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Condensed Matter Physics, 2005, Vol. 8, No. 2(42), pp. 335–356

Simulating an electrochemical interface

using charge dynamics

C.G.Guymon, R.L.Rowley, J.N.Harb, D.R.Wheeler

Department of Chemical Engineering,

BYU, Provo UT 84602

Received November 29, 2004

We present a simple classical method for treating charge mobility in met-

als adjacent to liquid solutions. The method, known as electrode charge

dynamics, effectively bridges the computational gap between ab initio cal-

culations on small metal clusters and large-scale simulations of metal sur-

faces with arbitrary geometry. We have obtained model parameters for a

copper (111) metal surface using high-level quantum-mechanical calcu-

lations on a 10-atom copper cluster. We validated the model against the

classical image-charge result and ab initio results on an 18-atom copper

cluster. The model is used in molecular dynamics simulations to predict

the structure of the ﬂuid interface for neat water and for aqueous NaCl so-

lution. We ﬁnd that water is organized into a two-dimensional ice-like layer

on the surface and that both Na

+

and Cl

−

are strongly bound to the cop-

per. When charging the metal electrode, most of the electrolyte response

occurs in the diffuse part of the double layer.

Key words: simulation, double layer, molecular dynamics, ab initio,

potentials, copper (111) surface, water

PACS: 61.20.Qg, 61.20.Ja

1. Introduction

Increased understanding of the interface between metals a nd liquid solutions (i.e.

the electrochemical interface) is vital for the design and improvement of catalytic,

corrosive, and electrochemical systems. The conﬁguration of liquid species at the in-

terface is intimately connected with mechanistic pathways of ion adsorption, metal

dissolution, and metal deposition. Atomically detailed simulation techniques such

as molecular dynamics have yielded insight into these kinds of problems. Simulati-

ons have been used to predict the adsorbed water orientation, io nic distributions,

ionic transport coeﬃcients, atomically resolved electrostatic potential proﬁles, and

interfacial diﬀerential capacitance [1–6].

Accurate molecular simulations of an electrochemical interface depend on the

quality of intermolecular potentials between species. Prior work has resulted in ad-

c

C.G.Guymon, R.L.Rowley, J.N.Harb, D.R.Wheeler 335

C.G.Guymon et al.

equate interaction potentials between species in bulk liquid electrolytes, however

the po t entials suitable at a metallic interface are less well established. For an inter-

molecular po t ential to be suitable for use in large-scale MD simulations, it should be

computationally inexpensive. In this work we focus on obtaining and using accurate

but inexpensive potentials between solution species and the metal. Two common

ways of treating a charged species interacting with a polarizable metal are (1) the

image-charge method and (2) the use of pairwise additive interatomic potentials

obtained from ab initio calculations. Both of these approximations have been used

to simulate large atomically discrete systems [7–10].

The image-charge method is based on an analytic solution to Poisson’s equation.

When a charged particle approaches a conducting surface, it polarizes the surface,

resulting in an attractive force. The induced potential ﬁeld outside the metal is

perfectly mimicked by placing an image point charge beneath the surface of the

metal. For a planar interface the image charge is equal and opposite to the “real”

charge and is equidistant to the interface. In a molecular simulation that uses im-

age charges, all real and image charges interact with each other. Due t o its classical

continuum derivation, at long range the method correctly reproduces the Coulombic

potential between a point charge and a semi-inﬁnite metallic surface. However, when

the charge is within about 3

˚

A of the surface the method overpredicts the strength

of interaction and does not resolve impo r t ant atomic-level details and spatial non-

uniformities. For example, the image-charge method predicts the same interaction

energy for positive or negative charges, and does not diﬀerent ia t e between surface ad-

sorption sites. Moreover, the method requires an ambiguous selection of the location

of the image mirror plane. Spohr and coworkers partially remedy these diﬃculties

by incorporating an additional species-surface interatomic potential to account for

the surface corrugation [7].

An alternative is to assume that the p otential energy between a molecule and

the metal surface is well represented by the interaction between the molecule and a

small metal cluster (typically 10–20 metal atoms). The advantage of this method is

that the molecule-cluster interaction energy can be obtained by quantum chemical

calculations. In a reﬁnement to this approach, the molecule-cluster interaction is

decomposed into pairwise atomic interactions that can then be used for the molecule

interacting with a much larger metal surface during simulation [3,8]. The primary

drawback of this method is tha t it neglects multibody charge induction in the metal,

which will be signiﬁcant in liquid simulations due to the high density of charges next

to the metal. This is in addition to concerns about the assumption that a small metal

cluster accurately represent s a surface [11].

In addition to the two above highlighted treatments of charge dynamics near a

metal surface, Allen et al. [12] and Boda et a l. [13 ] have developed and implemented

induced charge computation methods. These recent methods allow for image-charge-

type results with geometries other than a ﬂat surface, and t hus parallel some of the

advantages we claim for this work.

We present here a simple method, termed electrode charge dynamics (ECD), that

reproduces the image-charge result at long range and matches quantum chemical

336

Simulating an electrochemical interface

calculations at short range. ECD is superior to using only pairwise additive potentials

or image charges or a combination thereof. In particular, ECD is easy to use for

irregular surface g eometries. In the sections that follow we ﬁrst describe the method

and then validate it by comparing ECD with other calculations. We conclude with

MD simulation results for an aqueous NaCl electrolyte and neat water next to a

planar (111) copper surface.

