Polyelectrolyte adsorption on heterogeneously charged surfaces

Article abstract of DOI:10.1063/1.481474

The adsorption transition of a uniformly charged polyelectrolyte onto heterogeneously charged surfaces has been investigated using off-lattice Monte Carlo simulations. Each of these surfaces contains both positive and negative charges. In addition to the usual case of adsorption of a polyelectrolyte to a surface with net charge opposite to that of the polymer, we show that a polyelectrolyte can adsorb onto a surface with net surface charge density similar to that of the polyelectrolyte. This adsorption is caused by the spatial inhomogeneity of the surface charges, which creates attractive regions with charge density different from the overall charge density of the surface. The spatial inhomogeneity of the surface charges also leads to differences in the conformation of the adsorbed polyelectrolyte. The critical conditions of strength and range of electrostatic interactions and chain length necessary for adsorption of a polyelectrolyte to a heterogeneously charged surface are demonstrated.

Full text of DOI:10.1063/1.481474

Polyelectrolyte adsorption on heterogeneously charged surfaces
M. Ellis, C. Y. Kong, and M. Muthukumar
Citation: The Journal of Chemical Physics 112, 8723 (2000); doi: 10.1063/1.481474
View online: http://dx.doi.org/10.1063/1.481474
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JOURNAL OF CHEMICAL PHYSICS
VOLUME 112, NUMBER 19
15 MAY 2000
Polyelectrolyte adsorption on heterogeneously charged surfaces
M. Ellis, C. Y. Kong, and M. Muthukumara)
Polymer Science and Engineering Department, and Materials Research Science and Engineering Center,
University of Massachusetts, Amherst, Massachusetts 01003
͑
Received 1 November 1999; accepted 23 February 2000͒
The adsorption transition of a uniformly charged polyelectrolyte onto heterogeneously charged
surfaces has been investigated using off-lattice Monte Carlo simulations. Each of these surfaces
contains both positive and negative charges. In addition to the usual case of adsorption of a
polyelectrolyte to a surface with net charge opposite to that of the polymer, we show that a
polyelectrolyte can adsorb onto a surface with net surface charge density similar to that of the
polyelectrolyte. This adsorption is caused by the spatial inhomogeneity of the surface charges,
which creates attractive regions with charge density different from the overall charge density of the
surface. The spatial inhomogeneity of the surface charges also leads to differences in the
conformation of the adsorbed polyelectrolyte. The critical conditions of strength and range of
electrostatic interactions and chain length necessary for adsorption of a polyelectrolyte to a
heterogeneously charged surface are demonstrated. © 2000 American Institute of Physics.
͓
S0021-9606͑00͒51019-8͔
I. INTRODUCTION
experimentally relevant situations, the surfaces often have a
nonuniform charge distribution. For example, a micelle or a
vesicle may have a net opposite charge in comparison with
that of the polyelectrolyte, but the surface may contain iso-
lated regions of charges with neutral material in between.13
When a polyelectrolyte is near the surface, there may be
competition between these differently charged regions. In the
case of a protein, the surface can have both positive and
negative charges. The overall charges on the surface can be
tuned by changing the PH of the solvent. It has been ob-
served that complexation can still occur even when the net
charge on the protein is similar to that of the
The interaction of polyelectrolytes and charged surfaces
is an important phenomenon in industrial and biological pro-
cesses. Some examples of the former include enhanced oil
recovery by polymer–micelle flooding, flocculation of cel-
lulosic fibers in the manufacturing of paper, and drug re-
lease from pharmaceutical tablets. Examples in biological
systems include signaling, immobilization of enzymes in
polyelectrolyte complexes, and purification of proteins by
selective precipitation. As a result, the adsorption of poly-
1
2
3
4
5
6
electrolytes to charged particles has been the subject of ex-
tensive experimental investigation.7,8 There have been stud-
ies of isolated polyelectrolytes complexing with latex
1
0
polyelectrolyte. There have been theoretical attempts to
study adsorption of homopolymers onto random surfaces by
9
10–12
13–15
16
using
analytical
methods22
and
Monte
Carlo
spheres, proteins,
micelles,
vesicles, and micro-
1
7
simulations.2
3–25
However, these studies only considered
gel particles. These particles have distinct surface charac-
teristics which give rise to different binding behaviors with
the polyelectrolytes. They may be different in the charge
density, the sign of the surface charge ͑in comparison with
that of the polyelectrolyte͒, surface curvature, and heteroge-
neity of the surface charge distribution.
neutral polymer with short-range interactions. There have
also been recent simulation attempts2
6–28
to understand ad-
sorption of neutral heteropolymers to neutral heterogeneous
surfaces.
