# Investigations on the chain conformation of weakly charged polyelectrolyte in solvents by using efficient hybrid molecular simulations

**ABSTRACT** We have investigated the microstructural properties of a weakly charged polyelectrolyte modeled with both Hookean spring and

Debye-Hückel potential, by employing a novel hybrid scheme of molecular dynamics (MD) and Monte Carlo (MC) simulations. Although

the off-lattice pivot step facilitates the earlier computations stage, it gives rise to oscillations and hinders the stable

equilibrium state. In order to overcome this problem, we adopt the MC off-lattice pivot step in early stage only, and then

switch the computation to a pure MD step. The result shows that the computational speed-up compared to the previous method

is entirely above 10 to 50, without loss of the accuracy. We examined the conformations of polyelectrolyte in solvents in

terms of the end-to-end distance, radius of gyration, and structure factor with variations of the screening effects of solvent

and the monomer charges. The emphasis can favorably be given on the elongation behavior of a polyelectrolyte chain, with observing

the simultaneous snapshots.

Keywordspolyelectrolyte–molecular dynamics–Monte Carlo simulation–chain conformation–structure factor

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**ABSTRACT:**By applying a configurational-bias Gibbs ensemble Monte Carlo algorithm, priority simulation results regarding the conformation of non-dilute polyelectrolytes in solvents are obtained. Solutions of freely-jointed chains are considered, and a new method termed strandwise configurational-bias sampling is developed so as to effectively overcome a difficulty on the transfer of polymer chains. The structure factors of polyelectrolytes in the bulk as well as in the confined space are estimated with variations of the polymer charge density.Macromolecular Research 01/2003; 11(5):393-397. · 1.64 Impact Factor

Page 1

Macromolecular Research, Vol. 10, No. 6, pp 297-303 (2002)

Macromolecular Research, Vol. 10, No. 6, pp 297-303 (2002)

*e-mail : mschun@kist.re.kr

1598-5032/12/297-07?2002 Polymer Society of Korea

Macromolecular Research

Volume 10, Number 6 December 31, 2002

? Copyright 2002 by The Polymer Society of Korea

Investigations on the Chain Conformation of Weakly Charged Polyelectrolyte

in Solvents by Using Efficient Hybrid Molecular Simulations

Myung-Suk Chun* and Hyun Su Lee

Complex Fluids Team, Korea Institute of Science and Technology(KIST),

PO Box 131, Cheongryang, Seoul 130-650, Korea

Received Aug. 8, 2002; Revised Dec. 2, 2002

Abstract : We have investigated the microstructural properties of a weakly charged polyelectrolyte modeled with

both Hookean spring and Debye-Hückel potential, by employing a novel hybrid scheme of molecular dynamics

(MD) and Monte Carlo (MC) simulations. Although the off-lattice pivot step facilitates the earlier computations

stage, it gives rise to oscillations and hinders the stable equilibrium state. In order to overcome this problem, we

adopt the MC off-lattice pivot step in early stage only, and then switch the computation to a pure MD step. The result

shows that the computational speed-up compared to the previous method is entirely above 10 to 50, without loss of

the accuracy. We examined the conformations of polyelectrolyte in solvents in terms of the end-to-end distance,

radius of gyration, and structure factor with variations of the screening effects of solvent and the monomer charges.

The emphasis can favorably be given on the elongation behavior of a polyelectrolyte chain, with observing the simul-

taneous snapshots.

Keywords : polyelectrolyte, molecular dynamics, Monte Carlo simulation, chain conformation, structure factor.

