Editor’s Choice
Electrophoresis of neutral oil in water
Volker Knecht a,*, Zachary A. Levine b
, P. Thomas Vernier b
a
Max Planck Institute of Colloids and Interfaces, Science Park Golm, 14424 Potsdam, Germany
b
University of Southern California, Marina del Rey, USA
a r t i c l e i n f o
Article history:
Received 27 May 2010
Accepted 1 July 2010
Available online 12 August 2010
Keywords:
Electrophoresis
Molecular dynamics
MD
Computer simulation
Water
Oil
Interface
Hydronium
Hydroxide
Isoelectric point
a b s t r a c t
Negative electrophoretic mobilities of oil in water are widely interpreted in terms of adsorption of
hydroxide leading to negative surface charge. Challenging this traditional view, an increasing body of evidence
suggests surface depletion of hydroxide and surface accumulation of hydronium leading to a positive
surface charge. We present results from molecular dynamics (MD) simulations showing
electrophoretic mobilities of oil in water with the same sign and magnitude as in experiment but in
the absence of ions. The underlying mechanism involves interfacial roughness leading to gradients in
dielectric permittivity in field direction and, thus, local elevation of the applied electric field. Although
all molecules have zero net charge, their partial charges are distributed non-uniformly such that oil
exhibits negative and water positive excess charge in regions of high field intensities; this induces a
net force on the oil or the water against or in field direction, respectively. Our results indicate that deducing
net charges from electrophoretic mobilities as widely done can be misleading. Our findings suggest
that pH dependent electrophoretic mobilities in experiment being negative above and positive below
pH 2.5 arise from a competition between the negative mobility of the ion-free interface and the positive
mobility from adsorbed hydronium ions.
Ó 2010 Elsevier Inc. All rights reserved.
1. Introduction
It has been known for two centuries [1] that a suspended
particle exposed to a homogeneous static electric field may start
to migrate, an effect denoted as electrophoresis. The drift velocity
is proportional to the applied field. In general, the electrophoretic
mobility, i.e., the migration rate normalized by the field intensity,
is assumed to be proportional to the net charge on the particle.
Based on this assumption, electrophoresis experiments are widely
used to determine the charge of colloidal particles [2]. Oil droplets
show electrophoretic mobilities that are negative above and
positive below pH 2.5, denoted as isoelectric point (iep) [3,4]. This
behavior is commonly explained in terms of adsorption of hydroxide
(OHÀ
) or hydronium (H3O+
) giving rise to a negative or positive
surface charge depending on the pH [5,6]. Hydroxide adsorption at
water/hydrophobe interfaces has been also inferred from measurements
of disjoining pressures for thin water films as a measure for
the interaction between the two interfaces of the film [5,7]. Furthermore,
strong surface activity for hydroxide has been invoked
to explain titration experiments on oil-in-water emulsions. Here
it is observed that upon increasing the interfacial area by homogenization
of the emulsion, a base needs to be added to maintain
the apparent pH in the water.
The titration experiments rely on the measurement of the pH of
an inhomogeneous oil-in-water emulsion. From the relatively large
oil fraction used in these experiments, 2 vol.%, possible interference
of the oil with the pH sensor counterfeiting the measurements
cannot be excluded. In contrast, substantially lower oil
fractions (e.g., 0.05 vol.% in studies by Marinova et al. [6]) are used
in electrophoresis experiments, which thus may not suffer from
this problem. The disjoining pressure measurements interpreted
in terms of hydroxide adsorption require the presence of surfactants.
Therefore, those measurements do not probe the bare but
only the surfactant-covered aqueous interface.
Indeed, whereas titration, disjoining pressure, and electrophoresis
studies are widely interpreted in terms of higher interfacial
affinity for hydroxide than for hydronium, other investigations
suggest that hydroxide might in fact be less surface active than
hydronium if not even repelled from interfaces. Surface repulsion
of hydroxide is suggested from surface tension measurements
[8,9], second harmonic generation spectroscopy [10], and synchrotron
photoelectron spectroscopy of NaOH in aqueous microjets [9].
Vibrational sum frequency generation spectra indicate strong surface
enhancement of hydronium for strong acid solutions but no
sign of surface enhancement of hydroxide for equally strong base
solutions [11]. Classical MD simulations using polarizable force
fields indicate that water/hydrophobe interfaces repel hydroxide
and attract hydronium [11–13]. If hydronium exhibits higher or
lower surface affinity than hydroxide and if the hydroxide
0021-9797/$ - see front matter Ó 2010 Elsevier Inc. All rights reserved.
doi:10.1016/j.jcis.2010.07.002
* Corresponding author. Fax: +49 331 567 9612.
E-mail address: vknecht@mpikg.mpg.de (V. Knecht).
Journal of Colloid and Interface Science 352 (2010) 223–231
Contents lists available at ScienceDirect
Journal of Colloid and Interface Science
www.elsevier.com/locate/jcis
concentration at interfaces is enhanced or reduced compared to
the bulk water is a matter of ongoing debate [9,10,14–20].
Recent MD simulations showed negative electrophoretic mobilities
of oil in water although ions were absent but the underlying
mechanism remained unclear [21]. Later this effect was observed
to depend on the treatment of the dispersion interactions using
Lennard-Jones (LJ) potentials. Electrophoretic drift was observed
when the LJ interactions were truncated abruptly such that the
force was discontinuous at the cut-off distance and when a cutoff
distance of 1 nm was chosen. When the LJ interactions were
modified such that the force smoothly reached zero at the cut-off
boundary or the cut-off distance was increased to 2 nm, the electrophoretic
drift vanished [22].