2. Methodology

The electrode charge dynamics method is based on a simple model of charge

density in the metal, shown in ﬁgure 1. Metallic conduction-band electrons are free

to move in response to Coulombic forces originating within and without the metal,

resulting in regions of increased and decreased electron density. We represent the

conduction o r valence electrons throughout the metal with a diﬀuse negative charge

located at each metal atomic center of variable magnitude q

v

i

. A ﬁxed positive point

charge q

c

i

is co-located at each center to represent the nucleus and core electrons.

Figure 1. Charge on each metal atom.

Each diﬀuse charge is modeled with a Gaussian charge-density distribution,

ρ

i

(r) = q

v

i

γ

3

i

π

−3/2

exp

−γ

2

i

r

2

, (1)

where r is the distance from the atom center and γ

i

is an inverse-width parameter.

We treat charge mobility in t he metal by allowing each diﬀuse charg e magnitude

to ﬂuctuate in response to its environment. More speciﬁcally, each q

v

i

changes so

as to minimize the electrostatic energy or equivalently to equilibrate the chemical

potential for q

v

i

at each atom. The ﬂuctuation in q

v

i

is subject to two constraints. The

ﬁrst constraint is that q

v

i

6 0, corresponding to a limit on the depletion of electron

density from a given atom. The second constraint is a ﬁxed total charge Q

tot

in the

simulated metal:

n

X

i=1

(q

v

i

+ q

c

i

) = Q

tot

, (2)

where the sum is over all metal atoms.

337

C.G.Guymon et al.

At this point we note the similarity between our model and the charge equili-

bration schemes developed by Rapp´e and Goddard [14] and Rick, Stuart, and Berne

[15]. In these models as well a s in our model there are two adjustable parameters per

unique charge site. In this case, since the metal surface is composed of only one kind

of atom, there are only two parameters (here q

c

and γ) that characterize the polari-

zation behavior of the metal. The explicit division of the charge into core and valence

charges is unique to this work and leads to diﬀerent behavior for short-range inter-

actions. We also note o ur model’s classical resemblance to the quantum-mechanical

approximations used by Price and Halley [16].

To obtain the magnitude of the diﬀuse charge at each electrode atom, we mini-

mize the Coulombic potential energy U of t he metal subject t o the constraints. This

can be done analytically using Lagrang e’s method o f undetermined multipliers. The

Lagrangian to be minimized is

L = U − λ

n

X

i=1

(q

v

i

+ q

c

i

) − Q

tot

−

n

X

i=1

µ

i

q

v

i

, (3)

where λ and µ

i

are undetermined multipliers, and

U =

1

2

n

X

i=1

n

X

j=1

q

v

i

q

v

j

C

ij

+

n

X

i=1

n

X

j=1

q

v

i

q

c

j

C

∗

ij

+

1

2

n

X

i=1

n

X

j=1,

j6=i

q

c

i

q

c

j

1

r

ij

+

n

X

i=1

q

v

i

φ

set

i

+ U

ext

. (4)

C

ij

and C

∗

ij

are Coulomb overlap integrals given by

C

ij

=

erf(γ

ij

r

ij

)

r

ij

, C

∗

ij

=

erf(γ

i

r

ij

)

r

ij

, (5)

with γ

ij

= (γ

−2

i

+ γ

−2

j

)

−1/2

. U

ext

is the Coulomb energy from interactions of the

electrode charges with charges external to the electrode. φ

set

i

is a user speciﬁed oﬀset

potential or voltage that permits a voltage diﬀerence to be maintained between two

portions of the electrode.

According to the Kuhn-Tucker conditions for Lagrangia n problems with inequali-

ty constraints, the solution should satisfy

∂L

∂q

v

i

=

∂U

∂q

v

i

− λ − µ

i

= 0, (6)

with µ

i

positive when q

v

i

is not negative or µ

i

zero when q

v

i

is negative. In this

scheme, µ

i

acts as a switch t o maintain the inequality constraint that q

v

i

should

be negative, and λ corresponds to the to t al-charge equality constraint (equation 2).

This is a linear minimization problem and can be solved by matrix inversion at

each time step throughout the simulation. If there are very many metal atoms, the

computational cost of inversion can become excessive. (For a thorough discussion of

solving large linear sets of equations see reference [17].) Therefore, we have chosen

an alternative implementation.

338

Simulating an electrochemical interface

To avoid the CPU-costly matrix inversion at each step, we treat the diﬀuse

charges as dynamic degrees of freedom that respond to forces that decrease the

value of L. This is very similar to the way in which the solution particles seek out

the energy minima of phase space. We assign each diﬀuse charge a ﬁctitious mass

m

q

, as well as a charg e “velocity” v

i

. Each q

v

i

is forced down the gradient in the

Lagrangian L while the “temperature” of the diﬀuse charges is kept near zero. In

this way the diﬀuse charges in the electrode are eﬃciently held near the electrostatic

energy minimum at a signiﬁcant computational savings over matrix inversion. Note

that the particle and charge degrees of freedom are coupled as the charge in the

metal responds to the structure of the ﬂuid. For this reason separate thermostats

must be applied to the particle and charge degrees of freedom in order to maintain

them at diﬀerent temperatures.

