percent. ^A still better accuracy can of _course be achieved for shorter

chains,

which _can be simulated

during

_many renewal times.

The

global

chain structure _was characterized

by computing

the _mean

squared

end to end

distance,

R2(N

m

(ri

_r~

)2)

,

(8)

the radius of

gyration

R( (N

m

( ^r, (

rj

l.

~

(9)

i=i

~j=1

The intemal structure of the chain is

usually

described

by

the structure factor

l~

^~

s(q)=~ zexp(-q.r,)

,

(lo)

N

.=1

which _was obtained

by averaging

_over several q ^vectors with different orientations. Other

quantities

^which we found _to be

extremely

useful in the

description

of the local structure were

the _« intemal distances _» of the

chain,

the intemal distance

R~(n) being

defined as the _mean

squared

distance between two monomers that _are

separated by

_n bonds

along

the chain, I.e.

R~(n)m z ((r;

~~_i

r;)~) ⁽ⁱⁱ⁾

N _-n

+1~.)~~

3. Variational

approaches.

The

general

variational

approach

in statistical mechanics makes _use of the

Gibbs-Bogoliubov inequality relating

the free energy F of the system ^under consideration to the free energy

Fj

of _a reference system ^described

by

the trial Hamiltonian H~:

F _w

F~

₊

(H _H~)~ (12)

The brackets denote _an average taken with the statistical

weight

_exp

(- H~/kB T).

^The « best _» reference system ^{is determined}

by minimizing

the left hand side of

(12)

w.r.t. the

parameters

^of H~.

In the absence of

screening,

_{a very} natural choice for the trial Hamiltonian is that of _an Hookean chain whose ends _are

pulled by

a force

l~

H~(F)=~~~~~~j~(r,~i-r~)~+F.(rN-ri). (13)

2a

,~j

The structure of such _a chain is well known

[2]

to be a linear

string

of _« Gaussian blobs _» of size

f(F)

= kB ^{T/F. This structure is} very close _to the _{one we} expect ^{for the unscreened}

polyelectrolyte chain,

and the stretched chain Hamiltonian

(SCH) (13)

should therefore

provide

a

good

variational ansatz. In

fact,

this SCH _was used in

previous

work _on the _same

subject [6],

and it _was shown that in the

large

N limit the variational calculation

yields

the

following scaling

law for the end-to-end distance _:

R

~Nu~'~ _fin (Nu~'~)]~'~ (14)

This

prediction

differs

only by

_a

logarithmic

factor from _a

simple Flory

estimate. Here _we shall be interested in

comparing

the variational results _to simulations

performed

for finite chains. In this _context, the SCH has the

advantage

of

yielding

_very

simple

relations for the

quantities

of

interest,

such _as the _structure factor

(averaged

_over ^{all force}

directions)

ssc~ (q )

m

'~ z (e~

^{~~' ~~' 5}_~)

N

, ,j

~

i

z

_~~~

(_ ^{Ii -j q~}

^Sin

^{( Ii -j qF/3) ~i~~}

N2

6 I

j qF/3 )

and the intemal distances,

~ ~

~2

j/ ²

~~~~~~~

_{~~ ~ ~}

3

k~

T ~~~~

The

scaling

law

(14)

makes it clear that the stretched chain

picture

becomes

inadequate

when the electrostatic interactions _are

screened,

since in that _case the

swelling exponent

^is

expected

to take the

Flory

value _v

=

0.6. A variational method that results in _more reasonable values of the

swelling

exponent is the Gaussian variational

(GV)

method of des Cloizeaux

[3].

This

method is based _{on a} trial Hamiltonian which is the most

general quadratic

form of the bond vectors

uj

= r~ ~ i

rj.

^Such an Hamiltonian is tractable for

cyclic

chains

only.

In that _case the

quadratic

form is

diagonalized by

the variables

p ₌

(

~~

j.

²

^wjq

~

$

j

~ N

' (17)

and _can be written in the form

~ ^N-I

~t

~

~2 l~ _{~q ~q}

~-

q

(18)

q=I

The S~ ^{are related to the intemal} distances

R~(n) _by

:

R~(n)

=

j~

l _cos ~ "~~

(19)

~

=o

~q

~

with

H~

₌

2S~sin~(grq/N).

The minimization of the variational free energy ^w.r.t. the parameters S~ ^{results in}

[3]

:

2 grq

~

l

~~

_{2 "q~}

I'(R~( _{)) ~~°~}

H~

₌ ^I CDS

~ p

^~" _N ^~

where

I(A)

_is

a Gaussian

weighted

_average ^of the interaction

potential v(r),

~ _~~

^~

~

