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We present results of the hybrid Monte Carlo/molecular dynamics simulations of the osmotic pressure of salt solutions of polyelectrolytes. In our simulations, we used a coarse-grained representation of polyelectrolyte chains, counterions and salt ions. During simulation runs, we alternate Monte Carlo and molecular dynamics simulation steps. Monte Carlo steps were used to perform small ion exchange between simulation box containing salt ions (salt reservoir) and simulation box with polyelectrolyte chains, counterions and salt ions (polyelectrolyte solution). This allowed us to model Donnan equilibrium and partitioning of salt and counterions across membrane impermeable to polyelectrolyte chains. Our simulations have shown that the main contribution to the system osmotic pressure is due to salt ions and osmotically active counterions. The fraction of the condensed (osmotically inactive) counterions first increases with decreases in the solution ionic strength then it saturates. The reduced value of the system osmotic coefficient is a universal function of the ratio of the concentration of osmotically active counterions and salt concentration in salt reservoir. Simulation results are in a very good agreement with osmotic pressure measurements in sodium polystyrene sulfonate, DNA, polyacrylic acid, sodium polyanetholesulfonic acid, polyvinylbenzoic acid, and polydiallyldimethylammonium chloride solutions.

The osmotic pressure of polyelectrolyte solutions is controlled by contribution from counterions and salt ions [

In salt-free solutions, the osmotic pressure of the polyelectrolyte solutions can be described in the framework of the so-called Katchalsky’s cell model [_{B}_{B}_{cell}) [

In salt solutions, the situation is more complex. In their classical paper, Alexandrowicz and Katchalsky [

The exact analytical solution of the nonlinear Poisson-Boltzmann equation describing distribution of the electrostatic potential around a rod-like polyion only exists in the special case of the infinite cell size which is in contact with a salt reservoir [

At high salt concentrations, the osmotic pressure of polyelectrolyte solutions increases quadratically with polymer concentration [

While there is a significant number of experimental studies [

To obtain osmotic pressure of polyelectrolyte solutions we have performed hybrid Monte Carlo/molecular dynamics simulations [

Snapshots of the simulation boxes containing salt solutions of polyelectrolyte chains (

All particles in the system interacted through truncated-shifted Lennard-Jones (LJ) potential:
_{ij}_{cut} = 2.5σ, was set for polymer–polymer interactions, and _{cut} = 2^{1/6}σ was selected for all other pair wise interactions. The interaction parameter ε_{LJ} was equal to _{B}_{B} is the Boltzmann constant and _{B}

The connectivity of monomers into polymer chains was maintained by the finite extension nonlinear elastic (FENE) potential:
_{spring} = 30_{B}^{2} and the maximum bond length , _{max} = 1.2σ. The repulsive part of the bond potential was represented by the truncated-shifted LJ potentials with ε_{LJ} = 1.5 _{B}_{cut} = 2^{1/6}σ.

Interaction between any two charged particles with charge valences _{i} and _{j,} and separated by a distance _{ij}, was given by the Coulomb potential:
_{B} = ^{2}/ε_{B}_{B} was equal to 1.0σ. The Particle-Particle Particle-Mesh (PPPM) method implemented in LAMMPS [^{−5} was used for calculations of the electrostatic interactions between all charges in each simulation box.

The snapshots of the simulation boxes are shown in ^{−5} σ^{−3} and 0.1 σ^{−3}. Conversion of the reduced units to moles per liter is given at the end of this section.

Simulations were carried out with periodic boundary conditions for each simulation box. The total number of particles in both simulation boxes and their volumes remained constant during the entire simulation runs. We have only allowed exchange of the small ions between two simulation boxes. To perform these simulations, we have modified LAMMPS code by calling molecular dynamics steps as subroutines between Monte Carlo steps that were used to perform the exchange of small ions between two simulation boxes.

