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Breakdown of linear dielectric theory for theinteraction between hydrated ions and

graphene

Philip Loche,† Cihan Ayaz,† Alexander Schlaich,†,‡ Douwe J. Bonthuis,† andRoland R. Netz∗,†

†Fachbereich Physik, Freie Universität Berlin, Arnimallee 14, 14195 Berlin, Germany‡Laboratoire Interdisciplinaire de Physique, CNRS and Université Grenoble Alpes, UMR

CNRS 5588, 38000 Grenoble, France

AbstractMany vital processes taking place in elec-trolytes, such as nanoparticle self-assembly, wa-ter purification and the operation of superca-pacitors, rely on the precise many-body interac-tions between surfaces and ions in water. Herewe study the interaction between a hydratedion and a charge-neutral graphene layer usingatomistic molecular dynamics simulations. Forsmall separations, the ion-graphene repulsion isof non-electrostatic nature and for intermediateseparations Van-der-Waals attraction becomesimportant. Contrary to prevailing theory, weshow that non-linear and tensorial dielectric ef-fects become non-negligible close to surfaces,even for monovalent ions. This breakdown ofstandard isotropic linear dielectric theory hasimportant consequences for the understandingand modeling of charged objects at surfaces.

Graphical TOC Entry

Simulation snapshot of a chloride ionclose to a graphene layer; the insetpresents the free energy decompositioninto electrostatic and non-electrostaticcontributions.

1

The anomalous properties of electrolytes inthe vicinity of hydrophobic surfaces have a pro-found influence on a diverse range of applica-tions, such as graphene-based water purifica-tion1, double-layer capacitance2 osmotic powerharvesting3 and electrowetting4, as well as forbiological functions, such as molecular recog-nition5 and DNA condensation6. In the sim-plest theory, ions are assumed to be point-likeand embedded in a structureless, linear dielec-tric medium. At an interface to a low-dielectricmedium the resulting image charge repulsiondiverges inversely with the ion-interface sepa-ration. For the air-water interface, Onsagerand Samaras used these assumptions to de-rive the ion surface depletion and from thatthe roughly linear increase in surface tensionwith salt concentration, in good agreement withexperiments7. This theory does not accountfor the ion size and other ion specificities thatgive rise to Hofmeister effects. In fact, largehalide ions such as iodide are slightly less re-pelled from the air-water interface than smallhalide ions such as fluoride or chloride, whichin experiments leads to ionic-specific differencesof electrolyte interfacial tensions in agreementwith atomistic simulations that include explicitwater8–11. Similar ion-specific effects are alsopresent at solid surfaces12,13. In coarse-grainedmodels that do not explicitly account for molec-ular water effects ion-specific effects can be in-cluded by adding heuristic ion-surface interac-tions14–19. In alternative approaches the inter-action of a finite-size ion with a step-like dielec-tric boundary was derived from linear dielec-tric theory20,21,23 and the ion radius and excesspolarizability were estimated from experimen-tal data, yielding good agreement of predictedionic surface affinities with simulations and ex-periments22,24,25.Yet such heuristic approaches are problem-

atic for a number of reasons: First, the dielec-tric response of water at interfaces varies withdistance and is not isotropic but tensorial anddescribed by significantly different parallel andperpendicular profiles26. This becomes particu-larly important in nano-confinement as recentlyshown in simulations and experiments2,27. Sec-ondly, the electric field produced even by mono-

valent ions is strong and linear dielectric the-ory breaks down in the ionic vicinity; how suchnon-linear effects modify the dielectric interfa-cial behavior is far from clear. Thirdly, whileit is clear that van-der-Waals and steric inter-actions are present at solid surfaces in additionto dielectric image-charge interactions, the rel-ative magnitude of all these interactions is notknown, even though that information is criti-cal for constructing an accurate theory of ion-surface interactions.Here we determine the free energy profile for

