КОНТИНУАЛЬНО-ЕЛЕКТРОСТАТИЧНЕ ДОСЛІДЖЕННЯ ЙОННИХ МІЦЕЛ ІЗ ВИКОРИСТАННЯМ АТОМІСТИЧНИХ МОДЕЛЕЙ
The key parameter related to the structure of the electric double layer of ionic surfactant micelles – electrostatic potential – is considered. A brief overview of experimental methods and theoretical models for estimating electrostatic potential- is given. The calculating method for the electrostat...
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| Дата: | 2021 |
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| Автори: | , , |
| Формат: | Стаття |
| Мова: | Англійська |
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V.I.Vernadsky Institute of General and Inorganic Chemistry
2021
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Репозитарії
Ukrainian Chemistry Journal| _version_ | 1871465714380963840 |
|---|---|
| author | Farafonov, Vladimir Lebed, Alexander Mchedlov-Petrossyan, Nikolay |
| author_facet | Farafonov, Vladimir Lebed, Alexander Mchedlov-Petrossyan, Nikolay |
| author_institution_txt_mv | [
{
"author": "Vladimir Farafonov",
"institution": "V. N. Karazin Kharkiv National University"
},
{
"author": "Alexander Lebed",
"institution": "V. N. Karazin Kharkiv National University, Svobody sq., 4, Kharkiv, 61022, Ukraine"
},
{
"author": "Nikolay Mchedlov-Petrossyan",
"institution": "V. N. Karazin Kharkiv National University, Svobody sq., 4, Kharkiv, 61022, Ukraine"
}
] |
| author_sort | Farafonov, Vladimir |
| baseUrl_str | https://ucj.org.ua/index.php/journal/oai |
| collection | OJS |
| datestamp_date | 2026-07-22T08:23:46Z |
| description | The key parameter related to the structure of the electric double layer of ionic surfactant micelles – electrostatic potential – is considered. A brief overview of experimental methods and theoretical models for estimating electrostatic potential- is given. The calculating method for the electrostatic potential based on a numerical solution of the Poisson-Boltzmann equation using an atomistic model of anionic surfactant micelle - is proposed. The parameters necessary for the construction of atomistic models - are obtained from molecular dynamic modeling.  The electrostatic potentials for the micelles of sodium dodecyl sulfate and cetyltrimethylammonium bromide at different ionic strengths - were calculated by this method. The results are discussed in comparison with the values calculated in the simplified model, the Ohshima – Healy – White equation. |
| doi_str_mv | 10.33609/2708-129X.87.06.2021.55-69 |
| first_indexed | 2025-09-24T17:43:39Z |
| format | Article |
| fulltext |
55
УДК 544.77.022.532+537.213+537.222.1+537.226.1+004.942 doi: 10.33609/2708-129X.87.06.2021.55-69
CONTINUUM ELECTROSTATICS INVESTIGATION OF IONIC MICELLES
USING ATOMISTIC MODELS
V. S. Farafonov,* A. V. Lebed, N. O. Mchedlov-Petrossyan
V. N. Karazin Kharkiv National University, Svobody sq., 4, Kharkiv, 61022, Ukraine
*e-mail: farafonov@karazin.ua
The key parameter related to the structure of the electric double layer of ionic surfactant
micelles – electrostatic potential – is considered. A brief overview of experimental methods
and theoretical models for estimating electrostatic potential is given. The calculating meth-
od for the electrostatic potential based on a numerical solution of the Poisson-Boltzmann
equation using an atomistic model of anionic surfactant micelle is proposed. The parameters
necessary for the construction of atomistic models are obtained from molecular dynamic
modeling. The electrostatic potentials for the micelles of sodium dodecyl sulfate and cetyl-
trimethylammonium bromide at different ionic strengths were calculated by this method.
The results are discussed in comparison with the values calculated in the simplified model, the
Ohshima – Healy – White equation.
Keywords: surfactant micelle, Stern layer, electrostatic potential, Poisson – Boltzmann
equation, solvent-accessible surface.
INTRODUCTION. Micelles of colloidal
surfactants are widely used in different fields
of chemistry and related sciences and techno
logies [1–3]. Aqueous micellar solutions are
almost transparent and thermodynamically
stable (i.e., reversible) liquid systems [3–5].