~~~

ld~r

^v

^(r)

^{exp 3 ~~}

~ 2 A

(21)

The

coupled equations (19, 20)

_were studied in the

asymptotic

limit of

large

N

by

des Cloiseaux for short range v(r). The method

predicts

in that case a

swelling

_exponent

v =

2/3, slightly larger

than the correct value. Bouchaud _et al.

[10]

on the other hand

proved

that for

long

_range

interactions,

the method

correctly predicts

_v

= I. In order to compare the

predictions

of the Gaussian method to _our simulation

results,

_we

numerically

solved

equations (19, 20)

for _a

cyclic

chain of 16384 _monomers. The

resulting

intemal distances

R~(n)

and structure factors _were then

compared

_to the simulation results. This

comparison

is

however

slightly ambiguous,

since the systems ^{considered in the simulation and in the}

variational calculation _{are not}

strictly speaking

identical. The structure factor that _was

compared

to the simulation results is for

example

not that of _a short linear

chain,

but that of _a subchain of the

long cyclic

chain. The results

presented

in 4.3 _seem to indicate that this

approximation

is _a reasonable _one.

4. Results.

4.I A QUANTITATIVE DEFINITION OF THE BLOB SIZE. In its

original formulation,

the concept of _an electrostatic blob is _a

semi-quantitative

_one, the blob size

being

^defined

only through scaling

relations. A more

quantitative

^{definition is desirable in order} to

analyse

the simulation

results. Such _a definition _can be obtained

by using

the

analogy

between the structure of _an ideal stretched chain and that of _a

charged

chain

developed

in the

previous

section.

Equation (16)

can be rewritten _as

n 2

R(c~ (n )

=

na~

₊

_{f (F ) (22)}

9

2

=

na~

₊

n

~

f(F

_)~

where g is the number of _monomers in _a blob. In this

form,

it is clear that the intemal distances in _a stretched chain of blobs consists in _{a sum} of _a random walk

component, na~,

and of _a

quadratic

_component whose coefficient is related to the blob size. That this second

component

varies like

n~

is _moreover

intimately

related to the fact that the chain is _a linear

string

of blobs.

Therefore the structure of the chain at short

length

scales _can be obtained from the

knowledge

of intemal distances

by computing

the

quantity

:

h

(n

=

(R~ (n )la~

_n ^~'~

(23 )

According

to

equation (23),

the

slope

of

h(n)

at the

origin

_can be

interpreted

as the inverse of the blob size.

Moreover,

_a linear behaviour of h

(n)

with _n will indicate that the

string

of blobs is in _a rodlike

configuration.

We shall therefore

interpret

the

point

at which deviations from linear behaviour appear as

providing

_a

rough

estimate for the

persistence length

of the chain of blobs.

4.2 RESULTS _{IN THE ABSENCE OF SCREENING.} The results for the variation of the chain

radius of

gyration

_as a function of chain

length

for _a fixed

coupling

u and of

coupling

at fixed chain

length (N

=

loo)

are

presented

in

figures

la and 16. These results _are in close

agreement

^{with those obtained from the} variational calculation

using

the stretched chain model. This is _an indication of the

validity

of the

scaling results,

since the stretched chain model at

large

N

reproduces

these results. The structure factors _are also found _to be in excellent

agreement ^{with the}

predictions

of variational

computations (Fig. 2).

The

q~~

behaviour exhibited in the intermediate _wavevector range ^is characteristic of _a rodlike

object.

