2. Applied Clay Science 212 (2021) 106212
2
study the transport properties of clay minerals (Leroy et al., 2008). The
shear plane is also an essential parameter that characterizes the hy
drodynamic properties of colloidal systems. The shear layer (the space
between the clay surface and shear plane) greatly impacts the hydro
dynamic motion of pore fluid and suspended particles (Xu, 1998). As a
boundary condition, the shear plane is used to calculate the electroos
motic flow velocity (Wang et al., 2006; Vasu and De, 2010). In soil
mechanics, water in the shear layer is regarded as tightly bound water
(Li, 2016). Compared to several measurement techniques such as elec
trophoresis, electroosmosis and electroacoustics of the zeta potential
determination (Greenwood, 2003), to date, the measurement of the
shear layer thickness mainly follows two methods. The primary method
is based on the PB equation or modified PB equation. The potential
profile in the EDL is obtained, and then the shear plane's position is
calculated according to the experimental measurement of zeta potential
(Ding et al., 2015; Liu et al., 2017). In another way, only a few in
vestigations directly measured the shear layer thickness according to its
hydrodynamic definition. In electrophoresis experiments, a surface of
shear beyond the particle interface can be observed due to the electro
static interactions between the nanoparticle and the applied electric
field (Hunter, 1981). Besides, diffusion coefficients of the colloidal
particles, counterions or water were measured to determine the shear
plane thickness without applying the electric field (Xu, 1998; Jain et al.,
2021). Bourikas et al. (2001) and Panagiotou et al. (2008) referred to the
position by those two methods as the shear plane uniformly in common.
However, the obtained position in the first method is actually the po
sition of zeta potential rather than that of the shear plane. Henceforth, to
distinguish the two positions, this study defines the zeta position
measured by the first method, which indicates the potential at the
calculated position corresponding to the zeta potential, and shear plane
position measured by the second method, at which the streaming ve
locity is zero.
Although a number of observations have been achieved, based on the
mentioned methods, the accurate position of the zeta potential in the
EDL remains controversial. To simplify, many researchers have assumed
that the zeta position is close to the Stern plane or even locate at it;
namely, the capacity value between two planes is much large in the TLM
(Hiemstra and Van Riemsdijk, 2006). However, Li et al. (2003) and Liu
et al. (2017) demonstrated that the zeta position is much closer to the
Gouy plane than the Stern plane after comparing the zeta potential with
surface potential and Stern potential in montmorillonite-water systems.
Molecular dynamics (MD) is a computational method that models
the molecular structure and predicts the behaviors of materials at the
atomic scale (Zhang and Pei, 2020). MD was used to explore the micro-
mechanism of clay minerals (Moussa et al., 2017; Zhang et al., 2017; Liu
et al., 2021). Chang et al. (1998), Parsons and Ninham (2010), and
Bourg and Sposito (2011) successfully applied the MD technique to
investigate the structure of EDL for clay-water systems. MD simulations
can determine the profiles of species in the diffuse layer and bulk elec
trolyte. Furthermore, MD can be utilized to simulate the electroosmotic
flows between two planar-charged surfaces. Based on the electroosmosis
simulation, scholars have also proposed methods to determine the zeta
potential of materials in different electrolyte solutions (Předota et al.,
2016; Biriukov et al., 2020). Therefore, MD is an appropriate tool to
study the zeta potential and shear plane position of clay.
In this paper, MD was performed to determine the zeta potential and
position of the shear plane in NaCl-montmorillonite systems with
different concentrations. The electric potential profiles, particularly for
the Stern potential and zeta potential, were calculated based on MD and
the EDL model. The shear layer thickness and zeta position were dis
cussed based on several measurement methods in the paper. The ca
pacitances in the TLM model were finally discussed.
2. Theoretical background
2.1. Stern model
In the classical EDL theory, the Poisson-Boltzmann equation de
scribes the distributions of ions in electrolyte solutions (Lyklema, 1995).
d2
φ
dz2
= −
4π
ε0εr
∑
i
c0
i viFexp
(
− viFφ(z)
RT
)
(1)
where φ(z) is the potential distribution in the diffuse layer; ε0 is vacuum
dielectric constant; εr is the relative permittivity of bulk water; ci
0
is the
ion concentration within the bulk solution; vi is the valence of the ith
cation species. F is the Faraday constant; R is the gas constant. In the
Stern model, by solving Eq. (1), the distribution of electric potential
away from the clay surface in the Stern layer and diffuse layer can be
expressed as (Shang et al., 1994):
φ(z) =
⎧
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎩
φ0 −
φ0 − φs
ds
z, z ≤ ds
2RT
vF
ln
eκ(z− ds)
+ tanh
vFφs
4RT
eκ(z− ds)
− tanh
vFφs
4RT
, z > ds
(2)
φ0 =
σ
Cs
+ φs (3)
where Debye-Huckel parameter κ =
̅̅̅̅̅̅̅̅̅̅̅̅
2v2F2c0
ε0εrRT
√
(m− 1
) represents the
reciprocal of the thickness of the diffuse layer; ds is the thickness of the
Stern layer; φsrepresents the Stern layer potential; φ0is the surface po
tential and can be calculated for a given surface charge density σ and
constant capacitance of the Stern layer Cs. Once ds and φs are deter
mined, the potential at any distance from the clay surface in the EDL can
be calculated. In other words, given a zeta potential, the position of the
zeta potential can be located by the Stern model.
Previous investigations (Bourg and Sposito, 2011; Ricci et al., 2013;
Bourg et al., 2017; Hocine et al., 2016) have demonstrated that the Stern
plane is located at the position of the OSSC. Therefore, the Stern layer
thickness can be determined from the density profiles of cations in MD
simulation. Li et al. (2004) and Hou et al. (2009) have proposed a
method to calculate the potential at the top end of the diffuse layer based
on the Poisson-Boltzmann equation. The potential is only determined by
the mean concentration of counterions in the diffuse layer ci and the
concentration in the equilibrium solution (Hou et al., 2009):
Fφs = − 2RTln
1 − 0.5
[
− A
(
ci
)
+
̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅
A
(
ci
)2
− 10.873
√ ]
1 + 0.5
[
− A
(
ci
)
+
̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅
A
(
ci
)2
− 10.873
√ ] (4)
where
A
(
ci
)
= 3.7183 +
6.8731c0
i
ci − c0
i
(5)
The derivation process of Eq. (4) is detailed in the Supporting In
formation. Liu et al. (2015, 2016) have applied the method to calculate
the Stern potential. For the first time, this method is combined with MD
simulations to solve the potential distribution in the EDL. The average
concentration of cations in the diffuse layer can be calculated from the
density profiles in MD simulation after locating the onset of the diffuse
layer. Thus, by means of Eqs. (2)-(5), the potential distribution is
determined using only the cations density profile near the clay surface.
H. Pei and S. Zhang
3. Applied Clay Science 212 (2021) 106212
3
2.2. Zeta potential
The electroosmotic velocity profile is governed by the Navier-Stokes
(NS) equation together with the PB equation (Lorenz and Travesset,
2007):
u(z) =
ε0εr
η
Ex[φ(z) − ζ ] (6)
where ζ is the zeta potential. When the fluid far away from the surface,
the electroosmotic velocity is linearly related to zeta potential, namely
the Helmholtz-Smoluchowski (HS) equation (Delgado et al., 2007):
ζ = -
ueoη
ε0εrEx
(7)
where ueo represents the streaming velocity of the bulk solution under
the electric field; η is the viscosity of the solution; Ex is the external
electric field along the x-direction. In Eq. (6), the electroosmotic flow is
defined as a plug-like flow under the no-slip condition. However, MD
simulations of electroosmotic flows showed a significant slippage phe
nomenon in previous literature (Rezaei et al., 2018; Xie et al., 2020).
Dufreche et al. (2005) have investigated the electro-osmosis in Na-
montmorillonite by MD simulations, which shows a slip length should
be taken into account to describe the profile of electroosmotic flow. An
overestimated value of the zeta potential may be obtained when
considering the no-slip condition. A Navier-type slip condition is intro
duced to describe the slip velocity of electroosmotic flow at the interface
(Celebi and Beskok, 2018):
u‖(z0) = b
du
dz
⃒
⃒z=z0
=
ε0εr
η
Exb
dφ
dz
⃒
⃒z=z0
(8)
where u‖(z0) is the slip velocity at the slip wall of z = z0; b is the slip
length. Thus, zeta potential can be calculated as (Celebi and Beskok,
2018):
ζ =
η
ε0εrEx
(
− ueo + b
dφ
dz
⃒
⃒z=z0
)
(9)
Nonequilibrium molecular dynamics (NEMD) can be applied to
simulate the electroosmotic flow in montmorillonite nanopore, and the
zeta potential can be calculated by Eq. (9). Finally, the zeta position can
be determined by the zeta potential together with the potential profile
from the Stern model.
