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Conformational behaviors of a charged-neutral star micelle
in salt-free solution
Mingge Deng,ab Ying Jiang,ab Xuejin Li,*ab Lei Wangab and Haojun Liang*ab
Received 18th November 2009, Accepted 5th March 2010
First published as an Advance Article on the web 20th April 2010
DOI: 10.1039/b924281c
The conformational behaviors of charged brushes on a micelle self-assembled by charged-neutral
diblock copolymers in salt-free solution are extensively analyzed using a coarse-grained dissipative
particle dynamic (DPD) simulation. When only monovalent counterions exist, the brush
conformation of the corona in the micelle is exactly consistent with the predictions from the
blob-scaling theory based on the spherical polyelectrolyte brush model, which differentiates the
system into three distinct regimes: (I) quasi-neutral regime, (II) ‘‘Pincus’’ regime, and (III) osmotic
regime. For multivalent counterions such as divalence and trivalence, however, the strong
electrostatic correlations lead the micelle structures to deviate obviously from those of scaling
predictions. The collapse of the brush appears to be due to the drop in the osmotic pressure
inside the corona region of the micelle.
1 Introduction
The conformations and physical properties of polyelectrolyte
brushes are of great interest because of their fundamental
significance and potential applications in the field of bio- and
nanotechnology.1–3 In particular, spherical polyelectrolyte
brushes (SPBs) with long polyelectrolyte chains grafted onto
a solid core in the size of colloidal dimensions have been
widely utilized as novel carrier particles for functional bio-
molecules.4–6 Generally, photo-emulsion polymerization7 and
controlled radical polymerization8 are two normal methods of
manufacturing the SPB. Recently, an efficient and easy ap-
proach has been developed by simply dissolving charged-
neutral diblock copolymer in appropriate solvents,9 whereby
micelles with neutral cores and charged hairs, so-called
charged-neutral star micelles, were produced. The distinctively
responsive properties of this type of polymer brush and its
potential applications in industry have been extensively
investigated previously in terms of transition mechanisms
in a controlled environment.10 It is very likely that the
comprehensive investigations on the SPBs, in comparison with
neutral polymeric brushes, may offer us an opportunity to
understand deeply the manner of conformational transformation
of a polymeric brush. In the past decades, there has been active
interest in this field, and it has attracted the attention of many
experimental and theoretical scientists.11–20 However, owing
to fact that the performances of the charged chains are
governed mutually by multiple parameters such as ionic
strength, electrostatic interaction, and valence of counterions,
the system is expected to respond in a complicated manner
during stimulations emanating from the circumstance. Up to this
date, an understanding of this sort of system is far from
complete, and many problems are still left to challenge us. To
comprehend the conformational behaviors of this brush system,
we studied the conformational transitions of charged-neutral star
micelles built with charged-neutral diblock copolymers using a
dissipative particle dynamics (DPD) simulation.
2 Model and method
2.1 Dissipative particle dynamics formulation
We study the conformational transitions of charged-neutral
star micelles in salt-free solution with the help of the DPD
simulation technique. DPD is a simple but intrinsically
promising simulation method that allows the study of the
conformational behaviors of charged-neutral block copolymers.21
In DPD simulation, a particle represents the center of mass of
a cluster of atoms, and the position and momentum of the
particle is updated in a continuous phase but spaced at discrete
time steps. Particles i and j at positions ri and rj interact with
each other via a pairwise additive force, consisting of three
components: (i) a conservative force, FCij ; (ii) a dissipative
force, FDij ; and (iii) a random force, FR
ij . All forces are
non-zero within a cut-off radius rc. Hence, the total force on
particle i is given by
Fi ¼Xiaj
FCij þ FD
ij þ FRij ð1Þ
where the sum acts over all particles within rc. Specifically, in
our simulations
Fi ¼Xiaj
aijoðrijÞnij � go2ðrijÞðnij � vijÞnij þ soðrijÞzijDt�1=2nij
ð2Þ
where aij is a maximum repulsion between particles i and j, rij is
the distance between them, with the corresponding unit vector
a CAS Key Laboratory of Soft Matter Chemistry, Department ofPolymer Science and Engineering, University of Science andTechnology of China, Hefei, Anhui 230026, People’s Republic ofChina. E-mail: [email protected], [email protected]
bHefei National Laboratory for Physical Sciences at Microscale,University of Science and Technology of China, Hefei,Anhui 230026, People’s Republic of China
This journal is �c the Owner Societies 2010 Phys. Chem. Chem. Phys., 2010, 12, 6135–6139 | 6135
PAPER www.rsc.org/pccp | Physical Chemistry Chemical Physics
nij, vij is the difference between the two velocities, zij is a
random number with zero mean and unit variance, and gand s are parameters coupled by s2 = 2gkBT. The weight
function o(rij) is given by
oðrijÞ ¼1� rij=rc rijorc0 rij � rc
�ð3Þ
The standard values s=3.0 and g=4.5 are used in our study.