a single ion (modeled with chloride Lennard-Jones (LJ) parameters) close to a neutralgraphene/water interface. We choose graphenebecause it is both simple and highly relevantfor nanotechnological applications1,4,28–32. Notethat the graphene surface that we use is hy-drophobic, but our results are not necessarilylimited to hydrophobic surfaces.By splitting the ionic free energy into elec-

trostatic and non-electrostatic LJ contributionswe show that LJ contributions dominate closeto the surface. By thermodynamic integration(TI) we furthermore demonstrate that the non-linear dielectric response becomes significant asthe ion approaches the surface. Finally, to accu-rately describe the linear dielectric contributionto the ion-surface interaction, which only makesup a small fraction of the total interaction, ten-sorial dielectric theory must be used. Thus sim-ple isotropic linear dielectric theory misses thefundamental physics of ion-surface interactionsand the associated parameters require carefulinterpretation.We simulate two parallel neutral graphene

layers at a fixed separation L = 9.95 nmwith 8701 SPC/E water molecules using theGROMOS54A7 force field33 (see Figure 1a for asnapshot), results for the alternative TIP4PEW

water model are shown in the SupplementaryInformation. All simulation are performed us-ing the 2016 version of the GROMACS simu-lation package34. Figure 1b exhibits the wa-ter mass density profile ρm(z) with oscillationstypical for rigid surfaces. The system containstwo ions, the so-called test ion is located at avariable separation d from the carbon atomsforming one graphene sheet while the so-called

2

0 2 4 6 8 10z / nm

0

2000

ρm/( kg

m−

3)

a

b

Figure 1: a Simulation snapshot showingthe two parallel neutral graphene layers ata fixed distance L = 9.945 nm, SPC/E wa-ter molecules, the test ion at a separationd = 0.75 nm (green) and the counter ion at fixedposition z = 5 nm (blue). bWater mass densityprofile.

counter ion is fixed in the slab center. The totalsimulation system is charge neutral to avoid ar-tifacts from the periodic boundary conditions35.The LJ parameters of test and counter ions cor-respond to the chloride force field by Weeras-inghe and Smith36. During the simulation runsthe charge neutral graphene layers and the ionsare fixed in space. The d-dependent free en-ergy F (d) is obtained by TI. For each d wefirst create a neutral LJ cavity and thereby ob-tain the solvation free energy of a neutral LJsphere FLJ(d), followed by a charging procedurewhich yields the Coulomb free energy contribu-tion FCoul(d). To obtain the free energy of thetest ion, we subtract the solvation energy of theisolated counter ion (from separate bulk simu-lations) and the analytically calculated electro-static interaction between test and counter ions(see Supplemental Information for technical de-tails).Figure 2a shows a magnification of the water

mass density profile in the interfacial region, inFigure 2b the ion-surface interaction free en-

0

2000

ρm/( kg

m−

3)

zGDS

ρbulkm

0.2 0.4 0.6 0.8 1.0d / nm

0

10

20

30

∆F/

(kB

T)

LJ LJ + Coul Coul

∆FLJ

∆FLJ +∆FCoul

∆FCoul

0.0 0.5 1.0

10−1101103105107

a

b

Figure 2: a Water mass density profile withGibbs-dividing surface (GDS) position (red ver-tical line) and buk density (horizontal dottedline). b Ion-graphene interaction free energy∆F (d) split into the LJ and Coulomb contribu-tions ∆FLJ(d) and ∆FCoul(d). The inset showsresults on a log scale.

ergy, obtained by subtracting the solvation freeenergy in the slab center,

∆F (d) = F (d)− F (L/2) = ∆FLJ(d) + ∆FCoul(d),(1)

is presented together with its LJ and Coulombcontributions. For reference, the free energycontributions in the slab center are FLJ(L/2) =8.9 kBT and FCoul(L/2) = −151.1 kBT, whichagree perfectly with simulations in a bulk sys-tem (Supplementary Information). The resultsfor the total free energy ∆F (d) agree with ear-lier works12,37. Our decomposition into LJ andCoulomb parts shows that the LJ contributioncompletely dominates for small d < 0.3 nm andthus at separations considerably larger thanthe Gibbs-dividing surface (GDS) position atd ≈ 0.2 nm [red line in Figure 2b]. In the inter-mediate distance range 0.35 nm < d < 0.5 nmthe Coulomb repulsion is significantly weak-ened by LJ attraction. We conclude that theCoulomb contribution ∆FCoul(d) by itself does

