1118
Langmuir 2005, 21, 1118-1125
Molecular Dynamics Simulations of Polyelectrolyte
Multilayering on a Charged Particle
Venkateswarlu Panchagnula,† Junhwan Jeon,‡ James F. Rusling,†,§ and
Andrey V. Dobrynin*,‡,|
Polymer Program, Institute of Materials Science, Departments of Chemistry and Physics,
University of Connecticut, Storrs, Connecticut 06269, and Department of Pharmacology,
University of Connecticut Health Center, Farmington, Connecticut 06032
Received September 9, 2004. In Final Form: November 2, 2004
Molecular dynamics simulations of polyelectrolyte multilayering on a charged spherical particle revealed
that the sequential adsorption of oppositely charged flexible polyelectrolytes proceeds with surface charge
reversal and highlighted electrostatic interactions as the major driving force of layer deposition. Far from
being completely immobilized, multilayers feature a constant surge of chain intermixing during the deposition
process, consistent with experimental observations of extensive interlayer mixing in these films. The
formation of multilayers as well as the extent of layer intermixing depends on the degree of polymerization
of the polyelectrolyte chains and the fraction of charge on its backbone. The presence of ionic pairs between
oppositely charged macromolecules forming layers seems to play an important role in stabilizing the
multilayer film.
1. Introduction
Layer-by-layer assembly of charged molecules is a
versatile route to structured and robust ultrathin films,
which have evoked tremendous interest in a variety of
potential applications ranging from biosensors to nanoengineering and drug delivery.1 The simplicity of the
electrostatic assembly technique with practically no
limitations on the shape or identity of the macromolecular
charge-bearing species allows the fabrication of multilayer
films to user design specifications from synthetic polyelectrolytes,2 DNA,3 proteins,4 nanoparticles,5 and so forth.
First reported in 1966 and rediscovered in the early
1990s,6,7 this technique of film assembly is based on the
long-range electrostatic attraction of oppositely charged
molecules. The key to successful deposition of multilayer
assemblies in a layer-by-layer fashion is the inversion
and subsequent reconstruction of surface properties. A
typical experimental procedure involves immersing a solid
* Corresponding author. E-mail: [email protected].
† Department of Chemistry, University of Connecticut.
‡ Institute of Materials Science, University of Connecticut.
§ University of Connecticut Health Center.
| Department of Physics, University of Connecticut.
(1) (a) Lvov, Y. In Protein Architecture: Interfacing Molecular
Assemblies and Immobilization Biotechnology; Lvov, Y., Möhwald, H.,
Eds.; Marcel Dekker: New York, 2000; pp 125-167. (b) Lvov, Y. In
Handbook Of Surfaces And Interfaces Of Materials, Vol. 3. Nanostructured Materials, Micelles and Colloids; Nalwa, R. W., Ed.; Academic
Press: San Diego, CA, 2001; pp 170-189. (c) Rusling, J. F.; Zhang, Z.
In Handbook Of Surfaces And Interfaces Of Materials, Vol. 5. Biomolecules, Biointerfaces, And Applications; Nalwa, R. W., Ed.; Academic
Press: San Diego, CA, 2001; pp 33-71. (d) Rusling, J. F.; Zhang, Z. In
Biomolecular Films; Rusling, J. F., Ed.; Marcel Dekker: New York,
2003; pp 1-64.
(2) Ruths, J.; Essler, F.; Decher, G.; Riegler, H. Langmuir 2000, 16,
8871-8878. Arys, X.; Laschewsky, A.; Jonas, A. M. Macromolecules
2001, 34, 3318-3330.
(3) Sukhorukov, G. B.; Mohwald, H.; Decher, G.; Lvov, Y. Thin Solid
Films 1996, 285, 220-223.
(4) Rusling, J. F. In Protein Architecture: Interfacing Molecular
Assemblies and Immobilization Biotechnology; Lvov, Y., Mohwald, H.,
Eds.; Marcel Dekker: New York, 2000; p 337.
(5) Lvov, Y.; Ariga, K.; Onda, M.; Ichinose, I.; Kunitake, T. Langmuir
1997, 13, 6195-6203.
(6) Iler, R. K. J. Colloid Interface Sci. 1966, 21, 569-594.
(7) Decher, G. Science 1997, 277, 1232-1237.
substrate into dilute solutions of anionic or cationic
polyelectrolytes for a period of time optimized for adsorption followed by a rinsing step to remove any unadsorbed
material. Further film growth is achieved by alternating
the deposition of polyanions and polycations from their
aqueous solutions. After a few dipping cycles, experiments
show a linear increase of multilayer thickness, indicating
that the system reaches a stationary regime.