The diﬀuse charge and its velocity are propag ated using the following equations

of motion:

˙q

v

i

= v

i

− ξ, (7a)

˙v

i

=

F

i

m

q

− ζv

i

, (7b)

where over-dots indicate time derivatives. ξ, given by

ξ =

1

nτ

n

X

j=1

(q

v

j

+ q

c

j

) − Q

tot

, (8)

is an additional parameter to correct long-term numerical drift in the total charge

of the electrode. τ is the time constant equal to about 100 timesteps. Lo ng -term

drift is possible b ecause it is the derivative of the total charge constraint that is

incorporated into the form of the force F

i

on each charge. Since we use the Ewald

sum to account for long-ra nge Coulombic interactions, it is important to control

charges closely in o r der to maintain cell neutrality. ζ is an integral-feedback control

variable that maintains the average charge temperature at a set point

¯

T

q

. Using a

Nos´e-Hoover temperature-control scheme [18], the equation of motion for ζ is

˙

ζ =

T

q

−

¯

T

q

¯

T

q

τ

2

, (9)

where

¯

T

q

is the temperature setpoint – we ﬁnd that 5 K works well. The instanta-

neous charge temperature is

T

q

=

m

q

nk

B

n

X

j=1

v

2

j

, (10)

where k

B

is Boltzmann’s constant. In fact, the choice of mass m

q

is tied to the

choice of

¯

T

q

and the timestep, in order to maintain numerical stability. We use

m

q

= k

B

¯

T

q

(10 ps/|e|)

2

.

339

C.G.Guymon et al.

The force on each valence charge is

F

i

=

1

n

n

X

j=1

φ

j

− φ

i

, (11)

where φ

i

, the chemical potential for valence charge i, is

φ

i

=

∂U

∂q

v

i

− µ

i

=

n

X

j=1

C

ij

q

v

j

+

n

X

j=1

C

∗

ij

q

c

j

+

∂U

ext

∂q

v

i

+ φ

set

i

− µ

i

. (12)

Recall that µ

i

is the Lagr ange multiplier that acts to constrain q

v

i

to be negative or

zero. This inequality constraint acts like a hard wall for charge degrees of freedom.

When using continuous dynamics, a softer repulsion is desirable and so we set

µ

i

= −

k

B

¯

T

q

Q

µ

exp

q

v

i

Q

µ

, (13)

which acts as an exponential barrier with Q

µ

= 0.01 |e| being a stiﬀness parameter.

To summarize, when charged species approach the electrode, the diﬀuse charges

in the electrode adjust to minimize the energy, or to equilibrate the chemical po-

tential φ

i

for valence charge at each metal atom. To this point the discussion has

focused only on charge-charge interactions that represent gross pola r ization of the

metal conduction-band electrons. In addition to this, electron exchange and corre-

lation eﬀects, manifested at short range as a Pauli repulsion and at long range as

a disp ersion attraction, must be included to properly represent the solution-metal

interaction. We therefore incorporate pairwise van der Waals potentials ﬁtted from

quantum-mechanical calculations. That procedure is discussed in t he next section.

Validation of the model is given in section 4 and model predictions are given in

section 5. Nevertheless, for eﬃcient use of space some of the ﬁgures include at the

same time the ﬁt, validation, and predictions of ECD.

3. ECD parameter ﬁt to Cu

10

ab initio data

We have applied the ECD method to a copper (111) electrode surface. In this

section we report t he van der Waals pa rameters f or water, chloride, and sodium ion

interacting with a copper surface, in order to supplement the Coulombic interactions

from electrode charge dynamics.

It should be clearly understood that in the ﬁtting of the van der Waals po t entials,

Coulombic ECD interactions were not ﬁt. That is, γ and q

c

are characteristic of the

metal and not a function of the approaching species. γ and q

c

for Cu were chosen

to be 0.816

˚

A

−1

and 1 |e| respectively, based on the following rationales. We chose

q

c

= 1 |e| as this results in q

v

i

= −1 | e|, on average, corresponding to 1 conduction

electron per copper atom. If the electrode, a face-centered-cubic copper crystal with

lattice constant a

o

= 3.61496

˚

A, is modeled as a free-electron gas, the Fermi wave

vector k

F

is equal to 1.36

˚

A

−1

. At the Fermi level the pro bability that an available

340

Simulating an electrochemical interface

electron energy state is occupied is one-half. If we transform the ECD diﬀuse charge

density ( equation 1) to Fourier space and equate the probability that a given wave

vector is occupied t o the magnitude of the Fourier coeﬃcients, we can relate γ

to k

F

according to 1/2 = exp[−k

2

F

/(4γ

2

)]. The resulting value of γ appears to be

reasonable: the overall valence charge density undulates smoothly over the electrode

and yet large local charge variations ar e po ssible due to the fact that the diﬀuse

charge on each copper is conﬁned mostly to the volume within the Weigner-Seitz

radius (1.41

˚

A) [19].