The simulations in [21,22] used a simplified description where
hydrogens of CHn groups were treated implicitly using compound
atoms and molecules were non-polarizable. Here we show that for
a more detailed model in which not only the water but also CHn
groups are described in full atomic detail and molecules are polarizable,
uncharged oil in water may show electrophoretic drift even for
continuous LJ forces. The electrophoretic mobility of the oil is of the
same sign and size as the mobilities of oil droplets in distilled water
measured experimentally. The mechanism underlying the electrophoretic
drift in our simulations is revealed. In particular, we also
show why this effect is not observed for the simplified description.
Our findings suggest that electrophoretic mobility does not always
reflect the charge of an oil droplet. Our results render the negative
electrophoretic mobilities of oil droplets in water consistent with
a small positive surface charge indicated from the results from the
spectroscopic experiments and the MD simulations.
2. Methods and theory
2.1. Simulation setup
The system mainly considered in this work is shown in Fig. 1
and specified as system DEC in Table 1. It consisted of a decane slab
in water parallel to the xy plane under periodic boundary conditions.
An initial configuration for this system as well as the force
field parameters and the simulation protocol as used in [13] were
provided by Vacha. Hence, the initial configuration of a larger slab
denoted as system DECL was modeled as a 2 Â 2 array of the configuration
of DEC. In addition, a cubic box of bulk water or bulk
water containing a single methane denoted as system WAT or
MET, respectively, in Table 1, were simulated. System DEC was
simulated at equilibrium and exposed to external fields in the
range E0 = 0.05–0.5 V/nm in x direction. Here, for system DEC at
equilibrium and for each out of four different field intensities, a
single 10 ns simulation was conducted. For this system, in addition,
thirty 1.6 ns simulations at equilibrium and different field
intensities were started from the same initial configuration but different
sets of initial velocities. Likewise, for system DECL, thirty
200 ps simulations at E0 = 0.5 V/nm were started from the same
initial configuration but different sets of initial velocities. System
WAT and MET were each simulated for 4 ns at equilibrium.
The decane was described using a polarizable potential derived
from the general Amber force field (GAFF) [23]. The force field for
the methane (CH4) molecule was derived from that for a CH3 group
by adapting the partial charges for the hydrogens keeping them as
close as possible to the values for CH3 but yielding a zero net
charge for CH4. The water was described using the polarizable
POL3 water model [24,25] in conjunction with the SETTLE algorithm
to keep the O–H and H–H distances at ideal values. Polarizability
was simulated by using the shell model of Dick and
Overhauser [26]. Here, a shell particle representing the electronic
degrees of freedom is attached to a nucleus by a spring and the
potential energy is minimized with respect to the shell position
every step. Lennard Jones (LJ) interactions were treated unchanged
for interatomic distances r with 0 < r < r1 = 0.7 nm; for r1 < r
< rc = 1 nm, a third degree polynomial S(r) denoted as switch function
was added to the force such that the modified force and its
derivative were continuous at r = r1 and r = rc [27].
The neighbor list for non-bonded interactions considering all
pairs of atoms separated by less than 1.1 nm was updated every
10 steps. Full electrostatic interactions were considered using the
particle mesh Ewald (PME) technique [28] using tinfoil boundary
conditions [29] with (i) a cut-off distance of 1.1 nm in direct space
and (ii) a 0.12 nm grid spacing and a 4th order polynomial for
interpolation for the reciprocal sum. Decane and water were separately
coupled to an external temperature bath of 300 K using a
Berendsen thermostat [30] with a relaxation time of 0.1 ps. For systems
DEC or DECL the pressure in xy direction and for systems
WAT or MET the pressure in all directions were coupled to 1 bar
using a Berendsen barostat [30] with a coupling constant of 1 ps.
All simulations were performed using GROMACS [31], version
3.3.1. The simulations required 40500 CPU hours on AMD Opteron
3.0 GHz dual processor/dual core nodes. Configurations were saved
every 5 ps for further analysis.
For system DECL at an external field of 0.5 V/nm, thirty 200 ps
simulations were started from the same initial configuration but
different sets of initial velocities using modified LJ interactions
with r1 = 1.7 nm and rc = 2 nm and modified electrostatic interactions
with a cut-off distance of 2.1 nm in direct space. For comparison
to previous studies of oil slabs in water, an additional 100 ns
simulation at an external field of 0.5 V/nm was carried out using
a non-polarizable force field. Here, the (non-polar) hydrogen atoms
of the decane molecules were described using united atoms using
the GROMOS-87 force field [32] and the water was treated using
the three-site simple point charge (SPC) model [33]. All other conditions
were kept identical to the other setup.
Fig. 1. The system simulated was a decane slab in water in the absence of ions. The
system was described with an all-atom polarizable force field and LJ forces were
smoothed at the cut-off distance. An electric field applied parallel to the water/
decane interface induces a tangential movement between the phases; the decane
phase migrates against and the water in field direction. The water is shown in white
and the decane in gray. The figure was prepared using VMD [39].
Table 1
The systems simulated consisted of a decane slab in water of two different sizes
denoted as DEC and DECL, pure bulk water (WAT), and bulk water containing a single
methane molecule (MET). Given are the number of decane (Nsys,a) or water molecules
(Nsys,w), and the initial size of the simulation box, a  b  c.
