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General solution to the electric double layer with discrete interfacial charges
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10.1063/1.4739300
/content/aip/journal/jcp/137/6/10.1063/1.4739300
http://aip.metastore.ingenta.com/content/aip/journal/jcp/137/6/10.1063/1.4739300

Figures

Image of FIG. 1.
FIG. 1.

(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.

Image of FIG. 2.
FIG. 2.

Number density distribution as a function of 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 ( = 100 Å2) (a), and 0.08 M ( = 361 Å2) (b).

Image of FIG. 3.
FIG. 3.

(a) The function (, ) for fixed values of (in Å). The molecular area is = 361 Å2. (b) The potential shifted by a constant , as defined in Eq. (11) . Results are for monovalent counterions at concentration 0.08 M.

Image of FIG. 4.
FIG. 4.

(a) and (b) Number density distribution Eq. (7) as a function of 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.

Image of FIG. 5.
FIG. 5.

(a) The function (, ) at fixed values of (in Å). The molecular area is = 361 Å2. (b) The potential shifted by a constant , as defined in Eq. (11) . Results are for divalent counterions at concentration 0.04 M.

Image of FIG. 6.
FIG. 6.

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.

Image of FIG. 7.
FIG. 7.

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.

Image of FIG. 8.
FIG. 8.

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.

Image of FIG. 9.
FIG. 9.

The renormalized Gouy-Chapman length vs bulk ion concentration for divalent ions, molecular area = 361 Å2.

Image of FIG. 10.
FIG. 10.

(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 = 361 Å2.

Image of FIG. 11.
FIG. 11.

The function for different divalent counterion concentrations at fixed molecular area = 361 Å2.

Image of FIG. 12.
FIG. 12.

Disordered lattices, see Eq. (13) , with divalent counterions (a) function for different values (b) for different values. Results for divalent counterions at = 361 Å2, conc = 0.02 M.

Image of FIG. 13.
FIG. 13.

Systems with vdW attractions. (a) for a system with (λ = 3) and without (λ = 1) vdW attraction. (b) () for a system with and without vdW attractions. Results for divalent counterions at = 361 Å2 and conc = 0.02 M.

Image of FIG. 14.
FIG. 14.

(a) Comparison of the different and (b) for different counterion valence at molecular area = 361 Å2 (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).

Image of FIG. 15.
FIG. 15.

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

Image of FIG. 16.
FIG. 16.

Comparison of the potential of mean force of the 2d OCPFL (markers ⋄, □, ) against (blue solid line). The results are for divalent ions at molecular area = 361 Å2. ( is the ratio of mobile/fixed charges, see Fig. 15 .)

Image of FIG. 17.
FIG. 17.

Comparison of the potential of mean force of the 2d OCPFL (markers ⋄, □, ) against (blue solid line) for (a) monovalent (b) divalent ions with deformed lattice, at molecular area = 361 Å2. ( is the ratio of mobile/fixed charges, see Fig. 15 .)

Image of FIG. 18.
FIG. 18.

Comparison of the potential of mean force of the 2D OCPFL (markers ⋄, □, ) against (blue solid line) for (a) divalent with vdW attraction (b) trivalent ions, at molecular area = 361 Å2. ( is the ratio of mobile/fixed charges, see Fig. 15 .)

Image of FIG. 19.
FIG. 19.

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 = 100 Å2.

Image of FIG. 20.
FIG. 20.

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

Image of FIG. 21.
FIG. 21.

(a) and (b) The counterion distribution () (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 /3. (For a magnified plot showing the difference between the PB predictions and results from simulation for the region σ < < /3, see the supplementary material. 28 )

Image of FIG. 22.
FIG. 22.

(a) Number density distribution as a function of for different values of the azimuth coordinate θ ranging from 0 to π/4. Case of divalent counterions at concentration 0.08 M ( = 40 Å2). (b) Counterion density (, , = 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) = 70 Å2 and (d) = 100 Å2, 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 , ≈ σ.

Image of FIG. 23.
FIG. 23.

Results for = 70 Å2 (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 ν = ν is also shown (green), and (b) corresponding ν values as a function of .

Image of FIG. 24.
FIG. 24.

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

Image of FIG. 25.
FIG. 25.

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 = 361 Å2 at concentration 0.026 M.

Tables

Generic image for table
Table I.

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

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/content/aip/journal/jcp/137/6/10.1063/1.4739300
2012-08-10
2014-04-17
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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: General solution to the electric double layer with discrete interfacial charges
http://aip.metastore.ingenta.com/content/aip/journal/jcp/137/6/10.1063/1.4739300
10.1063/1.4739300
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