Article
pubs.acs.org/Macromolecules
Osmotic Pressure of Polyelectrolyte Solutions with Salt: Grand
Canonical Monte Carlo Simulation Studies
Rakwoo Chang,*,† Yongbin Kim,† and Arun Yethiraj‡
†
Department of Chemistry, Kwangwoon University, Seoul 139-741, Republic of Korea
Theoretical Chemistry Institute and Department of Chemistry, University of Wisconsin, Madison, Wisconsin 53706, United States
‡
ABSTRACT: The thermodynamic properties of polyelectrolyte
solutions are of long-standing interest. Theoretical complexity arises
not only from the long-ranged electrostatic interaction but also from the
multicomponent nature of the solution, In this work, we report grand
canonical Monte Carlo simulations for the effect of added salt on the
osmotic pressure of a primitive model of polyelectrolyte solutions. The
polymer chains are freely jointed charged hard spheres, and counterions
and co-ions are charged hard spheres. We use an ensemble that allows
us to calculate directly the osmotic pressure for a solution in equilibrium
with a bulk salt solution. As the bulk salt concentration is increased, the concentration of salt in the polyelectrolyte solution
decreases and for semidilute solutions the salt concentration is very low. In dilute solution, the salt contribution to the osmotic
pressure arises from electrostatic screening and excluded volume interactions. Semidilute solutions behave like salt-free solutions.
The simulations show that both polymer molecules and small ions make a significant contribution to the osmotic pressure, thus
questioning theories that ignore the polymer contribution. The latter effect results in the decrease in magnitude and a strong
concentration dependence of the osmotic pressure. The simulation results are in qualitative accord with experiments on DNA.
Scaling theories for the osmotic pressure, however, are not in agreement with the simulations or experiments.
■
INTRODUCTION
Polyelectrolyte solutions have interesting structural and
thermodynamic properties because of the subtle interplay
between electrostatic, excluded volume, and solvent-induced
interactions. For example, dilute polyelectrolyte solutions show
liquid-like ordering arising from electrostatic correlations,
manifested by a scattering peak in the static structure factor.1−3
In semidilute concentrations, the properties are dominated by
the segmental correlation length, ξ, and ξ ∼ ρm−1/2 (where ρm is
the monomer density or concentration). In concentrated
solutions, the properties are dominated by excluded volume
interactions and become similar to solutions of neutral
polymers.
The differences in the correlation length in these three
concentration regimes impact the thermodynamic properties
such as the osmotic pressure, Π. Scaling theory would predict
that Π ∼ ξ−3, i.e., Π ∼ ρm3/2 in semidilute concentrations but
this is not seen in simulations of rod-like4,5 and flexible
polyelectrolytes,6−8 where Π ∼ ρm2 and Π ∼ ρm9/4, respectively.
It is often assumed that the counterion contribution to the
osmotic pressure is much larger than the polyion contribution9−11 and is given by the ideal gas equation of state, i.e., Π =
ϕρckBT where ρc is the number density of counterions, kB is
Boltzmann’s constant, T is the temperature, and ϕ is the
osmotic coefficient, which corrects the ideal gas law with the
ansatz that a fraction 1 − ϕ of counterions are “condensed” on
the surface of the polyion (and thus do not contribute to the
osmotic pressure).10−13 Such an ansatz has been used in the
analysis of experiments14,15 but the validity of such an
© 2015 American Chemical Society
approximation has been questioned by computer simulations
of semidilute solutions.6−8
Experiments present a more complicated picture compared
to the simple Manning condensation idea. For example,
experiments11 on solutions of double-stranded B-DNA find Π
∼ ρm2.5, and other experiments16 suggest that the polymeric
contribution to the osmotic pressure scales as Π ∼ ρm2.2. These
exponents are consistent with that of 2.25 for neutral semidilute
polymer solutions and also in agreement with simulations of
primitive models of polyelectrolyte solutions.6−8
Liquid state theories have been presented for the osmotic
pressure of polyelectrolyte solutions. Jiang et al.17−19 obtained
an analytical solution of thermodynamics properties for the
restrictive primitive model of polyelectrolyte solutions using
Wertheim’s thermodynamic perturbation theory.20−23 Wertheim’s Ornstein−Zernike approach20−24 has also been
used25−28 to derive both thermodynamic and structural
properties of polyelectrolyte solutions. Chang and Yethiraj8
compared the above theories and the PRISM integral equation
theory to simulation results for the osmotic pressure of salt-free
polyelectrolyte solutions. The theories were in quantitative
agreement with simulations in concentrated solutions but not
in dilute or semidilute solutions. The theories were in better
agreement for weakly coupled systems than for strongly
coupled systems.
Received: July 21, 2015
Revised: September 14, 2015
Published: September 24, 2015
7370
DOI: 10.1021/acs.macromol.5b01610
Macromolecules 2015, 48, 7370−7377
Article
Macromolecules
temperature. At T = 298 K, lB = 0.7 nm in aqueous solution. In
this study we used lB/σ = 1.0 and 3.0 to study the effect of the
strength of the electrostatic interaction.
The system consists of Np polymer chains with degree of
polymerization Nm, Nc counterions, and Ns co-ions. We use Nm
× Np = 1024, e.g. Np = 64 for Nm = 16 and Np = 16 for Nm = 64.