Introduction

Polyelectrolytes, one of the typical complex fluids, are

polymers bearing ionizable groups which can dissociate

into charged polymer chains in polar solvents. Due to their

fascinating conformational changes in solvents, polyelectro-

lytes have always stimulated interests from a fundamental as

well as from a technological point of view. One of the main

technologically important properties of polyelectrolytes is

that they dissolve in water due to the electrostatic repulsion

between charged monomers,1-3 even though water is a poor

solvent for most of synthetic polymers. Especially, the

behavior of polyelectrolytes in confined spaces is connected

with numerous applications including separations with gel

electrophoresis, size-exclusion chromatography, and porous

membrane as well as transports within vascularized spaces,

renal glomerular channel, and other biological media.4,5 Very

Page 2

M. -S. Chun and H. S. Lee

298 Macromol. Res., Vol. 10, No. 6, 2002

recently, an understanding of polyelectrolytes has become

increasingly important in the design of microchannels for

micro-biochips and micro/nanofluidic devices.

However, the theoretical understanding of polyelectrolytes

is less developed than that of the properties of neutral polymers.

In addition to the occurrence of long-range interactions, the

large number of degrees of freedom of the counter-ions and

their fluctuations form more difficult problem. A comparison

between experiment and theory is often very difficult because

of different regions of validity concerning the density of the

solution. Scattering experiments need a certain contrast and

therefore it is almost impossible to study extremely dilute

solutions, which are the topic of most theories. In such a sit-

uation, computer simulations have the important possibility

to build a bridge between theory and experiment as they can

test theoretical aspects as well as experimentally measurable

quantities under well controllable conditions.6,7

There exist a good number of papers on the properties of

polyelectrolyte in solvents. In late 1990s, Dobrynin et al.2

developed a scaling theory that describes how, with varying

solvent quality or charge on the chain, the polyelectrolyte in

poor solvents undergoes a cascade of abrupt transitions

between pearl-necklace configurations with different numbers

of beads.8 They addressed that the length of the necklace

globule is proportional to the total polymer charge by using

Monte Carlo (MC) simulations. Micka and Kremer7 exam-

ined the polyelectrolyte conformation by exploiting hybrid

of Molecular Dynamics (MD) and MC algorithm. In their

study, polyelectrolytes were described with both the

Hookean model and the Debye-Hückel (D-H) potential so

as to include the screening effect resulting from salts as well

as counter-ions. They predicted the persistence length and

important conformational properties (e.g., radius of gyra-

tion, end-to-end distance, and structure factor) for various

chain lengths, Debye lengths, and so on.9,10 Recently, the

hybrid algorithm was utilized for more complicated problems

reported by Lyulin et al..11 Besides the D-H potential, the

finitely extendable nonlinear elastic (FENE) potential and

generalized Lennard-Jones (L-J) potential were adopted for

bonding attraction and non-electrostatic interaction, respec-

tively. They estimated the theta transition point and accounted

for the conformational results of Dobrynin et al.2 consistently

acquired with variations of the monomer charge fraction.

The purpose of the present study lies on a development of

the efficient hybrid architecture of MD and MC scheme to

quantitatively predict the microstructural properties of poly-

electrolytes by modifying the previous algorithm reported by

Micka and Kremer.7 Our scheme uses the Langevin dynamics

and the off-lattice pivot algorithm to properly equilibrate

both scales of short and long-ranges. We are faithfully guar-

anteed both rightness and effectiveness of the newly devel-

oped algorithm by comparing with the previous results.7,11

Simulation results on the conformational properties of poly-

electrolytes are presented according to the physicochemical

variations of, inter alia, the Debye length and the Bjerrum

length that we consider to be the important quantities for the

experimental scientists.

It would be noted that an explicit treatment of counter-ion

condensation originally proposed by Manning12 has been

studying as a recent work. Since this work is beyond the

present study, we do not address more unambiguous consid-

eration of the charging mechanism to describe the counter-ion

condensation.

Hybrid Architecture of Molecular Dynamics (MD)

and Monte Carlo (MC) Algorithm

MD Coupling to a Heat Bath. We used a standard velocity

Verlet algorithm in order to integrate the equations of

motion. Considering the frictional force and random force

, the equation of motion for monomer i is given by6,13-15

ξ ξit ( )

(1)

where m is the mass of monomer, ri the position vector of i-th

monomer, the Hamiltonian of i-th monomer, and Γ the

frictional coefficient that couples the monomers to the heat

bath. The random force is related to the frictional coe-

fficient via a fluctuation-dissipation theorem as,

, assuming Gaussian white noise.6,15 Here,

kB is the Boltzmann constant, T the absolute temperature, δij

the Kronecker delta, and δ(t-t?) the Dirac delta function

which shows the autocorrelation of the time scale.