3

not represent the total ion-surface interactionwell.In standard ionic hydration approaches wa-

ter is modeled as a symmetric linear homoge-neous dielectric medium38, according to whichthe ionic hydration free energy scales ∝ q2 withthe ionic valency q (measured in units of theelementary charge). It is well known that sincewater is charge asymmetric the ionic hydrationfree energy is also asymmetric with respect toq and in fact favors anion versus cation solva-tion39–41. Moreover, non-linear dielectric con-tributions have been shown to be significant forion solvation in bulk42,43. We thus introducea non-linear dielectric model for the Coulombpart of the ion-surface interaction

FCoul(d) = φ(d)q + A(d)q2 +B(d)q3 + C(d)q4.(2)

The potential φ primarily accounts for thecation/anion symmetry breaking and reflectsthe presence of a non-zero electrostatic poten-tial inside a LJ cavity due to water orientationeffects. The quadratic coefficient A is the lin-ear dielectric Born coefficient while B and Creflect non-linear dielectric effects. Figure 3ashows FCoul as a function of the ion charge q forseveral ion-surface separations d (blue symbols)as well as for a single ion in a periodic bulk wa-ter box (red symbols). The good agreement be-tween bulk and slab simulations for d = 1.0 nmdemonstrates that boundary condition effectsare properly accounted for in our analysis. Alldata show the cation/anion asymmetry aroundq = 039. Lines show fits according to eq. (2)(see Supplementary Information for details).Figure 3b–e presents the separation-

dependent dielectric coefficients defined in eq.(2), all rescaled by elementary charge and ther-mal energy kBT. The horizontal red dashedlines show bulk results, which agree well withthe slab simulations for large d. For the elec-trostatic potential in the LJ cavity φ in Figure3b we add a Volt scale, in the slab center weobtain φ(L/2) = 0.38 V in agreement with pre-vious bulk simulations43,46. At the interface thepotential φ only changes slightly, whereas the

sizable electrostatic potential shows large oscil-lations and even a different sign, as shown inthe Supplemental Information. This meansthat the sizable electrostatic potential of aplanar interface, which in previous works wassuggested to influence the interfacial ion trans-fer44,45, is largely compensated by the LJ cavitypotential. The quadratic coefficient A in Fig-ure 3c denotes the linear dielectric responseof water. The relative contribution of thelinear to the overall dielectric solvation re-sponse is for a monovalent negative ion given byA/(−φ+A−B+C); far away from the interface,i.e. for large d, we obtain 71 % and can thussay that the linear contribution is dominant.For the Coulomb contribution to the interac-tion energy, ∆FCoul(d) = FCoul(d)−FCoul(L/2),we obtain that at the surface (d = 0) the rela-tive linear contribution only amounts to 53 %.We conclude that non-linear dielectric effectsbecome significant at surfaces.We next demonstrate how the linear dielec-

tric contribution Aq2 in eq. (2) is related tocontinuum dielectric linear response theory. Aswe have shown previously, at a planar interfacethe dielectric response is tensorial and describedby perpendicular and parallel dielectric profilesε⊥(d) and ε‖(d)26, which can be straightfor-wardly extracted from our simulations and areshown in the Supplementary Information. Toarrive at an analytically manageable form, thedielectric profiles can be replaced by step func-tions of the form ε⊥/‖(d) = 1 for d < zDDS

⊥/‖and ε⊥/‖(d) = εbulk for d > zDDS

⊥/‖26, where the

bulk water dielectric constant for SPC/E wa-ter is εbulk = 70 and the dielectric dividingsurface (DDS) positions follow from the dielec-tric profiles via electrostatic integrability con-ditions as zDDS

⊥ = 0.18 nm and zDDS‖ = 0.03 nm.

