^{1}and A. Travesset

^{1}

### Abstract

We provide extensive molecular dynamics simulations of counterion and coion distributions near an impenetrable plane with fixed discrete charges. The numerical results are described by an explicit solution that distinguishes the plasma ( ) and the binding regime ( ) where σ is the ion diameter and A c = |e/ν| (ν is the surface charge density). In the plasma regime, the solution consists of a product of two functions that can be computed from simpler models and reveals that the effects of the discreteness of the charge extends over large distances from the plane. The solution in the binding regime consists of a Stern layer of width σ and a diffuse layer, but contrary to standard approaches, the strong correlations between ions within the Stern layer and the diffuse layer require a description in terms of a “displaced” diffuse layer. The solution is found to describe electrolytes of any valence at all concentrations investigated (up to 0.4M) and includes the case of additional specific interactions such as van der Waals attraction and other generalizations. We discuss some open questions.

We are indebted to Monica H. Lamm for discussions as well as for providing us with computer facilities. We also acknowledge discussions and interest from C. Calero, J. Faraudo, and D. Vaknin. This work is supported by National Science Foundation through Grant No. CAREER DMR-0748475.

I. INTRODUCTION

II. MODEL AND OBSERVABLES

A. Simulation methods

B. Units of length and temperature

III. RESULTS

A. Monovalent ions

B. Dilute solutions, divalent ions

C. Beyond dilute concentrations, divalent ions

D. Deviations from perfect lattice with divalent counterions

E. Divalent ions with van der Waals attraction

F. Trivalent counterions

IV. GENERAL SOLUTION TO THE PDDL

A. Solution in the plasma regime (large molecular area)

1. Physical interpretation of

2. The diffuse layer

B. The binding regime

V. CONCLUSIONS

A. Summary

B. Outlook

### Key Topics

- Photon density
- 12.0
- Surface charge
- 10.0
- Numerical solutions
- 8.0
- Electrolytes
- 7.0
- Electrostatics
- 7.0

##### H05F

## Figures

(a) The planar discrete double layer model. The charges at the interface are in gray, counterions in red and coions in blue. (b) Cross sectional view.

(a) The planar discrete double layer model. The charges at the interface are in gray, counterions in red and coions in blue. (b) Cross sectional view.

Number density distribution n cc as a function of r for different values of the azimuth coordinate θ ranging from 0 to π/4. Results in the figure are for monovalent counterions at concentration 0.05M (A c = 100 Å^{2}) (a), and 0.08 M (A c = 361 Å^{2}) (b).

Number density distribution n cc as a function of r for different values of the azimuth coordinate θ ranging from 0 to π/4. Results in the figure are for monovalent counterions at concentration 0.05M (A c = 100 Å^{2}) (a), and 0.08 M (A c = 361 Å^{2}) (b).

(a) The function w(r, z) cc for fixed values of z (in Å). The molecular area is A c = 361 Å^{2}. (b) The potential shifted by a constant , as defined in Eq. (11) . Results are for monovalent counterions at concentration 0.08 M.

(a) The function w(r, z) cc for fixed values of z (in Å). The molecular area is A c = 361 Å^{2}. (b) The potential shifted by a constant , as defined in Eq. (11) . Results are for monovalent counterions at concentration 0.08 M.

(a) and (b) Number density distribution n cc Eq. (7) as a function of r for different values of the azimuth coordinate θ ranging from 0 to π/4. Results in the figure are for divalent counterions at concentration 0.02 M.

(a) and (b) Number density distribution n cc Eq. (7) as a function of r for different values of the azimuth coordinate θ ranging from 0 to π/4. Results in the figure are for divalent counterions at concentration 0.02 M.

(a) The function w(r, z) cc at fixed values of z (in Å). The molecular area is A c = 361 Å^{2}. (b) The potential shifted by a constant , as defined in Eq. (11) . Results are for divalent counterions at concentration 0.04 M.

(a) The function w(r, z) cc at fixed values of z (in Å). The molecular area is A c = 361 Å^{2}. (b) The potential shifted by a constant , as defined in Eq. (11) . Results are for divalent counterions at concentration 0.04 M.

The function (blue dashed line) for different molecular areas. Also shown is the actual number density (red solid line) obtained from simulation, as well as independently obtained from Eq. (9) (cyan markers). Counterions are divalent and the bulk ion concentration is 0.02 M.

The function (blue dashed line) for different molecular areas. Also shown is the actual number density (red solid line) obtained from simulation, as well as independently obtained from Eq. (9) (cyan markers). Counterions are divalent and the bulk ion concentration is 0.02 M.