The polyelectrolyte is represented as N freely-jointed

beads-chain to take into account the excluded volume effect.

The bonds between neighboring beads are described by the

Hookean potential with harmonic springs. The Hamiltonian

appeared in eq. (1) includes the Hookean model and D-H

(equivalently, Yukawa) potential, which yields

(2)

where b =

monomers, rij the distance between i-th and j-th monomers,

and the Bjerrum length λB(i.e., in dimensional, e2/4πεkBT) is

a measure of the strength of the electrostatic interaction.

The dynamics of the solvent (i.e., water) is based on a con-

tinuum approach, and its dielectric constant ε is computed

using a relative permittivity as 78.5 taken at 298 K. Although

free ions are not included explicitly in the simulation, their

effect is described via the dependence of the inverse Debye

screening length κ (i.e., in dimensional,

the electrolyte concentration. Here, e represents the elemen-

tary charge, Zi the amount of charge on monomer i, and ni

the ionic concentration.

The second term of the right-hand side of eq. (1) models a

is the bond length between the adjacent

) on

md2ri

dt2

------- -

i

∇

–=

Γdri

dt

----- -

–

ξ ξit ( )

+

i

ξ ξit ( )

ξ ξit ( ) ξ ξjt' ( )⋅〈〉

6kBTδijδ t t'

–

()Γ

≡

3kBT

2b2

-------------- - ri ri 1

(

+

–

)2

λBkBT

1

κrij

rij

– ()

exp

----------------------- -

j

=

i 1–

∑

i

2=

N

∑

+

i

1=

N 1–

∑

=

b2

〈〉

Page 3

Hybrid Molecular Simulations of Polyelectrolyte Chain

Macromol. Res., Vol. 10, No. 6, 2002299

frictional damping of the solvent, which is proportional to

the temporal change of the position vector. The third one

mimics the random collisions with solvent molecules.15,16

The equation of motion is then integrated with velocity Verlet

algorithm,6,17 can be written

, (3)

,(4)

.

(5)

MC Scheme with Off-lattice Pivot Algorithm. We

employed the off-lattice pivot algorithm in MC simulations

with the canonical ensemble. This method is to allow the

simulation to be quite efficient for a single chain, and offers

all possible self-avoiding walks conformation with equal

probability.6 We choose some site ωk along the walk as a

pivot point, and apply some symmetry operation of the lat-

tice to the part of the walk subsequent to the pivot point.

The proposed new walk is

(6)

where Ξ is the chosen symmetry operation. In the off-lattice

pivot move, a pivot point among the chain monomers is

chosen at random, and three dimensional rotating angles are

also determined randomly to elaborate the new conforma-

tion, as follows

.

(7)

Here, A is the rotational transformation matrix that includes

the sinusoidal functions of random rotation angles α1, α2, and

α3. After pivoting, a classical Metropolis algorithm17 deter-

mines the adoption or rejection of the new conformation by

comparing a screened Coulomb (i.e., Debye-Hückel) inter-

action energy.