They may be considered as the most simple and
at the same time most important and unique
representatives of the so-called organized solu-
tions [6, 7]. Hereafter, we shall consider ion-
ic surfactants, which are diphilic molecules
consisting of a hydrophobic portion and ionic
group. In fact, they are colloidal electrolytes
with a hydrocarbon chain containing from 8 to
18 carbon atoms.
Micelles of a given colloidal surfactant are
formed in aqueous solutions upon reaching
a certain concentration, the so-called critical
micelle concentration (CMC) and at a tempe
rature equal or above the so-called Krafft point.
The driving force of micellization in water is
the hydrophobic interaction [3–5]. The hydro
philic parts of the surfactants are directed to-
wards the bulk (aqueous) phase, while the
hydrophobic tails form the internal hydrocar-
bon core. According to the generally accepted
model, these surfactant aggregates are highly
porous, highly hydrated, disordered clusters in
a state of dynamic equilibrium with monomers
in the bulk phase. In the concentration range
CONTINUUM ELECTROSTATICS INVESTIGATION OF IONIC MICELLES USING ATOMISTIC MODELS
56 ISSN 2708-129X. Укр. хім. журн., 2021
PHYSICAL CHEMISTRY
about the CMC, the micelles are several nm-
sized and practically sphere-shaped.
Although many models of the structure of
micelles of ionic surfactants can be found in
the literature, they all agree on the main point.
Namely, an electric double layer (EDL) appears
at the micelle-water interface. The ionic head
groups form the inner layer characterized by
surface electrostatic potential Ψ0. It attracts the
counterions from solution, which form the outer
layer. Different approaches were proposed
for its description, starting from the Goüy –
Chapman model, where the counterions are
considered size-less and adopting Boltzmann
distribution of concentration immediately af-
ter the inner layer, and then the Stern model,
where some portion of counterions is assumed
to adsorb strongly and form a thin dense lay-
er (subsequently called Stern layer), while
Boltzmann distribution starts from its outer
boundary. The thickness of the Stern layer is
usually denoted as δ, and the electrostatic po-
tential at its boundary is hence denoted as Ψδ.
Various departures from this model were de-
veloped beginning with the classical papers by
Stigter [8, 9]. For the counterions in the Stern
layer, mobile and localized adsorption mo
dels were considered by Rathman and Scame-
horn [10]. Treating micelle formation by small
system thermodynamics, Gilányi suggested
a diffuse monolayer structure instead of the
idealized Goüy – Chapman model [11]. This
listing can be continued up to the latest works
utilizing molecular dynamics (MD) simula-
tions [12–14].
In many cases, the electrostatic potential is
considered as a key parameter governing the
properties of the micellar solutions of ionic
surfactants, affecting their behavior in photo-
physical and analytical processes, chromatog-
raphy, shifting of equilibrium states and acce
leration of a variety of reactions, binding vari-
ous compounds, in synthesis of nanoparticles,
etc. [1–4,7,15]. Consequently, there are many
approaches to determine its value for ionic mi-
celles.
Among the experimental methods, the
most popular and universal one is based on the
application of colored or fluorescent acid-base
indicators [3,15–20]. There, Ψ is determined
using the so-called apparent dissociation con-
stant, app
aK , of the indicator bound to the mi-
celle and the so-called intrinsic constant, i
aK ,
Eq. 1.,
i app
a a
ln10 (p p )RT K K
F
Ψ= × − (1)
Here R, T, F have their usual meanings.
The apparent constant is in fact a two-phase
equilibrium constant because the pH value is
determined in the bulk phase, whereas the ac-
id-base couple of the indicator dye is located
in the micellar palisade. As the i
apK value, the
app
apK value of the same indicator in micelles
of nonionic surfactant is usually used. Some
more sophisticated assumptions to determine
these values are also used [3, 15, 16, 19, 20].
A similar approach is based on using acid-base
equilibria of stable free radicals and the elec-
tronic spin resonance method [21]. A method
of monitoring the fluorescence changes along
with electrostatic potential alterations was de-
veloped for membranes and vesicles [22]. Uti-
lization of the small-angle neutron scattering
method for determination of the surface elec-
trical charge and potential of surfactant mi-
celles is also reported [23].