Ab initio scans (energy vs. distance) of molecular and ionic interactions with

copper clusters having an exposed (111) plane have been performed to obtain accu-

rate Cu-species interactions. The calculations were performed with GAUSSIAN98

[20] at the MP2 level of theory for a ten-copper-atom (Cu

10

) cluster interacting re-

spectively with sodium cation, chloride anion, and water. Details can be found in

reference [21] on the water-copper scans as well as the split-valence basis set used

for copper for all the calculations here. The basis set 6 −31 + G* was used fo r both

sodium and chloride. On-top, bridge, and hollow routes [21] were probed in the case

of chloride, whereas only the on-top site was calculated with sodium. In all of the

ab in i tio results presented here the counter-poise (CP) correction is not included

because the basis-set-superposition error (BSSE) is large, particularly for chloride,

and the CP method may overcorrect BSSE in the case of metal clusters.

Three parameters were regressed fo r each site (H, O, Cl

−

, and Na

+

) interacting

with copper atoms by ﬁtting the modiﬁed-Morse potential,

U(r) = −

1 − {1 − exp [−A(r − r

∗

)]}

2

,

to the diﬀerence between the ab initio and ECD Coulombic interactions. The SPC/E

water geometry and charges [22] are used for ECD calculations and simulations.

The resultant modiﬁed-Morse parameters and the atomic partial charges are given

in table 1.

Table 1. Interaction parameters with Cu metal.

O H Cl

−

Na

+

, kJ/mol 5.000 0.8430 29.42 10.98

A,

˚

A

−1

1.350 1.303 1.409 0.9872

r

∗

,

˚

A 2.890 3.302 2.767 3.741

q

site

, |e| –0.8476 0.4238 –1.000 1.000

Shown in ﬁgure 2 are the ab initio data points and the ECD results for Cl

−

approaching a Cu

10

cluster over the on-top, bridge, and hollow sites. The sum of

the ECD Coulombic and ﬁtted van der Waals potentials agree very well with the ab

initio results in both the repulsive and attractive regions (such an agreement is not

guaranteed even with three adjustable parameters). The agreement is also good f or

the calculations of Na

+

approaching the on-top site of the Cu

10

cluster.

341

C.G.Guymon et al.

Figure 2. ECD and ab initio potential energy as a fun ction of distance from the

Cu

10

(111) plane for Cl

−

.

-50

0

U

, kJ/mol

54321

z

, Å

-50

0

-50

0

Cu

10

ECD

Cu

10

ab initio

Cu

18

ECD

Cu

18

ab initio

Cu

160

ECD

Cu

430

ECD

(a) hydrogens up, top

(b) hydrogens down, top

(c) hydrogens parallel, hollow

Figure 3. Interaction energy of water with copper clusters of diﬀering size for

three unique water appr oaches. Distances are measured from the oxygen normal

to the copper centers in the exposed (111) plane.

Figure 3 shows the sum of Coulombic and ﬁtted van der Waals potentials versus

the ab initio data for three orientations of wat er approaching the Cu

10

cluster.

Nine unique routes were calculated, but only three representative routes are shown.

342

Simulating an electrochemical interface

Figure 3 (a), (b), and (c) represent the average, best, and worst ﬁts, respectively, of

our model to the water-Cu

10

potential scans. (Also shown are validation results with

Cu

18

clusters and predicted results for larger clusters; these will be discussed shortly.)

We suspect a source of the diﬃculty for ECD here is our use of the r ig id (with respect

to geometry and dipole moment) SPC/E model of water, for the sake of simplicity.

While the ﬁts are not as good with water as with the io ns, we still conclude that

ECD combined with a van der Waals potential satisfactorily reproduces the ab initio

data.

Of the nine routes (hydrogens up, down, and parallel approaching the on-top,

bridge, and hollow sites), the hydrogens parallel over the on-top site are t he most

energetically favorable as predicted by the ab initio uncorrected MP2 results [21].

The ECD and van der Waals ﬁt gives the most favorable route as the hydrogens

parallel over the bridge site. Both of these results agree with recent DFT calculations

on platinum [23] and silver [24] that show the most favorable water orientation to

be the hydrogens parallel to the surface. Authors of both studies state that the

high-energy p atomic orbital on the oxygen originating from the lone pairs interacts

most strongly with the metal surface. The p orbital’s highest density is lateral to

the plane containing the oxygen and two hydrogens.

4. ECD validation

The ECD model for copper correlates reasonably well the quantum-mechanical

interaction po tentials of Na

+

, Cl

−

, and water with a small Cu

10

cluster. In essence

we use the ECD method and associated van der Waals potentials as an extrapolation

scheme to obtain inexpensive interactions between species and large metal surfaces.

We seek to validate and test the ECD copper-surface model by comparing its pre-

dictions to image-charge results at long range and to larger cluster calculations.

Ab in itio data have also been obtained for a Cu

18

cluster [25 ]. We compare the

ab initio and ECD predicted energies of chloride, sodium, and wat er near a Cu

18

cluster. We also show ab initio and ECD results of a hydrated sodium atom near

the Cu

10

and Cu

18

(111) cluster surfaces. ECD reproduces the image-charge result

at long range and successfully predicts the a b initio energies at the larger cluster

size.