Simulation Scheme and Runs. In the previous study, the

local structure was initially equilibrated by 105 MD steps,

and then the overall structure was relaxed by adding 5? 104

pivot moves. And subsequently, both MD and MC were

completely mixed to generate the final conformation via

1~3? 107 steps. In the process of the hybrid of MD and MC

proposed by Kremer and coworkers,7,11 the MC pivot step

rendering the initial bond stretching helps to generate the

large-scale structure. However, we found that the random

large-scale pivot moves surely result in a difficult procedure

in finding an accurate equilibrium point. For the present

simulations, the variables are nondimensionalized by the

monomer diameter σ = 1.0 for length scale as well as the

characteristic time τ = (mσ2/Λ)1/2 for time scale at constant

temperature kBT = 1.0Λ, where both σ and Λ denote the L-J

potential parameters.6

In Figure 1, we present two cases of totally different results

obtained from different accepting tolerances for the pivot

moves while the MD computation is running. In the process

of MD steps, the MC computations are performed if the ran-

dom number is smaller than the tolerance. Note that the

equilibrium configuration can be more sustained as the tol-

erance decreases (i.e., narrow acceptance bandwidth). A coil-

to-rodlike transition parameter β (

in Figure 1 involves the conformational information, where

both the end-to-end distance and the radius of gyration will

be explained in the next section. In Figure 1, we restarted the

hybrid of MD and MC step with two accepting tolerances of

MC pivot moves from the fully converged equilibrium state as

an initial condition. This result presents that after equilibrium

state achieved by MD computations, the MC off-lattice pivot

step rather hinders the convergence. Therefore, we reasonably

start with 103 MD steps to allow for initial bond stretching

and the 104 pivot steps equilibrate the large-scale structure.

Then pure MD time integration step of about 2? 105~106 is

carried for final conformation which corresponds to 1/10~1/50

configuration number of the previous study.11

Two initial conformations of the random walks and the

totally stretched states are used to set for comparison, and

then the equilibrium state is determined when both confor-

mations finally produce the same β for several quantities.

We performed each of 6 independent computations for each

) estimated

〉

dr

dt

---- -

n 1 2 ⁄

+

dr

dt

---- -

n

=

m

1

t ∆

2

–

---- -

rn

( )∇

–

Γdr

dt

---- -

n

–

ξn

++

rn 1+

rn

=

t ∆

+

dr

dt

---- -

n 1 2 ⁄

+

dr

dt

---- -

n 1+

dr

dt

---- -

n 1 2 ⁄

+

=

m

1

t ∆

2

–

---- -

rn 1+

()∇

–

Γdr

dt

---- -

n 1+

–

ξn 1+

++

ωi′

ωi

ωk Ξ ωi ωk

(

+

for

for

0 i k

≤ ≤

k 1+–

)

i N

≤ ≤

=

r

∆

A α1α2α3

,(,)

rx

∆[

ry

∆

rz

∆,,]T

=

REnd

2

〈〉

RG

2

〈⁄≡

Figure 1. Characteristic mean square ratio of end-to-end distance

and radius of gyration as the hybrid simulation proceeds for N =

128, b = 2.0, κ-1= 100, and λB= 1.0, restarting from the equilib-

rium value. For the case of tolerance of 1?10-3, equilibrium state

is not sustained.

Page 4

M. -S. Chun and H. S. Lee

300 Macromol. Res., Vol. 10, No. 6, 2002

of two initial conformations, from which total 12 independent

states are acquired with statistical errors less than 4%. Our

program yields the run time of about 5 hours on a Pentium

III processor for each case.

Results and Discussion

Conformation Properties and Structure Factor. The

characteristic quantities for the conformation of polyelectro-

lyte chains can be the end-to-end distance REnd and the

radius of gyration RG, defined as follows,

,(8)

(9)

where rCM is the center-of-mass position vector. A character-

istic relative stretching ratio β is 6 for the ideal chain with

random walks, 12 for the totally stretched state, and a value

around 6.3 for the self-avoiding walks. Figure 2 shows a

relationship between β and the chain dimension N. As the

number of monomers increases, a polyelectrolyte chain

tends to stretch because once the chain length increases the

screening effect decreases for the given value of Debye

length. Our simulation results are in a good agreement with

those of previous study.7

The structure factor represents an important microstructural

information that gives all length scale data, and makes the

theoretical prediction comparable to the experimental data.3,7,18

Considering the scattering from two particles, the incident

beam propagates along the vector ki and the scattered beam

along ks. The scattering wave vector q(? ?ks-ki) has a mag-

nitude

wavelength in the dispersion medium, and θ the scattering

angle.19 We can then define the spherically-averaged structure

factor SSP(q) for the scattering wave vector q in the spherical

coordinate, which is the Fourier transform of the pair corre-

lation function g(r), expressed as

, where λ is the

.