Nc and Ns vary depending on the salt concentration. The box
length L is chosen to achieve the desired monomer
concentration and the periodic boundary condition is applied
in all directions. The monomer concentration ranges from ρmσ3
= Np Nm σ3/L3 = 0.005 to 0.30 which spans the entire range of
concentrations from dilute to concentrated. The bulk salt
concentrations are ρbs σ3 = 7.3 × 10−3, 6.2 × 10−2, and 1.3 ×
10−1, which correspond to βμ* = −10, −5, and −2,
respectively, where β = 1/kB T and βμ* = βμ − ln(Λ+/σ)3 −
ln(Λ−/σ)3; μ (=μ+ + μ−) is the chemical potential of the salt
ion pair and Λ± (≡h/(2π m±kB T) 1/2) (h is Planck constant)
the thermal de Broglie wavelength of positively (or negatively)
charged salt ions with mass m±.
Grand Canonical Monte Carlo Simulations. Grand
canonical Monte Carlo (GCMC) simulations for primitive
model electrolytes have been performed by Valleau and
Cohen.36 In GCMC, the temperature (T), volume (V), and
chemical potential (μ) of co-ions and counterions are fixed, and
the concentration of each species is allowed to fluctuate via
insertion and deletion of particles. In case of electrolyte
systems, pairs of positively and negatively charged species
(depending on their charge valence) are added or removed to
preserve electroneutrality.
To mimic osmotic pressure measurements, we allow only the
co-ion and counterion concentrations to fluctuate while the
polymer concentration is fixed. This corresponds to the
experimental condition where solvent and salt ions can
permeate the membrane but the polyions cannot. Note that
the salt concentration is, in general, different in the
polyelectrolyte solution and “bulk” electrolyte solution with
which the salt ions are at equilibrium, which is known as the
Donnan equilibrium.29−31
There are three moves in GCMC: particle addition, deletion,
and displacement. The acceptance ratio between state i and
state j for 1−1 electrolytes is given by36
The effect of added salt on the osmotic properties of
polyelectrolyte solutions is an interesting and unavoidable
problem. Polyelectrolyte solutions in experiments are almost
never salt-free because of dissolved carbon dioxide and other
impurities. A low ionic strength of 10−4 M can have a significant
effect on polyelectrolyte properties. Furthermore, biological
systems are usually at fairly high ionic strength with salt
concentrations of the order of 150 mM.
Including added salt in theories adds another component to
an already complicated system, and in most cases salt is
incorporated implicitly by modifying the Coulomb interaction
with a Debye−Hückel screened Coulomb interaction with an
electrostatic screening length κ−1 (so-called Debye length).
Recently, Carrillo and Dobrynin performed hybrid Monte
Carlo/molecular dynamics simulations to study the osmotic
pressure of polyelectrolyte solutions with salt and reported
interesting universal behavior of the osmotic coefficient as a
function of the ratio of the osmotically active counterion and
salt concentrations.13 However, because of the high fluctuation
of the system pressure in both polyelectrolyte and salt systems,
the resulting osmotic pressure data, obtained by subtracting
from the pressure of the polyelectrolyte solution the pressure of
the salt-only solution, were not statistically satisfactory.
On the other hand, the osmotic properties of charged
colloids have been studied extensively mostly in the framework
of Donnan equilibrium and Poisson−Boltzmann theories.29−35
Although they are in close analogy with polyelectrolyte systems,
the charged colloids are mostly treated as structureless spheres
or rods, and hence, there exists no intimate correlation between
colloid conformation and their osmotic behavior.
In this work, we study the effect of added salt on the osmotic
pressure of polyelectrolyte solutions using grand canonical
Monte Carlo simulations. By decomposing the virial into
several contributions, we are able to identify the dominant
contribution in the osmotic pressure in different polymer
concentration regimes with high precision, and quantify the
effect of salt in various concentration regimes. Our work is an
extension of the methodology previously employed for salt-free
solutions.8 We find that both the polyions and small ions
contribute to the osmotic pressure significantly. The contribution of the small ions comes from both electrostatic
screening as well as excluded volume interactions. The
simulation results for the scaling behavior are consistent with
experiments, but not with scaling theories.
The paper is organized as follows. The Molecular Model and
Simulation Methods section describes the molecular model and
simulation methods, and the Results and Discussion presents
results for the osmotic pressure, followed by the Summary and
Conclusions.
fij
f ji
=
+
−
V 2 Ni ! Ni !
exp[βμ − β(Uj − Ui)]
+
Λ+3Λ−3 N j ! N j−!
(1)
where f ij is the probability of acceptance of a trial step i to j, N+i
(or N−i ) the number of counterions (or co-ions) in the state i,
and Ui the configuration energy of state i. Note that N+j = N+i +
1 and N−j = N−i + 1.
Eq 1 can be simplified into
■
MOLECULAR MODEL AND SIMULATION
METHODS
Molecular Model. Polymer chains are modeled as chains of
negatively charged hard spheres with diameter and bond length
equal to σ, which is the unit of length in this paper.
Counterions (or co-ions) are positively (or negatively) charged
hard spheres of the same size as the monomers of the polymer
chains. The solvent is treated as a dielectric continuum. The
strength of the electrostatic interaction is commonly controlled
2
by the Bjerrum length (lB) defined as e , where e, ϵ0, ϵ, kB,
⎡
⎛ N+ + 1⎞
⎛ N− + 1 ⎞
= exp⎢βμ* − ln⎜ i 3 ⎟ − ln⎜ i 3 ⎟
⎢⎣
⎝ V /σ ⎠
f ji
⎝ V /σ ⎠
fij
⎤
− β(Uj − Ui)⎥
⎥⎦
(2)
where βμ* = βμ − ln(Λ+/σ)3 − ln(Λ−/σ)3. An addition move
⎛ fij ⎞
is accepted with probability of min⎜1, f ⎟, and similarly, a
⎝ ji ⎠
⎛ f ji ⎞
deletion move is accepted with probability of min⎜1, f ⎟.