2.3. Triple-layer model
The triple-layer model is a popular model proposed to characterize
the electrochemical properties of colloidal systems, especially for clay-
solution systems. Taking the example of montmorillonite in a NaCl
electrolyte, the main equations of the TLM are expressed as follows
(Tournassat et al., 2009; Leroy et al., 2015):
Q0 + Qβ + Qd = 0 (10)
Qd = −
̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅
8ε0εrkbTci0NA
√
× sinh
(
Fφd
2RT
)
(11)
φβ = φd −
Qd
C2
(12)
φ0 = φβ +
Q0
C1
(13)
KNa = -
Q0 + Qβ
Qβ
aNaexp
(
−
Fφβ
RT
)
(14)
where Q0, Qβ and Qd are the surface charge densities at the 0, β-plane
and d-plane respectively and Q0is equal to the surface density of the
basal surface of montmorillonite minus that of ISCC. φ0, φβ and φd are
the corresponding potential. Notably, φd corresponds to zeta potential in
TLM. kb is the Boltzmann constant.NA is Avogadro number. KNa is the
equilibrium constant at β-plane. C1 and C2 are the capacitances within
β-plane and between β-plane and d-plane. The TLM can be simplified as
the basic Stern model by assuming φβ equal to φd.
2.4. Simulation methodology
In this paper, the zeta potential and the shear plane of montmoril
lonite in NaCl solution were investigated. The molecular model of
montmorillonite sheets was constructed with the unit cell formula of
Si8(Al3.25Mg0.75)O20(OH)4(Greathouse et al., 2015). Isomorphous sub
stitution occurs in the octahedral sheet, that is, Mg2+
substitution for
Al3+
. The clay layer used in the simulations consists of 4 × 4 unit cells
and has the size 20.66 Å × 35.85 Å × 6.54 Å and a surface charge density
of − 0.13C/m2
due to 12 Mg for Al substitutions. The layer was divided
in two halves located at the top and bottom of the simulation box to form
a slit-like pore with a width of 5.0 nm, which prevents overlapping of the
EDL. Water molecules with the number of 1187 were added in the pore
to reach a bulk density of 0.997 g/cm3
. Na+
was chosen as the coun
terion. Counterions were randomly distributed in the pore to compen
sate for the charge deficit of the clay layer. Besides, Na+
and Cl−
atoms
with the number of 2, 5, 11 and 22 were inserted in the water box,
leading to concentrations in the center of 0.20, 0.37, 0.70 and 1.30 mol/
L. The initial configuration of montmorillonite in 0.20 mol/L NaCl is
shown in Fig. 1. Periodical boundaries were set in all directions. The
interactions of minerals-minerals and minerals-water were described by
the ClayFF force field in this study (Cygan et al., 2004). The extended
simple point charge (SPC/E) water model was utilized to calculate the
water-water interactions. LAMMPS was used to perform all the MD
simulations (Plimpton, 1995).
The MD simulation was conducted in two parts. First, an equilibrium
molecular dynamics (EMD) simulation was performed under NVT
ensemble at 300 K. After a relaxation time of 10 ns, the trajectories of
atoms were recorded every 1 ns for another 200 ns. To prevent un
wanted movements of the clay particles, Al and Mg atoms in the clay
layer were fixed during the simulations. The density profiles and mean
square displacement of ion species and water molecules were obtained
in EMD. An electric field of 5 × 108
V/m was then applied parallel to the
clay surface acting on water molecules, cations and mobile surface
atoms (surface O atoms) in the NEMD simulation. Notably, the intensity
of the electric field is higher than the strength used in experiments, while
it is in the common range in MD studies. The purpose of the high electric
Fig. 1. Snapshot of MD model for a 50 Å mesopore constructed by the parallel
montmorillonite surfaces containing 0.20 mol/L NaCl. The width of the pore is
the distance between siloxane oxygens on each side of the clay layer.
H. Pei and S. Zhang
4. Applied Clay Science 212 (2021) 106212
4
field is to avoid excessive noise and obtain a reasonable accuracy. The
influence of electric field intensity on NEMD was discussed in Sup
porting Information. The system was relaxed by 20 ns under NVT
ensemble at 300 K to reach the equilibrium state. The streaming velocity
of water along the clay surface was recorded for 200 ns. The relative
permittivity and viscosity of the SPC/E water model (72.4 and 7.29 ×
10− 4
Pa⋅s) were used in Eq. (9) to calculate the zeta potential (Balasu
bramanian et al., 1996; Gereben and Pusztai, 2011). In this study, a
standard MD thermostat (Nosé-Hoover thermostat) was used in both
EMD and NEMD simulations. However, the thermostat used in NEMD
simulations is coupled only to the degrees of freedoms perpendicular to
the flow direction. The Lennard Jones (LJ) interactions were truncated
to 12 Å, and the Lorentz-Bertholet mixing rule was adopted for the LJ
interactions between different atoms. The Coulomb interactions were
calculated by the PPPM method (Hockney and Eastwood, 1988) with
99.99% accuracy. The time step of 1 fs was set to integrate Newton's
motion equations in the simulations.
3. Results and discussion
3.1. Ion distributions and stern potential
The density distributions of water along the aperture direction in the
montmorillonite mesopore containing different NaCl solutions present
similar tendencies, as shown in Fig. S2. It is noted that the location of
surface O atoms in the montmorillonite layer is the origin of the z-axis.
Pronounced water layers occur near the clay surface and dissipate
beyond 10 Å, which is in agreement with water on the clay surfaces from
previous investigations (Park and Sposito, 2002). The first peak of water
density is located at 2.7 Å, close to the value for other MD simulations
(2.7 ± 0.5 Å) (Bourg and Sposito, 2011) and X-ray experiments of mica
(2.5 ± 0.2 Å) (Cheng et al., 2001). For each NaCl-montmorillonite sys
tem, the density profiles of Na+
and Cl−
as a function of distance from
the clay surface are presented in Fig. 2. The convergence and error
analysis of ions density profile were discussed in Supporting Informa
tion. The density profile of Na+
close to the surface shows the first two
peaks corresponding to the plane of ISSC and OSSC, respectively. The
stern plane positions for 0.20, 0.37, 0.70 and 1.30 mol/L are determined
as the distance of 4.3 Å away from the clay surface. The thickness of
Stern layer in MD simulations of montmorillonite in mixed NaCl-CaCl2
solutions (4.35–4.55 Å) (Bourg and Sposito, 2011) and measurement
values of divalent cations OSSCs on mica (4.52 ± 0.24 Å) by X-ray (Park
et al., 2006) are close to the results in this study. The simulations show
that the z-coordination of OSSC is independent of the ionic
concentration.
The diffuse layer region is determined based on the Debye length
after locating the Stern plane. The mean concentration of cations in the
diffuse layer can be calculated from its density profiles, which are listed
in Table 1. The Stern potential of 0.20–1.30 mol/L NaCl solutions are
calculated as − 80.2, − 67.0, − 52.0 and − 39.7 mV by Eq. (4), respec
tively. The Stern potential decreases with increasing ion concentration,
which is consistent with the EDL theory. Miller and Low (1990) used
several methods combined with experiments and theories to determine
the Stern potential of montmorillonite in distilled water. The Stern po
tentials of Na-montmorillonite with the cation exchange capacity of 90
meq/100 g were measured in the range of − 55.5 to − 59.1 mV. The Stern
potential of montmorillonite with − 0.1 to − 0.15C/m2
surface charge
density in 0.1 mol/L NaCl solution was calculated in the range of − 79.5
to − 96.8 mV in theory (Liu et al., 2016). Compared with their results,
the calculated Stern potential is within a reasonable range.
