Supporting Information:
Adsorption of a polyelectrolyte chain at
dielectric surfaces: Effects of surface charge
distribution and relative dielectric permittivity
Ruochao Wang,
†
,
‡
Valeriy V. Ginzburg,
¶
Jian Jiang,
∗
,
†
,
‡
and Zhen-Gang Wang
∗
,
§
†
Beijing National Laboratory for Molecular Sciences, State Key Laboratory of Polymer
Physics and Chemistry, Institute of Chemistry, Chinese Academy of Sciences, Beijing
100190, P. R. China
‡
University of Chinese Academy of Sciences, Beijing 100049, P. R. China
¶
Chemical Engineering and Materials Science Department, Michigan State University,
East Lansing, Michigan, 48224, USA
§
Division of Chemistry and Chemical Engineering, California Institute of Technology,
Pasadena, California 91125, USA
E-mail: jiangj@iccas.ac.cn; zgw@caltech.edu
Here we start with a description of the local lattice Monte Carlo algorithm, then give
the density profiles of charged monomers, neutral monomers, and counterions when surface
charge density
σ
0
= 25
/
81 e
/
nm
2
and PE charge fraction of
f
c
= 21
/
121
. In addition,
a simple ion system is constructed to verify the existence of the bridging effect. We next
supplement the density profiles of monomers for surfaces with different charge distributions
in the absence and presence of dielectric contrast. The details of size frequency distribution of
three adsorbed conformations (tail, train, and loop) will be discussed. Finally, the adsorbed
S-1
amounts of surface counterions on the three charged surfaces are shown for comparison with
PE.
Description of the local lattice Monte Carlo algorithm.
S1
In a spatially varying
dielectric medium, the energy is given by
U
=
∫
D
2
2
(
r
)
d
3
r
,
(S1)
where the electric displacement
D
is constrained by Gauss’s law
∇·
D
=
ρ
f
,
(S2)
with
ρ
f
being the density of free charges. The general solution to Eq. (S2) is
D
=
−
∇
φ
+
∇×
Q
,
(S3)
where
Q
can be an the arbitrary vector potential. The field
D
is a fluctuating field, which
becomes the physical field upon taking its average at any fixed particle configuration. There-
fore, the partition function for a given configuration of charges is
Z
(
{
r
i
}
) =
∫
D
D
exp
[
−
β
∫
D
2
2
(
r
)
d
r
3
]
∏
r
δ
[
∇·
D
−
ρ
f
]
.
(S4)
After substituting
D
in Eq. (S4) with Eq. (S3), the cross term in the energy,
∫
∇
φ
·
(
∇×
Q
)d
r
3
,
is 0. Therefore, the partition function can be divided into a Coulomb term and a fluctuation
term.
Z
(
{
r
i
}
) = exp
[
−
β
2
∫
(
r
)(
∇
φ
)
2
d
r
3
]
×
∫
D
D
t
∏
r
δ
(
∇·
D
t
) exp
[
−
β
∫
D
2
t
2
(
r
)
d
r
3
]
,
(S5)
where
D
t
=
∇×
Q
is transverse part of the field
D
. The full partition function is obtained
S-2
by further integrating over the particle coordinates,
Q
(
N,V,T
) =
1
N
!
v
N
∫
∏
i
d
r
i
Z
(
{
r
i
}
)
,
(S6)
where
v
is a volume scale. Therefore, the thermodynamic average of a property
A
is com-
puted as
〈A〉
=
∫
∏
i
d
r
i
A
exp[
−
βU
]
∫
∏
i
d
r
i
exp[
−
βU
]
.
(S7)
According to Eq. (S6), the particle position
{
r
i
}
and the field
D
need to be sampled. For
the particle update, Jiang and Wang
S2
employed the electric field of ghost particles super-
posed around a charged particle to replace its movement, reducing the energy penalty and
improving the acceptance rate for the particle moves. For the field update, the open-circuit
moves based on the worm update method were supplemented to ensure ergodicity.
S2
Density profiles of charged and neutral monomers.
According to Figs. S1a and
S1b, the density profiles of charged monomers
ρ
∗
c
(
z
)
and neutral monomers
ρ
∗
n
(
z
)
near the
surfaces with uniform and regular charge distributions coincide roughly in the absence of
dielectric contrast. Compared with
ρ
∗
n
(
z
)
,
ρ
∗
c
(
z
)
in the vicinity of the randomly-charged sur-
face exhibits a more obvious enhancement than the other two types of charged surfaces. As
mentioned in the main paper, this is due to the charge distribution heterogeneity on the
randomly-charged surface and the self-adaptive adsorption of the PE. For cases with dielec-
tric contrast, there is a large increase of
ρ
∗
c
(
z
)
and
ρ
∗
n
(
z
)
on the surfaces with regular and
random charge distributions owe to the bridging effect (Figs. S1c, d). In contrast,
ρ
∗
c
(
z
)
and
ρ
∗
n
(
z
)
near the uniformly-charged surface are both suppressed by image repulsion.