In this paper, we consider uniformly charged polyelec-
trolytes placed above planar surfaces containing both posi-
tive and negative charges. While surface complexation is
governed by many factors, some of which are mentioned
above, we focus on the net charge of the surfaces. The posi-
tions of the charges are fixed but the distribution is otherwise
random. We also consider the role of salt concentration of
the solution on polyelectrolyte behavior. We have performed
Monte Carlo simulations to estimate the critical adsorption
conditions for polyelectrolyte chains to heterogeneous sur-
faces. The simulation data illustrate the effects of the surface
charges and salt concentration on polyelectrolyte behavior.
Regions of repulsive and attractive potential, whose intensi-
ties are governed by the screening length of the electrostatic
interactions between the chain and the surface, exist above
the surface and play a major role in determining the likeli-
The simplest example of polyelectrolyte adsorption deals
with a uniformly charged polymer adsorbing to an oppositely
charged planar surface with uniform charge density in a neu-
tralizing medium containing a known amount of added salt.
The use of the ground-state dominance approximation and
the separability assumption allows us to produce a simple
equation whereby the adsorption conditions of a polyelectro-
lyte can be predicted.18 This theory has been extended to
1
9
adsorption of a polyelectrolyte chain onto a spherical sur-
face and a patterned planar surface.20 These theories have
been validated using Monte Carlo simulations.2
0,21
However, the heterogeneity of the surface charge distri-
bution is not completely addressed in these theories. In some
a͒Electronic mail: muthu@polysci.umass.edu
0021-9606/2000/112(19)/8723/7/$17.00
8723
© 2000 American Institute of Physics
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1

8724
J. Chem. Phys., Vol. 112, No. 19, 15 May 2000
Ellis, Kong, and Muthukumar
hood of adsorption. This paper is organized as follows: the
model and simulation technique are detailed in Sec. II; the
results are presented in Sec. III; conclusions are summarized
in Sec. IV.
Here, q is the charge ͑ϩ1 or Ϫ1͒ of the unit making the jth
j
element of the lattice on the surface and rij is the distance
between the ith segment of the polymer and the jth element
of the lattice.
It must be pointed out that the electrostatic interaction
between the polymer and surface is described only through
the Debye–H u¨ ckel potential of Eqs. ͑2.2͒ and ͑2.6͒. The ac-
II. MODEL AND SIMULATION TECHNIQUE
Consider a semi-infinite three-dimensional space with an
impenetrable planar surface at zϭ0. The surface is decorated
by a square lattice of size 50ϫ50. At the center of each
square, we randomly put a ϩ1 or Ϫ1 charge. The total frac-
tion of positive charges on the lattice is defined as p. The
positions of the charges are fixed during the simulation. The
polymer chain charged uniformly with each bead bearing a
unit positive charge is contained in zϾ0 space. Let us as-
sume that the solution which contains the polyelectrolyte
chain can be treated as a continuum with an effective dielec-
tric constant ⑀ at a given temperature T.
The polymer chain is represented as a freely jointed
chain comprising N segments connected by freely rotating
bonds of Kuhn length l. Each segment is ascribed a hard-core
diameter a in order to account for excluded volume interac-
tions. In addition, each segment is assigned a charge of ϩ1.