⎝ ij ⎠
4πϵ0ϵkBT
and T are respectively elementary charge, vacuum permittivity,
solvent dielectric constant, Boltzmann’s constant, and absolute
7371
DOI: 10.1021/acs.macromol.5b01610
Macromolecules 2015, 48, 7370−7377
Article
Macromolecules
interactions, the third and fourth terms to polymer−ion
interactions, and the last three terms to ion−ion interactions.
The interaction potential is the sum of the hard sphere and
Coulomb interactions, i.e., β uij(r) = ∞ for r < σ and β uij(r) =
lB zi zj /r for r > σ where zi is the charge valence on a site of
species i.
The pressure, P, is calculated from the virial equation38,39
The displacement move is the same as that in canonical
Monte Carlo simulations. For polyelectrolyte chains, standard
moves such as reptation, crank-shaft, continuum configuration
bias, and translation moves are used and for small ions, only the
translation move is used. The trial displacement move is
accepted depending on the potential energy difference, ΔU ≡
Uj −Ui, between the original and trial configurations. Since
there exist hard-spherical and electrostatic interactions in this
system, a two step acceptance procedure is implemented: first,
we check for particle overlap from the trial configuration and
next, if there is no overlap, we calculate the electrostatic energy
difference using the tabulated Ewald summation method. The
trial move is accepted with probability of min(1,exp(−ΔU/
kBT)).37
Initial configurations are generated by randomly inserting
monomer beads of polyions and small ions. At high
concentrations, this random insertion method is not efficient
because almost every insertion trial move is rejected. Therefore,
for ρmσ3 > 0.1, counterions and polyion beads are randomly
located at some of the 4 M3 (M = 2, 3, ...) lattice points of a
face-centered cubic structure. The distance between nearest
neighboring lattice points is set to σ. In the case of polyions, Nm
adjacent lattice points are chosen for each molecule. The
GCMC simulations are then performed to equilibrate the
system until the potential energy of the system, the number of
small ions, and the size of polymer chains fluctuate about a
constant value. Starting with each of these equilibrated
configurations, trajectories of polyions and counterions are
generated using the GCMC simulations, and saved every 1000
moves. The properties reported in this study are the average of
3000−10000 trajectories. Usually, one standard deviation of the
average is obtained by block-averaging and in the figures this is
usually smaller than the size of the symbols.
We test for finite size effects by performing simulations with
doubling the number of polymer chains to Np = 128 at the
lowest two monomer concentrations (ρmσ3 = 0.005 and 0.01)
for Nm = 16 and lB/σ = 1.0, and found no difference.
Osmotic pressure calculation. The configuration integral,
ZN, of the polyelectrolyte solution is given by
ZN =
∫ exp(−βU ) dR1 dω1 ··· dR N dωN
p
p
∞
2πρmol
βP
=1−
3
ρmol
Nm
+ 2xpxc ∑
α=1
where ρmol =
U=
Np
Np
+
Nm
≡
N
αγ
xνxμρmol gνμ
(r) =
Nc
z νzμ
dβuEL(r )
= −lB 2
dr
r
i>j
(8)
2πρmol σ 3 ⎡ 2 2
βPHS
=1+
Nm xp f pp (σ +) + 2Nmxp
ρmol
3 ⎣
{xcf pc (σ +) + xsf ps (σ +)} + xc 2fcc (σ +) + xs 2fss (σ +)
+ 2xcxsfcs (σ +)⎤⎦
rfνμ (r ) =
Ns
1
NνmNμm
m
Nνm Nμ
i=1 j=1
rαij γ
(9)
where
∑ uij(rij11) + ∑ uij(rij11) + ∑ ∑ uij(rij11),
i>j
(7)
where zν is the charge valence on a site of species ν.
For hard chains eq 5 simplifies to
Ns
Nc
(6)
and
i=1 α=1 j=1
Ns
and r
Nμ
ν
1
⟨∑ ∑ ′δ(r + r γj − r iα)⟩
Nmol i = 1 j = 1
dβuHS(r )
= −exp(βuHS)δ(r − σ +)
dr
∑ ∑ ∑ uij(rijα1) + ∑ ∑ ∑ uij(rijα1)
i=1 α=1 j=1
+
Nm
Nν
,
Nmol
where the prime indicates that the terms for which i = j are
omitted when ν = μ. Note that the intramolecular interaction
term disappears in the virial expression.
The interaction potential uνμ(r) (ν,μ = p,c, and s) is the sum
of the hard-sphere (uHS(r)) and electrostatic (uEL(r))
contributions. The derivatives of these contributions are given
by40
Nm
Np
(Nmol is the total number of
V
The site−site correlation function, gαγ
νμ(r) (ν,μ = p,c, and s
and α,γ = 1,2,···,Nm for chains or 1 for ions), is given by
i=1 α>γ
Nc
=
Np + Nc + Ns
|rα12γ|.