System Nsys,a Nsys,w a  b  c (nm3
)
DEC 64 723 2.5 Â 2.5 Â 7.3
DECL 256 2892 5.0 Â 5.0 Â 7.3
WAT – 461 2.8 Â 2.8 Â 2.8
MET – 457 2.8 Â 2.8 Â 2.8
224 V. Knecht et al. / Journal of Colloid and Interface Science 352 (2010) 223–231
2.2. Analysis of water/decane system
2.2.1. Electrophoretic motion
Analyses of the systems at equilibrium or at steady states were
performed omitting the initial 100 ps for relaxation. If not stated
otherwise, standard errors were obtained from block averages
dividing the trajectories into four fragments. For system DEC or
DECL, the central observable was the position r = (x,y) of the center
of mass of the decane slab relative to the center of mass of the
water parallel to the interface (xy plane). Here we distinguished
between the position in field direction, x, and normal to the field
direction, y. For each field intensity, migration rates v  Dx/Dt
and corresponding standard errors were determined from the
initial and final configurations of the sampling period of the
thirty 1.6 ns simulations for system DEC and the thirty 200 ps
simulations for system DECL. For migration rates from the 10 ns
simulations of the polarizable all-atom and the non-polarizable
united atom model at E0 = 0.5 V/nm, standard errors were obtained
by dividing the trajectory into four segments. Electrophoretic
mobilities were obtained from fitting the function v = lEEapp to
the data from the thirty 1.6 ns simulations of system DEC, with
the effective electric field Eapp taken equal to the electric field in
the water bulk. The latter was determined from [34]
Eapp ¼
np
ðw À 1Þ0
: ð1Þ
Here, p is the average dipole moment of a water molecule in field
direction in the water phase at least 1 nm away from the Gibbs
dividing surface (position at which the water density equals half
the density in the water bulk). Furthermore, n = 33/nm3
is the bulk
number density of the water molecules and 0 is the permittivity of
vacuum. The value w = 122 for the relative permittivity of POL3
was obtained from the fluctuations of the dipole moment M of a
box of water (system WAT) according to [34]
w ¼ 1 þ
1
30
hM2
i
VkBT
ð2Þ
Here, V is the volume of the box, kB Boltzmann’s constant, T = 300 K
the temperature, and h. . .i denotes an average over the simulation.
The spatial flow pattern was studied by comparing the average
velocity vloc of atoms with the mass density q of the system both
as a function of the position z normal to the interface at an effective
field of 0.2 V/nm. The bin widths chosen for the distance scale
where 0.6 nm for vloc and 0.018 nm for q.
2.2.2. Drive versus friction
To dissect drive and friction governing electrophoretic motion
in our system, the flow of matter normal to the interface is considered
to undergo a discontinuous transition at the interface. If Eapp is
the applied electric field and F the force driving electrophoretic
movement, the quantity Q  F/Eapp has units of a charge and is denoted
as pseudo charge. If Ai is the interfacial area, the quantity
r  Q/Ai is denoted as surface pseudo charge. The friction force is
given by FR = b AiDv where Dv = v/2 and b denotes the friction coefficient
[35]. At steady state, F = FR, yielding
lE ¼ 2r=b: ð3Þ
The friction coefficient was determined from the mobility l  v/F
of the decane slab according to
b ¼
2
lAi
; ð4Þ
where Ai = 2Ab with Ab being the average area of the box parallel to
the interface (xy plane). The mobility was obtained via the Einstein
relation
l ¼
D
kBT
ð5Þ
where D denotes the diffusion constant. The latter was evaluated
from the thermal movement between the decane and the water slab
parallel to the interface in the 10 ns simulations by subtracting the
drift along x as follows. From the quantity ~rðtÞ ¼ ðxðtÞ À vt; yðtÞÞ, the
mean square displacement, defined as
msdðtÞ ¼ h½~rðt0
þ tÞ À ~rðt0
ފ2
it0 ð6Þ
was determined. Here hÁ Á Á it0 denotes an average over the times t
0
.
Error bars for msd(t) give the standard error from estimates from
the simulations at different field strengths. The diffusion constant
is determined from D = msd(t0)/t02d with t0 = 400 ps and d = 2
denoting the dimension of space. The pseudo charge density is
determined from
r ¼
1
Ai
lE
l
ð7Þ
Standard errors for lE, r and b are obtained from the standard errors
for migration rates v and mean square displacements msd(t)
via error propagation.
2.2.3. Equilibrium properties
Equilibrium properties for a water/decane interface were determined
from the 10 ns simulation of system DEC omitting the initial
100 ps for equilibration. As a measure for the linear extension of a
decane molecule, the quantity d = 2rg with rg denoting the radius of
gyration of a decane molecule defined by
rg ¼
P
ikrik2
mi
P
imi
!1
2
ð8Þ
was determined by averaging over all decane molecules. Here, mi
denotes the mass of atom i and ri the position of atom i with respect
to the center of the molecule. Furthermore, the average surface area
of the decane phase was analyzed. Here, the total ‘‘solvent accessible”
surface area (SASA) of a given water/decane configuration, denoted
as As, was determined. The SASA is defined as the area traced
out by the center of a probe sphere representing a water molecule
as it is rolled over respective groups of the solute and was calculated
based on an algorithm by Connolly [36] using the program
g_sas from the GROMACS suite [31]. The SASA per projected interfacial
area, a, was obtained from a = As/Ai. For each configuration,
the number of contacts between decane carbons and water oxygens
considering all carbon–oxygen pairs separated by less than 0.5 nm,
and the respective time average, NCO, were calculated. The number
of contacts per unit area, c0, was obtained from
c0 ¼ NCO=Ai: ð9Þ
The mass densities of water or oil, qw or qa, respectively, were
determined as a function of the position z normal to the interface
choosing a bin width of 0.018 nm and smoothed using a Gaussian
filter with a width of 0.073 nm. Hence, the respective normalized
densities
vi ¼ qi=qi;b; i ¼ w; a; ð10Þ
were evaluated. Here, qi,b denotes the mass densities in the bulk of
the respective phase. The charge density qa(z) of the oil was obtained
from
qaðzÞ ¼
X
i;jziÀzj<Dz=2
qi=AbDz
* +
: ð11Þ
Here,
P
is a sum over respective atoms or shell particles of the oil, zi
denotes the position of atom i normal to the interface, qi is the
V. Knecht et al. / Journal of Colloid and Interface Science 352 (2010) 223–231 225
partial charge of the atom or shell particle i, Dz = 0.0061 nm, and
qa(z) was smoothed using a Gaussian filter of 0.0244 nm width.