Once the Stern potential is determined, the electric potential distri
bution in the diffuse layer can be calculated by Eq. (2). The density
profiles of ion species beyond the Stern plane predicted by the Stern
model are consistent with MD results, as shown in Fig. 2. The Stern
model with a simple way of the Stern potential determination can well
match the molar density of diffuse swarm (DS) species, however, slightly
underestimates the density of the OSSC. The difference at the Stern
plane may be attributed to the PB equation neglecting some effects such
as ion polarizability. However, this study still focuses on the classical
EDL theory in this paper. A popular model named the modified Gouy-
Chapman (MGC) model has been proposed to describe quantitatively
the ion concentration profiles in the EDL (Tournassat et al., 2009; Le
Crom et al., 2020). The main equations are expressed as:
⎧
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎩
Fφ(a)
RT
= − 2 × sinh− 1
(
F|σ|
2κε0εrRT
)
Fφ(z)
RT
= 4 × tanh− 1
[
tanh
(
Fφ(a)
4RT
)
× exp( − κ(z − a) )
] (15)
where a is a distance from the surface. σ is the surface charge minus the
ions charge within a. For montmorillonite in 0.20 M NaCl solution, the
comparison of the GC, MGC and Stern model is shown in Fig. 3. The Na+
density profile predicted by the GC model has the maximum molar
density at the clay surface, which shows a distinct difference from MD
simulation. Although the profiles by the MGC model with different a are
close to the MD result beyond 10 Å, the values of calculated profiles are
in disagreement with each other near the clay surface. In addition, the
predicted profiles of the MGC model at the Stern plane are less accurate
than the Stern model. The density profile of the Stern model is slightly
larger than that of the GC and MGC model beyond 10 Å. The fitted
0 5 10 15 20 25
0
1
2
3
4
5
6
7
8
Na+
by MD
Na+
by Stern model
Na
+
concentration
(mol/L)
Distance from the clay surface (Å)
0
1
2
Cl- by MD
Cl- by Stern model
Cl
-
concentration
(mol/L)
Stern plane
(OSSC)
1.30 mol·L-1
Fig. 2. Molar density profiles of Na, Cl at the clay surface as obtained from the
MD model of montmorillonite in 1.30 mol/L NaCl electrolyte. The light dash
lines are the density profiles of Na and Cl predicted by the Stern model. The
density profiles of 0.70, 0.37 and 0.20 mol/L NaCl electrolytes are shown
in Fig. S3.
Table 1
Calculated parameters in the Stern model for montmorillonite in NaCl solutions.
The capacitance in the Stern layer is 1.0 F m− 2
.
0.20
mol/L
0.37
mol/L
0.70
mol/L
1.30
mol/L
Mean concentration of Na+
in
diffuse layer, c(mol/L)
1.43 1.94 2.55 3.50
Electroosmotic mobility of bulk
water, μeo/Ex (m2
/V⋅s)
11.02 10.13 8.77 7.90
Thickness of Stern plane, ds (Å) 4.3 4.3 4.3 4.3
Position of zeta potential, dζ (Å) 7.2 6.8 6.5 5.8
Thickness of Gouy plane, dg (Å) 11.1 9.3 7.9 7.0
Surface potential, φ0 (mV) − 210.2 − 197.0 − 182.0 − 169.7
Stern potential, φs (mV) − 80.2 − 67.0 − 52.0 − 39.7
Zeta potential, ζ (mV) − 46.8 − 38.8 − 26.9 − 22.3
H. Pei and S. Zhang
5. Applied Clay Science 212 (2021) 106212
5
parameter a seems to lack a clear physical significance resulting in an
uncertain origin of the density profile, compared to the peak located at
the Stern plane in the Stern model. Therefore, the Stern model combined
with a calculated Stern potential has good accuracy in predicting ion
density profiles in the EDL.
3.2. Electroosmotic flow and zeta potential
Fig. 4 shows the electroosmotic velocity profile of water in mont
morillonite pore containing 0.20 mol/L NaCl solutions. The streaming
velocity follows the hydrodynamical Navier-Stokes equation (Eq. (6))
beyond the first peak of water density profile, whereas the velocity drops
rapidly within that. A clear slip velocity can be observed at the first peak
of water density profile. This is because only a small number of water
molecules before the first peak of water density profile that strongly
absorbed by the clay surface have different dynamics properties from
the bulk water. The same observation has also been presented in the
NEMD simulation of an electroosmotic flow of CsCl solution in a charged
silica slit by Siboulet et al. (2017). Celebi and Beskok (2018); Celebi
et al. (2019) have considered the shear plane is located at the first peak
of the water density profile. Although the streaming velocity disobeys
the NS equation, there is still a non-zero velocity within 2.7 Å in this
simulation as observed in Fig. 4. Therefore, the first peak of the water
density profile cannot be regarded as the shear plane simply. Siboulet
et al. (2017) also demonstrated that the stagnant layer close to the
surface belongs to the ion hydration sphere. This paper confirms that the
flow domain starts at the first water density peak which is 2.7 Å away
from the clay surface in the simulation. A function was proposed to fit
the streaming velocity profile from MD simulations, which the deriva
tion is shown in Supporting Information. Then the slip velocity and slip
length are obtained based on the fitting velocity profile. Both slip length
and slip mobility decreases with the concentration increases of NaCl,
which the calculated values are listed in Table 2. Rezaei et al. (2018)
also proved such tendency of slip length and slip velocity. The electro
osmotic mobility (velocity profile normalized by field strength) along z-
direction is exhibited in Fig. 5. The plug-like profiles for NaCl-
montmorillonite systems are observed as expected. The average
mobility at the plateau is significantly affected by the ionic concentra
tion, decreasing with the increasing of concentration. The zeta potential
of 0.20–1.30 mol/L NaCl solutions calculated by Eq. (9) are − 46.8,
− 38.8, − 26.9 and − 22.3 mV respectively. Tournassat et al. (2009)
measured the zeta potential of − 38 mV for montmorillonite (σ= −
0.11C/m2
) in 0.12 molL− 1
NaCl solution. Mészáros et al. (2019)
measured the zeta potential of bentonite (σ= − 0.14C/m2
) in 0.1 mol/L
NaCl solution as − 37.4 to − 47 mV in the range of 2–12 pH. The zeta
potential of Na-montmorillonite (σ= − 0.16C/m2
, pH = 6.5) at 1.0 M
was determined as − 17.9 mV (Sondi et al., 1996). It shows that the
values of zeta potential in the NEMD simulation are slightly larger than
to the measurements. The zeta potential of 0.20–1.30 mol/L NaCl so
lutions are calculated as − 126.7, − 116.5, − 100.8 and − 90.9 mV
respectively, without considering the slip length (Eq. (7)), which are
significantly larger than the measured values. Therefore, the slip length
should be considered when calculating the zeta potential to avoid an
overestimated value.
3.3. Zeta position and shear plane position
After obtaining potential distribution based on the Stern model and
the zeta potential in the EDL, the zeta position can be calculated, as
shown in Fig. 6. The zeta positions are calculated as 7.2, 6.8, 6.5 and 5.8
Å for montmorillonite at 0.20–1.30 mol/L. Results show that the posi
tion of zeta potential cannot be identified as the Stern plane simply. The
zeta position is closer to the Gouy plane than the Stern plane. For
montmorillonite in NaCl solution, the zeta position from the clay surface
decreases as the ionic concentration increases because of the Debye
screening length.
The TLM can also obtain the zeta position after the determination of
zeta potential. The charge density at d-plane (zeta position) Qd can be
calculated by Eq. (11). Besides, Qd is equal to subtracting the compen
sated charge within the d-plane from the surface charge of the
0 5 10 15 20 25
0
1
2
3
4
5
6
0 5 10 15 20 25
0.00
0.05
0.10
0.15
0.20
Cl
-
concentration
(mol/L)
Na
+
concentration
(mol/L)
Distance from the clay surface (Å)
MD
Stern model
GC model
MGC a=0.21 nm
MGC a=0.31 nm
MGC a=0.43 nm
MGC a=0.57 nm
Fig. 3. The comparison among Na+
(and Cl−
in the inserted graph) density
profile for montmorillonite at NaCl concentration of 0.20 mol/L in MD simu
lation and the predictions of the Gouy-Chapman model, Stern model and
modified Gouy-Chapman model with fitted parameter a of 0.21, 0.31, 0.43 and
0.57 nm.
25 20 15 10 5 0 -5
0
5
10
15
20
25
30
35
40
45
50
55
60
0.0
0.5
1.0
1.5
2.0
2.5
w
u||
(z0
)
Water
density
(g/cm
3
)
ueo
b=2.9 Å
z=z0
0
|z z
du
k
dz
0
0
|
r
x z z
d
E
dz
uMD
uslip
ustick
u(z)
(m/s)
z (Å)
Fig. 4. Electroosmotic velocity profiles for montmorillonite in 0.20 mol/L NaCl
electrolytes. uMD is the velocity of the fluid along the aperture direction from
MD, uSlip from fitting uMD in a PB equation form as shown in Supporting In
formation, and uStick by assuming a vanishing velocity at z = 2.7 Å. u‖(z0) is the
slip velocity. b is the slip length.
Table 2
The slip length and slip mobility for montmorillonite in NaCl electrolytes ob
tained by MD and Eq. (8).