By joining consecutive particles with a spring force, we
can construct coarse-grained models of polymers.22,23 The
harmonic spring force with a spring constant ks = 10.0 and
an equilibrium bond length a0 = 0.86 in our simulations has
the form,
FSij = ks(1 � rij/a0)nij. (4)
The total force can also have an electrostatic contribution,
which is derived from the electrostatic field solved locally on a
grid. For the dimensionless Poissons equation which is scaled
with DPD length and energy,
r(p(r)rc(r)) = �lBr(r), (5)
where lB = e2/(kBTe) is the Bjerrum length that measures the
distance at which two charged particles interact with each
other with thermal energy kBT, and p(r) is the dielectric
permittivity related to the value in pure water. With an
iterating algorithm,24,25 we can solve the field equation
successfully, then the electrostatic force on charged particle i
is given by
FEi ¼ �qi
Xj
rcðrjÞ1� jrj � rij=RePj0 ð1� jrj0 � rij=ReÞ
" #( )ð6Þ
where Re is the smearing radius and rj is the grid position for
smearing out this point charge. Details on the grid method
that we used are available elsewhere.25
The simulations are performed using a modified version of the
DPD code named MYDPD.26,27 Time integration of motion
equations is calculated by a modified velocity–Verlet algorithm22
with l = 0.65 and time step Dt = 0.04.
2.2 Mesoscopic model for charged-neutral block copolymers
Within the DPD approach, some molecules of the system are
coarse-grained by a set of particles. In our simulations,
the polyelectrolyte is modeled as a block copolymer with
section of hydrophilic and hydrophobic blocks. Specifically,
we considered a subsystem containing m diblock copolymer
chains in a salt-free solution, each with a hydrophilic block A
built with NA charged monomers and a hydrophobic block B
with NB neutral monomers. Each of the charged monomers
carries one unit of positive charge q = +1, and the total
charges carried by block copolymers are Q = mNAq. The
valence n of counterions is constrained to n = Q/NC (where
NC is the number of counterions) with the electro-neutrality
requirement in this presumed salt-free system. In our specific
case, polyelectrolyte chains each having twelve charged
monomers and four neutral monomers are encapsulated into
a cuboid cell (subsystem) of length d = 40. These diblock
polyelectrolytes self-aggregate into a spherical micelle with a
neutral core radius of RB and a charged corona thickness of
RA, as shown in Fig. 1.
We used the simple model to characterize the dilute micelle
solution with volume fraction f ¼ 43pðRA þ RBÞ3=d3. The
scaling approaches based on a simple SPB model5,11 are briefly
elucidated in the following. Following Shusharuna et al.,5 the
elastic force per chain related to the conformational entropy
losses in the stretched chain is,
Felast
kBT� R
3=2A
N3=2A a
5=20
ð7Þ
In our simple SPB model, the neutral monomers (hydrophobic
particles) are collapsed into the core of the charged-neutral
micelle, the elastic force of this part can be balanced by the
hydrophobic interactions between the hydrophobic particles
and the hydrophilic particles/solvent particle. Thus, in our
opinion, it can be ignored in the simple SPB model. In our
simulations, the charged monomer A and neutral monomer B
are set to have the same bond length. When most of the
counterions are outside the micelle, the elastic force Felast is
balanced mainly by the unscreened electrostatic force
Felec
kBT� lBQ
2
R2Am
ð8Þ
This is referred to as the ‘‘Pincus’’ regime identified by
Shusharina.5 Then, the equilibrium thickness of the corona is
RA B NA3/7au2/7m�2/7Q4/7, (9)
where u= lB/a is the relative electrostatic interaction strength.
As we have stated earlier, the total charges carried by the block
copolymers are Q = mNAq, thus, the equilibrium thickness of
the corona can be rewritten as:
RA B NAau2/7m2/7. (10)
In the osmotic regime,28 most of the counterions are inside the
corona, and the elastic force Felast is balanced mostly by the
osmotic pressure of the counterions inside the corona
Fosm
kBT� NA
nRAð11Þ
Fig. 1 Schematic representation of a charged-neutral star micelle. d is
the length of the simulation box, RA is the width of the corona formed
by the charged blocks, and RB is the radius of the core formed by
neutral blocks.
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And the equilibrium thickness of the corona is
RA B n�2/5NAa. (12)
Together with the scaling approaches, we carried out DPD
simulations to study the conformational behaviors of a
charged-neutral star micelle in salt-free solution.