The function (black diamond markers) for different molecular areas together with the corresponding exponential fit (green solid line). Counterions are divalent and the bulk ion concentration is 0.02 M.

The function (black diamond markers) for different molecular areas together with the corresponding exponential fit (green solid line). Counterions are divalent and the bulk ion concentration is 0.02 M.

Function for different molecular areas and divalent counterions at concentration 0.02 M. Each plot is shifted by a constant so that the value at the minimum is the same for all molecular areas.

Function for different molecular areas and divalent counterions at concentration 0.02 M. Each plot is shifted by a constant so that the value at the minimum is the same for all molecular areas.

The renormalized Gouy-Chapman length vs bulk ion concentration for divalent ions, molecular area A c = 361 Å^{2}.

The renormalized Gouy-Chapman length vs bulk ion concentration for divalent ions, molecular area A c = 361 Å^{2}.

(a) Number density distribution for different bulk concentration values. (b) , and the corresponding exponential fit (Eq. (12) ) for M. The results are for divalent ions at molecular area A c = 361 Å^{2}.

(a) Number density distribution for different bulk concentration values. (b) , and the corresponding exponential fit (Eq. (12) ) for M. The results are for divalent ions at molecular area A c = 361 Å^{2}.

The function for different divalent counterion concentrations at fixed molecular area A c = 361 Å^{2}.

The function for different divalent counterion concentrations at fixed molecular area A c = 361 Å^{2}.

Disordered lattices, see Eq. (13) , with divalent counterions (a) function for different d values (b) for different d values. Results for divalent counterions at A c = 361 Å^{2}, conc = 0.02 M.

Disordered lattices, see Eq. (13) , with divalent counterions (a) function for different d values (b) for different d values. Results for divalent counterions at A c = 361 Å^{2}, conc = 0.02 M.

Systems with vdW attractions. (a) for a system with (λ = 3) and without (λ = 1) vdW attraction. (b) n cc (z) for a system with and without vdW attractions. Results for divalent counterions at A c = 361 Å^{2} and conc = 0.02 M.

Systems with vdW attractions. (a) for a system with (λ = 3) and without (λ = 1) vdW attraction. (b) n cc (z) for a system with and without vdW attractions. Results for divalent counterions at A c = 361 Å^{2} and conc = 0.02 M.

(a) Comparison of the different and (b) for different counterion valence at molecular area A c = 361 Å^{2} (and about the same bulk concentration, 0.07–0.08 M). The markers represent the calculated values and solid lines represent the exponential fits. The corresponding values are 10.22 Å(monovalent), 10.08 Å (divalent), 13.88 Å (trivalent).

(a) Comparison of the different and (b) for different counterion valence at molecular area A c = 361 Å^{2} (and about the same bulk concentration, 0.07–0.08 M). The markers represent the calculated values and solid lines represent the exponential fits. The corresponding values are 10.22 Å(monovalent), 10.08 Å (divalent), 13.88 Å (trivalent).

A representation of the two-dimensional one component plasma in a fixed square lattice, OCPFL model, characterized by e m given by ratio of mobile charges (red spheres) to fixed charges (gray spheres).

A representation of the two-dimensional one component plasma in a fixed square lattice, OCPFL model, characterized by e m given by ratio of mobile charges (red spheres) to fixed charges (gray spheres).

Comparison of the potential of mean force of the 2d OCPFL (markers ⋄, □, o) against (blue solid line). The results are for divalent ions at molecular area A c = 361 Å^{2}. (e m is the ratio of mobile/fixed charges, see Fig. 15 .)

Comparison of the potential of mean force of the 2d OCPFL (markers ⋄, □, o) against (blue solid line). The results are for divalent ions at molecular area A c = 361 Å^{2}. (e m is the ratio of mobile/fixed charges, see Fig. 15 .)

Comparison of the potential of mean force of the 2d OCPFL (markers ⋄, □, o) against (blue solid line) for (a) monovalent (b) divalent ions with deformed lattice, at molecular area A c = 361 Å^{2}. (e m is the ratio of mobile/fixed charges, see Fig. 15 .)

Comparison of the potential of mean force of the 2d OCPFL (markers ⋄, □, o) against (blue solid line) for (a) monovalent (b) divalent ions with deformed lattice, at molecular area A c = 361 Å^{2}. (e m is the ratio of mobile/fixed charges, see Fig. 15 .)

Comparison of the potential of mean force of the 2D OCPFL (markers ⋄, □, o) against (blue solid line) for (a) divalent with vdW attraction (b) trivalent ions, at molecular area A c = 361 Å^{2}. (e m is the ratio of mobile/fixed charges, see Fig. 15 .)