Concentration
(mol/L)
Slip length
(Å)
Slip mobility
(×10− 8
m2
/V s)
0.20 2.90 6.90
0.37 2.50 6.73
0.70 1.95 6.44
1.30 1.90 5.96
H. Pei and S. Zhang
6. Applied Clay Science 212 (2021) 106212
6
montmorillonite basal surface (Eq. (10)). Therefore, by combining two
equations, the d-plane positions are 6.0, 5.5, 5.3 and 4.8 Å for mont
morillonite at 0.20, 0.37, 0.70 and 1.30 mol/L, as listed in Table 3.
Results show that the position of the d-plane is located beyond the
β-plane, in accordance with the TLM. The same tendency is exhibited in
the TLM that the distance of zeta position from the clay surface de
creases with electrolyte concentration. However, the distance between
the zeta position and the Stern plane in the TLM is less than in the Stern
model. Fig. 7 shows the relationship between the zeta position from the
interface and diffuse layer, zeta position and ionic strength from MD
simulations. The results from the Stern model and the TLM indicate that
the zeta position from the clay surface has a strictly linear relationship
with the Debye length, which agrees with the conclusion demonstrated
by Charlton and Doherty (1999) through experiments. Moreover, the
results also demonstrate that the zeta position (log scale) is linearly
dependent on the log value of the ionic strength, in accordance with the
conclusions of Bourikas et al. (2001) and Panagiotou et al. (2008).
After the zeta position determination, the position of the shear plane
was also investigated in this paper. In the Stern model, the position of
zeta potential is located at the shear plane, at which the flow has a nil
velocity under no-slip conditions. In the simulation, however, there is no
stagnant layer within the zeta potential position. Similarly, more and
more MD simulations have proved that the stagnant layer near a solid
surface cannot be observed in the EDL (Lylema et al., 1998; Freund,
2002; Hartkamp et al., 2005; Lorenz et al., 2008; Zhang et al., 2011).
The distinction between the standard hydrodynamics and the MD results
may contribute to the following reason. In standard hydrodynamics and
Stern model, the concept of shear plane strictly separating two neighbor
regions with an infinite and a constant viscosity of water respectively is
in contradiction with the findings that the viscosity changes continu
ously near the interface by experiments and MD simulations (Schmickler
and Henderson, 1986; Evans and Wennerström, 1999; Bazant et al.,
2009; Hartkamp et al., 2018). Therefore, although the Stern model can
predict well the static potential and ion distributions on the global EDL,
the local physical properties in electrokinetic phenomenon cannot be
accurately described by the NS equation and Stern model.
Xu (1998) has proposed a method to measure the shear plane
thickness of micellar particles based on their self-diffusion coefficients.
The hydrodynamic radius Rh can be calculated by self-diffusion coeffi
cient Ds using the Stokes-Einstein relationship:
Ds =
kBT
6πηRh
(16)
The thickness of the shear layer can be obtained by subtracting the
particle radius from the hydrodynamic radius. Jain et al. (2021) deter
mined the shear layer thickness of nano-particles by calculating the self-
diffusion coefficient of cations in MD simulations. Therefore, the diffu
sion coefficients of water oxygen and cations in the slit pore divided into
slices of 2 Å thickness were calculated in the simulations to evaluate
whether the coefficients can be used to determine the shear plane. The
time interval is an important parameter in the calculation of the diffu
sion coefficient. The influence of the time interval was discussed in
Supporting Information. After comparison, the length of time intervals
of 10 ps was chosen used in mean-square displacement calculation in
MD simulations. The diffusion coefficients of water oxygen and Na+
parallel to the clay surface along z-direction are illustrated in Fig. 8,
normalized by the values in the bulk 0.20 mol/L NaCl solution. Diffusion
coefficients decrease linearly within 15 Å from the interface and reach a
stable value beyond that, in agreement with the other MD results of
montmorillonite nanopore (Bourg and Sposito, 2011). It can be noted
that both coefficients of water and Na+
slightly decrease as ionic con
centration increases. The Shear plane position is assumed located at the
turning point with the diffusion value equals to the bulk value, which is
15 Å away from the clay surface. However, the calculated position is
0 10 20 30 40 50
0
3
6
9
12
15
u(z)/E
x
(m
2
/V·s)
z (Å)
0.20 mol/L
0.37 mol/L
0.70 mol/L
1.30 mol/L
×10-8
2.7 Å
Fig. 5. Electroosmotic velocity profiles normalized by the strength of the
external electric field for montmorillonite in 1.30, 0.70, 0.37 and 0.20 mol/L
NaCl electrolytes. The flow domains start at 2.7 Å away from the clay surface.
Fig. 6. The electric potential distributions of montmorillonite with different
ion concentrations. The position of the zeta position is obtained from the Stern
model and zeta potential.
Table 3
Calculated parameters in the triple-layer model for montmorillonite in NaCl
solutions. The capacitance C1 in the Stern layer is 1.0 F m− 2
.
0.20 mol/
L
0.37 mol/
L
0.70 mol/
L
1.30 mol/
L
Surface charge at 0-plane,
Q0 (C/m2
)
− 0.1293 − 0.1297 − 0.1294 − 0.1288
Surface charge at d-plane,
Qd (C/m2
)
− 0.0562 − 0.0613 − 0.0573 − 0.0614
Capacitance C2 (F⋅m− 2
) 3.34 4.49 5.09 8.04
Position of β-plane, zβ (Å) 4.3 4.3 4.3 4.3
Position of d-plane, zd (Å) 6.0 5.5 5.3 4.8
Potential at 0-plane,
φ0 (mV)
− 193.0 − 181.4 − 167.5 − 156.6
Potential at β-plane, φβ
(mV)
− 63.7 − 51.7 − 38.1 − 27.8
Potential at d-plane,
φd (mV)
− 46.8 − 38.8 − 26.9 − 22.3
H. Pei and S. Zhang
7. Applied Clay Science 212 (2021) 106212
7
beyond the Gouy plane in all concentrations of NaCl solutions, which
contradicts the Stern model. Although the diffusion coefficients of water
and cations have reduced mobility near the surface, a stagnant layer
with zero diffusion coefficient cannot be observed. To sum up, the re
sults confirmed that the shear plane position cannot be observed from
the electroosmotic flow and diffusion coefficient in MD simulations.
3.4. Discussion on the TLM
The TLM consists of several equations and parameters that make the
solution difficult, which simplex algorithm is commonly used to solve
the model (Leroy and Revil, 2004; Leroy et al., 2015). Parameters such
as equilibrium constants at corresponding planes and capacitances of the
inner and outer part of the Stern layer need to be estimated before the
solution. Previous studies have evaluated these parameters based on
experiments and MD simulations, especially for capacitances C1 and C2.
A basic agreement was achieved for C1 based on the proposed values of
0.6–1.3 F m− 2
by experiments (Machesky et al., 1998; Sverjensky, 2001)
and 0.8–1.2 F m− 2
by MD simulations (Tournassat et al., 2009). How
ever, the value of C2 in the range of 0.2–5.5 F m− 2
(Yates et al., 1974;
Hiemstra and Van Riemsdijk, 2006; Wolthers et al., 2008; Bourg and
Sposito, 2011) is still in dispute. In this paper, the capacitance C2
between β-plane and d-plane was investigated. As the obtained position
of β-plane and d-plane in the TLM (Table 2), the capacitance C2 can be
estimated by the relation (Sahai and Sverjensky, 1997):
Ci = εiε0/Δzi (17)
where εi and Δzi are the relative permittivity and thickness of each
“capacitor”. The spaces between β-plane and d-plane for montmoril
lonite in 0.20–1.30 mol/L NaCl solutions are 1.7, 1.2, 1.0 and 0.6 Å.
Wander and Clark (2008) have calculated the dielectric constants for the
quartz-water interface as a function of the distance from the surface. In
this paper, the permittivity of the outer Stern layer is calculated based on
the proposed curve (permittivity scaled using reference value of 78 for
bulk water), given as 64.2, 60.9, 57.5 and 54.5, respectively. Thus, the
capacitance C2 in the TLM can be obtained by Eq. (17) as 3.34, 4.49,
5.09 and 8.04 F m− 2
, respectively. The results show that the value of
capacitance between β-plane and d-plane has a large range depending
on the ionic concentration, which is in disagreement with the constant
value calculated by Bourg and Sposito (2011) but in agreement with the
conclusion of Nishimura et al. (2002). After obtaining the capacitance
between β-plane and d-plane, the TLM can be solved. Fig. S4 shows the
potential distribution from the clay surface based on the TLM. The
capacitance within the Stern layer is set as 1.0 F m− 2
. The potential
2 3 4 5 6 7 8
4
5
6
7
8
9
by Stern model
by TLM model
Zeta
position
from
the
surface
(Å)
-1(Å)
0.28 / 5.37
d
0.25 / 4.28
d
2
0.98
R
2
0.96
R
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80.9 1
4
5
6
7
8
9
10
2
0.96
R
2
0.93
R
log -0.112log 0.664
d I
log -0.111log 0.751
d I
By Stern model
By triple-layer model
logd
(Å)
logI (M)
Fig. 7. The distance of the zeta potential position from the clay surface calculated by the Stern model and triple-layer model as a function of the Debye length and
ionic strength.