2.3. Analysis of methane/water system
For each configuration of the single methane molecule in water
(system MET), the number of contacts between the methane carbon
and water oxygens considering pairs with a separation less
than 0.5 nm was determined, and the respective time average,
Nw, was calculated. The number density of water oxygens, nO,
and the number density of water hydrogens divided by two, nH,
was determined as a function of the distance r from the center of
mass of the methane molecule. In addition, the number density
of water oxygens nOO as a function of the distance from the center
of mass of a water molecule, averaged over all water molecules,
was evaluated. The charge density qw(r) of the water was obtained
from
qwðrÞ ¼ h
X
i;jriÀrj<Dr=2
qi=4pr2
Dri: ð12Þ
Here,
P
is a sum over respective atoms or shell particles of the
water, ri denotes the value of r for atom or shell particle i,
Dr = 0.003 nm, and qw(r) was smoothed with a Gaussian filter of
0.012 nm width. The dielectric permittivity in the methane molecule,
m, was determined from the fluctuations of the dipole moment
Mw of the methane molecule according to [37]
m ¼ 4w=
1
0ð2w þ 1Þ
hM2
wi
3VmkBT
À 1
!
: ð13Þ
Here, the volume of the methane molecule, Vm, was determined
from Vm ¼ ð4p=3Þr3
m with rm = dOC À 0.5 dOO where dCO or dOO denote
the position of the first maximum of nO(r) or nOO(r), respectively.
The value for the dielectric permittivity of the water,
w = 122, was determined as described above using Eq. (2). The value
for the permittivity obtained was m = 9.4 ± 0.2.
The electrostatics for a single methane molecule in water exposed
to an electric field was analyzed using a corresponding spatially
dependent dielectric permittivity (r) derived from the MD
simulations. Hence, the component of the local electric field E (r)
in the direction of an applied field was evaluated from continuum
electrostatics. The dielectric permittivity (r) as a function of the
distance from the center of mass of the methane molecule, r, was
determined by fitting a smoothed step function
ðrÞ ¼ m þ
w
expðÀðr À r0Þ=DrÞ þ 1
ð14Þ
to the normalized water density
~vðrÞ 
nOðrÞnHðrÞ
nO;bnH;b
 1
2
: ð15Þ
Here, nO,b or nH,b denote the respective values of nO(r) or nH(r) in the
bulk. The fit yielded r0 = 0.311 nm and Dr = 0.0088 nm.
This dielectric function was used to model the methane molecule
in water for the continuum calculations. Here, the electric conductivity
of 5.5E-6 S/m and the dielectric constant 122 for the bulk water
were chosen. The function Eq. (14) specified the dielectric constant
within a sphere of radius 1.0 nm. This sphere was embedded at the
center of a cylinder of length 30 nm and a radius of 3 nm. The planar
surfaces of the cylinder were chosen to differ in electrostatic potential
such as to obtain an electric field Ew = 20 V/cm in the high dielectric
region. This field strength is in the range of the field strengths
used in electrophoresis experiments [6]. The Poisson equation for
this system was solved numerically via the finite element method
using the electrostatics module of the software COMSOL [38]. Here,
a large number of ’mesh’ volume elements are created and Poisson’s
equation is solved individually for each element matching the
boundary conditions specified. Inside the sphere, the COMSOL
parameter ‘maximum element size’ was set to 1.0E-10 whereas
the maximum element size in water (outside the sphere) was set
to 1.0E-4.
Hence, using cylindrical symmetry, the local electric field E parallel
to x emerging in response to the applied field was calculated
as a function of the position along the axis, x, and the distance from
the axis, q. This yielded the normalized field
aðq; xÞ ¼ Eðq; xÞ=Ew ð16Þ
on the defined mesh. One-dimensional profiles a(0,x) and a(q,0)
with bin sizes 0.001 nm were determined via interpolation and
smoothed with a rectangular filter of 0.04 nm width.
2.3.1. Mean-field calculation of electrophoretic pseudo charge
Although the total charge of water or oil is zero each, a non-zero
force on the water or oil may result from a nonuniform distribution
of respective partial charges leading to excess charge qi(r) with
i = w,a for water or oil, respectively, with sign and magnitude
depending on the position r. An electric field Eapp parallel to the
interface will result in a position dependent local electric field denoted
as E(r) due to gradients in dielectric permittivities. This will
lead to a net force Fmf;i ¼
R
EðrÞqiðrÞd
3
r on the water or the oil, corresponding
to a pseudo charge Qmf,i = Fmf,i/Eapp. This pseudo charge
was estimated from a mean-field description. The pseudo surface
charge of water is calculated using the pseudo charge of the hydration
shell of the single methane molecule in water, Qw given by
Qw = Fw/Ew; here, Fw denotes the total force on the hydration shell
in the presence of an external field of intensity Ew in the bulk water
corresponding to a high dielectric environment. Qw was determined
from (i) the charge distribution qw(r) evaluated via Eq.