∑ ∑ uii(rijαγ) + ∑ ∑ uii(riiαγ)
i>j α ,γ
Nmol
V
chains plus ions and V is the system volume), xν ≡
(3)
Nm
N
(5)
where Ri and ωi are the position vectors of the center of mass
and the orientation vectors of chain i, respectively, and ri are the
position vectors of counterions and co-ions. Note that the
orientation vector, ω, has 2(Nm−1) orientational degrees of
freedom. The total potential energy, U, consists of seven
contributions:
Np
0
m
r11
r11·r ̂
12· r ̂ α1
g pc (r ) + 2xpxs ∑ 12 g psα1(r )
r
r
α=1
⎤ d β u (r )
νμ
r 3 dr
+ xc 2gcc11(r ) + xs 2gss11(r ) + 2xcxsgcs11(r )⎥
⎥⎦ dr
dr1 ··· drNc
drNc + 1 ··· drNc + Ns
∫
Nm
⎡
11
⎢x 2 ∑ r12·r ̂ g αγ (r )
p
pp
⎢⎣ α , γ
r
rαi
(4)
∑∑
rγj |,
α=1 γ=1
where u(r) is the site−site interaction potential,
≡| −
and rαi is the position vector of site α in species i. The first two
terms correspond to intermolecular and intramolecular polymer
αγ
αγ
, ω1 , ω2) dω1 dω2
̂ )g (r12
∫ (r11
12· r12
m
m
Ω(N1 + N2 − 2)
(10)
where Nν is the number of sites in species ν and ων the
molecular orientation with 2(Nν − 1) degrees of freedom, Ω =
7372
DOI: 10.1021/acs.macromol.5b01610
Macromolecules 2015, 48, 7370−7377
Article
Macromolecules
4π, and g(rαγ
12,ω1,ω2) is the angle-dependent site−site pair
correlation function. When both species ν and μ are ions, fνμ(r)
reduces to the pair correlation function, gνμ(r), which is given
by
1
gνμ(r ) =
NνNμ
Nν
Nμ
∑∑
∫ g (r12αγ , ω1, ω2) dω1 dω2
Ω(N1+ N2 − 2)
α=1 γ=1
(11)
The pressure of polyelectrolyte solutions is given by
2πlBρmol
βP
βP
= HS +
ρmol
ρmol
3
∫0
∞
⎡N 2x 2z 2f (r )
⎣ m p p pp
+ 2Nmxpzp{xczcf pc (r ) + xszsf ps (r )} +
Figure 1. Osmotic coefficient, ϕ ≡
(12)
where zp, zc, and zs are the charge valences of monomers in
chains, counterions, and co-ions, respectively. Since fνμ(r) →1
for large r, the integrand in eq 12 goes to zero at large r from
electroneutrality.
It should be noted that the contribution PHS does not only
come from the hard-sphere interaction alone but also from the
electrostatic interaction because fνμ(σ+) is a function of both
hard-sphere and electrostatic interactions.
We define the partial virial, Γνμ
⎡
⎛l ⎞
2π m m
Nν Nμ xνxμσ 3⎢fνμ (σ +) + z νzμ⎜ B ⎟
⎝σ ⎠
⎣
3
∞
⎤
(fνμ (σx) − 1)x dx ⎥
⎦
1
the simulations except for the slight overestimation at lB/σ =
3.0.
An interesting and important point is that the salt
concentration (ρs) in polyelectrolyte solutions differs from
the bulk salt concentration (ρbs ) in a manner that depends on
the polymer concentration. This behavior is known as the
Donnan equilibrium.29−31 The chemical potential of the salt
ions in the polyelectrolyte solution is the same as in the bulk
salt. As the polymer concentration is increased, the free volume
available in the polyelectrolyte solution is lower, and the work
required to insert salt ions (and hence chemical potential) is
therefore greater. Therefore, as the polymer concentration is
increased, the salt concentration in the polyelectrolyte solution
decreases for a fixed value of the bulk salt concentration. The
consequence is that the polyelectrolyte solution behaves like a
salt-free solution at high polymer concentrations.
Figure 2 depicts the salt concentration, ρ s , in the
polyelectrolyte solution with Nm = 16 as a function of the
Γνμ =
∫
and, therefore, the osmotic coefficient, ϕ ≡
ϕ = 1 + ρmol Γtot = 1 + ρmol
βP
,
ρmol
(13)
is given by
∑ ∑ Γνμ
ν
μ
as a function of salt
concentration, ρs at lB/σ = 1.0 and 3.0 for 1−1 electrolyte solutions.
Symbols are results from grand canonical Monte Carlo (GCMC)
simulations, and solid and dotted lines are predictions from the MSA
integral equation.41,42
xc 2zc 2fcc (r )
+ xs 2zs 2fss (r ) + 2xcxszczsfcs (r )⎤⎦r dr
βP
,
ρmol
(14)
where Γtot is the total virial of the solution. We can also
decompose Γ tot into contact (Γ contact ) and long-range
(Γlong−range) contributions with the following definitions:
Γcontact =
∑∑
ν
μ
2π m m
Nν Nμ xνxμσ 3fνμ (σ +)
3
Γlong − range = Γtot − Γcontact
(15)
(16)
For salt-free polyelectrolyte solutions, the pressure of the
solution is identical to the osmotic pressure of the solution
because of the electric neutrality condition. In case of
polyelectrolyte solutions with added salts, the osmotic pressure
can be obtained by subtracting the pressure of polylelectrolyte
solutions by the pressure of salt only solutions at the same
chemical potential (Donnan equilibrium):
Π(μ) = Ppolyelec(μ) − Psalt(μ)
Figure 2. Salt concentration, ρs, as a function of the monomer
concentration, ρm, for three different bulk salt concentrations, ρbs σ3 =
7.4 × 10−3, 6.2 × 10−2, and 1.3 × 10−1 for Nm = 16, and lB/σ = 1. The
counterion concentration for salt-free solutions, ρc, is also shown for
comparison. The dashed lines are corresponding bulk salt concentrations.
monomer concentration, ρm, for three different bulk salt
concentrations, ρbs . Similar results are observed for other values
of Nm (not shown). In all cases, ρs < ρbs , and ρs approaches ρbs
for low salt concentrations. There exists a polymer concentration where the salt concentration, ρs, becomes equal to the
counterion concentration in salt-free solutions, ρc (salt free) .