Figure

3

presents

^{the simulation results for} the function

h(n) (Eq. (23)),

for various chain

lengths

and _a fixed

coupling

_u

= 0, I. The linear behaviour _at small _n discussed in the

previous

section is

clearly displayed by

these results. The

slope

at the

origin

is almost

independent

of the chain

length,

_as

required by

the electrostatic blob

picture.

A deviation from linear behaviour is observed when _n becomes

larger

than about half of the chain

length

N. This deviation _can be

explained by

_a finite size effect. For

large

values of _n, the _monomers located close _to the chain ends

provide

the dominant contribution to the _sum

(11).

It has been

previously

observed

[15]

that the chain

stretching

_was

slightly

weaker in the extreme parts ^{of the chain than in its central}

part,

so that _a linear

extrapolation

of the small _n behaviour tends to overestimate

R~(n)

_at

large

_n. The result

presented

in

figure

3 for the intemal distances between the 200 innermost _monomers of _a 400 _monomer chains indicates that these end effects _are

insignificant

for _{n w} N/2.

The results for the electrostatic blob

size, computed using

the

procedure

^{outlined in section} 4,I, are

presented

in table I. The ratio of the simulation result to the

scaling prediction

u~ ^~/~ a is

roughly independent

of both chain

length

and

coupling,

thus

confirming

the

validity

of the

scaling picture.

Finally, figure

4 shows that except ^{for the end}

effects,

the variational calculations _account

fairly

well for the variation of

h(n)

with _n.

Reproducing

end effects is

clearly beyond

the

ability

of these methods _: the stretched chain model _assumes that the tension is constant

along

the

chain,

while the Gaussian method deals with

cyclic

chains.

60

m

~

40

zo

0

loo 200 300 400

N a)

30

m

zo

lo

0

0

u

b)

Fig.

1. a)

Gyration

radius _{as a} function of chabl

length

for _a

coup1blg

_u

=

0.1, in the absence of

screening.

The squares are simulation results, the full line

corresponds

to a variational calculation

using

the stretched chain model. b)

Gyration

radius _as a function of the

coupling

parameter ^for a chain of loo

monomers without

screening. Kcy

_a8 in

figure

la.

~

~'~

'

l

~0~

o.oi o.ooi

o.ooi

qa

Table I. Size

f

_«

of

the electrostatic blob _» in the absence

of screening,

obtained

from

the

slope of h(n)