(12), and (ii) the electric field normalized to the field in the water
bulk, a(q, x), obtained as described in Section 2.3, according to
Qw ¼ 2p
Z w
0
qwððq2
þ x2
Þ1=2
Þaðq; xÞqdqdx: ð17Þ
Here, w = 1.18 nm, and the integral was evaluated as a sum with the
integrand and respective volume elements calculated on a regular
grid with bin size 0.001 nm for q and x. To this aim, the profiles
qw((q2
+ x2
)1/2
) from Eq. (12) and a(q,x) from Eq. (16) defined on
other grids were interpolated accordingly.
If an external field is applied parallel to a water/decane interface,
the average electric field is the same in the bulk of the water
and in the oil phase as known from continuum electrostatics.
Hence, the average electric field is independent of the position z
normal to the interface; its value is denoted as Eapp as used in Eq.
(1). At a particular z position at the interface, both water and oil
molecules are present, and Eapp arises from an average over contributions
from the respective water and the oil fraction. If vw(z) or
va(z) denote the water or the oil fraction with vw(z) + va(z) = 1
and Ew(z) or Ea(z) are the respective intensities of the electric field,
the relation
Eapp ¼ vwðzÞEwðzÞ þ vaEaðzÞ ð18Þ
holds. We write
EaðzÞ ¼ amEwðzÞ: ð19Þ
Here, am denotes an effective factor by which the electric field in the
oil is enhanced compared to the electric field in the water fraction.
A range of values for this factor was estimated as described based
on findings from the analysis in Section 2.3 as described in the Results
section. With v(z)  vw(z) and va(z) = 1 À v(z) we obtain
EwðzÞ ¼
Eapp
vðzÞ þ am À amvðzÞ
: ð20Þ
226 V. Knecht et al. / Journal of Colloid and Interface Science 352 (2010) 223–231
Here we determine v(z) from the number density of water oxygens
along z, nw(z), and the respective density in the bulk water, nw,b,
according to v(z) = nO(z)/nO,b(z). The contribution of the water at
position z to the total pseudo charge of the interfacial water will
scale with the respective number of atomic contacts between water
and oil per unit volume, c(z), estimated from
cðzÞ ¼ avðzÞð1 À vðzÞÞ: ð21Þ
Here, a is chosen such that
Rþu
Àu
cðzÞdz ¼ c0, with c0 denoting the
number of oxygen–carbon contacts per unit interfacial area as obtained
from Eq. (9), and u = 1.8 nm. If Qw denotes the pseudo charge
on the hydration shell of a methane molecule determined from Eq.
(17), Nw denotes the number of interatomic contacts between water
and methane analyzed as described above, and
~qwðzÞ 
Qw
Nw
cðzÞ; ð22Þ
the electrophoretic force Fw,i on the water at a water/decane interface
with projected interfacial area Ab is
Fw;i ¼ Ab
Z þu
Àu
EwðzÞ~qwðzÞdz: ð23Þ
The surface pseudo charge of the water is rw = Fw,i/EappAb. Using Eq.
(20) we obtain
rw ¼
Qw
Nw
Z þu
Àu
cðzÞ
vðzÞ þ am À amvðzÞ
dz: ð24Þ
The surface pseudo charge on the oil, ra is defined by
ra ¼ Fa;i=EappAb; ð25Þ
where Fw,i denotes the electrophoretic force on the decane at a
water/decane interface with projected interfacial area Ab. The surface
pseudo charge on the oil obeys
ra ¼
Rþu
Àu
qaðzÞEaðzÞdz
Eapp
: ð26Þ
Eqs. (18) and (19) yield
EaðzÞ ¼
Eapp
1 À vðzÞ þ vðzÞ
am
: ð27Þ
Hence, the surface pseudo charge on the oil is obtained from
ra ¼
Z þu
Àu
qaðzÞ
1 À vðzÞ þ vðzÞ
am
dz: ð28Þ
3. Results and discussion
3.1. Polarizable oil in water exhibits negative electrophoretic mobilities
The system simulated is shown in Fig. 1 and consisted of a decane
(C10H22) slab in water under periodic boundary conditions.The
decane was described using a polarizable potential derived from
the general Amber force field (GAFF), and the water was treated
using the polarizable POL3 water model [24,25]. The surface tension
of the water/decane interface in the simulations at zero field
intensity was 43 ± 8 mN/m, and, hence, within the error equal to
the experimental value of 50.11 ± 0.04 mN/m, giving confidence
that the model gives a reliable description of the interface.
To compare with previous studies [21,22], simulations in which
decane was described using the GROMOS-87 force field [32] and
the water was treated using the simple point charge (SPC) model
[33] were carried out as well. LJ interactions were switched between
0.7 and 1.1 nm. An electric field of 0.28 V/nm was applied
parallel to the decane slab and the average drift v of decane relative
to water in field direction was measured. As given in Table 2, the
drift vanished for SPC/GROMOS but was non-zero for POL3/GAFF.
Increasing the size of the oil slab or increasing the cut-off distance
to 2 nm for POL3/GAFF did not affect the drift velocities significantly.
In the following we will focus on the results from simulations
using POL3/GAFF.