Several important features have been established for
the layer-by-layer film formation. The thickness of each
adsorbed layer shows almost a linear dependence on the
salt concentration.8,9 Flexible polyelectrolytes in twocomponent multilayers are known to intermix over several
adjacent layers. This layer intermixing can be suppressed
by using more rigid blocks for the assembly.7,10 Intrinsic
charge compensation by polyions accompanied by overcharging and the kinetically irreversible nature of deposition has also been reported.9 It was also shown that the
layer thickness and molecular organization of adsorbed
polymers could be tuned by adjusting the pH of the dipping
solutions.11 In this case, the solution pH controls not only
the adsorbing polyelectrolyte but also the previously
adsorbed layers. There is an ionization threshold below
which there is no multilayer formation.12 The measurements of the Young’s modulus of polyelectrolyte multilayers yielded an estimate that is very close to that
expected for ionomers.13 It could therefore be possible that
the structure of multilayer films is close to that of ionomers,
(8) Losche, M.; Schmitt, J.; Decher, G.; Bouwman, W. G.; Kjaer, K.;
Macromolecules 1998, 31, 8893-8906.
(9) Schlenoff, J. B.; Ly, H.; Li, M. J. Am. Chem. Soc. 1998, 120, 76267634. Dubas, S. T.; Schlenoff, J. B. Macromolecules 1999, 32, 81538160.
(10) Kleinfeld, E. R.; Ferguson, G. S. Science 1994, 265, 370-373.
Glinel, K.; Laschewsky, A.; Jonas, A. M. Macromolecules 2001, 34, 52675274.
(11) Shiratori, S. S.; Rubner, M. F. Macromolecules 2000, 33, 42134219.
(12) Steitz, R.; Jaeger, W.; Klitzing, R. v. Langmuir 2001, 17, 44714474. Voigt, U.; Jaeger, W.; Findenegg, G. H.; Klitzing, R. v. J. Phys.
Chem. B 2003, 107, 5273-5280.
(13) Vinogradova, O. I.; Andrienko D.; Lulevich, V. V.; Nordschild,
S.; Sukhorukov, G. B. Macromolecules 2004, 37, 1113-1117.
10.1021/la047741o CCC: $30.25 © 2005 American Chemical Society
Published on Web 01/07/2005
Polyelectrolyte Multilayering on a Charged Particle
reflecting strong ionic interactions between the polycations
and polyanions within multilayers.
Although layer-by-layer assembly has been widely used
on charged planar surfaces, there has been increasing
interest in their spherical counterparts. Multilayer assembly of polyelectrolytes on spherical particles has been
successfully performed to obtain hollow nano- and microsized structures.14-16 This process alters the physiochemical properties of the spherical substrates, which have
many potential applications in drug delivery, catalysis,
composites, surface coatings, and so forth.
Despite extensive experimental studies, the theoretical
models of electrostatic self-assembly are still in their
infancy. Netz and Joanny17 considered the formation of
multilayers in a system of semiflexible polymers, assuming
that the deposited layer structure was fixed, providing a
solid charged substrate for the next layer. Mayes et al.18
applied a similar idea of a solid substrate to flexible
polyelectrolytes. Flexible polyelectrolytes were assumed
to form a brushlike layer at the solid substrate of previously
adsorbed chains in which the loop size and layer thickness
are controlled by the strength of the ionic pair interactions
formed between oppositely charged chains in neighboring
layers and by electrostatic repulsion between polyelectrolytes within the same layer. Strong charge overcompensation occurs in the case of highly ionized flexible
polymer chains adsorbed in a brushlike manner, leading
to the formation of multilayers. This model predicts a
square root dependence of polymer surface coverage on
salt concentration. Both these models neglect interpenetration and chain complexation between the layers. An
opposite limit of strong intermixing of polyelectrolytes
between neighboring layers was considered by Castlenovo
and Joanny,19 by incorporating the complex formation
between oppositely charged polyelectrolytes into selfconsistent field equations. Analysis of these equations was
limited to solutions of high ionic strength, where the
electrostatic interactions are exponentially screened and
can effectively be treated as short-range interactions.
Another route for multilayer formation was shown by Solis
and de la Cruz.20 They found that stratification of
polyelectrolytes near charged surfaces could also be
achieved by increasing the incompatibility between oppositely charged polymers.
It is difficult to directly test all assumptions of theoretical
models in experiments. However, computer simulations
can help to elucidate factors involved in the multilayer
assembly process and verify theoretical assumptions made
in models of electrostatic film assembly. This, for example,
can be done by changing the interaction parameters, by
changing the length of the simulation runs, and by
analyzing the layer structure, which is easily accessible
through the polymer density profiles, counterion distributions, and so forth.
(14) (a) Caruso, F.; Carsuo, R. A.; Mohlwald, H. Science 1998, 282,
1111-1114. (b) Gittins, D. I.; Caruso, F. J. Phys. Chem. B 2001, 105,
6846-6852.
(15) Caruso, F.; Fiedler, H.; Haage, K. Colloids Surf., A 2000, 169,
287-293.
(16) Usha, A. S.; Caruso, F.; Rogach, A. L.; Sukhorukov, G. B.;
Kornowski, A.; Mohwald, H.; Giersig, M.; Eychmuller, A.; Weller, H.
Colloids Surf., A 2000, 163, 39-44.
(17) Netz, R. R.; Joanny, J.-F. Macromolecules 1999, 32, 9013-9025.
(18) Park, S. Y.; Rubner, M. F.; Mayes, A. M. Langmuir 2002, 18,
9600-9604.
(19) Castlenovo, M.; Joanny, J. F. Langmuir 2000, 16, 7524-7532.