The ith and jth segments separated by a distance rij interact
through an excluded volume interaction due to hard-core re-
pulsion
tual electrostatic potential between a charged monomer of a
polymer and a charged group of a surface in an ionic solution
is unknown at present. The counterions of the polyelectrolyte
and the surface, and dissociated salt ions are assumed to be
randomly distributed and consequently appear through the
value of the Debye length. If the solution conditions were to
allow counterion condensation, the values of the parameters
qi and ␬ would be modified. The potential used here is suf-
ficient in addressing the generic questions pertinent to the
adsorption of a polyelectrolyte on a charged heterogeneous
surface. But, care must be exercised in any quantitative com-
parison between the results given below and experimental
results.
The basic parameters in this model simulation are
l,a,l ,p, and ␬. The temperature T and the dielectric con-
B
stant, ⑀, appear through the Bjerrum length, l . In the simu-
B
lation performed in the present study, we have fixed l
ϭ1,a/lϭ 3/2,l /lϭ1.3, and Nϭ60. ␬l and p are the vari-
ables in the simulation.
ͱ
B
0
,
rijϾa
The chain is allowed to evolve by a dynamic Monte
Carlo algorithm. This algorithm changes the chain confor-
V ͑r ͒ϭ
͑2.1͒
29
1
ij
ͭ
ϱ, rijрa,
mation by local motion of segments. A segment chosen at
random is rotated through an angle ␣ about the axis defined
by adjacent segments to a new trial position. If the chosen
segment lies at either end of the chain, it is moved through
two randomly chosen angles ␽ and ␾ to a new trial position
͑For details, see Ref. 29͒. Appropriate statistical weights are
assigned to the configuration by Metropolis rules: attempts
are accepted if ͑1͒ the moved segment does not touch or
penetrate the surface or does not overlap with other seg-
and a screened Coulombic interaction
eϪ␬rij
V ͑r ͒ϭk Tq q l
i j B
,
͑2.2͒
2
ij
B
rij
where q is the charge per Kuhn length of the ith segment, k
is the Boltzmann constant, and l is the Bjerrum length,
i
B
B
e2
l ϭ
B
,
͑2.3͒
4
␲⑀ ⑀k T
0 B
ments, and if ͑2͒ exp͓Ϫ(VnewϪVold)/k T͔Ͼ⌫, where
B
where e is the electronic charge and ⑀ is the permittivity of
vacuum. ␬ is the inverse Debye length, determining the
range of the electrostatic interaction, and is given by
0р⌫р1 is a random number and V is the total potential
energy of a given chain conformation. Whether or not the
move is accepted, each move is counted as a step. N such
elementary steps compose one Monte Carlo time step.
The simulation is carried out as follows. First, we fix the
value of ␬. Then, a random surface is created for a given p.
The polyelectrolyte chain is created and equilibrated in the
absence of the surface using the above procedure for typi-
0
4
␲e2
␬2
ϭ
c z ,
2
͑2.4͒
͚i
i
i
⑀
kbT
where c and z are the concentration and the valence, re-
i
i
spectively, of the ith mobile ion.
5
cally 10 Monte Carlo time steps. Next, the polymer is trans-
The lattice on the surface is taken to be the x– y plane at
zϭ0. The polymer segments interact with the surface
through an excluded volume interaction
lated so that its center of mass is above the center of the
surface and the monomer closest to the surface is at a dis-
tance of 5l. The Monte Carlo clock is reset to 0 and data are
0
,
ziϾa/2
6
taken for 10 more time steps. For the next value of ␬, we
V ͑z ͒ϭ
͑2.5͒
3
i
ͭ
ϱ, ziрa/2,
create a new random surface and repeat the above procedure.
For the purposes of this investigation, we consider a polymer
to be bound to the surface if at least one of its monomers
with position (x,y,z) satisfies the following conditions:
where z is the z coordinate of the ith segment of the poly-
i
mer.
The polymer segments also interact with the charges on
the lattice with a screened Coulombic interaction
0
рxр50
exp͑Ϫ␬rij͒
0рyр50
boundϭ
͑2.7͒
ͭ
V ͑r ͒ϭϩk Tl q
B j
.
͑2.6͒
4
ij
B
rij
zрa.