These crossover monomer concentrations are 0.006, 0.04, and
0.08 for ρbs σ3 = 7.4 × 10−3, 6.2 × 10−2, and 1.3 × 10−1,
respectively. One would expect the effect of salt to be
prominent at concentrations lower than these crossover
(17)
where μ (=μ+ + μ−) is the chemical potential of salt ion pairs.
■
RESULTS AND DISCUSSION
For simple electrolytes the mean-spherical approximation
(MSA) integral equation theory41,42 is in near quantitive
agreement with the simulation results. Figure 1 compares the
MSA theory to GCMC simulations for 1−1 electrolytes and for
lB/σ = 1.0 and 3.0. The theory is in excellent agreement with
7373
DOI: 10.1021/acs.macromol.5b01610
Macromolecules 2015, 48, 7370−7377
Article
Macromolecules
function of ρm for Nm = 16 and various salt concentrations
(including the salt-free limit). In dilute solutions, RE decreases
with increasing salt concentration, which is due to the screening
of electrostatic interaction by salt ions. In semidilute solutions,
however, RE is independent of the salt concentration.
As the polymer concentration is increased, the scaling
exponent ν, defined by RE ∼ Nνm,varies from 1 to 1/2 (see
Figure 3b) marking a transition from rod-like behavior in dilute
salt free solutions, to ideal chain behavior in semidilute
solutions. In dilute solutions the apparent scaling exponent
depends on ρs (and ρsb) with ν decreasing as the salt
concentration is increased.
We decompose the virial into several contributions in order
to estimate the effect of different interactions on the osmotic
pressure. Note that these contributions are not truly
independent because all of them depend on the pair correlation
functions, which are functions of all the interactions. The
contact (Γcontact), long-ranged (Γlong−range), and total (Γtot)
contributions to the virial, Γ, are shown in Figure 4. In part
a, the results for low salt case (ρbs σ3 = 7.4 × 10−3) are shown
along with the salt free system (blue open circles) and the effect
of salt ions are most significant in dilute regime (ρm σ3 < ρ*m σ3 =
0.05) where they screen the electrostatic interaction between
polyions, and this results in a weak concentration dependence
of the virial with added salt, in contrast to the salt free system.
This effect is more pronounced with increasing salt
concentration, the results of which are shown in Figure 4b.
In medium and high salt concentrations (ρbs σ3 = 6.2 × 10−2 and
1.3 × 10 −1), Γtot is always positive in all monomer
concentrations and, as a result, the osmotic coefficient defined
in eq 14 is always greater than 1.
concentrations, with a transition to salt-free behavior at high
polymer concentrations.
The polymer overlap threshold concentration, ρm* is weakly
sensitive to the bulk salt concentration. We estimate the overlap
threshold concentration of the polymer from the relation:
0.64Nm
ρm* ≈
, where RE is the average end-to-end distance of
((π / 6)R 3)
E
polyelectrolyte chains, and is a function of monomer
concentration.8 The resulting overlap threshold concentrations,
ρm*, for ρbs σ3 = 7.4 × 10−3, 6.2× 10−2, and 1.3 × 10−1 are
respectively 0.051, 0.057, and 0.065. This implies the effect of
salt concentration on the overlap threshold concentration is not
significant. This is because at the concentration where the
polymers begin to overlap, the salt concentration is already
reduced to below the salt-free value.
Above the overlap threshold concentration the chain size is
independent of salt concentration. Figure 3a displays RE as a
Figure 3. Root mean-square end-to-end distance, RE, as a function of
monomer concentration, ρm, (a) for Nm = 16 and various salt
concentrations, and (b) for ρbs σ3 = 7.4 × 10−3 and Nm = 8, 16, 32, and
64.
Figure 4. Contact (Γcontact) and long-range (Γlong−range) contributions of Γtot for polyelectrolytes with Nm = 16 as a function of the monomer
concentration ρm at (a) low salt concentration (ρbs σ3 = 7.4× 10−3) and (b) at three different bulk salt concentrations. Results for the salt-free case
and the neutral case (where lB = 0) are also shown for comparison in part a.8 In parts c and d, all six contributions of Γtot are also shown for two
different salt concentrations.
7374
DOI: 10.1021/acs.macromol.5b01610
Macromolecules 2015, 48, 7370−7377
Article
Macromolecules
in the intermediate monomer concentration regime, the system
pressure deviates from that of the salt-free polyelectrolytes.
The scaling of the osmotic pressure with concentration is a
function of salt concentration in dilute solutions and similar to
that of neutral chains in semidilute solutions. We obtain the
scaling exponents using the data for ρm > ρs because data from
lower concentrations deviate from a simple power law. Figure
5b shows that the slope gets steeper with increasing the salt
concentration from 1.78 at low salt concentration to 2.48 at
medium and high salt concentrations. This implies that any
intermediate scaling exponent can be obtained by tuning the
bulk salt concentration. This is in contrast to scaling theoretical
predictions,9,43 where the osmotic pressure of various
polyelectrolytes with different salt concentrations is predicted
to collapse onto a single curve. However, this salt concentration
dependence of the osmotic pressure has been reported for
DNA solutions.11,44−46
The reason why the exponent, ∂[ln Π] , becomes higher with
All components in the solution contribute to the virial. The
sign of the contribution depends on the charge and there is a
significant cancellation in the resulting osmotic pressure. All 6
pair contributions, Γνμ(ν,μ = p, c, and s), to the virial for low
and medium salt concentrations are shown in Figure 4, parts c
and d. For low salt (Figure 4c),Γpc and Γcs among the six
contributions are negative because of the electrostatic attraction
between opposite charges and Γpc is dominant over Γcs, except
at ρmσ3 = 0.005. On the other hand, Γcc is the largest positive
contribution among the same charge pairs. However, in the
semidilute regime (ρm σ3 > 0.05), Γpp is as large as Γcc, which
implies that the polymeric contribution is not negligible in this
concentration regime. For high salt (Figure 4d), the magnitude
of all virial contributions becomes much smaller than that in the
low salt concentration. Especially, the polymeric contributions
Γpp and Γpc are highly damped because of the electrostatic
screening especially in dilute solution. Overall, it is clear that
the polymer contribution to the virial is not negligible.