(20) Solis, F. J.; de la Cruz, M. O. J. Chem. Phys. 1999, 110, 1151711522.
Langmuir, Vol. 21, No. 3, 2005 1119
Polyelectrolyte multilayering on a spherical particle was
studied using the Monte Carlo simulations21-23 based on
the assumption that multilayering occurs whether one
proceeds in a stepwise fashion, as envisaged in a real
experiment, or when the oppositely charged polyelectrolytes are added together. In such a situation, it was noted
that the polyelectrolyte multilayering on a spherical
surface requires a sufficiently strong extra short-ranged
attractive interaction between the macroion and the
polyelectrolyte. Such Monte Carlo simulations assume
that the multilayering is in an equilibrium state. Thus,
its preparation is independent of the route chosen to reach
equilibrium.
Although surface charge reversal is essential in the
assembled layer growth, several fundamental questions
bearing importance for understanding these systems
remain unanswered. Recently, we showed that electrostatic interactions are indeed the leading driving force for
the multilayer assembly.24 In this paper, we use molecular
dynamics simulations to address the following: (a) the
effect of the chain degree polymerization and charge
fraction on surface charge reversal, leading to polyelectrolyte multilayering around a charged spherical particle,
and (b) the intermixing between the layers, which has
been widely observed in experiments.1a,b The sequential
deposition process implemented in our simulations resembles a real experimental situation unlike the method
adopted earlier.23 Our simulation results emphasize the
importance of electrostatic interactions and short-range
attractive interactions as the driving force in multilayer
formation with surface overcharging accompanying layer
growth at every deposition step. The simulation model
and method are described in section 2. In section 3, we
present the simulation results along with a detailed
analysis of the layer structure and dynamics. Finally, in
section 4, we summarize our findings.
2. Model and Simulation Methodology
The molecular dynamics simulations of multilayer
assembly were performed from solutions of polyelectrolyte
chains with degrees of polymerization N ) 32, 16, and 8
monomers. For each chain length, the fractions of charged
monomers on a chain are equal to f ) 1, 0.5, and 0.25
corresponding to every one, second, and fourth bead in
the chain carrying a charge. Multilayers were formed
around a charged spherical particle that has Nsphere ) 80
negatively charged Lennard-Jones beads with the diameter 1σ (see Figure 1a). The central particle with 80
overlapping beads on its surface has a 3-D structure
similar to a bucky ball of C80. The radius of this particle,
measured as the distance between its center and centers
of mass of the forming particle beads, is equal to 1.5σ.
Such a structure is compact enough to restrict the
adsorption onto the outer surface of the particle. This will
enable multilayer growth only on the outside of the core
of the central charged particle. When the radius of this
central particle is increased beyond 1.5σ, the beads on the
surface no longer overlap and leave spaces in between.
These spaces are wide enough to permit chain penetration
to adsorb onto its interior. The connectivity of beads in
(21) Messina, R.; Holm, C.; Kremer, K. Langmuir 2003, 19, 44734482.
(22) Messina R. Macromolecules 2004, 37, 621-629.
(23) Messina, R.; Holm, C.; Kremer, K. J. Polym. Sci., Part B: Polym.
Phys. 2004, 42, 3557-3570.
(24) Panchagnula, V.; Jeon, J.; Dobrynin, A. V. Phys. Rev. Lett. 2004,
93, 037801-1.
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Langmuir, Vol. 21, No. 3, 2005
Panchagnula et al.
Figure 1. Evolution of the multilayer structure around the spherical macroion during the adsorption of fully charged (f ) 1)
polyelectrolytes with N ) 32. The snapshots were taken after the completion of deposition steps from 1 through 5 with a time
duration of 2.0 × 105 MD steps per each deposition step. The macroion is shown in black in part a, followed by the buildup of layers
1-5 sequentially color-coded as red, blue, magenta, cyan, and orange beaded chains (b-f).
the chains is maintained by the finite extension nonlinear
elastic (FENE) potential:25
UFENE(r) ) -
(
1
r2
kspringR2max ln 1 - 2
2
R
max
)
(1)
lBqiqj
rij
ULJ(rij) )
{
(2)
where the Bjerrum length, lB ) e2/kBT, defined as the
length scale at which the Coulomb interaction between
two elementary charges e in a dielectric medium with a
dielectric constant is equal to the thermal energy kBT,
describes the strength of the electrostatic interactions. In
our simulations, the value of the Bjerrum length lB is equal
to 1.0σ. All charged particles in our simulations are
monovalent ions with a valency qi ) (1. The particleparticle particle-mesh (PPPM) method implemented in
LAMMPS26 has been used to calculate the electrostatic
interactions with all periodic images of the system. Also,
(25) Stevens, M. J.; Kremer, K. J. Chem. Phys. 1995, 103, 16691690.
[( ) ( ) ( ) ( ) ]
4LJ
0
The spring constant kspring is equal to 15kBT/σ2, where kB
is the Boltzmann constant, T is the absolute temperature,
and the maximum bond length Rmax between the beads is
set to 2σ.
Counterions were explicitly included in our simulations.