4.1. ECD and image charges

As stated previously, the simple image-charge potential correctly reproduces the

energy of a charge interacting with a metal surface at long range. It is not possible

to use the ECD method for a true semi-inﬁnite metal. Therefore we compare the

ECD model result with the theoretical ( image-charge) result for a point charge

approaching a sphere of inﬁnite dielectric constant.

In the ECD calculation, a negative and a positive unit charge are separately

brought toward a neutral spherical cluster of simulated copper atoms. The distance

from the central copper in the spherical cluster to t he outer-most copper was less

343

C.G.Guymon et al.

than 7

˚

A with 135 coppers in the cluster. The nearest neighbor distances in the

cluster correspond to the bulk fcc lattice, namely a

o

/

√

2. O n the other hand, the

energy of interaction of a point charge q with a neutral sphere of inﬁnite dielectric

is

U(r) = −

q

2

a

3

2r

2

(r

2

− a

2

)

, (14)

where a is the sphere radius and r is the distance to the external charge from the

center o f the sphere.

-100

-50

0

U

, kJ/mol

20151050

r - a

, Å

image

pos. charge ECD

neg. charge ECD

Figure 4. Comparison of ECD with image-charge method for a spherical copper

cluster of eﬀective radius a = 7.27

˚

A. In s et shows negative charge at position

r − a = 5.73

˚

A with cluster atoms shaded according to net charge: black is

positive, white is n egative, and gray is neutral.

Figure 4 shows that the predicted ECD Coulombic energy agrees with the image-

charge result at r −a distances greater than about 5

˚

A. Radius a was adjusted until

the ECD and image-charge results agreed at the furthest data point from the sphere,

resulting in a = 7.27

˚

A. This value is consistent with a cutoﬀ radius that could

generate the corresponding cluster of 135 discrete ato ms. Shown in the ﬁgure is the

fact that the ECD routine predicts unique energies for the positive and negative

unit charges at close range. Inset in the ﬁgure is a representation of the copper

cluster when a negative charge is at distance r −a = 5.73

˚

A. Each copper is shaded

according to its net charge, q

c

+ q

v

i

, where black is positive, white negative, and gray

is neutral.

4.2. Cluster-size results with ECD

Figures 3 and 5 show ( among other things) the ab initio results for a Cu

18

cluster

[25] with water, Cl

−

, and Na

+

. Due to the large computational expense, only a few

points could be calculated. The same level of theory and basis sets as with the Cu

10

cluster are used. Distances are measured normal from the surface Cu nuclei to the

344

Simulating an electrochemical interface

-300

-200

-100

0

U

, kJ/mol

54321

z

, Å

-300

-200

-100

0

Cu

10

ECD

Cu

10

ab initio

Cu

18

ECD

Cu

18

ab initio

Cu

160

ECD

Cu

430

ECD

(a) Cl¯

(b) Na

+

Figure 5. Interaction energy of chloride and sodium with copper clusters of dif-

ferent size. The ions approach the on-top site of the (111) plane in all cases.

oxygen, chloride, or sodium. The corresponding ECD predictions compare well with

the ab initio results in terms of r elative displacement o f the potential from the Cu

10

results. In particular, there is excellent agreement between the Na

+

and Cl

−

–Cu

18

ab initio and ECD result. Note that this is a purely predictive result without a ny

re-parameterization. We conclude that the ECD model correctly predicts cluster size

dependence.

4.3. ECD and ab initio results for hydrated-sodium clusters near copper

We have also compared ECD model predictions t o ab initio results for hydrated

Na

+

near a Cu

10

and Cu

18

(111) surface, shown in ﬁgure 6. Calculations were per-

formed for sodium only due to computational ease and because the BSSE is much

smaller than with chloride.

For the Cu

10

cluster, sodium is hydrated with one to ﬁve waters. In the case

of Cu

18

, one o r two waters surround the sodium. A. Karttunen and T. Pakkanen

calculated the optimum geometry of the system at the Hartree-Fock level and the

ﬁnal energy at the MP2 level of theory [25]. In the geometry optimization, all waters

were allowed to move freely while Na

+

motion was restricted to the direction nor mal

to the surface. Na

+

was placed directly over an on-top and a hollow site f or the

Cu

10

and Cu

18

clusters respectively. The optimized hydration energy of the isolated

sodium-water clusters were also computed. The energies given in ﬁgure 6 are the

energies of the hydrated ion near a copper cluster, relative to an isolated copper

cluster and isolated sodium-wat er cluster.

345

C.G.Guymon et al.

-250

-200

-150

-100

-50

0

U

, kJ/mol

543210

number of hydrating waters

Cu

10

ab initio

Cu

10

ECD

Cu

18

ab initio

Cu

18

ECD

Figure 6. MP2 ab initio optimum and ECD energies as a function of the number

of waters near a sodium ion. Geometries were optimized using Hartree-Fock level

of theory in the ab initio case. The sodium is above the on-top and hollow site

for Cu

10

and Cu

18

respectively.