We thus see that zDDS⊥ > zDDS

‖ , the shift be-tween the DDS is in between the values previ-ously found for diamond26 and self-assembledsurfaces27. The coarse-grained dielectric modelfor the graphene-water interface thus consistsof three regions as shown in Figure 4a: ε⊥ =ε‖ = 1 in the half space z < zDDS

‖ , tensorialdielectric constants ε⊥ = 1 and ε‖ = εbulk ina thin slab defined by zDDS

‖ < z < zDDS⊥ , and

4

−1.0 −0.5 0.0 0.5 1.0q / e

−150

−100

−50

0

F Cou

l/

(kB

T)

d = 0.0 nmd = 0.1 nm

d = 1.0 nmbulk (single ion)

0.0 0.2−2

0

a

14

16

18

φe/

(kB

T)

10

20

Be3/(k

BT

)

0 1 2d / nm

−120

−100

−80

Ae2/(k

BT

)

0 1 2d / nm

−20

0

20

Ce4/(k

BT

)

0.35

0.40

0.45

φ/

V

b

c

d

e

Figure 3: a Coulomb part is for different separations from the graphene surface d represented bydifferent blue symbols. Red symbols show results for an ion in a periodic bulk water box. Linesare fits according to eq. (2). b–e Coefficients are defined in eq. (2) and obtained from fits of theCoulomb solvation contribution shown in a. The horizontal red dashed lines show results for an ionin a periodic bulk water box.

solid

ε‖ = 1

ε⊥ = 1

interface

ε‖ = 70

ε⊥ = 1

zGDS

water

ε‖ = 70

ε⊥ = 70

aσ

0.0 0.2 0.4 0.6 0.8d / nm

−110

−100

−90

−80

−70

(Flin C

oul/

q2)/

(kB

T/e2

)

simu-lation

isotropic point chargeisotropic sphereanisotropic sphereq =−0.2 eq =−0.5 eq =−0.8 eq =−1.0 eA(d)

zDDS⊥zDDS

‖

a

b

Figure 4: a Continuum model for the dielectricinteraction of a sphere with radius a and surfacecharge σ with an interface that exhibits a tenso-rial dielectric constant for zDDS

‖ < z < zDDS⊥ . b

Rescaled linear dielectric free energy contribu-tion ∆F lin

Coul/q2 based on eq. (5). Blue symbols

show simulation results, red crosses denote thelinear prediction Aq2 based on eq. (2), linesare different continuum model predictions (seetext).

ε⊥ = ε‖ = εbulk in the half space zDDS⊥ < z.

Within linear response theory, the Coulombfree energy associated with a charge densityρ(r) is given by

F linCoul =

1

2

∫∫ρ(r)G(r, r′)ρ(r′)dr dr′ . (3)

The Green’s function G follows from the Poissonequation

ε0 {∇ · [ε̂(r) · ∇G(r, r′)]} = −δ(r− r′) , (4)

where the anisotropic permittivity tensor ε̂(r)is diagonal and piecewise constant in the threeregions as defined in Figure 4a. The derivationof G is explicitly shown in the SupplementaryInformation.We model the ion as a sphere with radius a

and surface charge density σ = q/4πa2, yield-ing a total charge of q, so that the charge dis-tribution of an ion centered at position r0 isρ(r) = σδ(|r − r0| − a). Inside the sphere weassume the dielectric constant to be same as inthe surrounding medium, so that we can use eq.(3) to calculate the electrostatic energy. In theSupplementary Information we show by numer-ically solving the Poisson equation that using afixed dielectric constant ε = 1 inside the ion,the dielectric energy is very similar to the re-sults of eq. (3).