S-3
Fig. S1: Density profiles of (a) charged and (b) neutral monomers near surfaces with different
charged distributions in the absence of dielectric contrast. Density profiles of (c) charged and
(d) neutral monomers near surfaces with different charged distributions in the presence of
dielectric contrast. The surface charge density is
σ
0
= 25
/
81 e
/
nm
2
, the PE charge fraction
is
f
c
= 21
/
121
.
S-4
Density profiles of monomers with different surface charge densities in the
absence of dielectric contrast.
As shown in the Figs. S2, the uniformly- and regularly-
charged surface show similar adsorption strength for the PE, regardless of the changes in
surface charge density
σ
0
. The randomly-charged surface always has the strongest adsorption
capacity for the PE in the absence of dielectric contrast, except for the case of the highest
surface charge density
σ
0
= 100
/
81 e
/
nm
2
, for which the randomly-charged surface shows
similar adsorption with the two other surfaces because of the excessive charges diminish the
impact of surface charge distribution.
S-5
Fig. S2: Density profiles of monomers near surfaces with different charge density (a) 100/81,
(b) 25/36, (c) 4/9, (d) 100/729, (e) 1/9, (f) 25/324, (g) 4/81, (h) 1/36, (i) 1/81, (j) 4/729
e
/
nm
2
in the absence of dielectric contrast. The PE charge fraction is
f
c
= 21
/
121
.
S-6
The density profile of counterions.
According to Fig. S3a, the density profiles of
surface counterions
ρ
∗
−
(
z
)
near three charged surfaces coincide roughly in the absence of
dielectric contrast, except that the contact density on the randomly-charged surface is ob-
viously higher than the others. As discussed in the main paper, this is due to the charge
distribution heterogeneity on the randomly-charged surface. For cases with dielectric con-
trast, the density profiles of surface counterions
ρ
∗
−
(
z
)
in the vicinity of surfaces with regular
and random charge distributions are similar owe to the bridging effect (Fig. S3b). Near the
uniformly-charged surface,
ρ
∗
−
(
z
)
and
ρ
∗
+
(
z
)
are both suppressed by image repulsion.
Fig. S3: Density profiles of counterions with different charged distributions (a) in the
absence of, and (b) in the presence of dielectric contrast. The surface charge density is
σ
0
= 25
/
81 e
/
nm
2
, the PE charge fraction is
f
c
= 21
/
121
.
S-7
Comparison of density profiles of monomers for surfaces with different charge
distributions in the absence and presence of dielectric contrast.
Fig. S4: Comparison of density profiles of monomers
ρ
∗
p
(
z
)
for surfaces with (a) uniform,
(b) regular, and (c) random charge distributions in the absence and presence of dielectric
contrast.
Evidence for the existence of bridging effect.
Here we design a new system to
verify the existence of stable bridge structures between charged sites, charged particles and
their images. The simulation box is set
10
a
×
10
a
×
40
a
, and the substrates are considered at
−
10
≤
z/a
≤−
1
and
21
≤
z/a
≤
30
. We place an ion with a negative elementary charge
−
e
at a lattice site above the center of the bottom surface with a positive elementary charge
e
.
The vertical distance between the ion and the bottom surface is
d
. Another equivalent ion
is placed at the lattice site with a distance
d
perpendicular to the center of the top surface.
S-8
A positive elementary charge
e
is evenly distributed to each lattice site on one uniformly-
uniformly
-
charged surface
discretely
-
charged surface
z
d
z
d
Fig. S5: (a) The schematic diagram of the system that an ion with a negative elementary
charge
−
e
are fixed at a lattice site perpendicular to the uniformly- or discretely-charged
surface at different distance
d
. The dashed white sphere represents a virtual image charge,
and the red dotted lines represent a bridging structure formed by the ion, the surface charge,
and the image charge. For simplicity, we only show the bottom surface in the diagram. (b)
The electrostatic energy difference in uniformly- and discretely-charged surface systems as
a function of vertical distance
d
between the fixed ion and the surface in the presence and
absence of dielectric contrast, where the electrostatic energy of each system at
d
=
a
is the
fiducial value.
charged surface, or is allocated to the surface site facing to the ion on one discretely-charged
surface. The dielectric constants are considered to be the same as the PE systems. We
fix the position of ions and only perform the field updates during simulations to obtain the
equilibrium energy. The final electrostatic energy of the system is computed by Eq. (3)
shown in the main paper. As shown in Figs. S5, the energy required for ion desorption in
discretely-charged surface system with image repulsion is more than for other cases. In the
presence of image repulsion, the energy of the uniformly-charged surface system with
d
= 2
a
and
d
= 3
a
are sightly lower than that with
d
=
a
, indicating that bridge structures are not
stable enough to counter image repulsion.
