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.
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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 © XXXX 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
A
DOI: 10.1021/acs.macromol.5b01610 Macromolecules XXXX, XXX, XXX−XXX
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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
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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 B
DOI: 10.1021/acs.macromol.5b01610 Macromolecules XXXX, XXX, XXX−XXX
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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− ρmol 3 Nm
+ 2xpxc ∑
U=
Np
where ρmol =
Np
+
Nm
≡
∑ i>j
Ns
uij(rij11)
+
∑ i>j
+
(6)
(7)
and
z νzμ dβuEL(r ) = −lB 2 dr r
(8)
where zν is the charge valence on a site of species ν. For hard chains eq 5 simplifies to 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 +)⎤⎦
(9)
where rfνμ (r ) =
Ns
∑ ∑ uij(rij11),
1 NνmNμm
m Nνm Nμ
i=1 j=1
rαij γ
and r
Nμ
ν 1 ⟨∑ ∑ ′δ(r + r γj − r iα)⟩ Nmol i = 1 j = 1
dβuHS(r ) = −exp(βuHS)δ(r − σ +) dr
Ns
Nc
uij(rij11)
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
i=1 α=1 j=1
Nc
(Nmol is the total number of
V
N
αγ xνxμρmol gνμ (r) =
∑ ∑ ∑ uij(rijα1) + ∑ ∑ ∑ uij(rijα1) i=1 α=1 j=1
+
Nm
=
Np + Nc + Ns
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
Nmol V
|rα12γ|.
Nm
Np
α=1
r11 12· r ̂ α1 g ps (r ) r
chains plus ions and V is the system volume), xν ≡
∑ ∑ uii(rijαγ) + ∑ ∑ uii(riiαγ) i>j α ,γ
+ 2xpxs ∑
(5)
(3)
Nm
r
Nm α1 g pc (r )
⎤ d β u (r ) νμ + xc 2gcc11(r ) + xs 2gss11(r ) + 2xcxsgcs11(r )⎥ r 3 dr ⎥⎦ dr
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
r11 12· r ̂
α=1
dr1 ··· drNc
drNc + 1 ··· drNc + Ns
∫
Nm ⎡ 11 ⎢x 2 ∑ r12·r ̂ g αγ (r ) pp ⎢⎣ p 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 = C
DOI: 10.1021/acs.macromol.5b01610 Macromolecules XXXX, XXX, XXX−XXX
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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 gνμ(r ) =
1 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.
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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 D
DOI: 10.1021/acs.macromol.5b01610 Macromolecules XXXX, XXX, XXX−XXX
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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, ρ*m , 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. E
DOI: 10.1021/acs.macromol.5b01610 Macromolecules XXXX, XXX, XXX−XXX
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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. F
DOI: 10.1021/acs.macromol.5b01610 Macromolecules XXXX, XXX, XXX−XXX
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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 ρ*m 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
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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 G
DOI: 10.1021/acs.macromol.5b01610 Macromolecules XXXX, XXX, XXX−XXX
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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.
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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.
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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.
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DOI: 10.1021/acs.macromol.5b01610 Macromolecules XXXX, XXX, XXX−XXX