Fig. 8. Self-diffusion coefficients of water oxygen and Na+
parallel to the clay surface as a function of distance from the basal surface. Error bars show the values of
95% confidence interval.
H. Pei and S. Zhang
8. Applied Clay Science 212 (2021) 106212
8
calculated by the TLM is smaller than the Stern model, especially for the
potential between β-plane and d-plane. The positions of zeta potential
calculated by the TLM are also smaller than those by the Stern model, as
mentioned above.
4. Limitations
This paper only studies the EDL structure and electroosmosis prop
erties of the basal surface of montmorillonite. It is well known, mont
morillonite nanoparticle has two surfaces including basal and edge
surface. Due to the large specific surface area, the basal surface is the
domain part of the surface of montmorillonite, which indicates the re
sults in this paper can generally reflect the properties of montmoril
lonite. However, surface complexes on the edge site can also develop in
the EDL. Cations exchange on edge surfaces shows complex, thermo
dynamically non-ideal behavior much different from exchange on the
basal surface (Lammers et al., 2017). Besides, the orientation (Kraevsky
et al., 2020) and cleavage type (Shen and Bourg, 2021) of the edge in
fluence the stability of the edge structure and its adsorption character
istics. The investigations of surface complexation of edge surfaces can be
found in Newton et al. (2016, 2017) and Kraevsky et al. (2020). Besides,
the edge surface of montmorillonite also has an influence on the value of
zeta potential. Edges surfaces have a pH-dependent charge, while basal
surfaces bear a negative charge generated by isomorphous substitution
(Delgado et al., 1985; Durán et al., 2000). This is a significant reason
that the zeta potential of montmorillonite changes as a function of pH
value. The effect of particle morphologies on the zeta potential was also
not considered in this paper. Those reasons may cause the gap between
the zeta potential measurement and the MD simulation result.
This paper distinguished the zeta potential and the shear plane. The
latter one was confirmed not to exist from MD simulations. The zeta
potential position was measured and its properties were investigated in
the paper. Thus, two questions arise. What is the physical meaning of the
zeta potential if the shear plane does not exist? What is the physical
meaning of the zeta potential position? This paper attempts to answer
these questions by combing our thoughts with previous literature.
Because those are still arguments in collochemistry, more in-depth work
should be done in the future.
Firstly, although its definition is related to the shear plane, the
generation and physical meaning of zeta potential is independent of the
existence of the shear plane. The shear plane is an imaginary plane only
used to reconcile the Poisson-Boltzmann equation and the Navier-Stokes
equation (Eq. (6)). Předota et al. (2016) have demonstrated that the zeta
potential does not arise from the existence of a shear plane, but rather
precisely from the electrokinetically driven motion of ions within the
whole inhomogeneous interfacial region. Delgado et al. (2007) also said,
the zeta potential is fully defined by the nature of the surface, its charge,
the electrolyte concentration in the solution, and the nature of the
electrolyte and of the solvent. Therefore, the zeta potential is still
meaningful without the shear plane. Secondly, although the physical
interpretation of the zeta potential is still ambiguous, a widely accepted
meaning is proposed by Delgado et al. (2007). The zeta potential is the
observed electrokinetic signal crossing from the non-contributing region
for electrokinetic phenomenon to the double layer.
Following this idea, the physical meaning of the zeta potential po
sition is given as which is the boundary of electrokinetic phenomenon
non-contributing region. The charges located between the surface and
the zeta potential position are electrokinetically inactive exhibiting only
electrostatic properties and contribute to the excess conductivity of the
double layer (Delgado et al., 2007). This is the reason that the bulk
streaming velocity is related to the zeta potential rather than the surface
potential in the H–S equation. Although no electrokinetic effect is
generated in this region under an external electrical field, water behind
the zeta potential position has non-zero velocity in the electroosmotic
tangential flow because of the viscosity. This also shows the difference
between the shear plane and zeta potential position and proves the non-
existence of the shear plane.
5. Conclusions
A novel approach based on molecular dynamics was proposed in this
paper to determine the zeta potential and the position of the shear plane
for montmorillonite in NaCl electrolyte with a concentration of
0.20–1.30 mol/L. It combines an electrostatic surface complexation
model (Stern model or TLM) that determines the electric potential dis
tribution in the EDL from the ion density profiles in the EMD with the
electroosmotic flow NEMD simulation that calculates the zeta potential.
The conclusions can be summarized as follows:
(1) The density profiles of ion species clearly show the existence of
ion adsorption complexes. The Stern model has better accuracy in
predicting the ions density profiles after the Stern potential
determination, compare with the Gouy-Chapman and modified
Gouy-Chapman model.
(2) The zeta potential is calculated based on the electroosmotic ve
locity profile, considering the slip length, close to the experi
mental measurement. Methods including NEMD and self-
diffusion coefficients cannot observe the shear plane. The posi
tion of zeta potential was determined by the Stern model and
TLM, showing a certain distance from the Stern plane. The dis
tance of the zeta position from the clay surface has a strictly linear
relationship with the Debye length and the ionic strength in the
log scale, both in agreement with the experimental observation.
(3) The capacitance between the Stern and shear plane in the TLM
was calculated indicating that the capacitance is variable
depending on the ionic strength rather than a constant value.
Declaration of Competing Interest
The authors declare that they have no known competing financial
interests or personal relationships that could have appeared to influence
the work reported in this paper.
Acknowledgements
This research has been supported by the China National Key R&D
Program during the 13th Five-year Plan Period (Grant No.
2018YFC1505104 and 2017YFC1503103), National Natural Science
Foundation of China (Grants No. 51778107) and Liao Ning Revitaliza
tion Talents Program (Grants No. XLYC1807263).
Appendix A. Supplementary data
Supplementary data to this article can be found online at https://doi.
org/10.1016/j.clay.2021.106212.
References
Balasubramanian, S., Mundy, C.J., Klein, M.L., 1996. Shear viscosity of polar fluids:
Molecular dynamics calculations of water. J. Chem. Phys. 105, 11190–11195.
https://doi.org/10.1063/1.472918.
Bazant, M.Z., Kilic, M.S., Storey, B.D., Ajdari, A., 2009. Towards an understanding of
induced-charge electrokinetics at large applied voltages in concentrated solutions.
Adv. Colloid Interface 152, 48–88.
Biriukov, D., Fibich, P., Předota, M., 2020. Zeta potential determination from molecular
simulations. J. Phys. Chem. C 124, 3159–3170. https://doi.org/10.1021/acs.
jpcc.9b11371.
Bourg, I.C., Sposito, G., 2011. Molecular dynamics simulations of the electrical double
layer on smectite surfaces contacting concentrated mixed electrolyte (NaCl-CaCl2)
solutions. J. Colloid Interface Sci. 360, 701–715. https://doi.org/10.1016/j.
jcis.2011.04.063.
Bourg, I.C., Lee, S.S., Fenter, P., Tournassat, C., 2017. Stern layer structure and energetics
at mica-water interfaces. J. Phys. Chem. C 121, 9402–9412. https://doi.org/
10.1021/acs.jpcc.7b01828.
Bourikas, K., Hiemstra, T., Van Riemsdijk, W.H., 2001. Ion pair formation and primary
charging behavior of titanium oxide (anatase and rutile). Langmuir 17, 749–756.
H. Pei and S. Zhang
9. Applied Clay Science 212 (2021) 106212
9
Celebi, A.T., Beskok, A., 2018. Molecular and continuum transport perspectives on
electroosmotic slip flows. J. Phys. Chem. C 122, 9699–9709. https://doi.org/
10.1021/acs.jpcc.8b02519.
Celebi, A.T., Cetin, B., Beskok, A., 2019. Molecular and continuum perspectives on
intermediate and flow reversal regimes in electroosmotic transport. J. Phys. Chem.
C. https://doi.org/10.1021/acs.jpcc.9b02432.
Chang, F.R.C., Skipper, N.T., Sposito, G., 1998. Monte Carlo and molecular dynamics
simulations of electrical double-layer structure in potassium-montmorillonite
hydrates. Langmuir 14, 1201–1207. https://doi.org/10.1021/la9704720.