In order to maintain a constant temperature of the system during molecular dynamics simulation steps, we coupled each simulation box to the Langevin thermostat. The motion of particles (monomers, counterions, and salt ions) was described by the following equations:
_{LJ}, where τ_{LJ} is the standard LJ-time τ_{LJ =} σ(m/_{B}^{1/2}. The velocity-Verlet algorithm with a time step ∆t = 0.01τ_{LJ} was used for integration of the equations of motion Equation 4. The duration of the molecular dynamics simulation runs between Monte Carlo steps was 100 integration steps.

During Monte Carlo simulation steps, acceptance or rejection of the Monte Carlo moves followed a standard particle exchange procedure [

The simulations were performed according to the following procedure. At the beginning of each simulation run, both boxes contained salt ions at the same concentrations. For initial configuration of the simulation box with polyelectrolyte chains, we have used one of the system configurations from our simulations of polyelectrolytes in salt solutions [^{4} τ_{LJ} and 10^{5} τ_{LJ} depending on how long it required for the mean-square radius of gyration of a polyelectrolyte chain to reach equilibrium. Note that the system’s pressure equilibrates much faster, ~10^{4} τ_{LJ}, than it requires for the chain equilibration (see SI). We used data collected during the final 2 × 10^{4} τ_{LJ} for the data analysis. It is important to point out that the equilibrium bulk (reservoir) salt concentration was obtained after the equilibrium ion concentrations in both simulation boxes were reached.

Evolution of concentration of the negatively charged salt ions in simulation box with (red line) and without (black line) polyelectrolyte chains during the simulation run. Simulations were performed at polymer concentration _{p} = 0.01 σ^{−3}.

In our simulations, we used a coarse grained representation of the polymers and salt ions. One of the possible mappings of our systems to a real system can be done by making a connection through the value of the Bjerrum length, _{B}. In aqueous solutions at normal conditions, the value of the Bjerrum length is 0.7 nm. In our simulations, we set the value of the Bjerrum length to be equal to the LJ-diameter σ. Therefore, for the studied salt concentration range between 10^{−5} σ^{−3} and 0.1 σ^{−3}, the value of the Debye screening length due to only salt ions, _{D} ≡ (8π_{B}c_{s})^{−1/2}, is varied between 44 nm and 0.44 nm. Using the relation between the solution ionic strength and the Debye radius, ^{−5}_{s} < 0.5

Dependence of the pressure in polyelectrolyte box on salt concentration in this box _{s,p} for polyelectrolyte solutions with polymer concentrations _{p} = 10^{−4} σ^{−3} (red squares), _{p} =5 × 10^{−4} σ^{−3} (blue rhombs), _{p} = 10^{−3} σ^{−3} (green triangles), _{p} = 5 × 10^{−3} σ^{−3} (black circles), _{p} = 10^{−2} σ^{−3} (gray hexagons), _{p} = 2.5 × 10^{−2} σ^{−3} (purple triangles), _{p} = 5 × 10^{−2} σ^{−3} (open red squares), and _{p} = 10^{−1} σ^{−3} (open blue rhombs). The dashed lines correspond to the best fit to the data obtained in simulations without ion exchange using Sigma plot’s smooth spline method [

We first will discuss pressure dependence on the polymer and salt concentrations in a simulation box containing polyelectrolyte chains. These data are presented in _{p} in polyelectrolyte solutions depends on equilibrium salt concentration _{s,p} at different polymer concentrations, _{p}. The error bars for all our figures were calculated by using standard definition of the variance σ_{A}

As we have already pointed out in our previous paper, the definition of the fraction of osmotically active counterions given by Equation 6 is only warranted in the concentration range where the pressure of the polyelectrolyte solutions is dominated by the linear term in the virial expansion. Analysis of the pressure data shows that we can represent the fraction of the condensed counterions (osmotically inactive counterions) 1 _{D}(c_{p},c_{s,p})/b − 1))]

Now let us turn our attention to the salt reservoir. _{s,s}. It follows from this figure that almost all our simulation data correspond to ideal solution regime where system pressure scales linearly with salt concentration. The deviation from the linear regime occurs at salt concentrations _{s,s} > 5 × 10^{−2} σ^{−3}. At these salt concentrations, the short range interactions between ions (excluded volume terms) begin to contribute to the system pressure. Our simulations also indicate that similar behaviour of the osmotic pressure should be expected for a wide concentration range up to salt and polymer concentrations of 0.1 M (see unit conversion at the end of

Dependence of fraction 1 − _{D}/_{p} ≤ 0.05σ^{−3} and salt concentrations c_{s},_{p} < 0.05σ^{−3}. Notations are the same as in

Dependence of the pressure in salt box on salt concentration. The dashed line is given by the following equation _{s} = 2_{B}_{s,s}.