As shown in the ﬁgure, the same calculations were performed using the ECD

method, but the oxygen and sodium positions were taken from the ab initio geometry

optimization. We used SPC/E [22] water in the ECD calculations and located the

SPC/E hydrogens as close as possible to the ab initio hydrogens. For the energy of

Na

+

interacting with the water we used the para meters of sodium cation found in

reference [26].

Although not shown in ﬁgure 6, the sodium hydration energy of the ECD and the

uncorrected ab initio calculations agree within 4% in all cases. While the energies in

the presence of t he copper surface for the two techniques compare less well, the ECD

model correctly predicts the energetic trend with increasing water. The qualitative

change in t he ECD curve when 4 and 5 waters are present can be ascribed to the

fact that 1 and 2 waters, respectively, are located between the sodium and the metal

surface. The ab initio geometries show these waters to be distorted (HOH bisector

angle is ∼ 105

◦

) from the geometry of the other hydrating waters, which more

closely resemble SPC/E water. The fact that the ECD model does not reproduce

this geometry distortion, in addition to discrepancies in the wat er-copper interaction

are the sources of the inaccuracies in ﬁgure 6.

5. ECD predictions

In the previous section we showed that ECD does reasonably well in predicting

the energies for situations in which it was not parametrized. Here we discuss the

ECD predictions of Na

+

, Cl

−

, and water on the (111) surface of copper clusters

with 1 60 and 430 atoms-cluster sizes that are far from accessible to fully ab initio

calculations. We also show molecular dynamics simulation results using ECD to

simulate pure water and an electrolyte solution between two copper electrodes.

346

Simulating an electrochemical interface

5.1. ECD large-cluster predictions

Also shown in ﬁgures 3 and 5 are the ECD results with clusters of size 160 and

430 atoms for Na

+

, Cl

−

, and water. These large clusters are circular-disk in shape

and have two metal layers with the (1 11) plane exposed. At these sizes, t he interac-

tion energy at the energy minimum is about 60% Coulombic for the ions and about

30% Coulombic for water (hydrogens up, on-top), with the remainder of the energy

due to van der Waals site-site interactions. Note that the ECD interaction energy

for water converges at a cluster size around 160 atoms while a much larger cluster

size is necessary for the ions. ECD predicts the most favorable water orientation

to be hydrogens parallel for the Cu

10

cluster, but for Cu

430

there is slightly greater

preference for hydrogens up with oxygen on the bridge site. This prediction is consis-

tent with the conventional picture that water chemisorbs to catalytic metal surfaces

preferentially through the oxygen. Interestingly, this shift in behavior with cluster

size, as predicted by ECD, could help resolve discrepancies between experimental

results for extended surfaces and small-cluster ab initio calculations.

5.2. MD simulation details and results

The ECD model developed for the copper interface was employed in a series of

preliminary molecular dynamics simulations to gain insight into the adsorbed water

structure on the (111) copper surface as well a s the surface so dium and chloride

densities.

Figure 7. 3D representation of simulation cell with metal atoms on both sides

and a single water molecule, shown to indicate scale, located in the middle of the

ﬂuid region.

5.2.1. Simulation details

We simulated a sodium-chloride aqueous solution between two copper electrodes

at various voltages and two diﬀerent ionic concentrations. The copper electrodes

were each 5 layers thick, with the (111) plane exposed. SPC/E water is used a s the

solvent and the ion-ion and ion-water potentials are given in reference [26]. The cell

dimensions are 97.7 , 20.45, and 22.14

˚

A in the x, y, and z directions respectively,

for simulations with no ions and 1200 water molecules. Simulations with ions were

347

C.G.Guymon et al.

conducted in a cell of dimensions 101 .5 , 28.1 2, 30.99

˚

A with 80 sodium cations, 80

chloride anions, and 23 40 water molecules. Metal atomic positions were ﬁxed during

the simulations, with 800 and 1540 total copper atoms fo r the smaller and larger

cell, respectively. Shown in ﬁgure 7 is the larger cell geometry with a representative

water molecule shown in the middle of the cell.

In fact, because of periodic boundary conditions, the two electrodes are joined

across the boundary and form one metal slab. The oﬀset potential φ

set

in equation 4

allows us to set the voltage of the two electrode halves at diﬀerent relative va lues.

In every simulation t he total charge on the combined electrode is zero. The total

charge in t he ﬂuid region is also zero. The charges in the electrodes adjust so as to

maintain the voltage diﬀerence, as well as to respond to local ﬁeld changes due to

the liquid. Since t hey constitute constant-potential surfaces, the electrodes behave

as a Faraday cage. That is, they shield the usual interaction in a 3-D Ewald sum

between perio dic images of the unit cell in the direction normal to the electrodes.

Eﬀectively, we obtain a 2-D Ewald sum at no a dditional cost. Here, we use the P3M

method due to Hockney and Eastwood [27] to perform the Ewald reciprocal sum.

Each simulation run was equilibrated for 50,000 time steps. Data were collected

after equilibration for 200,000 or 400 ps for each run unless otherwise stated. With

no ions present, 5 runs were simulated with 0, 1 and a 5 volt potential diﬀerence

between the perfectly polarizable electrodes (i.e., no reactions are permitted) for

a total of 15 runs. For simulations with ions, 6 runs were simulated with 0, 0.5

and a 1 volt potential diﬀerence between the electrodes for a total of 18 runs. The

density of the water in the middle of the solution-ﬁlled region is 54 M for electrolyte

simulations and 55 M for neat water. Ion concentrations in the middle of the cell

are around 1.7 M.