3 Results and discussion
For simplicity, the charged-neutral micelle in monovalent-
counterion solution is firstly chosen as our model. Three
typical regions corresponding to the conformations of brush
block as a function of relative electrostatic interactions u are
displayed in Fig. 2 and Fig. 3. Within the region of the small
value of u, most of the counterions distribute into the solution
without entering into the interior of the corona region because
of the extremely weaker attractions of charged segments on
counterions, which are incapable of overcoming the large
entropy aroused by fluctuation of the ions (see Fig. 2a).
The conformation of the charged blocks on the micelles is
essentially dominated by the elastic force similar to those in the
neutral micelles made of the amphiphilic diblock copolymers.
The swelling of the PE corona of the charged-neutral micelle is
not remarkable in comparison with the neutral corona. The
region is referred to as the quasi-neutral regime. The remarkable
character of this quasi-neutral regime is that the thickness of
corona RA is independent of u. Our simulation results soundly
demonstrated this point, as indicated in u o 0.01 in Fig. 3.
In the intermediate regimes, i.e., the ‘‘Pincus’’ regime, with
the enhancement of attractive force on the counterions by
charged segments, a part of the ions are adsorbed into the
micelles, while a large number of ions still remain in solution
due to the entropy effect. Owing to the lack of sufficient
quantities of counterions within the corona region of the
micelles, only a small part of the charges on the brush block
is screened, and the majority of charges remain unscreened.
The conformation of the block in this circumstance is balanced
by two factors: repulsion among charges on the block and
recoiling force with the requirement of maximum conforma-
tional entropy of chain. The scaling theory indicates the
exponent relation of RA B u0.28 (eqn (10)) for the growth of
corona thickness with u value. Our present calculation results
are consistent with the scaling theoretical prediction (Fig. 3).
As for the system having larger values of u, more ions are
absorbed into the corona of the micelle, which results in the
rise of osmotic pressure and screening of the charges on the
polyelectrolyte. The two effects are mutually responsible for
the conformation of charged corona blocks. As indicated in
Fig. 3, the growth of the thickness of the corona starts to
deviate from the ‘‘Pincus’’ regime elucidated by eqn (10).
When the fluctuation entropy of counterions are effectively
hurdled, the ions involved in the micelle inside of the micelle
reach saturation, and most parts of the charges on the polyion
(polyelectrolyte) are screened by counterions. The charged
blocks gain a maximum extension in length and then remain
constant (Fig. 3), coinciding with the scaling prediction in
eqn (12) This region is designated in terms of osmotic regime
(Fig. 2c).
Despite agreement of scaling theory with our calculations in
the case of the monovalent counterion, the behavior of RA for
divalent and trivalent counterion is perceived to deviate from
the scaling predication in both ‘‘Pincus’’ and osmotic regimes.
For instance, the corona thickness of the charged blocks in
a di- or trivalent counterion circumstance exhibits a tiny
collapse in osmotic regime for very large u value, unlike in the
monovalent counterion where the thickness remains steady. A
detailed explanation of this phenomenon will be given later on.
Next, the details on the conformation and behavior of the
micellar corona are elucidated in Fig. 4 via a calculation of the
Fig. 2 Density profile of polyions (black line), counterions (red line)
and net charges (blue line) as a function of the radius from the
center of the micelle R with different relative electrostatic interaction
(a) u = 0.001 ,(b) u = 0.1 and (c) u = 2.5, respectively. The
morphologies on the right (solvent are omitted for clarity; Block A,
green; Block B, blue; Counterions, red) correspond to each density
profile.
Fig. 3 Micelle corona thickness RA as a function of relative electro-
static interaction u.
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pair correlation functions, g(r), between charged segments and
the counterions of mono-, di-, and tri-valence, respectively. In
a simulation, it is straightforward to measure g(r), which is the
ratio between the average particle number density r(r) at a
distance r from a reference particle and the density at a
distance r from a particle in an ideal gas at the same overall
density. It has been previously proposed that the following
equation can be used to calculate the g(r),29
gðrÞ ¼ rðrÞVN¼ nðrÞV
4pr2DrNð13Þ
where n(r) is the number of particles that are a distance
between r and r + Dr away from the reference particle, and
N and V are the total number of particles in the system and the
volume of the system, respectively. The relative higher peak
of g(r) for trivalent counterions reveals the existence of a
relatively strong correlation between charged segments and
the ions with multiple valences. Furthermore, an obvious
appearance of the second peak in the cases of di- and
trivalence indicates the establishment of somewhat ordered
structures of the ions around the charged segments, implying a
strong condensation of multivalent counterions yielded by the
strong electrostatic correlations. The drop of osmotic pressure
in the system can be attributed primarily to the condensation
of the counterions.15–17 Moreover, the reduction in the
number of multivalent ions with the charged-neutral requirement
in the system is another subordinate reason for the drop of
osmotic pressure. This clarifies why the brushes formed by
these charged blocks collapse in multivalent ion solution for
the larger value of u (Fig. 3).