Fig. 2a shows that the data are consistent with a linear scaling of v
with the field intensity, indicating electrophoretic motion. A linear
fit yields an electrophoretic mobility of À1.21 ± 0.04 Â 10À8
m2
/Vs
as given in Table 3. This is of the same sign and similar magnitude
as electrophoretic mobilities in the range À2 to À5.5 Â 10À8
m2
/Vs
for micron-sized oil droplets in distilled water measured experimentally
[3]. The electrophoretic mobilities, as well as other properties
of a water/oil interfaces given below, are summarized in Table 3.
Fig. 2 b and c shows steady state properties for a water/decane
interface parallel to an effective field of 0.21 V/nm. The mass densities
of decane, water, and the total system (Fig. 2b) and the average
velocity vloc in field direction (Fig. 2c) are plotted as a function
of the position z normal to the interface. The data suggest that
vloc(z) is constant in the bulk phases and exhibits a transition at
the interface. Thus, the interface is subjected to shear forces. Note
that the transition in vloc(z) occurs within a distance range of
0.6 nm. For comparison, the effective linear size of a decane molecule,
d = 2rg, given by its radius of gyration, rg, is found to be
0.7 nm. The length scale over which vloc(z) changes from the value
in the oil to that in the water phase is thus comparable or even
smaller than the molecular dimensions.
The current theory describes the flow profile v(z) as a continuous
transition using continuum hydrodynamics. In general, continuum
theory is appropriate for processes whose typical length
scales in the system are large compared to the molecular dimensions
which is not the case here. Although continuum theory has
been successfully used in explaining the mobilities of ions in water
[40] or the hydrodynamic properties of proteins based on detailed
models for the macromolecular surfaces [41], it is not clear a priori
if continuum theory would be suitable for the process considered
here. Hence, instead of describing v(z) as a continuous transition
using continuum hydrodynamics, we approximate the flow profile
as a discontinuous transition associated with a friction coefficient
b. The friction coefficient is determined from the thermal motion
of decane parallel to water yielding b = (0.74 ± 0.02) Â 106
Pa m s.
This is similar to the coefficient of b = 0.7 Â 106
Pa m s for the friction
between the two leaflets of a fluid lipid bilayer from previous
MD simulations [42]. The driving force normalized by the field
intensity shall be denoted as electrophoretic pseudo charge [21].
The pseudo charge normalized by the projected interfacial area is
the surface pseudo charge r. The surface pseudo charge is related
to the friction coefficient and the electrophoretic mobility according
to Eq. (3). We obtain r = À0.028 ± 0.001 e/nm2
.
Table 2
Electric field induced drift of decane slabs in water in molecular dynamics
simulations. The initial projected area of the decane slab, Ab(nm2
), the force field,
the cut-off distance for van der Waals interactions, rvdw, and the migration rate in the
direction of an external electric field of 0.5 V/nm parallel to the interface, v, are given.
The two different values for Ab correspond to the systems DEC or DECL given in
Table 1. The interaction potentials used were the non-polarizable united atom
GROMOS/SPC and the polarizable all-atom GAFF/POL3 force field. Van der Waals
interactions were switched between rvdw À 0.3 nm and rvdw. The drift is zero for
GROMOS/SPC and non-zero for GAFF/POL3.
Ab (nm2
) Force field rvdw (nm) v (m/s)
6.25 GROMOS/SPC 1 À0.1 ± 0.2
6.25 GAFF/POL3 1 À3.6 ± 0.2
25.00 GAFF/POL3 1 À4.2 ± 0.3
25.00 GAFF/POL3 2.0 À4.1 ± 0.5
V. Knecht et al. / Journal of Colloid and Interface Science 352 (2010) 223–231 227
3.2. Mechanism for charge-free electrophoresis
The electrophoretic mobility of a particle in an electrolyte solution
can be affected by ion relaxation and polarization of the double
layer and could be even non-zero for a particle with zero net
charge but finite quadrupole moment [43–45]. However, a theory
explaining the finite electrophoretic mobility of neutral oil in water
in the absence of an electrolyte as observed here is not available.
The physical mechanism underlying this phenomenon will be revealed
in this section. Analysis shows that the water accessible surface
area of the oil is 2.9 times larger than the projected interfacial
area. This indicates considerable interfacial roughness and solvent
exposure of hydrophobic groups. The number of contacts between
water molecules and CHn groups per projected interfacial area is
36.3(±0.07)/nm2
 c0. For comparison, we find that the first hydration
shell of a single methane (CH4) molecule in water with an outer
shell diameter of 1.3 nm contains 16.8 ± 0.1 water molecules.
The structure of this hydration shell will give rise to an electrophoretic
pseudo charge as shown in the following.
Fig. 3 shows the density of water oxygens and hydrogens (a)
and the excess charge qw(r) from the water (b) as a function of
the distance r from the center of mass of the methane molecule
at equilibrium. The distance at which the water density equals half
the density in the bulk is indicated with a dotted line. This distance
corresponds to a spherical shell which we shall denote as the dividing
surface between methane and water. The water oxygens and
hydrogens exhibiting negative or positive partial charges, respectively,
are distributed unequally, leading to positive excess charge
close to methane where the water density is low. Here, the dielectric
permittivity, m, is lower than in the bulk water, w. Analysis of
the dipole fluctuations yields w = 122 from Eq. (2) and m = 9.4
from Eq. (13).
The dielectric permittivity at the interface between methane
and water between the two bulk values was interpolated according
to Eq. (14). Hence, the electrostatics around a methane molecule in
water placed at the origin and exposed to an external field in x
direction was evaluated using continuum theory. The corresponding
position dependent electric field normalized by the electric
field denoted as a as a function of x and the distance from the x
axis, q, is shown in Fig. 4. In the methane molecule, a = 1.54. Normal
to the direction of the applied field, a monotonically decreases
to one for increasing distance r from the center of the methane
molecule. Parallel to the applied field there is a minimum at the
dividing surface (r = 0.32 nm).
