Simulating an electrochemical interface using charge dynamics

Abstract

We present a simple classical method for treating charge mobility in metals adjacent to liquid solutions. The method, known as electrode charge dynamics, effectively bridges the computational gap between ab initio calculations on small metal clusters and large-scale simulations of metal surfaces with arbitrary geometry. We have obtained model parameters for a copper (111) metal surface using high-level quantum-mechanical calculations 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 solution. 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 copper. When charging the metal electrode, most of the electrolyte response occurs in the diffuse part of the double layer.

Full text

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