Comparison of the potential of mean force of the 2D OCPFL (markers ⋄, □, o) against (blue solid line) for (a) divalent with vdW attraction (b) trivalent ions, at molecular area A c = 361 Å^{2}. (e m is the ratio of mobile/fixed charges, see Fig. 15 .)

Potential of mean force for a OCPFL model, see Fig. 15 , where particles interact via a Yukawa potential. The filled ⋄ markers are simulation results, solid lines are the approximation given by Eq. (15) . The function is shown in magenta (o markers). Case of monovalent counterions at molecular area Ac = 100 Å^{2}.

Potential of mean force for a OCPFL model, see Fig. 15 , where particles interact via a Yukawa potential. The filled ⋄ markers are simulation results, solid lines are the approximation given by Eq. (15) . The function is shown in magenta (o markers). Case of monovalent counterions at molecular area Ac = 100 Å^{2}.

(a) and (b) The counterion distribution n cc (z) (red markers) and the Poisson Boltzmann distribution (blue) for the monovalent counterion systems. As described in Sec. ??? , z match ≈ σ.

(a) and (b) The counterion distribution n cc (z) (red markers) and the Poisson Boltzmann distribution (blue) for the monovalent counterion systems. As described in Sec. ??? , z match ≈ σ.

(a) and (b) The counterion distribution n cc (z) (red markers) and the diffuse layer (DL) distribution (blue), which is described by PB as discussed in the main text, for divalent counterion systems, with z match ≈ a L /3. (For a magnified plot showing the difference between the PB predictions and results from simulation for the region σ < z < a L /3, see the supplementary material. ^{28} )

(a) and (b) The counterion distribution n cc (z) (red markers) and the diffuse layer (DL) distribution (blue), which is described by PB as discussed in the main text, for divalent counterion systems, with z match ≈ a L /3. (For a magnified plot showing the difference between the PB predictions and results from simulation for the region σ < z < a L /3, see the supplementary material. ^{28} )

(a) Number density distribution n cc as a function of r for different values of the azimuth coordinate θ ranging from 0 to π/4. Case of divalent counterions at concentration 0.08 M (A c = 40 Å^{2}). (b) Counterion density n cc (x, y, z = 0) around an interfacial charge for the same case as in (a), as a function of cartesian coordinates x,y (counterion density increases from dark to light shades of brown). (c) A c = 70 Å^{2} and (d) A c = 100 Å^{2}, counterion distribution (red markers) and the counterion distribution for the case of a uniformly charged interface (blue) for the divalent counterion systems. As described in Sec. IV B , z match ≈ σ.

(a) Number density distribution n cc as a function of r for different values of the azimuth coordinate θ ranging from 0 to π/4. Case of divalent counterions at concentration 0.08 M (A c = 40 Å^{2}). (b) Counterion density n cc (x, y, z = 0) around an interfacial charge for the same case as in (a), as a function of cartesian coordinates x,y (counterion density increases from dark to light shades of brown). (c) A c = 70 Å^{2} and (d) A c = 100 Å^{2}, counterion distribution (red markers) and the counterion distribution for the case of a uniformly charged interface (blue) for the divalent counterion systems. As described in Sec. IV B , z match ≈ σ.

Results for A c = 70 Å^{2} (concentration 0.047 M), (a) counterion distribution (red markers) and the counterion distribution for the case of a uniformly charged interface (blue dashed line) for the divalent counterion systems. Result for the uniform surface charge simulation with ν0 = ν equiv is also shown (green), and (b) corresponding ν values as a function of z.

Results for A c = 70 Å^{2} (concentration 0.047 M), (a) counterion distribution (red markers) and the counterion distribution for the case of a uniformly charged interface (blue dashed line) for the divalent counterion systems. Result for the uniform surface charge simulation with ν0 = ν equiv is also shown (green), and (b) corresponding ν values as a function of z.

Summary of the results. DL is the diffuse layer (description provided in the main text).

Summary of the results. DL is the diffuse layer (description provided in the main text).

The radial potential of mean force for counterions (left) and coions (right). The blue markers represent simulation results and the red solid line is calculated using Eq. (A2) . Results are for divalent ions at A c = 361 Å^{2} at concentration 0.026 M.

The radial potential of mean force for counterions (left) and coions (right). The blue markers represent simulation results and the red solid line is calculated using Eq. (A2) . Results are for divalent ions at A c = 361 Å^{2} at concentration 0.026 M.

## Tables

Summary of functions used in this paper (α = cc (counterions) or co (coions)).

Summary of functions used in this paper (α = cc (counterions) or co (coions)).

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