Thus, positive excess charge of the water resides in a region
where the electric field is enhanced compared to the water bulk.
On the contrary, negative excess charge of the water partially resides
in a region where the electric field is decreased compared
to the water bulk. Thus, although the hydration shell has a total
charge of zero, the total force on the water, being the spatial integral
over the local excess charge times the local electric field as expressed
in Eq. (17), will be positive. The pseudo charge of the
hydration shell obtained from Eq. (17) is Qw = 0.8 e.
(a)
(b)
(c)
Fig. 2. Electrophoresis of uncharged decane in water. (a) Migration rate v of decane in water as a function of the applied electric field, Eapp. The data set and a fit of the
function v = lE Eapp are shown. (b and c) Steady state properties for water/decane interface parallel to an effective electric field of 0.20 V/nm with (b) mass densities and (c)
average velocity in field direction as a function of the position normal to the interface. In (c), horizontal lines indicate the average velocity in the respective bulk phase and the
respective standard errors.
Table 3
Parameters for water/oil interfaces from MD simulation at finite (E – 0) or zero field
intensities (E = 0) as well as from experiment. The electrophoretic mobility, lE, the
friction coefficient for tangential movement between the phases, b, the electrophoretic
pseudo charge, r, and the isoelectric point, pI, are given.
Quantity MD, E – 0 MD, E = 0 Experiment
lE(10À8
m2
/Vs) À1.21 ± 0.04 À0.3 to À1.5 À2 to À5.5 [3]
b (106
Pa s) 0.74 ± 0.02 0.8 ± 0.1 –
r (e/nm2
) À0.028 ± 0.001 À0.008 to À0.037 –
pI 2.50 ± 0.04 2.4 to 3.0 2.5 [4]
228 V. Knecht et al. / Journal of Colloid and Interface Science 352 (2010) 223–231
Hence, the surface pseudo charge for the water at a water/decane
interface can be estimated from Eq. (24). The missing parameter
here is am denoting the effective factor by which on average the
electric field in the oil is enhanced compared to the bulk water.
Here we assume that the electrostatics around the methyl(ene)
groups from the decane exposed to interfacial water is locally similar
to the electrostatics for the methane molecule in water. Fig. 4
shows that the electric field in the water is not constant but the
methane affects the electric field beyond the dividing surface over
a distance comparable to the radius of the methane molecule. Parallel
to the applied field, as noted above, the electric field shows a
local minimum corresponding to a = 0.29  amin at the dividing
surface. At the water/oil interface, the water region around interfacial
methyl(ene) groups will not be as extended as for the single
methane molecule in water; hence, the electric field will most
likely not reach its asymptotic value corresponding to a = 1. Thus
the average electric field in the water will correspond to an average
a-value denoted as aeff between 0.29 and 1. The effective factor am
will thus lie in the range between am/1 and am/amin. Hence,
1:54 < am < 5:32: ð29Þ
This yields a surface pseudo charge for the water at a water/decane
interface in the range between 30 and 60 e/nm2
.
A negative surface pseudo charge on decane in water is revealed
from the properties of a water/decane interface at equilibrium
shown in Fig. 5. The figure shows (a) the densities of water or decane
normalized to the respective densities in the bulk phases, and (b) the
excess charge per unit volume of the oil. It is evident that the oil
exhibits negative excess charge in the region of low oil density at
z > 0. In this region, the local electric field in the oil phase is larger
than for z < 0 for the following reason. The electric field parallel to
the interface, E, will be the same in the water and in the oil phase
as known from continuum electrostatics, hence E(z) = const  Eapp.
Microscopically, E(z) arises from an average over the electric field
Ew(z) in the water and Ea(z) in the oil fraction according to Eq. (18).
The corresponding surface pseudo charge of the oil can be obtained
from Eqs. (28) and (29), yielding À0.018 e/nm2
< ra < À0.004 e/nm2
.
The force on the oil phase in the presence of an electric field Eapp
will be Fa = raAbEapp, where Ab denotes the projected interfacial area
(area of the simulation box in xy direction). This will be opposite in
sign but smaller in magnitude than the respective force on the water,
Fw = rwAbEapp. The latter may be written as Fw = Fw,rel + Fw,abs with
Fw,rel  ÀFa. The force Fw,abs will act on the center of mass of the system.
For our simulations the center of mass motion is removed while
in experiment Fw,abs is counterbalanced by an opposing force from
the electrodes; hence Fw,abs will not lead to motion in either case.
In contrast, Fw,rel and Fa will lead to a total force F = Fa À Fw,rel = 2Fa
acting on the oil relative to the water. With Eqs. (7) and (25) this
yields
r ¼ 2ra; ð30Þ
resulting in À0.037 e/nm2
< r < À0.008 e/nm2
, as given in Table 3.
This range includes the surface pseudo charge r determined from
the electrophoretic motion in the MD simulations.
When oil and water are treated using the SPC/GROMOS model,
the structure of water close to CHn groups is similar to that observed
here for CH4 [21]. However, there is no partial charge on
the oil, so the respective surface pseudo charge must be zero and
the electrophoretic mobility lE will vanish. This is consistent
with the absence of electrophoretic motion for water/oil interfaces
(a)
(b)
Fig. 3. Equilibrium properties of a single methane molecule in water. (a) The number densities of water oxygens or hydrogens and (b) the charge density from the water are
shown as a function of the distance from the center of mass of the methane molecule.