Yet, all these experimental methods are
based on the use of molecular probes, which
makes their results depending on the chosen
V. S. Farafonov, A. V. Lebed, N. O. Mchedlov-Petrossyan
57https://ucj.org.ua
UCJ № 6 / Vol. 87
probe. One of the reasons is the variation of the
probe location in the micelle. In general, the
probes are considered to be located within the
Stern layer, which implies the Ψ in Eq. 1 should
be somewhere between Ψ0 and Ψδ. Another
reason is the possibility of specific interactions
between the probe and surfactant. In total,
for sodium dodecyl sulfate (SDS) micelles the
variation of Ψ, as found using Eq. 1, reaches
about 200 mV [15].
This makes the demand for approaches that
consider the sole micelles and, hence, are free
from this issue. In this connection, the most
relevant approach is the electrophoretic meas-
urement of the electrokinetic (or ζ-) potential
of colloid particles. No foreign probes are re-
quired, and the obtained ζ values characterize
solely the particles and the solution. However,
ζ-potential is a priori of a lesser magnitude
than Ψ because corresponds not to the surface,
but to some slipping plane located ~1 nm far
from it. Hence it is just a lower estimate of |Ψ0|.
It was supposed that the rough and dynamical
character of the micelle surface due to mono-
mer motion reduces the potential fade near the
surface smearing the difference between the Ψ
and ζ values [24]. However, this assumption
was later considered improbable because the
slipping plane is located about 0.8 nm farther
than the Stern layer [25].
So that, at present, only theoretical methods
to the stated problem are available. The mostly
used ones are based on solving the Poisson –
Boltzmann (PB) equation, Eq. 2. This equation
establishes the relation between the charge
density of the system and the electrostatic po-
tential created by it. The density is represented
as a sum of two contributions: the fixed one,
which is constant and belongs to the solute, and
the mobile one, which depends on the electro-
static potential distribution in a self-consistent
manner and represents the coions and coun-
terions in the solution [26]. It is this equation
that grounds the Goüy – Chapman model of
EDL, as well as the Debye – Hückel theory of
strong electrolytes.
0 0 0
( , , )( , , ) ( , , )( ( , , ) ( , , )) sinh
4 4 2
f x y zx y z Fc F x y zx y z x y z
RT
∇ ⋅ ∇ = − = − + −
ρρ ψε ψ
πε πε πε
(2)
Here ψ(x, y, z) is the electrostatic poten-
tial, 0ε has its usual meaning, ρ(x, y, z) is total
charge density, ε(x, y, z) is the relative permit-
tivity (in the general case it is a function of co-
ordinates, too), ρf is the fixed charge density, c
is the total concentration of the electrolyte (as-
sumed here to consist of single-charged ions).
Overall, the solvent is treated here as a con-
tinuum characterized with uniform relative
permittivity, thus, disregarding the discrete
nature of solvent molecules and ions. The so
lute is also treated as a continuum with some
other value of relative permittivity. The ions are
originally considered to be point charges oc-
cupying no volume. Nevertheless, by adjusting
boundary conditions the ions may be attribu
ted with a radius Ri for interaction with the so
lute making them unable to approach it tightly,
as is assumed in the Stern model of EDL.
The PB equation is a non-linear differential
equation of the second order, which makes its
analytical solving possible just for the simplest
geometries of charge distribution like uni-
formly charged planes, spheres, or cylinders.
It can be linearized to facilitate solving, but
this procedure is correct for relatively weakly
CONTINUUM ELECTROSTATICS INVESTIGATION OF IONIC MICELLES USING ATOMISTIC MODELS
58 ISSN 2708-129X. Укр. хім. журн., 2021
PHYSICAL CHEMISTRY
charged solutes only. The approximate analy
tical solution of this equation in its non-linear
form for a uniformly charged spherical particle
with radius r was deduced by Ohshima, Healy,
and White, Eq. 3 [27]. It will be called Ohshi-
ma – Healy – White (OHW) equation further
in the text.
The authors also derived the formula for cy-
lindrical particles [27]. If the qs value is known,
Ψ0 can be calculated and vice versa. In summa-
ry, there are two parameters for a micelle rep-
resented as a homogeneously charged sphere:
radius and molecular area of a head group. In
this respect, it is worth to mention the work
by Lukanov and Firoozabadi who numerically
solved the PB equation modified to incorpo-
rate non-electrostatic solute – ions interactions
for similar geometry [28].