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J. Chem. Phys., Vol. 112, No. 19, 15 May 2000
Polyelectrolyte adsorption
8725
FIG. 1. ͑a͒ and ͑b͒ are the time (t) evolutions of the net polymer–surface interaction energy per monomer Es and the number of monomers bound,
respectively, for pϭ0.0. ͑c͒ and ͑d͒ are the time (t) evolutions of the net polymer–surface interaction energy per monomer and E the number of monomers
s
bound, respectively, for pϭ0.4. The value of ␬l is 0.45 for both pϭ0.0 and 0.4.
The number of contacts with the surface, the adsorption en-
the time evolutions of the polymer–surface interaction en-
ergy per segment, E , and the density profile, ␳(z), are
ergy E and the number of monomers bound are given in
s
s
sampled. The adsorption energy per segment is defined as
Figs. 1͑c͒ and 1͑d͒, respectively. In comparison with Figs.
1
͑a͒ and 1͑b͒, the adsorption energy is weakened by the re-
1
pulsive charges on the surface, and the number of monomers
bound also decreases. As a comparison of the adsorbed poly-
mer’s confirmations, the monomer density profiles for p
ϭ0.0 and 0.4 are presented in Fig. 2. At ␬lϭ0.45, the poly-
electrolyte can adsorb onto both surfaces. The adsorbed
polymer has a stronger surface interaction energy with the
pϭ0.0 surface. The adsorbed layer is confined closer to the
E ϭ
V ͑r ͒.
͑2.8͒
͑2.9͒
s
͚ ͚
4
ij
N
i
j
The density profile is defined as
͑z͒ϭ ␦͑zϪz ͒,
␳
͚i
i
where z is the z position of the ith segment.
i
III. RESULTS
A. Adsorption to net attractive surfaces
We first consider the adsorption of a single polyelectro-
lyte to an oppositely charged surface, where pϭ0. For ␬l
ϭ0.45, the time evolutions of the polymer–surface interac-
tion energy E and the number of monomers bound are given
s
in Figs. 1͑a͒ and 1͑b͒, respectively. The polymer is placed
near the surface at zero time. As the chain is allowed to
undergo dynamics under the influence of the potential from
the surface in addition to the intrachain interaction, the sur-
face interaction energy starts to decrease, and eventually the
polymer is adsorbed onto the surface. The polymer chain
stays adsorbed onto the surface until the end of the simula-
tion. Now, we consider the effect of increasing the number of
repulsive charges on the surface. For ␬lϭ0.45 and pϭ0.4,
FIG. 2. The density profile ␳(z) in the z direction for pϭ0.0 and 0.4. The
value of ␬l is 0.45.
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8726
J. Chem. Phys., Vol. 112, No. 19, 15 May 2000
Ellis, Kong, and Muthukumar
͗ ͘
FIG. 3. The average polymer–surface interaction energy per segment Es
as a function of ␬l for pϭ0.0 and 0.4.
FIG. 5. The distribution of positive and negative charges on a patterned
surface (pϭ0.516). The region where the binding occurred in outlined on
the pattern. The value of ␬l is 0.05 and peff is 0.43 ͑gray and white squares
represent positive and negative charges, respectively͒.
surface. For the pϭ0.4 surface, the repulsive charges on the
surface weaken the adsorption and the adsorbed layer ex-
pands.
The adsorption transitions can be monitored by measur-
͗ ͘
ing the time average of Es , Es , which is plotted against ␬l
and repeatedly observed even for net neutral and repulsive
surfaces (0.5рpр0.52), in addition to attractive surfaces
in Fig. 3 for pϭ0.0 and 0.4. There are critical values of ␬c
above which the polymer does not bind to the surfaces and
the adsorption energy is zero. Adsorption of the polymer
chain by a surface occurs when the energy of adsorption
overcomes the entropy loss of the polymer due to surface
confinement. By increasing the salt concentration, the
surface–polymer interaction is screened. At the critical val-
(
pϽ0.5). For a given surface, the polymer chain is also ob-
served to reside on a certain region on the surface. In order to
quantify properties of this region, we calculate the effective
ues ␬ , these two opposing tendencies exactly cancel each
c
͗ ͘
other. Near ␬ , the Es may be negligible even though the
c
polymer may be adsorbed. In order to insure that we deter-
mine the adsorption transitions accurately for all simulations,
we also examine the average number of monomers bound to
the surface. When both the total number of monomers bound
to the surface and the average adsorption energy per mono-
mer go from zero to nonzero, we define that point as the
adsorption point.