Osmotic pressure is defined as a difference between pressure of polyelectrolyte solution and salt reservoir:
_{p} − _{s}
_{p}_{s}

Dependence of the osmotic coefficient on the reduced polymer concentration. Notations are the same as in

Since the majority of our simulations corresponds to a ion concentration range where the contribution of ions can be taken into account in the framework of the ideal gas approximation, we can test how well the classical Donnan equilibrium approach [

Note that in our simulations, polyelectrolyte chains were positively charged (see _{s}^{+}_{s}^{−}

In writing Equation (11), we set the value of the electrostatic potential in salt reservoir to zero. It follows from Equation (11) that the product of the concentrations of salt ions stays constant at each point of the solution. Our simulations are done at constant chemical potential which requires that this product stays constant in the salt reservoir as well. This leads to:

By solving Equations (10) and (12), we can find the average concentration of the positively and negatively charged salt ions in polyelectrolyte solution as a function of the average salt concentration _{s,s} in salt reservoir and polymer concentration _{p}. The ionic contribution to the osmotic pressure is equal to the difference between the ideal gas pressure of salt ions in the polyelectrolyte solution and in the salt reservoir:

In the limit of low salt concentrations, _{s,s} << ^{*}c,_{B}_{p}. At higher salt concentrations, _{s,s} >> _{p}

Equation (13) can be transformed into universal form as follows:

This allows us to collapse all simulation data sets shown in

Dependence of salt concentration in polyelectrolyte solution on salt concentration in reservoir. Notations are the same as in

Universal plot of the system osmotic coefficient as a function of the reduced polymer concentration. Dashed line corresponds to Equation (15) with no adjustable parameters. Notations are the same as in

Despite the large error bars, our simulations show universality of the reduced osmotic coefficient as a function of the ratio of the concentration of the osmotically active counterion to the salt concentration in salt reservoir. In _{s} = 7.6 × 10^{−3} M for NaPAA, _{s}^{−2} M for NaPASA, and _{s} = 5.2 × 10^{−3} M for PDADMA-Cl. For Kakehashi et al. [_{s} = 9.7 × 10^{−3} M and _{s} = 1.3 × 10^{−2} M, respectively. As one can see from

Dependence of the reduced osmotic coefficient on the ratio of concentration of osmotically active counterions and salt ions. NaPSS data are from Takahashi _{w} = 4 × 10^{5} g/mol (brown squares with filled left top corner), _{w} = 6.5 × 10^{5} g/mol (brown squares with filled top), and _{w} = 1.2 × 10^{6} g/mol (brown squares with filled bottom). DNA data (purple rhomb with filled right side) are from Raspaud

We applied a hybrid Monte Carlo/molecular dynamics simulation method to model salt ion exchange between salt reservoir and polyelectrolyte solution of chains with the degree of polymerization

It is important to point out that one can also model Donnan equilibrium in ionic systems by performing semi-grand canonical simulations [

In our simulations of the osmotic pressure of polyelectrolyte solutions, we only study the effect of monovalent salts in theta solvent conditions for the polymer backbone. The addition of the multivalent salts [

The authors are grateful to the National Science Foundation for the financial support under the Grant DMR-1004576. Jan-Michael Y. Carrillo’s contribution was sponsored by the Office of Advanced Scientific Computing Research; U.S. Department of Energy and performed at the Oak Ridge National Laboratory, which is managed by University of Tennessee-Battelle, LLC under Contract No. DE-AC05-00OR22725.

The authors declare no conflict of interest.