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 10/2003; 11(5):393-397. · 1.68 Impact Factor
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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
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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
〈〉
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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
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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.
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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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