Interaction between any two charged particles, bearing
charge valences qi and qj and separated by a distance of
rij, is described by the Coulomb potential:
UCoul(rij) ) kBT
all the charged and uncharged particles interact through
the truncated/shifted Lennard-Jones (LJ) potential:
σ
rij
12
-
σ
rij
6
-
σ
rc
12
+
σ
rc
6
r e rc
r > rc
(3)
where rij is the distance between any two interacting beads
i and j and σ is the bead diameter chosen to be the same
regardless of the type of beads. A cutoff distance rc ) 2.5σ
was chosen for the macroion/polymer-polymer interactions and rc ) 21/6σ for the polymer-counterion as well as
counterion-counterion interactions. The interaction parameter LJ is set to 0.75kBT. As the depth of the energy
well, given by LJ, cannot be directly compared with
experimental values, we investigated a range of values
for LJ from good to poor solvent conditions that are known
to be realistic for soft matter systems.27 The effective
interaction for θ-solvent condition is given by LJ ) (0.34
( 0.02)kBT for the uncharged system. The value LJ )
0.75kBT, thus, is that of a poor solvent condition and is
sufficiently strong for the assembly of multilayers. We
also found that the multilayers did not form under good
solvent conditions. The combination of FENE and LJ
potentials prevents the chains from crossing each other
during the simulation run.
We can map our system onto an experimental system,
such as sodium poly(styrene sulfonate) (NaPSS), in water.
(26) Plimpton, S. J. Comput. Phys. 1995, 117, 1-19. LAMMPS
website: http://www.cs.sandia.gov/∼sjplimp/lammps.html.
(27) Micka, U.; Holm, C.; Kremer, K. Langmuir 1999, 15, 40334044.
Polyelectrolyte Multilayering on a Charged Particle
Langmuir, Vol. 21, No. 3, 2005 1121
Table 1. Parameters Used in the Simulations
rinsing step. To separate adsorbed chains from the rest
of the polyelectrolytes we used a cluster algorithm with
the cutoff radius equal to 1.2σ. A chain is considered to
belong to a cluster if it has at least one monomer within
a distance of 1.2σ from any monomer belonging to a chain
forming the cluster. The cluster analysis was performed
by analyzing the matrix of distances between all monomers
in the system. After each simulation run, only the
counterions needed for the compensation of excess charge
of the particle-adsorbed chains in the aggregate were kept
in the simulation box to maintain system electroneutrality.
At the beginning of the second layer deposition step,
the simulation box is refilled with M2 ) M1 oppositely
charged polyelectrolytes together with their counterions.
This is followed by the simulation run (dipping cycle). We
repeated the dipping and rinsing cycles to simulate the
buildup of nine layers and performed two different
simulations with durations of 2.0 × 105 and 2.0 × 106
integration steps for every deposition cycle. This translates
to about 2.0 Rouse relaxation times for short simulation
runs and 20 relaxation times for the longer runs. Before
collecting the data, the system was allowed to equilibrate
for a half time of the total simulation run during each
dipping cycle.
parameter
value
N
Nsphere
f
qi
σ
Rmax ) 2σ
rc ) 2.5σ
kspring )
15kBT/σ2
T
Lbox ) 26σ
lB ) 1.0σ
32, 16, and 8
80
1, 0.5, and 0.25
(1
7.14 Å
14.28 Å
17.85 Å
0.729 kJ/mol/Å2
degree of polymerization of the chain
no. of beads on the macroion
fraction of charged monomers
charge valence on a monomer/bead
diameter of each bead
maximum bond length
cutoff distance
spring constant
298 K
78.5
184 Å
7.14 Å
temperature
dielectric constant of water
length of the cubical simulation box
Bjerrum length
If we assume that our Bjerrum length, lB ) 1.0σ, is equal
to the Bjerrum length in aqueous solutions at room
temperature (T ) 298 K), lB ) 7.14 Å, the monomer size
is equal to σ ) 7.14 Å. This corresponds to approximately
2.9 monomers of NaPSS with a monomer size 2.5 Å.
Compared with a fully charged chain (f ) 1), this
corresponds to a fraction of charged monomers f ) 0.35.
As we are interested in the general class of polyelectrolyte
multilayer adsorption, each bead in the coarse-grained
bead spring model employed in our simulations represents
several chemical units. This method of mapping has been
routinely used in the literature earlier.28 The values of
parameters and their relation to the corresponding real
units used in the simulation are summarized in Table 1.
For simplicity, reduced units will be used for discussion
in the rest of the manuscript.
Simulations were carried out in a constant number of
particles, volume, and temperature ensemble (NVT) with
periodic boundary conditions. The constant temperature
is achieved by coupling the system to a Langevin thermostat. In this case, the equation of motion of the ith
particle is
m
dv
bi
(t) ) F
Bi(t) - ξv
bi(t) + F
BiR(t)
dt
(4)
Bi is the net deterministic
where b
vi is the bead velocity and F
force acting on the ith bead of mass m. F
BiR is the stochastic
force with zero average value 〈F
BiR(t)〉 ) 0 and δ-functional
BiR(t′)〉 ) 6ξkBTδ(t - t′). The friction
correlations 〈F
BiR(t) F
coefficient ξ was set to ξ ) m/τLJ, where τLJ is the standard
LJ time τLJ ) σ(m/LJ)1/2. The velocity-Verlet algorithm
with a time step of ∆t ) 0.01τLJ was used for integration
of the equations of motion (eq 4). The solvent effect is
modeled by using a dielectric medium with an effective
dielectric constant. The dielectric continuum captures the
solvent effects on the strength of electrostatic interactions.