at the

origin (see text), for different

values

of

the chain

length

N and

of

the

coupling

parameter u. The last column

gives

the ratio

of

this size to the

scaling

estimate

~- ^lJ3 ~

N _u

50 0.01 6.7 1.45

50 0.05 3.6 1.28

50 0.1 2.6

1.20

50 0.5 1.3 1.03

100 0.01 6,3 1.35

100

0.05 3.3

1.16

100 0.1 2A 1,ll

100 0.5 1.2 1.0

100 1.0 1.1

0.95

200 0.1 2.2

1.01

400 0.1 2.0

zso

zoo

~~~,

iso

,,,"""

Gi ^"'

li

loo

50

0

O loo ZOO 300 400

n

Fig.

4. h(n) as a function of _n for _u

= 0. I, K = O and N

= 400. The full line is the simulation result, the

long

dashes _are the result of the stretched chain model and the short dashes those from the Gaussian

variational calculation.

4.3 RESULTS _FOR SCREENED COULOMB INTERACTIONS. As discussed in section

4.I,

the

variation with _n of the intemal distances

R~(n)

_{will be used here to estimate the}

persistence length

of the chain of

blobs,

which is _our

primary

interest in this

study.

The function

h(n)

for _a chain of N

= 400 _monomers is

displayed

in

figure

5 for _a

coupling

_u

=

0.I and

iso

-

~

loo

50

O loo 200 300 400

n

Fig.

5. h(n) _{as a} function of _n for _u

= 0.1, N

= 400, and various values of the

screening

parameter

Ka. From top to bottom _Ka

= 0, _Ka

= 0.025 and _Ka

=

0.05.

various values of the

screening _parameter

_«a. A similar

plot

^is

presented

in

figure

6. As

expected,

the

slope

at the

origin,

_or

equivalently

the blob

size,

is _not affected

by

the

screening provided

the

screening length

is

larger

than f. The linear behaviour that in the absence of

screening

_was observed up ^{to n} = N/2 is on the other hand

strongly

affected. A deviation from

lo

a

6

4

z

~o

_{5 lo 15 zo}

n

Fig.

6. h(n) _{as a} function of _n for _u

= 0.1, N

= 200, and various values of the

screening

_parameter

Ka. From top to bottom _Ka

= 0, _Ka

= 0.025 and _Ka

= 0.05. Only ^{the small} _n part ^{of the} curve is presented in order _to emphasize ^the

rapid

deviation from linear behaviour.

this behaviour is observed _{as soon as} h

(n)

becomes of the order of

magnitude

of the inverse of the

screening

_parameter. This result is at variance with the

prediction

of

[8],

both

qualitatively

and

quantitatively.

A numerical estimate of the

persistence length predicted by (6)

for _a

screening

_{parameter «a}

=

0.05 is e.g.

i~~

₌₁₉₀

a. The

typical

end-to-end distance of the chain in the absence of

screening

is 170 _a. Therefore

(6)

would

suggest

^that for such _a weak

screening,

the chain remains in the _same rodlike

configuration

it had in the absence of

screening.

This is

clearly

not what is observed in _our simulation~

The variation with

screening

of the end-to-end distance of _a 200 _monomers chain and its structure factor _are

compared

_to the

predictions

of the Gaussian

approximation

in

figures

7 and

8. The numerical agreement between the simulation and variational results is fair. The

structure factor, like in the unscreened _case, exhibits _a q~ ^{I behaviour in} the intermediate q

region.

An estimate of the

persistence length

could in

principle

be obtained

by analysing

this

q~ regime.

Such _an

analysis

however is much less informative than the _{one we} did of the intemal

distances, mainly

because the _structure factor is obtained

by sampling

_a

relatively

small number of wavevectors and

consequently

is not

extremely

accurate.

60

e~

k

zo

o

5

l(<a)

Fig.

7. End-to-end distance of _a 200 _monomer chain _{as a} function of the

screening

_K ~. The value of the

coupling

parameter ^{is 0. I. The} squares are simulation results, the full line

corresponds

to the Gaussian

vwiational method.

5. Discussion and summary.

From the results

presented

in the above section, _we conclude that the

picture

of _an

«electrostatic

persistence length»

of the flexible

polyelectrolyte

that would scale like

«

~ is incorrect for the model _we consider.

Rather,

_we find that

screening

starts

affecting

the chain

configuration

at

length

scales

larger

^than _{« ~,} in agreement ^with

Pfeuty's theory.

This conclusion is based _on the

qualitative

observation that deviations from the unscreened

m

o i _.

m

m

~ _w

~

10~~

"

m

~

~i _{_3}

10

" ~

fl m

_~

m lo

lo

qa

Fig. 8. - tructure factor f

results,

slope -1.

the OSF

argument only

considers the influence of electrostatic interactions _on chain

configurations

that _are rodlike _over scales

longer

than «~ _~. In _a very flexible chain, ^these

configurations

form in the absence of

charges

_{a very} small subset of all

possible configurations.

Therefore it _seems unreasonable to take

only

^these

configurations

into _{account to}

analyze

the effect of

introducing

the screened Coulomb interaction.

In summary, our simulations

clearly

validate the

scaling picture

of _a

weakly charged

polyelectrolyte being,

in the absence of

screening,

_a linear

string

of _« electrostatic blobs _».

Screening

_{causes a} deviation from

linearity,

which _we take _as indicative of _{a crossover to}

excluded volume

behaviour,

for

length

scales

larger

than the

screening length.

The

generalization

of the

Odjik-SkoInick-Fixman

^formula to flexible

polyelectrolytes proposed by

Khokhlov and Khachaturian _seems therefore not to be

justified.

The _reasons for this failure

were discussed

qualitatively. Finally,

_we have shown that variational methods

yield

quantitatively

correct results for short chains. For

long

_range

forces,

these methods _are also known to be accurate in the

asymptotic

limit of

long

chains.

They

could therefore prove to be useful tools for the

study

^of

charged polymer

systems.

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