Charlton, I.D., Doherty, A.P., 1999. Locating the micellar shear plane and its relationship
with the Debye screening length. J. Phys. Chem. B 103, 5081–5083. https://doi.org/
10.1021/jp9843914.
Cheng, L., Fenter, P., Nagy, K.L., Schlegel, M.L., Sturchio, N.C., 2001. Molecular-scale
density oscillations in water adjacent to a mica surface. Phys. Rev. Lett. 87 https://
doi.org/10.1103/PhysRevLett.87.156103, 156103-156103–4.
Cygan, R.T., Liang, J.-J., Kalinichev, A.G., 2004. Molecular models of hydroxide,
oxyhydroxide, and clay phases and the development of a general force field. J. Phys.
Chem. B 108, 1255–1266. https://doi.org/10.1021/jp0363287.
Delgado, A.V., Gonzalez-Cabllero, F., Bruque, J.M., 1985. On the zeta potential and
surface charge density of montmorillonite in aqueous electrolyte solutions. J. Colloid
Interface Sci. 13 (1), 203–211.
Delgado, A.V., Gonzalez-Cabllero, F., Hunter, R.J., Koopal, L., Lyklema, J., 2007.
Measurement and interpretation of electrokinetic phenomena. J. Colloid Interface
Sci. 309, 194–224.
Ding, W., Liu, X., Song, L., Li, Q., Zhu, Q., Zhu, H., Hu, F., Luo, Y., Zhu, L., Li, H., 2015.
An approach to estimate the position of the shear plane for colloidal particles in an
electrophoresis experiment. Surf. Sci. 632, 50–59. https://doi.org/10.1016/j.
susc.2014.08.024.
Dufreche, J.F., Marry, V., Malikova, N., Turq, P., 2005. Molecular hydrodynamics for
electro-osmosis in clays: from Kubo to Smoluchowski. J. Mol. Liq. 118, 145–153.
https://doi.org/10.1016/j.molliq.2004.07.076.
Durán, J.D.G., Ramos-Tejada, M.M., Arroyo, F.J., Gonzalez-Cabllero, F., 2000.
Rheological and electrokinetic properties of sodium montmorillonite suspensions I.
Rheological properties and interparticle energy of interaction. J. Colloid Interface
Sci. 229, 107–117.
Evans, D.F., Wennerström, H., 1999. The Colloidal Domain: Where Physics, Chemistry,
Biology, and Technology Meet, 2nd ed. Wiley-VCH, New York.
Freund, J.B., 2002. Electro-osmosis in a nanometer-scale channel studied by atomistic
simulation. J. Chem. Phys. 116, 2194–2200. https://doi.org/10.1063/1.1431543.
Gereben, O., Pusztai, L., 2011. On the accurate calculation of the dielectric constant from
molecular dynamics simulations: the case of SPC/E and SWM4-DP water. Chem.
Phys. Lett. 507, 80–83. https://doi.org/10.1016/j.cplett.2011.02.064.
Greathouse, J.A., Hart, D.B., Bowers, G.M., Kirkpatrick, R.J., Cygan, R.T., 2015.
Molecular simulation of structure and diffusion at smectite-water interfaces: using
expanded clay interlayers as model nanopores. J. Phys. Chem. C 119, 17126–17136.
https://doi.org/10.1021/acs.jpcc.5b03314.
Greenwood, R., 2003. Review of the measurement of zeta potentials in concentrated
aqueous suspensions using electroacoustics. Adv. Colloid Interf. Sci. 106, 55–81.
https://doi.org/10.1016/S0001-8686(03)00105-2.
Hartkamp, R., Siboulet, B., Dufreche, J.F., Coasne, B., 2005. Ion-specific adsorption and
electroosmosis in charged amorphous porous silica. Phys. Chem. Chem. Phys. 17
(2005), 24683–24695.
Hartkamp, R., Biance, A.L., Fu, L., Dufreche, J.F., Bonhomme, O., Joly, L., 2018.
Measuring surface charge: why experimental characterization and molecular
modeling should be coupled. Curr. Opin. Colloid Interface Sci. 37, 101–114.
Hiemstra, T., Van Riemsdijk, W.H., 2006. On the relationship between charge
distribution, surface hydration, and the structure of the interface of metal
hydroxides. J. Colloid Interface Sci. 301, 1–18. https://doi.org/10.1016/j.
jcis.2006.05.008.
Hocine, S., Hartkamp, R., Siboulet, B., Duvail, M., Coasne, B., Turq, P., Dufrêche, J.F.,
2016. How ion condensation occurs at a charged surface: a molecular dynamics
investigation of the Stern layer for water-silica interfaces. J. Phys. Chem. C 120,
963–973. https://doi.org/10.1021/acs.jpcc.5b08836.
Hockney, R., Eastwood, J., 1988. Computer Simulation Using Particles. CRC Press.
Hou, J., Li, H., Zhu, H., Wu, L., 2009. Determination of clay surface potential: a more
reliable approach. Soil Sci. Soc. Am. J. 73, 1658–1663. https://doi.org/10.2136/
sssaj2008.0017.
Hunter, R.J., 1981. Zeta Potential in Colloidal Science: Principles and Applications.
Academic Press, New York.
Jain, K., Mehandzhiyski, A.Y., Zozoulenko, I., Wågberg, L., 2021. PEDOT:PSS nano-
particles in aqueous media: a comparative experimental and molecular dynamics
study of particle size, morphology and z-potential. J. Colloid Interface Sci. 584,
57–66. https://doi.org/10.1016/j.jcis.2020.09.070.
Kraevsky, S.V., Tournassat, C., Vayer, M., Warmont, F., et al., 2020. Indentification of
montmorillonite particle edge orientations by atomic-force microscopy. Appl. Clay
Sci. 186, 105442.
Lammers, L.N., Bourg, I.C., Okumura, M., Kolluri, K., et al., 2017. Molecular dynamics
simulations of cesium adsorption on illite nanoparticles. J. Colloid Interface Sci. 490,
608–620.
Le Crom, S., Tournassat, C., Robinet, J.C., Marry, V., 2020. Influence of polarizability on
the prediction of the electrical double layer structure in a clay mesopore: a molecular
dynamics study. J. Phys. Chem. C 124, 6221–6232.
Leroy, P., Revil, A., 2004. A triple-layer model of the surface electrochemical properties
of clay minerals. J. Colloid Interface Sci. 270, 371–380. https://doi.org/10.1016/j.
jcis.2003.08.007.
Leroy, P., Revil, A., Kemna, A., Cosenza, P., Ghorbani, A., 2008. Complex conductivity of
water-saturated packs of glass beads. J. Colloid Interface Sci. 321, 103–117.
Leroy, P., Tournassat, C., Bernard, O., Devau, N., Azaroual, M., 2015. The electrophoretic
mobility of montmorillonite. Zeta potential and surface conductivity effects.
J. Colloid Interface Sci. 451, 21–39. https://doi.org/10.1016/j.jcis.2015.03.047.
Li, G.X., 2016. Advanced Soil Mechanics, second edition. Tsinghua University Press (in
Chinese).
Li, H., Wei, S., Qing, C., Yang, J., 2003. Discussion on the position of the shear plane.
J. Colloid Interface Sci. 258, 40–44. https://doi.org/10.1016/S0021-9797(02)
00077-2.
Li, H., Qing, C.L., Wei, S.Q., Jiang, X.J., 2004. An approach to the method for
determination of surface potential on solid/liquid interface: theory. J. Colloid
Interface Sci. 275, 172–176.
Liu, X., Hu, F., Ding, W., Tian, R., Li, R., Li, H., 2015. A how-to approach for estimation of
surface/Stern potentials considering ionic size and polarization. Analyst 140,
7217–7224. https://doi.org/10.1039/c5an01053e.
Liu, X., Ding, W., Tian, R., Li, R., Li, H., 2016. How ionic polarization affects Stern
potential: an insight into hofmeister effects. Soil Sci. Soc. Am. J. 80, 1181–1189.
https://doi.org/10.2136/sssaj2016.04.0095.
Liu, X., Ding, W., Tian, R., Du, W., Li, H., 2017. Position of shear plane at the clay-water
interface: strong polarization effects of counterions. Soil Sci. Soc. Am. J. 81,
268–276. https://doi.org/10.2136/sssaj2016.08.0261.
Liu, X.T., Yang, S., Gu, P.K., Liu, S., Yang, G., 2021. Adsorption and Removal of Metal
Ions by Smectites Nanoparticles: Mechanistic Aspects, and Impacts of Charge
Location and Edge Structure.
