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
DebyeHückel potential, by employing a novel hybrid scheme of molecular dynamics (MD) and Monte Carlo (MC) simulations. Although
the offlattice 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 offlattice pivot step in early stage only, and then
switch the computation to a pure MD step. The result shows that the computational speedup 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 endtoend 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
 Citations (1)
 Cited In (0)

Article: Partitioning and conformational behavior of polyelectrolytes confined in a cylindrical pore
[Show abstract] [Hide abstract]
ABSTRACT: We study the behavior of a uniformly charged polyelectrolyte confined in a circular cylindrical pore with constant surface potentials. From Green's function theory with an effective steplength renormalized by monomermonomer interactions, the partition coefficient and associated chain conformations are predicted in a variety of situations. Depending upon the ionic strength of surrounding fluids, the polyelectrolyte conformation may be stretched and follow a selfavoiding walk, yielding a significant reduction in the partition coefficient compared to the ideal chain result. The monomermonomer interaction is found to be as important as the polymerpore interaction in determining the partitioning behavior, especially in the longchain regime, due to the characteristics of confined spaces.Macromolecules. 01/2000; 33(23).
Page 1
Macromolecular Research, Vol. 10, No. 6, pp 297303 (2002)
Macromolecular Research, Vol. 10, No. 6, pp 297303 (2002)
*email : mschun@kist.re.kr
15985032/12/29707?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
MyungSuk Chun* and Hyun Su Lee
Complex Fluids Team, Korea Institute of Science and Technology(KIST),
PO Box 131, Cheongryang, Seoul 130650, 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 DebyeHückel potential, by employing a novel hybrid scheme of molecular dynamics
(MD) and Monte Carlo (MC) simulations. Although the offlattice 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 offlattice pivot step in early stage only, and then switch the computation to a pure MD step. The result
shows that the computational speedup 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 endtoend 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,13 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, sizeexclusion 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
microbiochips 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 longrange interactions, the
large number of degrees of freedom of the counterions 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 pearlnecklace 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 DebyeHückel (DH) potential so
as to include the screening effect resulting from salts as well
as counterions. They predicted the persistence length and
important conformational properties (e.g., radius of gyra
tion, endtoend 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 DH potential, the
finitely extendable nonlinear elastic (FENE) potential and
generalized LennardJones (LJ) potential were adopted for
bonding attraction and nonelectrostatic 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 offlattice pivot algorithm to properly equilibrate
both scales of short and longranges. 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 counterion
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 counterion
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 by6,1315
ξ ξit ( )
(1)
where m is the mass of monomer, ri the position vector of ith
monomer, the Hamiltonian of ith 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 fluctuationdissipation theorem as,
, assuming Gaussian white noise.6,15 Here,
kB is the Boltzmann constant, T the absolute temperature, δij
the Kronecker delta, and δ(tt?) the Dirac delta function
which shows the autocorrelation of the time scale.
The polyelectrolyte is represented as N freelyjointed
beadschain 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 DH
(equivalently, Yukawa) potential, which yields
(2)
where b =
monomers, rij the distance between ith and jth 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 righthand 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 Offlattice Pivot Algorithm. We
employed the offlattice 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 selfavoiding 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 offlattice
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., DebyeHückel) 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
largescale structure. However, we found that the random
largescale 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 LJ
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
torodlike transition parameter β (
in Figure 1 involves the conformational information, where
both the endtoend 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 offlattice 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 largescale 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 endtoend distance
and radius of gyration as the hybrid simulation proceeds for N =
128, b = 2.0, κ1= 100, and λB= 1.0, restarting from the equilib
rium value. For the case of tolerance of 1?103, 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 endtoend distance REnd and the
radius of gyration RG, defined as follows,
,(8)
(9)
where rCM is the centerofmass 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 selfavoiding 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(? ?kski) has a mag
nitude
wavelength in the dispersion medium, and θ the scattering
angle.19 We can then define the sphericallyaveraged 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 sphericallyaveraged 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
counterions in solvents. In the region of wave number q
ranged 2?102 to 2, the SSP(q) function depends on the
Debye length κ1 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 endtoend distance
and radius of gyration for different chain dimensions (N = 16, 32,
64, 128) with b = 2.0, κ1= 100, and λB= 1.0.
Figure 3. The comparisons of sphericallyaveraged structure
factor at different chain dimensions (N = 32, 64, 128) with b =
2.0, κ1= 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
κ1. 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 selfavoiding 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 POVRay 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 κ1), 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
monomertomonomer 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 longrange 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
monomertomonomer 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 offlattice 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 largescale 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) κ1= 2.1, (b) κ1= 5.0, (c) κ1= 20.0,
and (d) κ1= 100.0, with N = 128, b = 2.0, and λB= 1.0. Although
all of the monomers are charged, graycolored monomers are
applied to clearly enhance 3dimensional visualizations. The
counterions are virtually indicated as small darkgray spheres.
Figure 4. The variations of sphericallyaveraged structure factor
at different Debye lengths κ1, 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
freelyjointed beads and chain have been modeled by the
harmonic spring potential coupled with DH potential (cf.,
the solution of the linearized PoissonBoltzmann equation).
Simulation results of the polyelectrolyte conformation were
achieved with various Debye length κ1 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. R012001000004110)
from the KOSEF. M.S. C. especially thanks to Dr. C. Holm
and Director Professor K. Kremer at the Theory Group of
MaxPlanck Institute for Polymer Research at Mainz, for
valuable discussions during the visiting research funded by
the DFG as well as the KOSEF. Regarding the offlattice
pivot algorithm, the basic code developed by the Theory
Group was employed in our program.
Nomenclatures
A
b
e
g
: rotational transformation matrix []
: monomertomonomer 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 : endtoend distance []
RG
: radius of gyration []
rCM : dimensionless centerofmass vector of polyelectrolyte
chain []
ri
: dimensionless position vector of ith 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 κ1= 100.0.
The coloring is as in Figure 5.
Figure 6. The variations of sphericallyaveraged structure factor at
different Bjerrum lengths λB, with N = 64, b = 2.0, and κ1= 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 ith and jth monomers []
: sphericallyaveraged 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 []
: LJ parameter of interaction energy [J]
: wave length in the dispersion medium [m]
: Bjerrum length []
: dimensionless random force []
: LJ parameter of monomer diameter [m]
: characteristic time
Mathematical
δij
δ
: Kronecker delta
: Dirac delta function
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