There are three different regimes in the pressure of the
polyelectrolyte solution with added salt. These regimes are seen
in Figure 5a. The figure shows the system pressure, P, of
∂[ln ρm ]
higher salt concentration (or higher chemical potential) is
related to the pressure behavior of polyelectrolyte solutions:
∂[ln Π(ρs )]
∂[ln ρm ]
=
ρm (∂Ppolyelec(ρs )/∂ρm )
Ppolyelec(ρs ) − Psalt(ρs )
(18)
As shown in Figure 5(a), the pressure of salt-free polyelectrolyte solutions is the lower bound for those of polyelectrolyte
solutions with salt and therefore,
∂Ppolyelec(ρs )
∂ρm
decreases with
increasing salt concentration. However, the decrease in the
osmotic pressure (or the pressure difference between
polyelectrolyte and salt-only solutions) is stronger with
increasing salt concentration, which is due to both enhanced
excluded volume interaction and screened electrostatic
interaction. This leads to the overall increase in the exponent
of the osmotic pressure with increasing salt concentration. In
other words, more addition of salts increases the excluded
volume interaction and screens the electrostatic interaction
more efficiently, which leads to the less monomer concentration dependence of the pressure of polyelectrolyte solutions
and stronger salt concentration dependence of the corresponding osmotic pressure.
The effects of degree of polymerization on the osmotic
pressure are shown in Figure 6. For low salt and high monomer
concentrations (ρbs σ3 = 7.4 × 10−3, ρmσ3 ≥ 0.07) as seen in
Figure 6(a), the osmotic pressure collapses into a single line
with the scaling exponent of 9/4, independent of the polymer
size. However, for lower monomer concentrations, the osmotic
Figure 5. (a) System pressure, P, and (b) the corresponding osmotic
pressure, Π, of polyelectrolyte solutions with Nm = 16 as a function of
the monomer concentration, ρm, at three different bulk salt
concentrations (ρbs σ3 = 7.4× 10−3, 6.2× 10−2, and 1.3× 10−1 along
with that of salt-free polyelectrolyte solutions. The results from the
salt-free polyelectrolytes were taken from the previous study.8 In parts
a and b, vertical and horizontal dashed lines in red, blue, and purple
colors correspond to the bulk salt concentrations and the
corresponding pressures of the bulk salt solutions ρbs σ3 = 7.4× 10−3,
6.2× 10−2, and 1.3× 10−1, respectively. In part b, the slopes were
obtained by fitting the corresponding simulation data with a power
function: Π ∼ ραm.
polyelectrolyte solutions with the degree of polymerization Nm
= 16 as a function of the monomer concentration for three
different bulk salt concentrations ρbs σ3 = 7.4 × 10−3, 6.2× 10−2,
and 1.3× 10−1. In the first regime, when ρm < ρbs , the system
pressure of polyelectrolyte solutions is similar to that of the
corresponding bulk salt concentration. This implies that the
pressure of the polyelectrolyte solutions in this regime mainly
comes from salt ions and the resulting osmotic pressure is
expected to be very small. In the second regime, at high
monomer concentrations (ρm ≫ ρsb), the pressure of
polyelectrolyte solution with added salt collapses into that of
salt-free polyelectrolyte solutions regardless of the salt
concentration. This is because the salt concentration in the
polyelectrolyte solution phase is much smaller than the
corresponding counterion concentration, which is shown in
Figure 2. It implies that at these high concentrations the major
component of the osmotic pressure comes from the
interactions of polyelectrolytes and their counterions.8 Finally,
Figure 6. Osmotic pressures Π as a function of the monomer
concentration, ρm, for various degrees of polymerization at (a) ρbs σ3 =
7.4 × 10−3 and (b) ρbs σ3 = 6.2 × 10−2.
7375
DOI: 10.1021/acs.macromol.5b01610
Macromolecules 2015, 48, 7370−7377
Article
Macromolecules
Figure 7. Simulation results of primitive models of polyelectrolyte solutions with monomer size σm = σ and salt ion size σc = σs = 0.5σ for Nm = 16
and βμ* = −10: (a) salt concentration, (b) RE, (c) virial contributions (Γcontact and Γlong−range), and (d) osmotic pressure, Π, as a function of monomer
concentration (ρm). Results of the polyelectrolyte solutions with σm = σc = σs = σ are also shown for comparison. In part a, red and blue dashed lines
correspond to the concentrations of salt-only systems for σs = σ and 0.5σ, respectively, and the black solid line indicates the counterion concentration
in the corresponding salt-free polyelectrolyte solutions.
overlap threshold concentration from ρm*σ3 = 0.05 (σs = σ) to
ρm*σ3 = 0.06 (σs = 0.5σ). This polyelectrolyte compaction can be
attributed to the increase in the salt concentration as well as the
enhanced electrostatic screening because salt ions including
counterions can approach the polyelectrolyte backbone more
closely.