By coupling the system to the Langevin thermostat, the
stochastic effects of the solvent are taken into account.
Simulations were performed by the following procedure.
The spherical particle remains fixed at the center of a
cubic simulation box with a box size Lbox ) 26σ during the
whole simulation run (Figure 1). Counterions from the
charged particle were uniformly distributed over the box
volume. Then, M1 positively charged polyelectrolytes
together with their counterions were added into the
simulation box. For our longest chains with N ) 32, there
were 16 chains in the simulation box. During each
deposition step for chains of different lengths, we have
maintained the same polymer concentration c of newly
added polyelectrolyte chains at c ) 0.03σ-3. After completion of the first simulation run (dipping cycle), unadsorbed
polyelectrolyte chains were removed, corresponding to the
(28) Bright, N. J.; Stevens, M. J.; Hoh, J.; Woolf, T. B. J. Chem. Phys.
2001, 115, 4909-4918.
3. Results and Discussion
3.1. Adsorption of Polyelectrolytes on the Spherical Particle. We now describe the sequential deposition
process of fully charged polyelectrolytes with a degree of
polymerization N ) 32. Simulations corresponding to nine
deposition steps have been performed, and the snapshots
in Figure 1 represent the first five deposition steps. Each
layer of oppositely charged polyelectrolytes has been
displayed in a different color for clarity. In the first
snapshot (Figure 1a), the negatively charged spherical
macroion is seen placed in the center of a cubical cell
containing positively charged polyelectrolytes before the
start of the simulation run. Counterions are not shown in
the snapshots for clarity. Subsequent snapshots (Figure
1b-f) represent adsorption events as seen at the end of
each simulation run consisting of 2.0 × 105 MD steps each.
The simulation of the first layer results in an adsorption
of four polyelectrolyte chains (128 monomers) onto the
surface of the spherical particle made up of 80 negative
beads. Thus, in addition to charge neutralization, an excess
of 48 charges is accumulated around the particle that
allows further adsorption of oppositely charged polyelectrolyte chains. The four chains wrap around the charged
particle, covering it completely.
Simulation of the next layer deposition, as described
above in the simulation methodology, led to the second
layer reversing the surface charge. Further dipping cycles
are repeated to assemble layers of polyelectrolytes onto
one another onto the spherical core of adsorbed chains.
Unlike the first layer, subsequent layers tend to form what
can be considered as “patches” or “islands” of layers. This
difference arises as the size of the adsorbed core increases
while the charge density is less than that of the spherical
particle alone. As the deposition steps are repeated, the
incoming chains have access to interior chains which they
can displace or adsorb onto, depending on their respective
charges, causing desorption or intermixing to occur. The
snapshots in Figure 1e and f show intermixing among
various types of the adsorbed chains. Thus, at the outset,
the multilayer structure seems to be much more complicated than the simplistic, “stratified”, and frozen picture
that one is tempted to imagine. Simulations of much longer
1122
Langmuir, Vol. 21, No. 3, 2005
Figure 2. Dependence of the number of adsorbed monomers
Nads on the number of deposition steps N for a time duration
of 2.0 × 105 MD steps per each deposition cycle.
durations (2.0 ×106 MD steps each) resulted in a similar
overcharging and layer buildup.
The evolution of the layer structure can be monitored
by plotting the number of adsorbed monomers Nads during
the assembly as a function of the number of deposition
steps. This parameter shows strong dependence on the
chain degree of polymerization N and fraction of charged
monomers f.24 Figure 2 represents data for the charged
chains with f ) 1 and 0.5. The adsorbed amount is
consistently high for chains with N ) 32 (f ) 1 and 0.5)
followed by N ) 16 (f ) 1) over others. For chains with
degrees of polymerization N ) 8, the number of adsorbed
chains is significantly less and seems to approach a steady
state as the number of deposition steps increases. An initial
repetitive pattern of adsorption and desorption with no
gain in the number of adsorbed chains is found for these
chains. For short chains with N ) 8 (f ) 0.5) and weakly
charged chains with f ) 0.25 for all the chain lengths (N
) 32, 16, and 8), although the first layer is formed, it is
unable to achieve the charge reversal essential for growing
additional layers. This trend can be understood by
considering the concept of charge overcompensation due
to charge fractionalization.29
When a polyelectrolyte chain adsorbs electrostatically
on a charged surface, the stoichiomentry is greater than
1:1, more often than not, with the formation of loops and
tails. The charge on these loops and tails builds up the
excess charge essential for the growth of the multilayers.
This effect is more pronounced in longer chains, with
higher degree of ionization than in the ones that are either
short or partially charged. In the case of shorter chains,
the loose ends that form tails or loops are less probable,
and in chains that are not fully charged, the residual
charge is less even if loops and tails are formed.