In principle, Eq. 3 may be used to predict
the potential of Stern layer Ψδ instead of the
surface potential Ψ0. In such a case, the surface
charge density is reduced due to the presence
of adsorbed counterions. Hence, the qs value
must be multiplied by (1 – β) to reflect the par-
tial neutralization of the micelle charge, where
β is the degree of neutralization. Additionally,
the r value should be increased to match the
total size of the micelle together with its Stern
layer. It was shown that for well-defined col-
loidal systems like aqueous micellar solutions
of SDS or cetyltrimethylammonium bromide
(CTAB) such calculations give plausible results
[16, 17].
Yet, despite the usefulness of the described
equation, it is a severe simplification origina
ting from the inability to analytically solve the
PB equation for a case of complex geometry.
However, at present one is no more constrained
to analytical solutions due to the availability of
computers allowing the direct numerical sol
ving of this equation. Nowadays there is a va-
riety of software developed for this sake [29].
There, the input is the atomistic model of the
system of interest (consisting of the positions
of atoms, their van der Waals radii and atom-
ic point charges) and, the output is the corre-
sponding distribution of the electrostatic po-
tential in surrounding space.
[ ]
1
2
0
2 2 2
8ln cosh( / 4)2 2sinh( / 2) 1
cosh ( / 4) ( ) sinh ( / 2)s
YRTq Y
F r Y r Y
ε ε κ
κ κ
= + +
(3)
2
0
2
0
2; ;
4
aggr
s
N Fe F Iq e Y
s r RT RT
κ
π ε ε
Ψ
= = = =
Here, qs is the surface charge density, s is
the molecular area of a head group on the mi-
cellar surface, Naggr is the aggregation number,
e is the elementary charge, κ is the reciprocal
Debye length, I is the ionic strength of a solu-
tion. (represented as a three-dimensional grid
of ψ(x, y, z) values). The solution parameters
(relative permittivity, temperature, ionic
strength) are also required. The main benefit
of the numerical approach is that the charge
distribution is considered in all its complexity
with no need to simplify or impose some limi-
tations on its geometry. Additionally, the inter-
polation schemes can be applied for ε(x, y, z) at
the solute-solvent boundary to form a smooth
contact between the two continuums. Hence,
V. S. Farafonov, A. V. Lebed, N. O. Mchedlov-Petrossyan
59https://ucj.org.ua
UCJ № 6 / Vol. 87
this technique is widely used for studying va
rious biomolecules and biological assemblies
(lipid membranes, individual proteins, and
protein complexes including viruses) having
an elaborate structure, where it reveals the re-
gions of high positive or negative charge den-
sity [30–33]. However, to our knowledge, such
objects as surfactant micelles have not been in-
vestigated with this method up to date.
This work aims to fill this gap and perform
the numerical computations of the electrostatic
potential around ionic micelles based on their
accurate atomistic models instead of the sim-
plified ones. We chose SDS and CTAB micelles
for this study as the typical and well-defined
ones. The produced results will be contrasted
with the predictions of the simplified model
given by the OHW equation.
EXPERIMENT AND DISCUSSION OF
THE RESULTS. Computational procedure. The
parameters of micelles required by the OHW
equation (radius, surface area, degree of neu-
tralization) were calculated grounding on the
molecular dynamics simulations of SDS and
CTAB micelles composed from 60 (the former)
or 80 and 95 (the latter) surfactant monomers.
Unlike SDS, for CTAB there is some spread in
the experimental data about aggregation num-
ber, hence, two values of Naggr were considered.
We used the last 10 ns interval of the trajecto-
ries generated during our previous investiga-
tions of these systems, for details please refer
to the corresponding papers [13, 14]. The radi-
us was calculated in two ways: 1) as the gyra-
tion radius of hydrocarbon core multiplied by
5 / 3 , and 2) as the position of the maximum
of the radial distribution function (RDF) be-
tween micelle center of mass and S (N) head
groups atoms. Both ways provided similar val-
ues (within 0.05 nm).
The solution parameters submitted to the
OHW equation were: ε = 78.5; T = 298.15 K;
I = 0.05 M = 50 mol m–3.
To determine the degree of neutralization
we calculated the radial distribution func-
tion g(d) of counterions around the micelle,
Eq. 4. It shows how much the density of coun-
terions on distance d from the surface dif-
fers from the cell-average value. Preliminary,
the function N(d) was calculated as the total
number of ions laying within distance d from
the surface. The peaks on RDF’s indicate the
regions of increased ions density, hence, it is
reasonable to define the Stern layer as the peak
in g(d).