(10)

The spherically-averaged structure factor is eventually

derived as a function of scattering wave number q, given by 3,18

.(11)

An important fact to be noted here is that the chain length

dependency appears in the small q region as displayed in

Figure 3. This is because the correlation between monomers

increases with decreases of the wave number q, owing to the

long wave length of scattering wave vector. To the contrary,

the structure factor converges to unity because as the q

increases it does not depend on the monomer correlation,

which can easily be confirmed by eq. (11). The rightness of

our algorithm can be verified from a good agreement with

the previous results.7

Screening Effect of the Solvent. In Figures 4 and 5, we

investigate the screening effect caused by the salt and

counter-ions in solvents. In the region of wave number q

ranged 2?10-2 to 2, the SSP(q) function depends on the

Debye length κ-1 with a constant slope, which is related to

the chain conformations in terms of collapsed, totally

REnd

〈〉

rN r1

–

()2

〈〉

=

RG

〈〉

1

N

--- -

ri rCM

–

()2

i

1=

N

∑

〈〉

=

q

2 k

θ 2 ⁄()

sin

4π λ ⁄()θ 2 ⁄()

sin

==

SSPq ( )

1

N

--- -

iq

–

ri rj

–

()⋅()

exp

i j

<

N

∑

2

≡

1

N

--- - dr

∫

iq r ⋅()

exp

g r ( )

=

SSPq ( )

1=

1

N

--- -

qrij

qrij

()

sin

---------------------- -

i j ,

N

∑

+

Figure 2. Characteristic mean square ratio of end-to-end distance

and radius of gyration for different chain dimensions (N = 16, 32,

64, 128) with b = 2.0, κ-1= 100, and λB= 1.0.

Figure 3. The comparisons of spherically-averaged structure

factor at different chain dimensions (N = 32, 64, 128) with b =

2.0, κ-1= 100, and λB= 1.0. Solid curves correspond to results of

the present hybrid algorithm, and symbols indicate those of Micka

and Kremer.7

Page 5

Hybrid Molecular Simulations of Polyelectrolyte Chain

Macromol. Res., Vol. 10, No. 6, 2002301

stretched, and intermediate states. On the exterior of this

region of the q, however, structure factors do not depend on

κ-1. The predicted structure factor can properly be compared

with the scattering experiments. When a magnitude of the

slope between q and SSP(q) is small, a polyelectrolyte chain

is expanded with the self-avoiding walks conformation in

good solvent condition. As the slope increases a chain expe-

riences the random walks in theta solvent condition that is a

balanced state between the attraction and the repulsion, and

ultimately the poor solvent condition with collapsed globular

conformation is obtained.

Figure 5 visualized with a software POV-Ray Version 3.1

(cf., http://www.povray.org) provides the same trend as dis-

cussed above. As the screening effect decreases (i.e., larger

value of κ-1), a chain starts to elongate due to the strong

repulsion, and finally forms a stretched equilibrium state.

Monomer Charge Effect. The structure factors for various

Bjerrum lengths λB were estimated to investigate a relationship

between the chain conformation and the charge of monomers.

In Figure 6, as the Bjerrum length increases the magnitude

of the slope between q and SSP(q) decreases due to the

monomer-to-monomer repulsion. Figure 7 shows snapshots of

the conformational structure with different Bjerrum lengths,

and the almost spherical globule is observed in Figure 7(a).

In cases of λB with 0.25 and 0.5, a long-range structure

becomes elongated, with which local globules can also be

found amongst polyelectrolyte chain. Finally, a chain

becomes to the fully elongated state due to the stronger

monomer-to-monomer repulsion when λB equals unity.