It is worthwhile to discuss the effect of first layer
interactions with the macroion on the layer adsorption.
When purely repulsive nonbonded (Lennard-Jones) interactions between the central macroion and the polyelectrolyte chains were used in the simulations, multilayering was not observed, although the first layer forms
with charge reversal due to attractive electrostatic
interactions with the macroion. The first layer is subsequently peeled off during the second deposition cycle, as
the oppositely charged chains prefer complexation with
(29) Grosberg, A. Yu.; Nguyen, T. T.; Shklovskii, B. I. Rev. Mod.
Phys. 2002, 74, 329-345.
Panchagnula et al.
the first layer and ultimate desorption. This happens as
the chains from the second deposition cycle encounter
repulsive interactions with the macroion along with the
attractive electrostatic interactions with the first layer.
Therefore, the presence of short-range attractive interactions between the macroion and the polyelectrolytes is
essential for the first layer to be intact and the multilayer
assembly to progress. This aspect has also been observed
in previously reported simulations on charged spherical21
and planar surfaces,22 which underline the need of
sufficiently strong extra short-range interactions for the
multilayering.
The occurrence of chain exchange can be observed by
monitoring the evolution of the total number of adsorbed
monomers in the aggregate. For example, in the case of
the deposition of fully charged (f ) 1) polyelectrolyte chains
with a degree of polymerization N ) 16, after completion
of six deposition steps, there are 496 monomers in the
polymeric film covering the charged particle. After completion of the seventh deposition step, the total number of
monomers grows up to 608, out of which only 464
monomers were adsorbed during the first six deposition
steps. This difference of 32 monomers accounts for the
desorption of two chains. This indicates an exchange that
took place between the incoming polyelectrolytes and the
ones of similar charge previously present in the aggregate.
Nevertheless, the total number of adsorbed chains increased substantially, indicating a steady layer growth.
This dynamic exchange that occurred during the assembly
confirms the accessibility of interior layers to the newly
coming chains. It is worthwhile to note that Schlenoff and
co-workers reported slow displacement or exchange of
adsorbed chains along with the kinetically irreversible
nature of the deposited layers.9 Thus, desorption or chain
exchange evidenced in this simulation attempts to capture
what might transpire in an actual experiment and
indicates the possibly dynamic nature of the multilayer
assembly process.
The faster than linear increase in the total number of
adsorbed monomers on the charged particle observed in
Figure 2 can be attributed to the increase in the available
accessible area for the incoming chains as the mass of the
adsorbed layer increases. Assuming spherical layer structure around the macroion, the radius of gyration of the
adsorbed layers (Rg; Rg2 is defined as the mean square
distance between the monomers in a given conformation
and the center of mass of the aggregate) of polyelectrolyte
chains increases with the total number of adsorbed
monomers Nads. Thus, by plotting Nads/Rg2 as a function
of the number of deposition steps (Figure 3), the polymer
surface coverage shows linear dependence. For each
system, the steady state regime is reached after the
deposition of the first few layers.
Simulations of much longer durations (2.0 × 106 MD
steps each) result in overcharging and layer buildup, as
seen in shorter simulation runs. In these simulations, the
chain exchange occurs more frequently. However, variations in the chain degree of polymerization reveal different
regimes of the adsorption process at different time scales
(Figure 3). For the longest chains, N ) 32 with f ) 1, the
total amount of adsorbed monomers after nine deposition
steps differs only by two chains (64 monomers) for the
simulation durations of 2.0 × 105 and 2.0 × 106 MD steps.
In the case of N ) 16 (f ) 1), this difference in adsorbed
monomers for the two different simulation durations
increases to 13 chains containing 208 monomers. When
a longer chain (N ) 32) adsorbs, there are more available
adsorption sites than in the case of shorter chains. In this
case, desorption of chains is less likely even in the longer
Polyelectrolyte Multilayering on a Charged Particle
Langmuir, Vol. 21, No. 3, 2005 1123
Figure 4. Dependence of the overcharging fraction |∆Q|/fNads(()
on the deposition step for different fractions of charged
monomers, f ) 1 (filled symbols) and f ) 0.5 (open symbols),
with degrees of polymerization N ) 32 (diamonds), 16 (squares),
and 8 (triangles).
Figure 3. Dependence of the polymer surface coverage Nads/
Rg2 on chain degrees of polymerization N ) 32 (diamonds), 16
(squares), and 8 (triangles), for the fully charged chains (f ) 1),
as the number of deposition steps (Nl) increased for a time
duration of (a) 2.0 × 105 MD steps and (b) 2.0 × 106 MD steps
per deposition step.
simulation run. Thus, the ease of desorption increases as
the degree of polymerization decreases. As expected, when
N ) 8, desorption is facilitated even at shorter simulation
runs, as the adsorption sites per chain are far less than
the longer chain counterpart. Hence, the adsorption
scenario does not change much when the simulation
proceeds to 2.0 × 106 MD steps. Interestingly, N ) 16
presents the intermediate case of a trapped state at 2.0
× 105 MD steps and a well equilibrated state when
simulated for 2.0 × 106 MD steps, which accounts for the
chain desorption.