B. Adsorption to net neutral and repulsive surfaces
In Fig. 4, we plot ␬ l versus p. Adsorption is regularly
c
FIG. 6. The effective positive charge density peff for ͑a͒ pϭ0.25 and ͑b͒
pϭ0.516.
FIG. 4. Dependence of ␬ l on p. A line is included as a guide in the figure.
c
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J. Chem. Phys., Vol. 112, No. 19, 15 May 2000
Polyelectrolyte adsorption
8727
FIG. 7. ͑a͒ Pattern (pϭ0.52) investigated by Monte Carlo simulations. ͑b͒ Time (t) evolution of the net polymer–surface interaction energy per monomer Es
for ␬lϭ0.35. ͑c͒–͑d͒ correspond to the position of the polymer in the potential field.
Indeed, for all values of pр0.52, the binding region has
value of peff which is significantly lower than the p for the
entire surface. Figures 6͑a͒ and 6͑b͒ are graphs of peff for
surfaces of pϭ0.25 and 0.516, respectively. For cases where
the overall p is attractive and for cases where the overall p is
repulsive, peff is always lower than p and peff is always at-
tractive. This difference between peff and p is created by the
spatial inhomogeneity of the surface charges. Since the
charges are put on the surface randomly, there may be some
regions with more attractive or repulsive charges. When the
polymer chain approaches a net neutral or a slightly repul-
sive surface, adsorptions will occur on these attractive re-
gions. Since there may be more than one attractive region,
charge density peff , the fraction of positive charges in the
region within which adsorption occurs. We first identify the
region on the surface in the x– y plane where the polymer
chain forms contacts for more than 100 Monte Carlo time
steps. Then, within this region we compute peff by dividing
the total number of positive charges by the total number of
charges. If there are two or more disconnected attractive re-
gions on the surface, peff is taken by averaging these differ-
ent regions. Figure 5 shows a surface of pϭ0.516 and the
simulation was run with ␬lϭ0.05. The outlined area is the
region within which adsorption occurred. peff is found to be
0
0
.43, significantly lower than the p of the overall surface,
.516.
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8728
J. Chem. Phys., Vol. 112, No. 19, 15 May 2000
Ellis, Kong, and Muthukumar
cube depended on whether the potential is repulsive or at-
tractive: a large dark gray cube corresponds to repulsive po-
tential and a small light-gray cube corresponds to attractive
potential. After the values of potential at zϭ0.5 were
mapped, we performed the same procedure for
z
ϭ1.5,2.5,3.5,4.5,5.5, and 6.5.
For purposes of illustration, we present a typical case for
the distribution of charges illustrated in Fig. 7͑a͒. The frac-
tion of positive charges on the entire pattern p is 0.52, so that
the net charge of the surface is similar to that of the poly-
electrolyte. At ␬lϭ0.35, the outlined area is the region
within which adsorption occurred. peff was found to be 0.41,
significantly lower than the p of the overall surface, 0.52.
The pattern has a large region of negative charges which
create an attractive region. The time evolution of E is given
s
in Fig. 7͑b͒. Initially, a polymer is placed above the surface
and the adsorption energy fluctuates near the zero point. At
tϳ6400, the adsorption energy starts to decrease and reaches
a steady state near tϳ20 700. The snapshots of the polymer
chain at different times are shown in Figs. 7͑c͒–͑f͒. At t
ϭ6400, when the polymer is above the center of the pattern
where there is a large region of repulsive potential, the poly-
mer avoids the repulsive region, moves towards the region of
attractive potential, and eventually adsorbs.
The polymer is able to bind to the surface due to the
attraction of the charges in an attractive area on the surface.
Though that region contains some repulsive charges, the
polymer is able to bind to it because the overall potential
above that section is still attractive. This indicates that the
correlation of attractive charges on the surface is one of the
most important factors in determining whether or not a poly-
mer will bind to the surface. Identification of the regions of
potential above the surface helps to explain why peff is less
than p: the polymer is repelled from the repulsive potential
and guided towards the region of attractive charges by the
overall attractive potential of a region of highly correlated
opposite charges.