This enhanced electrostatic screening between oppositely
charged species induces significant decrease in Γlong−range as well
as slight increase in Γcontact, leading to the overall increase in Γtot
except at very high monomer concentration (ρmσ3 = 0.3),
where the excluded volume interaction is more important than
the electrostatic interaction (Figure 7c). As a result, the osmotic
pressure of the system with σs = 0.5σ becomes slightly higher
than that with σs = 0.5σ up to ρmσ3 < 0.2 and the scaling
exponent also gets slightly smaller from 1.78 (σs = σ) to 1.63
(σs = 0.5 σ).
pressure becomes smaller with increasing the degree of
polymerization. At medium bulk salt concentration (ρbs σ3 =
6.2 × 10−2) as seen in Figure 6(b), however, the scaling
exponent of 3 seen in the concentrated regime (ρmσ3 = 0.3) is
extended to in the semidilute concentration regime (ρmσ3 =
0.03), which implies that with increasing salt concentration, the
excluded volume interaction becomes more important in the
osmotic behavior of the polyelectrolyte solutions in both
semidilute and concentrated regimes.
Finally, we investigate the effects of salt size on the osmotic
behavior of polyelectrolyte solutions with added salts. Figure 7
summarizes simulation results of primitive models of
polyelectrolyte solutions with monomer size σm = σ and salt
ion size σc = σs = 0.5σ, along with those with σm = σc = σs = σ
for comparison. It is noted that the chemical potential used in
this simulation (βμ* = −10) corresponds to ρbs σ3 = 7.4× 10−3
for salt-only solutions with σs = σ and 8.1 × 10−3 for those with
σs = 0.5σ. The salt concentrations of the two salt-only systems
are also shown as red and blue dashed lines in Figure 7a.
As expected and seen in Figure 7a, salt ions with the smaller
size can be inserted more easily into the polyelectrolyte
solution at a given monomer concentration and chemical
potential especially at high monomer concentrations. However,
it should be noted that the amount of inserted salt ions is still
much smaller than that of counterions except at very low
monomer concentrations. For example, at ρmσ3 = 0.05, which
corresponds to the overlap threshold concentration for the
system with σs = σ, the ratio of salt ions to counterions is only
0.018 and 0.036 for σs = σ and σs = 0.5σ systems, respectively.
However, the polyelectrolyte size is slightly affected by the
salt size: with decreasing the salt size, RE becomes smaller by
about 3−5% (Figure 7b). This leads to slight shift in the
■
SUMMARY AND CONCLUSIONS
In this paper, we use grand canonical Monte Carlo simulations
to investigate the effect of added salt on the osmotic pressure of
polyelectrolyte solutions using the condition called as the
Donnan equilibrium. This system is in close analogy with
charged colloids except that the polyelectrolyte conformation is
correlated with the osmotic behavior. The simulation ensemble
enables a direct calculation of the osmotic pressure as a
function of the concentration of bulk electrolyte that is in
equilibrium with the solution. In dilute solution, the salt ions
affect the osmotic pressure not only by electrostatic screening
but also by increasing the excluded volume effect. This effect
results in decrease in magnitude and stronger concentration
dependence of the osmotic pressure. As the polymer
concentration is increased, since the chemical potential of the
7376
DOI: 10.1021/acs.macromol.5b01610
Macromolecules 2015, 48, 7370−7377
Article
Macromolecules
salt ions is fixed, the increased interaction between polymer/
counterions and salt ions is compensated for by a decrease in
salt concentration. Semidilute solutions therefore behave
essentially like salt-free polyelectrolyte solutions. In fact, the
osmotic pressure in this regime is more influenced by the
excluded volume interaction, compared to the salt-free systems
and the scaling behavior is similar to that of neutral polymer
solutions.
The simulation results show that the polymer contribution as
well as small ion contributions is not negligible in the osmotic
pressure of polyelectrolyte solutions refuting the claim that the
osmotic pressure of polyelectrolyte solutions mainly comes
from free small ions. Finally, we mention that the osmotic
behavior of polyelectrolyte solutions in the presence of added
salt depends on an interplay among various polymer,
counterions, and co-ion correlations. However, theories for
the osmotic behavior of polyelectrolyte solutions sometimes
emphasize only one component among those interactions. For
example, scaling theories by Odijk only take the polymeric
contribution into account,47 whereas theories by Dobrynin et
al. emphasize the role of free counterions except at high
polymer concentration.9,13 The resulting scaling behavior of the
osmotic pressure is qualitatively different from scaling theories:
the scaling exponent increases with increasing salt concentration. This scaling behavior is however in good agreements
with experiments with DNA.11,16
We are currently working on the effects of multivalent salts
and solvent quality on the osmotic behavior of polyelectrolyte
solution and will also evaluate various theories including scaling
theories and integral-equation theories with the simulation
results presented in this study.
■
(9) Dobrynin, A. V.; Colby, R. H.; Rubinstein, M. Macromolecules
1995, 28, 1859−1871.
(10) Kakehashi, R.; Yamazoe, H.; Maeda, H. Colloid Polym. Sci. 1998,
276, 28−33.
(11) Raspaud, E.; da Conceiçao, M.; Livolant, F. Phys. Rev. Lett. 2000,
84, 2533−2536.
(12) Manning, G. S. J. Chem. Phys. 1969, 51, 924.
(13) Carrillo, J.-M.; Dobrynin, A. Polymers 2014, 6, 1897−1913.
(14) Blaul, J.; Wittemann, M.; Ballauff, M.; Rehahn, M. J. Phys. Chem.
B 2000, 104, 7077−7081.
(15) Arh, K.; Pohar, C.; Vlachy, V. J. Phys. Chem. B 2002, 106, 9967−
9973.
(16) Verma, R.; Crocker, J. C.; Lubensky, T. C.; Yodh, A. G. Phys.