5

The ion radius a we determine by comparingBorn’s result for the dielectric hydration freeenergy of a charged spherical shell in bulk47

FBorn = q2(ε−1bulk − 1)/(8πε0a) with the linearcontribution of the simulated Coulomb free en-ergy

F linCoul(d) = FCoul(d)− φ(d)q−B(d)q3 −C(d)q4.

(5)For a chloride ion with q = −1 and in bulk ford = L/2 we obtain F lin

Coul = −108.9 kBT, whichgives an effective radius of a = 0.254 nm. Thisradius is significantly larger than the Paulingcrystal radius for chloride, 0.18 nm48, and alsolarger than the radius obtained from associat-ing the Born expression with the experimentalF = −142.1 kBT or the simulated total chloridehydration free energy F = −142.2 kBT, whichboth yield a radius of 0.19 nm40. This is notsurprising, since the total hydration free energycontains non-electrostatic as well as non-linearelectrostatic contributions, which are not cap-tured by the Born expression FBorn and there-fore should not be included in a comparison ofFBorn with hydration energies. The good agree-ment between the crystal radius and the fittedradii when using the total hydration free energyis purely coincidental and thus unphysical.Figure 4b compares simulation results for the

linear dielectric contribution according to eq.(5) for a few different values of q (open sym-bols) with the Green’s function result defined ineq. (3) (solid line). The agreement is excellentwithout any fitting parameters and shows thatthe tensorial dielectric coarse-grained model de-scribes the linear dielectric free energy contri-bution very well. Note that in Figure 4b allenergies are rescaled by the ion charge squared,q2, in order to eliminate the trivial quadraticcharge dependence. We also show the linear co-efficient A(d) taken from Figure 3b (red crosses)and again find good agreement, which recon-firms that the non-linear expansion of the di-electric response in eq. (2) is robust. Wealso compare with dielectric theories that ne-glect the tensorial character and instead use anisotropic dielectric profile with a jump locatedat the Gibbs dividing surface (whose locationzGDS = 0.19 nm is indicated by a red verti-

cal line in Figure 4). The dotted line showsthe free energy for a point charge, which corre-sponds to the simple image-charge constructionand which diverges as the GDS is approached.The dashed line shows the free energy for acharged sphere of finite radius calculated previ-ously21; note that this calculation corrected anerror in a previous publication20. It is seen thatthe finite-size sphere result for an isotropic di-electric model (broken line) is shifted away fromthe interface compared to the tensorial model.The divergence of the point-charge model isclearly not present in the simulation data, butthis model agrees quite well with the simula-tion data further away from the interface. Thisagreement again stems from the cancellation oftwo errors, namely the neglect of the finite ionsize and the wrong positioning of the dielectricdividing surface at the GDS, and thus is nota robust feature of such a simplified model de-scription.In summary, the interaction of a hydrated

ion with a neutral hydrophobic graphene sur-face is dominated by LJ interactions at smallseparation, while the Coulombic contribution(which is of dielectric nature) contains signifi-cant non-linear contributions. In order to ac-curately describe the linear contribution to thedielectric ion-surface interaction, finite ion sizeand tensorial interfacial dielectric effects mustbe taken into account. The failure of standardlinear dielectric theory at interfaces is typicallyhidden by a subtle cancelation of errors dueto using an unphysical ion size, the neglect oftensorial and non-linear dielectric effects andthe neglect of non-electrostatic forces. Finally,our results show that generalizing the Poisson-Boltzmann approach to electrolyte interfaces byincluding non-electrostatic ion-surface interac-tions, as has become widespread practice in re-cent years12,18,49,50, is meaningful.

Acknowledgement This work was sup-ported by the Deutsche Forschungsgemein-schaft within a grant from Sonderforschungs-bereich (SFB) 765 and by grant NE 810/11.

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Supporting Information Avail-ableThe following files are available free of charge.

• imagecharge_si.pdf: Contains details onthe simulation methods, a comparisonwith a different water model and the ana-lytical and numerical solutions of the ten-sorial Poisson equation.

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