It is reasonable to conceive that the strong correlation
between the charged segments and the counterions can
account for not only the localization of ions in the space
around the charged segments but the suppression of the
mobility of these ions in the solution as well. The self-diffusion
coefficient D0 of the ions is calculated based on the function of
the time dependence of average mean-square displacement
(MSD) hDr2i = 6D0t (inset of Fig. 4). Compared with the
mono- and di-valence cases, the mobility of trivalence is
significantly hindered, implying the ‘‘freeze’’ effect of ions
inside the corona of the charged-neutral micelles.
Based on the above analysis, conformations of the charged
blocks, stretched out or collapsed on the spherical core built in
hydrophobic chains, is intimately correlative to the strength of
electrostatic interaction of two charged segments and that
of ions and segments. Upon the increase in the relative
electrostatic interaction u, the thickness of brush RA increases
in an S style as the system goes from the quasi-neutral to the
‘‘Pincus’’ regime and then to osmotic regime (Fig. 3). To
provide a deeper insight into the process, the dependence of
electrostatic potential energy E on the relative electrostatic
interaction u is presented.
E ¼Z
rðrÞcðrÞ � pðrÞ8plB
jrcðrÞj2� �
dr: ð14Þ
To calculate the electrostatic potential energy, we follow the
iterating algorithm of Beckers et al.24 and the grid method of
Groot,25 where the electrostatic field is solved on a grid. This
method has been used to evaluate the electrostatic force FE
in our simulations, it is consistently used to evaluate the
electrostatic potential energy E. Calculation strategies adopted
here permit the consideration of the inhomogeneity of the
electrostatic permittivity in the system.
In the quasi-neutral regime, as the majority of the counter-
ions are distributed in the solution without being contained
inside of the micellar corona, positive and negative charges
do not screen each other at all, and the electrostatic potential
energy is expected as E p u, as shown in Fig. 5. In the
‘‘Pincus’’ regime, although the electrostatic potential energy
still increases with the relative electrostatic interactions u,
it deviates from the linear relationship of E p u, due to
the screening effect produced by the counterions adsorbed
inside of micelles. In the osmotic regime, more counterions
intrude into the coronal part of the micelle due to strong
relative electrostatic interactions. The screening effects are
also strengthened heavily, especially on the multivalent
counterions.30 Thus, we can observe an obvious decline of
Fig. 4 Pair correlation function g(r) between charged segments and
charged segments in osmotic regime at u = 2.5. Inset: MSD (hDr2i) ofcounterions as a function of reduced time at u=2.5. The self-diffusion
coefficient D0(n=+1) = 0.83, D0(n=+2) = 0.46 and D0(n=+3) =
0.21. (The black, red and blue lines represent the valence of the counterions
as +1, +2 and +3, respectively.)
Fig. 5 Electrostatic potential energy E as a function of relative
electrostatic interaction u.
6138 | Phys. Chem. Chem. Phys., 2010, 12, 6135–6139 This journal is �c the Owner Societies 2010
electrostatic potential energy E, as shown in the inset of Fig. 5,
and the transition from ‘‘Pincus’’ regime to osmotic regime
appears.
4 Conclusion
In this paper, we have made a systematical analysis of
the conformation transition of spherical polyelectrolyte
brushes in salt-free solution of charged-neutral micelles under
different valent counterions using the dissipative particle
dynamic simulation. Our calculation results indicate that the
scaling predictions can well match our simulation results for
the case of monovalent counterions in the systems but deviate
from those for multiple valence counterions. The deviation
implies that the scaling analysis fails for the treatment of
complex circumstances such as those existing in the strong
correlation between the charged segments and counterions in
the multiple valence counterions system. In our simulation, we
found that the trivalent counterions can condense to the
charged segments when electrostatic interactions are extremely
strong. This condensation may suppress the osmotic activity
of the trivalent counterions inside the micelle corona and lead
to the collapse of the corona. Moreover, the transitions
from quasi-neutral to ‘‘Pincus’’ regime and from ‘‘Pincus’’ to
osmotic regime are clearly understood in terms of electrostatic
potential energy.
Acknowledgements
We would like to thank the two anonymous referees whose
critical comments helped us in improving the quality of our
manuscript. We are grateful for the financial support provided
by the Program of the National Natural Science Foundation
of China (Nos. 20934004, 20874094 and 50773072), NBRPC
(Nos. 2005CB623800 and 2010CB934500). X. Li would like to
acknowledge the financial support provided by China
Postdoctoral Science Foundation (No. 20090460729) the
Fundamental Research Funds for the Central Universities,
the K. C. Wong Education Foundation, Hong Kong. Parts of
the simulations were carried out at the Shanghai Super-
computer Center.
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