Lorenz, C.D., Travesset, A., 2007. Charge inversion of divalent ionic solutions in silica
channels. Phys. Rev. E 75, 061202.
Lorenz, C.D., Crozier, P.S., Anderson, J.A., Travesset, A., 2008. Molecular dyncamics of
ionic transport and electrokinetic effects in realistic silica channels. J. Chem. Phys. C
112, 10222–10232.
Lyklema, J. 1995. Fundamentals of Interface and Colloid Science, Academic Press, 2:
1–232.
Lylema, J., Rovillard, S., de Coninck, J., 1998. Electrokinetics: the properties of the
stagnant layer unraveled. Langmuir 14, 5659–5663.
Machesky, M.L., Wesolowski, D.J., Palmer, D.A., Ichiro-Hayashi, K., 1998.
Potentiometric titrations of rutile suspensions to 250◦
C. J. Colloid Interface Sci. 200,
298–309. https://doi.org/10.1006/jcis.1997.5401.
Mészáros, R., Jobbik, A., Varga, G., Bárány, S., 2019. Electrosurface properties of Na-
bentonite particles in electrolytes and surfactants solution. Appl. Clay Sci. 178,
105127. https://doi.org/10.1016/j.clay.2019.105127.
Miller, S.E., Low, P.F., 1990. Characterization of the electrical double layer of
montmorillonite. Langmuir 6, 572–578. https://doi.org/10.1021/la00093a010.
Moussa, C., Xu, J., Wang, X., Zhang, J., Chen, Z., Li, X., 2017. Molecular dynamics
simulation of hydrated Na-montmorillonite with inorganic salts addition at high
temperature and high pressure. Appl. Clay Sci. 146, 206–215. https://doi.org/
10.1016/j.clay.2017.05.045.
Newton, A.G., Kwon, K.D., Cheong, D.K., 2016. Edge structure of montmorillonite from
atomistic simulations. Minerals 6 (25).
Newton, A.G., Kwon, K.D., Cheong, D.K., 2017. Na-montmorillonite edge structure and
surface complexes: an atomistic perspective. Minerals 7 (78).
Nishimura, S., Yao, K., Kodama, M., Imai, Y., et al., 2002. Electrokinetc study of synthetic
smectites by flat plate streaming potential technique. Langmuir. 18, 188–193.
Ou, C.Y., Chien, S.C., Yang, C.C., Chen, C.T., 2015. Mechanism of soil cementation by
electroosmotic chemical treatment. Appl. Clay Sci. 104, 135–142. https://doi.org/
10.1016/j.clay.2014.11.020.
Panagiotou, G.D., Petsi, T., Bourikas, K., Garoufalis, C.S., et al., 2008. Mapping the
surface (hydr)oxo-groups of titanium oxide and its interface with an aqueous
solution: the state of the art and a new approach. Adv. Colloid Interface 142, 20–42.
Park, C., Fenter, P.A., Nagy, K.L., Sturchio, N.C., 2006. Hydration and distribution of ions
at the mica-water interface. Phys. Rev. Lett. 97, 1–4. https://doi.org/10.1103/
PhysRevLett.97.016101.
Park, S.H., Sposito, G., 2002. Structure of Water Adsorbed on a Mica Surface. Phys. Rev.
Lett. 89, 8–10. https://doi.org/10.1103/PhysRevLett.89.085501.
Parsons, D.F., Ninham, B.W., 2010. Charge reversal of surfaces in divalent electrolytes:
the role of ionic dispersion interactions. Langmuir 26, 6430–6436. https://doi.org/
10.1021/la9041265.
Peng, J., Ye, H., Alshawabkeh, A.N., 2015. Soil improvement by electroosmotic grouting
of saline solutions with vacuum drainage at the cathode. Appl. Clay Sci. 114, 53–60.
https://doi.org/10.1016/j.clay.2015.05.012.
Plimpton, S., 1995. Fast parallel algorithms for short-range molecular dynamics.
J. Comput. Phys. https://doi.org/10.1006/jcph.1995.1039.
Předota, M., Machesky, M.L., Wesolowski, D.J., 2016. Molecular origins of the zeta
potential. Langmuir 32, 10189–10198. https://doi.org/10.1021/acs.
langmuir.6b02493.
Rezaei, M., Azimian, A.R., Pishevar, A.R., 2018. Surface charge-dependent
hydrodynamic properties of an electroosmotic slip flow. Phys. Chem. Chem. Phys.
20, 30365–30375. https://doi.org/10.1039/c8cp06408c.
Ricci, M., Spijker, P., Stellacci, F., Molinari, J.F., Voïtchovsky, K., 2013. Direct
visualization of single ions in the Stern layer of calcite. Langmuir 29, 2207–2216.
https://doi.org/10.1021/la3044736.
Sahai, N., Sverjensky, D.A., 1997. Solvation and electrostatic model for specific
electrolyte adsorption. Geochim. Cosmochim. Acta 61, 2827–2848. https://doi.org/
10.1016/S0016-7037(97)00127-0.
Şans, B.E., Güven, O., Esenli, F., Çelik, M.S., 2017. Contribution of cations and layer
charges in the smectite structure on zeta potential of Ca-bentonites. Appl. Clay Sci.
143, 415–421. https://doi.org/10.1016/j.clay.2017.04.016.
H. Pei and S. Zhang
10. Applied Clay Science 212 (2021) 106212
10
Schmickler, W., Henderson, D., 1986. New models for the structure of the
electrochemical interface. Prog. Surf. Sci. 22, 323–420.
Shang, J.Q., 1997. Zeta potential and electroosmotic permeability of clay soils. Can.
Geotech. J. 34, 627–631. https://doi.org/10.1139/t97-28.
Shang, J.Q., Lo, K.Y., Quigley, R.M., 1994. Quantitative determination of potential
distribution in Stern-Gouy double-layer model. Can. Geotech. J. 31, 624–636.
https://doi.org/10.1139/t94-075.
Shen, X.Y., Bourg, I.C., 2021. Molecular dynamics simulations of the colloidal interaction
between smectite clay nanoparticles in liquid water. J. Colloid Interface Sci. 584,
610–621.
Siboulet, B., Hocine, S., Hartkamp, R., Dufreche, J.F., 2017. Scrutinizing electro-osmosis
and surface conductivity with molecular dynamics. J. Phys. Chem. C 121 (12),
6756–6769.
Sondi, I., Bišćan, J., Pravdić, V., 1996. Electrokinetics of pure clay minerals revisited.
J. Colloid Interface Sci. 178, 514–522. https://doi.org/10.1006/jcis.1996.0146.
Sverjensky, D.A., 2001. Interpretation and prediction of triple-layer model capacitances
and the structure of the oxide-electrolyte-water interface. Geochim. Cosmochim.
Acta 65, 3643–3655. https://doi.org/10.1016/S0016-7037(01)00709-8.
Sverjensky, D.A., 2005. Prediction of surface charge on oxides in salt solutions: Revisions
for 1:1 (M+L-) electrolytes. Geochim. Cosmochim. Acta 69, 225–257. https://doi.
org/10.1016/j.gca.2004.05.040.
Tombácz, E., Szekeres, M., 2004. Colloidal behavior of aqueous montmorillonite
suspensions: the specific role of pH in the presence of indifferent electrolytes. Appl.
Clay Sci. 27, 75–94. https://doi.org/10.1016/j.clay.2004.01.001.
Tournassat, C., Chapron, Y., Leroy, P., Bizi, M., Boulahya, F., 2009. Comparison of
molecular dynamics simulations with triple layer and modified Gouy-Chapman
models in a 0.1 M NaCl-montmorillonite system. J. Colloid Interface Sci. 339,
533–541. https://doi.org/10.1016/j.jcis.2009.06.051.
Vasu, N., De, S., 2010. Electroosmotic flow of power-law fluids at high zeta potentials.
Colloids Surf. A Physicochem. Eng. Asp. 368, 44–52. https://doi.org/10.1016/j.
colsurfa.2010.07.014.
Wander, M.C.F., Clark, A.E., 2008. Structural and Dielectric Properties of Quartz – Water
Interfaces 19986–19994. https://doi.org/10.1021/jp803642c.
Wang, J., Wang, M., Li, Z., 2006. Lattice Poisson-Boltzmann simulations of electro-
osmotic flows in microchannels. J. Colloid Interface Sci. 296, 729–736. https://doi.
org/10.1016/j.jcis.2005.09.042.
Wolthers, M., Charlet, L., Cappellen, P.V., 2008. The surface chemistry of divalent metal
carbonate minerals; a critical assessment of surface charge and poential data using
the charge distribution multi-site ion complexation model. Am. J. Sci. 308, 905–941.