(a)
(b)
Fig. 4. Electrostatics for single methane molecule in water exposed to an electric
field in the x direction. The ratio of the local electric field over the intensity of the
field in the water bulk, a, is shown. (a) Two-dimensional representation. Here, the
methane molecule is centered at the origin, the direction of the electric field is
depicted, and a at each position (x,z) is color-coded. The system is cylindrically
symmetric around the x axis. (b) a along the solid and the dotted lines in (a).
V. Knecht et al. / Journal of Colloid and Interface Science 352 (2010) 223–231 229
parallel to an applied electric field in this model observed here and
by others [22].
3.3. Electrophoretic mobility reproduced from equilibrium properties
Due to the computational expense, only nm length scales and
time scales of tens of nanoseconds are accessible with maintainable
computational effort for the detailed model used here. To be
able to distinguish electrophoretic motion from diffusion for this
time and length scale, high electric fields in the range 0.05–
0.15 V/nm were applied (the values refer to the value of the electric
field in the water bulk). Although these exceed those typically used
in electrophoresis experiments [6] by several orders of magnitude,
the migration rates of the oil slabs observed in our simulations
scale linear with the field intensity, indicating that nonlinear effects
are not significant. This suggests that the simulations performed
at high electric field intensities can be directly related to
the available experimental data [21].
Moreover, we may predict the electrophoretic mobility for the
oil slab considered here even solely from the equilibrium properties,
that is, the properties at zero field strength. To this aim, we
use (i) the friction coefficient as determined from an equilibrium
simulation, and (ii) the surface pseudo charge from the mean-field
calculations, rmf. We iterate that our mean-field model was
parameterized based on results from an MD simulation at equilibrium
and continuum calculations for an applied field using linear
response theory. With b = 0.8 ± 0.1 Â 106
Pa m s and the range for
rmf from the mean-field theory as given in Table 3 we obtain
À1.5 Â 10À8
m2
/Vs < lE < À0.3 Â 10À8
m2
/Vs. This range includes
the value for the electrophoretic mobility determined from the
drift at non-zero field.
3.4. Role of ions
Experimental electrophoretic mobilities of oil droplets in water
are pH dependent being negative for pH > pI and positive for
pH < pI with pI denoting the isoelectric point (iep). The negative
electrophoretic mobilities above the iep are commonly explained
in therms of adsorption of hydroxide ions at the interface. However,
findings from surface tension measurements and spectroscopic
experiments as well as MD simulations indicate that
hydrophobic surfaces in water repel hydroxide and attract hydronium
[12–14]. For hydronium at a water/alkane interface an
adsorption free energy of DG = À1.9 kcal/mol is found from MD
simulations [13]. Our results suggest that the positive surface
charge from hydronium, r+(pH), may compete with the negative
pseudo charge ra of the oil yielding the total surface charge,
r(pH), according to
rðpHÞ=2 ¼ ra þ rþðpHÞ ð31Þ
such that r(pH) < 0 for pH > pI and r( pI) = 0. Note that Eq. (31) is a
generalization of Eq. (30). At pH 7, a hydronium or a hydroxide ion
in solution would be present in a system of the size simulated for
only 1.6 Â 10À6
of the time. The number of hydronium ions DN(pH)
adsorbed at an interface with a total area of DA is
DNðpHÞ ¼ 10ÀpH mol
L
DzDA expðÀDG=kBTÞ: ð32Þ
Here, kB is Boltzmann’s constant, T = 300 K the temperature, and Dz
the thickness of the layer in which hydronium is adsorbed. Simulations
suggest Dz = 0.3 nm [13]. With DA = 2Ab, Ab being the average
area of the box parallel to the interface as defined in Section 2.2.2
and pH = 7 we obtain DN % 5 Â 10À6
. With
rþðpHÞ ¼ eDNðpHÞ=DADz ð33Þ
and, hence, r+(7) = 4 Â 10À7
e/nm2
, the total surface charge from Eq.
(31) would be equal to the surface pseudo charge of the ion-free
interface within the error. Consequently, within the statistical accuracy
the electrophoretic mobility of decane, lE = 2r/b, would be the
same as that of the ion-free interface. Hence, not to include any ions
in our simulation system is a good approximation at neutral pH.
Increasing the pH would not change the electrophoretic mobility
significantly, in agreement with experiment [6]. At the iep,
r(pI) = 0, and, hence, r+(pI) = Àra. This yields
pI ¼ À
1
ln 10
DG
kBT
þ ln
Àra
e
L
mol
  
¼ 2:50 Æ 0:04: ð34Þ
This is in excellent agreement with the experimental value pI = 2.5
[4].
4. Conclusions
The current interpretation of electrophoresis experiments based
on continuum theory relies on the assumption that, in general,
electrophoretic mobility reflects net charge. Our MD simulations
indicate that this assumption is invalid for oil droplets in water.
A molecular mechanism for electrophoretic mobility of uncharged
oil in water is revealed. Remarkably, we find that the negative electrophoretic
mobilities of oil droplets in water are consistent with a
small positive surface charge indicated from previous spectroscopic
and simulation results.
(a)
(b)
Fig. 5. Equilibrium properties of a water/decane interface. (a) The density of water or decane normalized to the respective bulk densities and (b) the excess charge density of
the decane as a function of the position normal to the interface are depicted.
230 V. Knecht et al. / Journal of Colloid and Interface Science 352 (2010) 223–231
Acknowledgments
The authors thank R.R. Netz, D. Bonthuis, and A. Smith for stimulating
discussions and Y.G. Smirnova for technical help.
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