The size frequency distribution of three adsorbed conformations.
The details
of size frequency distribution of three adsorbed conformations (tail, train, and loop) will be
S-9
discussed here. As shown in Fig. S6a, the energy gain from the electrostatic neutralization
decreases with
m
tail
, resulting in the overall decline of
ν
(
m
tail
)
, which was also observed by
Beltrán et al.
S3
Fig. S6: The size frequency distribution of (a) tail, (b) train and (c) loop conformations as
a function of
m
β
for surfaces with different charge distributions in the absence of dielectric
contrast.
There are decreasing–increasing cycles of every 6 monomers due to the electrostatic at-
traction between charged monomers (
∆
p
= 6
) and surfaces. Compared to the randomly-
charged surface, there are less short and more long tails on surfaces with uniform and regular
charge distributions. This is because the ends of the PE form short tail conformations, while
the middle parts tend to be adsorbed on the randomly-charged surface in other more stable
S-10
conformations. For train and loop conformations, the size frequency distribution
ν
(
m
β
)
de-
creases with
m
β
due to the competition between electrostatic attraction and conformational
entropy (Figs. S6b and S6c). The self-adaptive adsorption results in the formation of more
train and loop conformations on the randomly-charged surface than on the uniformly- and
regularly-charged ones.
Fig. S7: The size frequency distribution of (a) tail, (b) train and (c) loop conformations as
a function of
m
β
for surfaces with different charge distributions in the presence of dielectric
contrast.
As shown in Fig. S7a, the trends in the size frequency distribution of tails on the
uniformly-charged surface and the discretely-charged surfaces are completely opposite. In
each cycle of the size frequency distribution near uniformly-charged surface,
ν
(
m
tail
)
initially
S-11
drops slowly, then experiences a rapid decrease (caused by image repulsion), and finally
increases to enter the next cycle. It is worth noting that the decreases at the end of each
cycle are smaller than the subsequent increments. This is because long tails appear more
often than short tails near multiples of
m
tail
= 6
as a consequence of repulsive interactions
between charged monomers in long tails and their images. In contrast,
ν
(
m
tail
)
gradually
increases with
m
tail
in each cycle near regularly- and randomly-charged surfaces due to the
bridging effect (see Fig. S7a). Because charged monomers can be attached to discretely-
charged surfaces through bridging attraction, a short tail is more likely to be formed near
a charged monomer (i.e., the one adsorbed on surfaces when
m
tail
is multiples of 6) than
a long tail. The slight drop when
m
tail
= 1
∼
3
is due to the image repulsion that tends
to keep charged monomers at the ends of the PE away from regions without charged sites
on discretely-charged surfaces. Similar to dielectric homogeneous systems, the average oc-
currence frequencies of train and loop conformations on the surfaces with dielectric contrast
decrease gradually with
m
β
. In the loop size frequency distribution diagram (Fig. S7c), cy-
cles appear more clearly in discretely-charged surface systems due to the bridging attraction,
while there are no apparent cycles in the uniformly-charged surface system due to the image
repulsion.
The adsorbed amount of surface counterions.
As shown in Fig. S8a, the adsorbed
amounts of surface counterions
ζ
−
on three charged surfaces increase parabolically with
surface charge density
σ
0
in the absence of dielectric contrast. The difference in adsorbed
amounts
ζ
−
on surfaces with uniform and regular charge distributions is subtle, and
ζ
−
on
the randomly-surface is slightly higher than that on the former two. Unlike the PE, the
difference in the adsorbed amount of surface counterions on surfaces with uniform or regular
charge distributions versus those with random charge distributions is not significant. This
indicates that randomly-charged surface has a better discrimination for the PE than surface
counterions. As dielectric contrast is considered, the bridging effect leads to much higher
ζ
−
on discretely-charged surfaces than that on the uniformly-charged one (see Fig. S8b), which
S-12
was also observed by Wang et al.
S4,S5
The one-to-one adsorption mode between counterions
and charged sites as a consequence of the bridging effect results in a consistent increase of
ζ
−
on surfaces with regular and random charge distributions.
Fig. S8: The adsorbed amount of surface counterions
ζ
ion
on uniformly-, regularly- and
randomly-charged surfaces with different surface charge density
σ
0
(a) in the absence of, and
(b) in the present of dielectric contrast.
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S-13
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S-14