Xie, Y., Fu, L., Niehaus, T., Joly, L., 2020. Liquid-solid slip on charged walls: the dramatic
impact of charge distribution. Phys. Rev. Lett. 125, 1–7. https://doi.org/10.1103/
PhysRevLett.125.014501.
Xu, R., 1998. Shear plane and hydrodynamic diameter of microspheres in suspension.
Langmuir 14, 2593–2597. https://doi.org/10.1021/la971404g.
Yates, D.E., Levine, S., Healy, T.W., 1974. Site-binding model of the electrical double
layer at the oxide/water interface. J. Chem. Soc. Farady Trans. 70, 1807.
Zhang, C., Liu, Z., Dong, Y., 2017. Effects of adsorptive water on the rupture of nanoscale
liquid bridges. Appl. Clay Sci. 146, 487–494. https://doi.org/10.1016/j.
clay.2017.07.002.
Zhang, H., Hassanali, A.A., Shin, Y.K., Knight, C., Singer, S.J., 2011. The water-
amorphous silica interface: analysis of the Stern layer and surface conduction.
J. Chem. Phys. 134 https://doi.org/10.1063/1.3510536.
Zhang, S., Pei, H., 2020. Rate of capillary rise in quartz nanochannels considering the
dynamic contact angle by using molecular dynamics. Powder Technol. 372,
477–485. https://doi.org/10.1016/j.powtec.2020.06.018.
Zhao, Q., Choo, H., Bhatt, A., Burns, S.E., Bate, B., 2017. Review of the fundamental
geochemical and physical behaviors of organoclays in barrier applications. Appl.
Clay Sci. 142, 2–20. https://doi.org/10.1016/j.clay.2016.11.024.
H. Pei and S. Zhang
![Applied Clay Science 212 (2021) 106212
2
study the transport properties of clay minerals (Leroy et al., 2008). The
shear plane is also an essential parameter that characterizes the hy
drodynamic properties of colloidal systems. The shear layer (the space
between the clay surface and shear plane) greatly impacts the hydro
dynamic motion of pore fluid and suspended particles (Xu, 1998). As a
boundary condition, the shear plane is used to calculate the electroos
motic flow velocity (Wang et al., 2006; Vasu and De, 2010). In soil
mechanics, water in the shear layer is regarded as tightly bound water
(Li, 2016). Compared to several measurement techniques such as elec
trophoresis, electroosmosis and electroacoustics of the zeta potential
determination (Greenwood, 2003), to date, the measurement of the
shear layer thickness mainly follows two methods. The primary method
is based on the PB equation or modified PB equation. The potential
profile in the EDL is obtained, and then the shear plane's position is
calculated according to the experimental measurement of zeta potential
(Ding et al., 2015; Liu et al., 2017). In another way, only a few in
vestigations directly measured the shear layer thickness according to its
hydrodynamic definition. In electrophoresis experiments, a surface of
shear beyond the particle interface can be observed due to the electro
static interactions between the nanoparticle and the applied electric
field (Hunter, 1981). Besides, diffusion coefficients of the colloidal
particles, counterions or water were measured to determine the shear
plane thickness without applying the electric field (Xu, 1998; Jain et al.,
2021). Bourikas et al. (2001) and Panagiotou et al. (2008) referred to the
position by those two methods as the shear plane uniformly in common.
However, the obtained position in the first method is actually the po
sition of zeta potential rather than that of the shear plane. Henceforth, to
distinguish the two positions, this study defines the zeta position
measured by the first method, which indicates the potential at the
calculated position corresponding to the zeta potential, and shear plane
position measured by the second method, at which the streaming ve
locity is zero.
Although a number of observations have been achieved, based on the
mentioned methods, the accurate position of the zeta potential in the
EDL remains controversial. To simplify, many researchers have assumed
that the zeta position is close to the Stern plane or even locate at it;
namely, the capacity value between two planes is much large in the TLM
(Hiemstra and Van Riemsdijk, 2006). However, Li et al. (2003) and Liu
et al. (2017) demonstrated that the zeta position is much closer to the
Gouy plane than the Stern plane after comparing the zeta potential with
surface potential and Stern potential in montmorillonite-water systems.
Molecular dynamics (MD) is a computational method that models
the molecular structure and predicts the behaviors of materials at the
atomic scale (Zhang and Pei, 2020). MD was used to explore the micro-
mechanism of clay minerals (Moussa et al., 2017; Zhang et al., 2017; Liu
et al., 2021). Chang et al. (1998), Parsons and Ninham (2010), and
Bourg and Sposito (2011) successfully applied the MD technique to
investigate the structure of EDL for clay-water systems. MD simulations
can determine the profiles of species in the diffuse layer and bulk elec
trolyte. Furthermore, MD can be utilized to simulate the electroosmotic
flows between two planar-charged surfaces. Based on the electroosmosis
simulation, scholars have also proposed methods to determine the zeta
potential of materials in different electrolyte solutions (Předota et al.,
2016; Biriukov et al., 2020). Therefore, MD is an appropriate tool to
study the zeta potential and shear plane position of clay.
In this paper, MD was performed to determine the zeta potential and
position of the shear plane in NaCl-montmorillonite systems with
different concentrations. The electric potential profiles, particularly for
the Stern potential and zeta potential, were calculated based on MD and
the EDL model. The shear layer thickness and zeta position were dis
cussed based on several measurement methods in the paper. The ca
pacitances in the TLM model were finally discussed.
2. Theoretical background
2.1. Stern model
In the classical EDL theory, the Poisson-Boltzmann equation de
scribes the distributions of ions in electrolyte solutions (Lyklema, 1995).
d2
φ
dz2
= −
4π
ε0εr
∑
i
c0
i viFexp
(
− viFφ(z)
RT
)
(1)
where φ(z) is the potential distribution in the diffuse layer; ε0 is vacuum
dielectric constant; εr is the relative permittivity of bulk water; ci
0
is the
ion concentration within the bulk solution; vi is the valence of the ith
cation species. F is the Faraday constant; R is the gas constant. In the
Stern model, by solving Eq. (1), the distribution of electric potential
away from the clay surface in the Stern layer and diffuse layer can be
expressed as (Shang et al., 1994):
φ(z) =
⎧
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎩
φ0 −
φ0 − φs
ds
z, z ≤ ds
2RT
vF
ln
eκ(z− ds)
+ tanh
vFφs
4RT
eκ(z− ds)
− tanh
vFφs
4RT
, z > ds
(2)
φ0 =
σ
Cs
+ φs (3)
where Debye-Huckel parameter κ =
̅̅̅̅̅̅̅̅̅̅̅̅
2v2F2c0
ε0εrRT
√
(m− 1
) represents the
reciprocal of the thickness of the diffuse layer; ds is the thickness of the
Stern layer; φsrepresents the Stern layer potential; φ0is the surface po
tential and can be calculated for a given surface charge density σ and
constant capacitance of the Stern layer Cs. Once ds and φs are deter
mined, the potential at any distance from the clay surface in the EDL can
be calculated. In other words, given a zeta potential, the position of the
zeta potential can be located by the Stern model.
Previous investigations (Bourg and Sposito, 2011; Ricci et al., 2013;
Bourg et al., 2017; Hocine et al., 2016) have demonstrated that the Stern
plane is located at the position of the OSSC. Therefore, the Stern layer
thickness can be determined from the density profiles of cations in MD
simulation. Li et al. (2004) and Hou et al. (2009) have proposed a
method to calculate the potential at the top end of the diffuse layer based
on the Poisson-Boltzmann equation. The potential is only determined by
the mean concentration of counterions in the diffuse layer ci and the
concentration in the equilibrium solution (Hou et al., 2009):
Fφs = − 2RTln
1 − 0.5
[
− A
(
ci
)
+
̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅
A
(
ci
)2
− 10.873
√ ]
1 + 0.5
[
− A
(
ci
)
+
̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅
A
(
ci
)2
− 10.873
√ ] (4)
where
A
(
ci
)
= 3.7183 +
6.8731c0
i
ci − c0
i
(5)
The derivation process of Eq. (4) is detailed in the Supporting In
formation. Liu et al. (2015, 2016) have applied the method to calculate
the Stern potential. For the first time, this method is combined with MD
simulations to solve the potential distribution in the EDL. The average
concentration of cations in the diffuse layer can be calculated from the
density profiles in MD simulation after locating the onset of the diffuse
layer. Thus, by means of Eqs. (2)-(5), the potential distribution is
determined using only the cations density profile near the clay surface.
H. Pei and S. Zhang](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)