Although the number of adsorbed monomers varies with
the fraction of charged monomers on the polymer backbone, the overcharging process is universal during the
steady state growth. This can be confirmed by plotting
the overcharging fraction |∆Q|/fNads(() as a function of the
deposition steps (Figure 4). The overcharging fraction gives
the ratio of the absolute value of layer overcharging |∆Q|
to the net charge carried by the adsorbed chains fNads(().
Nads(() represents the number of monomers adsorbed
during a deposition step Nl and takes the corresponding
desorbed chains into consideration. As the system reaches
a steady state, the overcharging fraction has a value close
to 0.5 and is independent of the chain’s degree of
polymerization and fraction of charged monomers on the
polymer backbone. However, the overcharging fraction
has more fluctuations in the case of shorter chains with
N ) 8 and f ) 1, which can be attributed to the fluctuations
in the number of adsorbed chains that are retained without
being desorbed in the subsequent deposition steps.
3.2. Monomer Density Distribution. The local structure of the polyelectrolytes adsorbed onto the spherical
particle can be analyzed using the radial monomer density
distribution function F(r). These density distribution
functions are shown in Figures 5-7, for different degrees
of polymerization and duration of simulation runs. These
distribution functions were averaged during deposition
of the last layer independently for each set of chains
adsorbed during different deposition steps. This separate
data collection allowed us to analyze the evolution of the
multilayer structure and the interpenetration of the layers
during the deposition process. The plots clearly indicate
a layered distribution of the polyelectrolyte chains around
the spherical particle of radius 1.5σ. The first layer has
more monomers present in it, and hence the sharp peak
in the density profile. A decrease in polymer density of
the subsequent layers is associated with the increase of
the effective radius of the adsorbing aggregate.
Figure 5a for the adsorption of chains with a degree of
polymerization N ) 32 shows a multilayer arrangement
with the peak for the seventh layer occupying a third region
in the monomer density profile for the simulation run of
2.0 × 105 MD steps. Multilayering can also be seen in the
case where the degree of polymerization is N ) 32 for a
simulation run of 2.0 × 106 MD steps in Figure 5b and N
) 16 (Figure 6). The insets in these figures show the
difference in radial density distribution of positively and
negatively charged chains, ∆F(r) ) F+(r) - F-(r), representing the radial distribution of net charge. The multilayer nature of the structure can be clearly seen in the
insets for N ) 32 and 16 in Figures 5 and 6. The asymmetric
layer growth with an overall increase in the surface areas,
discussed in the previous section, gives rise to a high
probability of mistakes, and hence, the multilayering is
limited to three layers only. Also, the appearance of chains
from the last deposition steps (for example, the fifth to
ninth layers) around the central particle can be attributed
to this asymmetric layer growth and the “mistakes”
propagating as the deposition proceeds. The layering is
better stratified on charged planar surfaces and extends
up to several layers.30 The adsorption on a spherical surface
(as compared to a planar surface) is less favorable due to
(30) Patel, P. A.; Jeon J.; Mather, P. T.; Dobrynin, A. V. Phys. Rev.
E, submitted for publication.
1124
Langmuir, Vol. 21, No. 3, 2005
Figure 5. Density profiles of the fully charged (f ) 1)
polyelectrolyte chains with N ) 32 after the completion of nine
deposition steps with a duration of (a) 2.0 × 105 MD steps and
(b) 2.0 × 106 MD steps each. The insets show the difference
between the corresponding radial monomer densities of positively and negatively charged chains, ∆F(r) ) F+(r) - F-(r).
the significant chain conformational changes needed in
the polymer to adsorb, resulting in a loss of entropy.
Chains with like charges from different deposition steps
exchange places, although this exchange is limited to
immediate neighbors in the former case, while the
penetration is almost complete and far reaching in longer
simulation runs. Chains with N ) 16 show similar trends
of intermixing. In the case of the shorter chains with N
) 8 (Figure 7), regardless of the duration of simulation
runs, the intermixing is complete and the multilayer
structure is not reached. Shorter chains have shorter
relaxation times. Thus, even the shortest simulation run
is sufficient to allow equilibration of the chains in a layer.
Our simulation procedure corresponds to a Rouse dynamics of polymers for which the chain relaxation time
increases with the chain degree of polymerization as N2.
The chain relaxation time is 4 and 16 times shorter for
chains with N ) 16 and 8, respectively, in comparison
with that for chains having N ) 32 repeat units. Therefore,
it is easier for shorter chains to penetrate into the interior
layers even during shorter simulation runs. On the other
hand, longer chains have a greater tendency to form
multiple loops and tails that inhibit access to inner layers,
at least temporarily. Therefore, multilayers are formed
when the chains are sufficiently long and when the system
is far from equilibrium. Thus, an intermediate partially
trapped state is reached.
Interestingly, this aspect of the simulation also gives
us some insight about experimental time scales during
the dipping cycles in an experimental situation. Ideally,
Panchagnula et al.
Figure 6. Same as Figure 5 for chains with a degree of
polymerization N ) 16.