Conclusions

An efficient hybrid algorithm of MD and MC was devel-

oped to predict the polyelectrolyte conformation. In the

present algorithm, the MC off-lattice pivot step was adopted

just in early stage and then a pure MD step followed. As a

consequence, we could reasonably exclude the unstable

oscillations due to random large-scale pivot moves which

caused a difficult procedure in the previous study. Comparing

Figure 5. Snapshots of the equilibrium chain conformations at dif-

ferent Debye lengths of (a) κ-1= 2.1, (b) κ-1= 5.0, (c) κ-1= 20.0,

and (d) κ-1= 100.0, with N = 128, b = 2.0, and λB= 1.0. Although

all of the monomers are charged, gray-colored monomers are

applied to clearly enhance 3-dimensional visualizations. The

counter-ions are virtually indicated as small dark-gray spheres.

Figure 4. The variations of spherically-averaged structure factor

at different Debye lengths κ-1, with N = 64, b = 2.0, and λB= 0.25.

Page 6

M. -S. Chun and H. S. Lee

302 Macromol. Res., Vol. 10, No. 6, 2002

with the previous scheme, it is evident that the computa-

tional configuration number is quite reduced, and a right-

ness of the algorithm is verified.

For a weakly charged polyelectrolyte in solvents, the

freely-jointed beads and chain have been modeled by the

harmonic spring potential coupled with D-H potential (cf.,

the solution of the linearized Poisson-Boltzmann equation).

Simulation results of the polyelectrolyte conformation were

achieved with various Debye length κ-1 and Bjerrum length

λB. We view the present investigation as a promising first

step, which we will extend to the problem of polyelectro-

lytes in confined spaces in the future.

Acknowledgement. This study was supported by the

Basic Research Fund (Grant No. R01-2001-000-00411-0)

from the KOSEF. M.-S. C. especially thanks to Dr. C. Holm

and Director Professor K. Kremer at the Theory Group of

Max-Planck Institute for Polymer Research at Mainz, for

valuable discussions during the visiting research funded by

the DFG as well as the KOSEF. Regarding the off-lattice

pivot algorithm, the basic code developed by the Theory

Group was employed in our program.

Nomenclatures

A

b

e

g

: rotational transformation matrix [-]

: monomer-to-monomer bond length [-]

: elementary charge [Coul]

: pair correlation function [-]

: Hamiltonian [J]

: wave vector [-]

: Boltzmann constant [J/K]

: dimensionless monomer mass [-]

: number of monomers [-]

: ionic concentration [1/m3]

k

kB

m

N

ni

q

REnd : end-to-end distance [-]

RG

: radius of gyration [-]

rCM : dimensionless center-of-mass vector of polyelectrolyte

chain [-]

ri

: dimensionless position vector of i-th monomer [-]

: scattering wave number [-]

Figure 7. Snapshots of the equilibrium chain conformations at dif-

ferent Bjerrum lengths of (a) λB= 0.0 (neutral), (b) λB= 0.25, (c)

λB= 0.5, and (d) λB= 1.0, with N = 128, b = 2.0, and κ-1= 100.0.

The coloring is as in Figure 5.

Figure 6. The variations of spherically-averaged structure factor at

different Bjerrum lengths λB, with N = 64, b = 2.0, and κ-1= 5.0.

Page 7

Hybrid Molecular Simulations of Polyelectrolyte Chain

Macromol. Res., Vol. 10, No. 6, 2002 303

rij

SSP

T

t

Zi

: distance between i-th and j-th monomers [-]

: spherically-averaged structure factor [-]

: absolute temperature [K]

: dimensionless time [-]

: amount of charge on monomer i [-]

Greek Letters

β

Ι

ε

θ

κ

Λ

λ

λB

ξ ξ

σ

τ

: characteristic relative stretching ratio [-]

: dimensionless frictional coefficient [-]

: dielectric constant or permittivity [Coul2/J ⋅ m]

: wave angle [deg]

: inverse Debye length [-]

: L-J parameter of interaction energy [J]

: wave length in the dispersion medium [m]

: Bjerrum length [-]

: dimensionless random force [-]

: L-J parameter of monomer diameter [m]

: characteristic time

Mathematical

δij

δ

: Kronecker delta

: Dirac delta function

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