FIG. 8. ͑a͒ Mean-squared displacement perpendicular to the surface of the
center of mass of the polymer versus t for different values of ␬l. Slope of
1
.0 is included as a guide in the figure. ͑b͒ Mean-squared displacement
parallel to the surface. Slopes of 1.0 and 0.5 are included as guides.
the exact location of the adsorbed polyelectrolyte strongly
depends on the initial conditions.
2
The time-dependent mean-squared displacements, R (t),
C. Polymer adsorption kinetics
of the center of mass of the polymer chain was monitored
during the calculation, where
The kinetics of adsorption were studied by following the
position of the polymer during the simulation. At every point
above the surface, a monomer will experience a potential
field generated by the heterogeneous surface. We can divide
the space above the surface into regions of attractive poten-
tial and regions of repulsive potential as governed by the
charge distribution of the surface and the long-range nature
of electrostatic potential. We observe the motion of the poly-
electrolyte through this field.
In order to visualize the potential, we discretized space
above the surface. At zϭ0.5 we imagined a 60ϫ60 lattice
centered over the center of the surface. The 50ϫ50 lattice
sites in a square region were located directly above surface
charges so that this lattice overhangs the surface by 5 sites on
each edge. We calculated and stored the potential at each
lattice site. We recorded the maximum value of the absolute
potential. After the potential of every lattice site was calcu-
lated, we went through the lattice again. If the absolute value
of the potential at a site was within 1.0% of the maximum
value, we put a cube at that site. The size and the color of the
2
͓RCM͑t͒ϪRCM͑0͔͒2
R ͑t͒ϭ
͗
͘
,
͑3.1͒
with RCM(t) being the position vector of the center of mass
of the chain at time t. Figure 8 contains the double-
2
logarithmic plot of R (t) perpendicular and parallel to the
surface of the polyelectrolyte versus Monte Carlo time for
the representative cases, viz. pϭ0.0 and 0.52 for ␬lϭ0.05
and 0.75, respectively.
For pϭ0, both situations ␬lϭ0.05and 0.75 correspond
to polyelectrolyte adsorption. As seen from Fig. 8͑a͒, the
chain is first confined to a two-dimensional layer and then
two-dimensional localization proceeds ͓Fig. 8͑b͔͒. This cor-
responds to the kinetics of adsorption of a chain. In the
crossover region of two-dimensional localization, an appar-
2
␯
͗ ͘
ent exponent ␯, ( R (t) ϳt ), can be tabulated as a function
of p, ␬, and time range. For pϭ0.52, at ␬lϭ0.75 ͑above the
adsorption threshold͒, the motion of the chain can be de-
2
scribed by classical diffusion, R (t)ϳt. When the range of
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J. Chem. Phys., Vol. 112, No. 19, 15 May 2000
Polyelectrolyte adsorption
8729
the electrostatic interaction is increased to ␬lϭ0.05, a cross-
over from diffusion to localization is shown in Figs. 8͑a͒ and
Molecular Biology of the Cell ͑Garland, New York, 1994͒.
A. Margolin, S. Sherstyuk, V. Izumrudov, A. Zezin, and V. Kabanov, Eur.
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P. Dubin, M. Strege, and M. West, in Large Scale Protein Purification,
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H. Dautzenberg, W. Jaeger, J. K o¨ tz, B. Philipp, Ch. Seidel, and D.
5
6
8͑b͒.
IV. CONCLUSIONS
7
8
We have performed Monte Carlo simulations to examine
the conformation, adsorption conditions, and kinetics of a
uniformly charged polyelectrolyte adsorbing onto heteroge-
neously charged surfaces. A heterogeneously charged sur-
face can create a complicated potential field above the sur-
face. Under the influence of this field, the polyelectrolyte
seeks out the area of lower potential. The region to which the
polymer adsorbs consistently is more attractive than the sur-
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ACKNOWLEDGMENTS
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Acknowledgment is given to the NSF for Grant No.
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