-80

-60

-40

-20

0

20

40

60

80

layer charge,

µ

C/cm

2

108642

layer

∆φ

= 0 V

∆φ

= 5 V

Figure 8. Average charge on the copper layers for two diﬀerent cases of ∆φ

between electrode halves. Layers 1 and 10 are in contact with the surrounding

neat liquid water. Error bars show the average of the standard deviation of the

charges, with the wider error-bar caps for case ∆φ = 0 V.

348

Simulating an electrochemical interface

5.2.2. Simulation results

Shown in ﬁgure 8 is a graphical representation of the average charge on the

electrode atoms as a function of the electrode layer when there is a 0 and 5 vo lt

electrostatic potential diﬀerence between electro des for water only in the liquid.

Notice the charge dipole in the metal center that is required to generate the potential

diﬀerence between the two electrode halves. This charge dipole found at the location

of the step change in potential is in agreement with Poisson’s equation (∇

2

φ =

−ρ/). When ions are present in the solution with a zero po tential drop across the

ﬂuid, the charge on the layer is equivalent to the charge without ions but the standard

deviation of that charge is twice as large. The large average standard deviations in

copper charges for the surface layers indicate that these layers respond most strongly

to the ﬂuctuating ﬁelds due to the ﬂuid, as we expect.

20

15

10

5

0

density, mol/L

806040200

x, Å

40

20

250

200

150

100

50

0

Cl

_

Na

+

oxygen

hydrogen

(a)

(b)

Figure 9. Average species density proﬁles for (a) water sites and (b) ions versus

cell position for zero potential diﬀerence between the electrodes.

Shown in ﬁgure 9 is the oxygen and hydrogen density proﬁles for water with

ions present and with a uniform electrostatic potential across the electrodes. When

a 1-volt potential diﬀerence is placed across the ﬂuid, the resulting changes in the

local water density are small and cannot be visibly distinguished. However, the

addition of ions to the solution does generate more noticeable diﬀerences in the

density of the water nearest the surface as the contact-adsorbed ions displace some

of the adsorbed water molecules. As for the ion distributions, notable in ﬁgure 9 are

strongly adsorbed chloride and sodium ions on the surface, in nearly equal numbers.

There is a second peak for chloride next to the ﬁrst, however, with no corresponding

sodium peak. This second peak is responsible for the small amount of excess chloride

on the surface of the charge-neutral electrode. As with the water density proﬁle, t he

ion surface density changes little with the imposed voltage between electrodes; most

of the change occurs in the diﬀuse part of the double layer.

349

C.G.Guymon et al.

7x10

-3

6

5

4

3

2

1

0

probability, arb. units

-1.0 -0.5

0.0 0.5 1.0

cosine

θ

without ions

with ions

Figure 10. Probability of the cosine of the angle between the surface normal and

the surface adsorbed (within 3.5

˚

A of the surface) water dipole moment.

Figure 10 gives the distributions of water dipole orientation in the surface-

adsorbed layer (within 3.5

˚

A of the surface) as indicated by the cosine of the angle

between the dipole vector and the surface normal. The electrode pot ential diﬀer-

ence is zero. The waters closest to the surface are strongly oriented with the dipole

moment parallel to the surface, with a slight hydrogen-up tendency. Although not

shown, this distribution is virtually unchanged under a 5 V range of potential diﬀer-

ence between the electrodes. To signiﬁcantly orient the dipole of the water t owards

the surface wo uld require a very strong electrode charge. As a consequence, the ECD

model predicts a much larger barrier to ﬂipping the surface-wat er dipole than does

prior wo r k with a nonpolarizable charged surface [1,5]. However, the presence o f

ions in solution is strong enough to create a noticeable disruption to the neat-water

dipole distribution.

The large surface density for water and narrow dipole distribution indicate that

the water is highly structured near the surface. To gain some insight into this phe-

nomenon, we rendered a snapshot of the ﬁrst and second layers of wa t er next to

the copper surface within a neat water solution. Figure 11 shows a very ordered

two-dimensional rhombus ice-like structure in the ﬁrst layer. Although other inves-

tigators[1,8,28] do not see such a pronounced ordering of the surface layer, X-ray

spectroscopy and DFT results [29] suggest that the surface adsorbed water forms

“ﬂat ice”. In our model’s ice-like layer, the oxygen-oxygen nearest neighbor distance

is about 2% smaller t han the bulk liquid value of 2.79

˚

A. While there is a mismatch

between the surface-water lattice and the underlying copper lattice, it appears that

the waters somewhat prefer to occupy the bridge and hollow sites. This energetic

preference is in agr eement with the results obtained by Price and Halley [16].

Figure 12 is a corresponding snapshot with ions present. Only one layer of ad-

sorbed water and ions is shown. Sodium cations are shown in black and chloride

anions are shown in white. As expected, the ions disrupt the regular array of the

water near the surface. This is consistent with the ion-induced changes in ﬁgure 10.

Somewhat unexpectedly, both ions readily contact the adsorb.