Figure 7. Same as Figure 5 for chains with a degree of
polymerization N ) 8.
we should dip a solid substrate in a polyelectrolyte solution
long enough so that the adsorption process reaches a
Polyelectrolyte Multilayering on a Charged Particle
Figure 8. Charge-charge correlation functions g((r) for the
fully charged (f ) 1) polyelectrolyte chains with N ) 32, 16, and
8 after completion of nine deposition steps of duration of 2.0 ×
105 steps per each deposition cycle.
steady state. In reality, some loosely adsorbed chains from
a deposition step are propagated onto the subsequent
layers even after the rinsing steps. Nevertheless, these
mistakes carry on and govern the multilayer assembly
process with an occasional loss of loose chains along its
course. Thus, desorption or the process of chain exchange
evidenced in this simulation closely resembles what
happens in an actual experiment1a,b and indicates the
dynamic nature of the assembly process.
On the basis of the above results, multilayer structure
can be understood as a partially trapped nonequilibrium
state that given sufficient time will undergo complete
intermixing. The idea of a frozen layer structure contradicts the results of our simulations. In our simulations,
we found a strong intermixing of the layers even in the
case of short deposition steps. The intermixing between
polyelectrolyte chains deposited during different deposition steps continues during the whole simulation run.
Thus, the assumption of the frozen layer structure17,18
should be reconsidered to obtain a more realistic theoretical model of electrostatic assembly.
3.3. Interchain Charge-Charge Correlation Functions. The reason for slow intermixing inside the adsorbed
layer for long chains with a degree of polymerization
N ) 32 and 16 is the formation of ionic pairs between
oppositely charged monomers compensating each other.
The local organization of such charged pairs in the
adsorbed structure can be described by the charge-pair
correlation function g((r)31 between positively and negatively charged monomers. Figure 8 shows the correlation
functions g((r) between oppositely charged monomers for
the duration of each deposition cycle equal to 2.0 × 105
MD steps. The correlation function g((r) is proportional
to the probability of finding a pair of oppositely charged
monomers at a distance of r. The correlation function
sharply decreases at about 4σ in the case of N ) 8,
indicating a very thin layer structure as compared to the
slowly decaying g((r) for N ) 32 and 16 where the chargedpair correlations extend much further, indicating a more
(31) Rapaport, D. C. The Art of Molecular Dynamics Simulation;
Cambridge University Press: New York, 1995; Chapter 4.
Langmuir, Vol. 21, No. 3, 2005 1125
dense layering. g((r) has peaks around σ and 2σ, indicating
the distance between the beads with predominant ionic
correlations. The closest possible approach distance
between the beads in any pair is σ. Such ionic pairs act
as effective friction centers slowing the motion of the
polymer chains. The polymer dynamics in this case is
controlled by the association and dissociation of ionic pairs
and can be described in the framework of the sticky Rouse
model for short chains and by the sticky reptation model
for chains above the entanglement threshold.32 Our
simulations show that the formation of ionic pairs could
lead to a 10 times increase of the chain relaxation time
in the adsorbed layer in comparison with that of a chain
in a solution. In the case of shorter chains, and also for
chains with a smaller charge fraction of charged monomers, the number of ion pairs would be considerably low,
allowing faster chain displacement than for the case of
the fully charged chains with a higher degree of polymerization.
4. Conclusions
In conclusion, multilayer film formation was found to
be electrostatically driven with sequential charge reversal,
giving rise to an assembly that is in a partially trapped
state. Only strongly charged chains with degrees of
polymerization N ) 32 and 16 having charge fractions of
f ) 1 and 0.5 were able to regenerate the surface properties
and to produce a steady buildup of multilayers. Chains
that did not have a critical threshold of charge, namely,
with degrees of polymerization N ) 32, 16, and 8 (f ) 0.25)
and N ) 8 (f ) 0.5), failed to adsorb beyond the first layer.
Among the chains that adsorb, a universal overcharging
behavior was observed regardless of charge fraction and
chain length. Our simulation results also emphasize the
importance of the first layer in multilayer formation with
a fine interplay of both electrostatic and nonbonded
interactions. Further, the degree of intermixing arising
from the exchange of chains during the course of the
deposition cycle also varies with the chain length as well
as the chain relaxation dynamics. Polyelectrolytes with
a higher degree of polymerization have greater intermixing
at longer simulation runs than in shorter runs. When the
degree of polymerization is sufficiently low, the chains
overcome the trapping barriers and equilibrate even
during short simulation runs. Oppositely charged monomers form ion pairs that slow the chain motion. Thus, a
successful theory of multilayer assembly should explicitly
take into account the dynamic nature of chain adsorption
and formation of ionic pairs.
Acknowledgment. Funding for this research from
the Petroleum Research Fund under grant PRF-39637AC7 (A.V.D.), from the National Science Foundation
(DMR-0305203, A.V.D., and CTS-0335345, J.F.R.), and
from NIEHS of the National Institutes of Health (PHS
Grant No. ES03154, J.F.R.) is gratefully acknowledged.
Supporting Information Available: A movie of the
polyelectrolyte multilayer assembly containing snapshots of the
simulation for five deposition steps. This material is available
free of charge via the Internet at http://pubs.acs.org.
LA047741O
(32) Rubinstein, M.; Dobrynin, A. V. Curr. Opin. Colloid Interface
Sci. 1999, 4, 83-87.