Rev. Lett. 1998, 81, 4004−4007.
(17) Jiang, J. W.; Liu, H. L.; Hu, Y.; Prausnitz, J. M. J. Chem. Phys.
1998, 108, 780−784.
(18) Jiang, J.; Liu, H.; Hu, Y. J. Chem. Phys. 1999, 110, 4952−4962.
(19) Jiang, J. W.; Blum, L.; Bernard, O.; Prausnitz, J. M. Mol. Phys.
2001, 99, 1121−1128.
(20) Wertheim, M. S. J. Stat. Phys. 1984, 35, 19.
(21) Wertheim, M. S. J. Stat. Phys. 1984, 35, 35.
(22) Wertheim, M. S. J. Stat. Phys. 1986, 42, 459.
(23) Wertheim, M. S. J. Stat. Phys. 1986, 42, 477.
(24) Kalyuzhnyi, Y. V.; Stell, G. Chem. Phys. Lett. 1995, 240, 157−
164.
(25) Blum, L.; Kalyuzhnyi, Y. V.; Bernard, O.; Herrera-Pacheco, J. N.
J. Phys.: Condens. Matter 1996, 8, A143.
(26) Bernard, O.; Blum, L. J. Chem. Phys. 2000, 112, 7227−7237.
(27) von Solms, N.; Chiew, Y. C. J. Chem. Phys. 1999, 111, 4839−
4850.
(28) von Solms, N.; Chiew, Y. C. J. Chem. Phys. 2003, 118, 4321−
4330.
(29) Hill, T. L. Discuss. Faraday Soc. 1956, 21, 31−45.
(30) Hill, T. L. J. Phys. Chem. 1957, 61, 548−553.
(31) Stigter, D.; Hill, T. L. J. Phys. Chem. 1959, 63, 551−555.
(32) Tamashiro, M. N.; Levin, Y.; Barbosa, M. C. Eur. Phys. J. B 1998,
1, 337−343.
(33) Philipse, A.; Vrij, A. J. Phys.: Condens. Matter 2011, 23, 194106.
(34) Denton, A. R. J. Phys.: Condens. Matter 2010, 22, 364108.
(35) Deserno, M.; von Grünberg, H. H. Phys. Rev. E: Stat. Phys.,
Plasmas, Fluids, Relat. Interdiscip. Top. 2002, 66, 011401.
(36) Valleau, J. P.; Cohen, L. K. J. Chem. Phys. 1980, 72, 5935−5941.
(37) Allen, M. P.; Tildesley, D. J. Computer Simulation of Liquids;
Oxford University Press: New York, 1987.
(38) Nezbeda, I. Mol. Phys. 1977, 33, 1287.
(39) Honnell, K. G.; Hall, C. K.; Dickman, R. J. Chem. Phys. 1987, 87,
664.
(40) Chang, J.; Sandler, S. I. Chem. Eng. Sci. 1994, 49, 2777−2791.
(41) Waisman, E.; Lebowitz, J. L. J. Chem. Phys. 1972, 56, 3086.
(42) Waisman, E.; Lebowitz, J. L. J. Chem. Phys. 1972, 56, 3093.
(43) Wang, L.; Bloomfield, V. A. Macromolecules 1990, 23, 194−199.
(44) Strey, H. H.; Parsegian, V. A.; Podgornik, R. Phys. Rev. Lett.
1997, 78, 895−898.
(45) Strey, H. H.; Parsegian, V. A.; Podgornik, R. Phys. Rev. E: Stat.
Phys., Plasmas, Fluids, Relat. Interdiscip. Top. 1999, 59, 999−1008.
(46) Hansen, P. L.; Podgornik, R.; Parsegian, V. A. Phys. Rev. E: Stat.
Phys., Plasmas, Fluids, Relat. Interdiscip. Top. 2001, 64, 021907.
(47) Odijk, T. Macromolecules 1979, 12, 688−693.
AUTHOR INFORMATION
Corresponding Author
*(R.C.) Telephone: +82 2 940-5243. Fax: +82 2 942-0108. Email: [email protected].
Notes
The authors declare no competing financial interest.
■
ACKNOWLEDGMENTS
We acknowledge supports from National Research Foundation
Grant funded by Korean government (MEST)
(2013R1A1A1A05009866), Korea CCS R&D Center
(2014M1A8A1049296), Education-Research Integration
through Simulation on the Net (EDISON)
(2012M3C1A6035363), Plasma Bioscience Research Center
(PBRC) (2010-0027963), Supercomputing Center/KISTI
(KSC-2015-C2-004), and Kwangwoon University (2015).
This research is also supported partially by National Science
Foundation through grant CHE-1111835.
■
REFERENCES
(1) Mandel, M. Polyelectrolytes; Riedel: Dordrecht, The Netherlands,
1988.
(2) Hara, M. Polyelectrolytes: Science and Technology; Dekker: New
York, 1993.
(3) Förster, S.; Schmidt, M. Adv. Polym. Sci. 1995, 120, 51−133.
(4) Yethiraj, A. J. Chem. Phys. 2003, 118, 3904.
(5) Yethiraj, A. J. Phys. Chem. B 2009, 113, 1539−1551.
(6) Stevens, M. J.; Kremer, K. Phys. Rev. Lett. 1993, 71, 2228−2231.
(7) Stevens, M. J.; Kremer, K. J. Chem. Phys. 1995, 103, 1669−1690.
(8) Chang, R.; Yethiraj, A. Macromolecules 2005, 38, 607−616.
7377
DOI: 10.1021/acs.macromol.5b01610
Macromolecules 2015, 48, 7370−7377