350

Simulating an electrochemical interface

Figure 11. Snapshot of the ﬁrst two water layers on the copper surface for a

simulation of neat water. The darker colored waters indicate the layer closest to

the surface.

Figure 12. Snapshot of the water orientation on the copper surface with ions

present. Chloride is shown in white and sodium in black.

Shown in ﬁgure 13 are the electrostatic potentials as a function of position, with

and without ions present when the electrodes are the same potential. The convex

351

C.G.Guymon et al.

shape o f the electrostatic potential proﬁle in the middle of the cell indicates that

there is an excess of sodium ion away from the surface, or a net positive ionic charge

in the middle of the ﬂuid (∇

2

φ = −ρ/). Apparently the simulation cell is not lar ge

enough to contain the entire diﬀuse double layer. This is in stark contrast to earlier

electrolyte simulations with static surface charges in that the distribution of ions

and solvent generates a ﬂat potential proﬁle in the liquid within several Angstroms

of the surface [5].

10

8

6

4

2

0

electrostatic potential, V

806040200

x, Å

NaCl soln.

water only

Figure 13. Electrostatic potential in the ﬂuid for NaCl solution (solid line) and

water only (dotted line). The two systems have diﬀerent cell lengths as indicated

at the right of the plot.

The reader may notice that the average ion densities in ﬁgure 9 and the elec-

trostatic potentials in ﬁgure 13 are not symmetric. This is a manifestation of slow

relaxation of t he double layer that requires longer simulation times than we used in

order to generate the expected symmetry in the average properties. Aga in, the hi-

ghly responsive charges in the electrode for the ECD model tend to stabilize charge

separation in the double layer to a much greater degree than has been observed

in prior simulation work. For example, with an electrolyte between electrodes at

the same potential, the average charge-density diﬀerence between the two electrode

halves averaged to +0.6 µC/cm

2

over a 2.4-ns simulation.

Since the diﬀuse region of the double layer is longer than half the length of

the ﬂuid-ﬁlled region, diﬀerential-capacitance calculations of the interface a r e no t

reliable. However, we did calculate a cell integral capacitance of 13 and 12 µF/cm

2

for the solutions with and without ions, respectively.

352

Simulating an electrochemical interface

6. Conclusion

We have developed a simple and robust method to treat the charge mobility in

a metal for use in molecular dynamics simulations. Electrode charge dynamics acts

as a bridge between small-scale ab initio metal-cluster calculations and large-scale

molecular dynamics simulations of metal surfaces of arbitrary geometry. Our use

of ECD to simulate a water-NaCl solution near copper electrodes has revealed be-

havior with, in some instances, marked diﬀerences from prior simulation work. Our

simulations show the presence of a dense 2D ice-like rhombus structure of water on

the surface that is relatively impervious to perturbation by typical electrode poten-

tials/charges. The model also predicts that sodium and chloride ions a re strongly

adsorbed to the copper surface at both positive and negative electrode charge, but

that there is generally an excess of chloride associated with the surface. Based on the

potential proﬁle in the cell, it appears that the diﬀuse part of the double layer ex-

tends beyond 40

˚

A. Further work is required to assess the veracity of the simulations

with regard to double-layer properties and to reﬁne the intermolecular potentials.

Nevertheless, we believe the approach described here for treating metal polarizability

will permit more accurate representations of the electrochemical interface.

7. Acknowledgements

We want to thank Dr. Tapani Pakkanen and his students at the University of

Joensuu for their work in performing much of the ab initio calculations presented

here. We’d also like to acknowledge ﬁnancial support from the NSF , grant number

CTS–0215786 as well as support from Brigham Yo ung University.

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Симуляція електрохімічної поверхні з

використанням зарядової динаміки

К.Дж.Гуймон, Р.Л.Роулей, Дж.Н.Гарб, Д.Р.В іхлер

Відділення хімічної інженерії, Прово, США

Отримано 29 листопада 2004 р.

Ми представляємо простий класичний метод для трактування за-

рядової мобільності у металах, які межують з рідкими розчинами.

Метод, відомий як електродна зарядова динаміка, ефективно

заповнює п рогалину між ab initio розрахунками на малих металічних

кластерах і велико-масштабними симуляціями металічних повер-

хонь з довільною геометрією.Ми отримали модельні параметри для

металічної (111) поверхні міді, використовуючи квантово-механічні

розрахунки в исокого порядку на 10-атомному мідному кластері.

Модель була перевірена шляхом порівняння з класичними резуль-

татами по методу відображень, а також з ab initio результатами для

18-атомного м і дного кластера. Модель використана в молекуляр-

но динамічних розрахунках для стр уктури флюїдної поверхні чистої

води і водного розчину NaCl. Ми побачили, що вода організовується

у двовимірний льодопо дібний шар на поверхні і що обидва, Na

+

i

Cl

−

, іони є сильно прив’язані до міді. Коли металічний електрод за-

ряджати, то основна реакція електроліту проявляється у дифузивній

частині подвійного шару.

Ключові слова: симуляція, подвійний шар, молекулярна динаміка,

ab initio потенціали, (111) поверхня міді, вода

PACS: 61.20.Qg, 61.20.Ja

356