0 0
2
0
( ) 1 ( )( )
( )
1 1 ( )
4 ( )
d dN dg d
dV d
dN d
r d dd
ρ
ρ ρ
ρ π
= = =
=
+
(4)
Where ρ0 is the cell-average number density
of counterions, ρ(d) is the number density of
counterions at distance d from micelle, dN is
the number of counterions within an infinitely
thin layer around micelle at distance d, dV is
the volume of this layer, which is approximat-
ed by the volume of a spherical layer, r is the
micelle radius.
The input for PB computations was ob-
tained in two stages. At first, the distribution
of electrostatic potential around micelles was
calculated using the Adaptive Poisson – Boltz-
mann solver (APBS) software [34,35]. The box
size was chosen similar to the MD cell size,
namely 8 nm for SDS and 10 nm for CTAB,
and grid dimensions were 257×257×257. The
micelle was centered in the box. Relative per-
mittivity was set to 78.5 for water and 2 for mi-
celle (matching higher alkanes like n-decane).
Radius of solvent molecules was set to the
CONTINUUM ELECTROSTATICS INVESTIGATION OF IONIC MICELLES USING ATOMISTIC MODELS
60 ISSN 2708-129X. Укр. хім. журн., 2021
PHYSICAL CHEMISTRY
default value that is 0.14 nm. Because we deal
with highly charged systems, non-linearized
version of Poisson-Boltzmann equation was
chosen. Both the cation and the anion of the
background electrolyte were single-charged
and were attributed with radius 0.1 nm for SDS
and 0.2 nm for CTAB matching the ionic radii
of Na+ and Br–, respectively [36]. The computa-
tions were carried out at a range of electrolyte
concentrations from 0.05 M to 1 M including
the case of no electrolyte.
The atomistic models were obtained from
the above stated MD simulations of SDS and
CTAB micelles. For each surfactant, 3 models
were prepared by extracting instantaneous
configurations from MD trajectory at 10 ns,
15 ns, and 20 ns. The atomic point charges
were taken from the used potential models,
which belong to the optimized potentials for
liquid simulations – all-atom (OPLS-AA) force
field [13, 14]. The atomic radii were provided
by gmx editconf utility executed with -mead
option specified.
The second stage was averaging the spa-
tial distribution of the electrostatic potential
ψ(x, y, z) over relevant regions to get meaning-
ful average values, which can be interpreted and
compared with predictions of other methods.
The quantities predicted by analytical models
of micelles are usually related to the distance
from the micelle surface, while the micelle
shape itself is considered as a simple geometric
form (sphere, cylinder etc.). However, in our
case it appears non-trivial to define the concept
of “distance from micelle surface” because the
atomistic models have rough, irregular shapes
preventing the application of simple geometric
criteria like distance from the micelle center of
mass. Consequently, it is difficult to outline the
regions for averaging the ψ(x, y, z).
A B
Fig. 1. A: The principle of calculation of solvent-accessible surfaces. Dark blue circles – CH2 groups
of hydrocarbon core, dotted orange circle – a head group, red line “vdW” – van der Waals surface of
hydrocarbon core, light blue outlined circles – solvent probes with radius RW, light green outlined circle –
solvent probe with radius (RW + x), solid and dashed violet lines – SAS of hydrocarbon core obtained with
the smaller and the bigger probes, respectively. The former SAS establishes the origin of the distance scale,
d = 0, and the points of the latter SAS are considered as located on distance x from the hydrocarbon core.
B: SDS micelle within its solvent-accessible surface. The regions of the surface located within 0.4 nm
of S atoms are highlighted orange, and the regions within 0.5 nm of two S atoms are highlighted white.
S and O atoms are shown as red and yellow spheres, respectively.
V. S. Farafonov, A. V. Lebed, N. O. Mchedlov-Petrossyan
61https://ucj.org.ua
UCJ № 6 / Vol. 87
We solved this problem by employing sol-
vent-accessible surfaces (SAS). According to
the definition, SAS is the locus of the center of
a spherical probe as it rolls over the accessible
regions of the van der Waals surface of object.
The probe radius R is usually set to the approx-
imate radius of water molecules RW that is 0.14
nm. In turn, the van der Waals (also called mo-
lecular) surface is the union of atom-centered
spheres each having the radius set to the van
der Waals radius of the corresponding atom
[37]. Here we generated the solvent-accessible
surface of the hydrocarbon core of micelles us-
ing the gmx sasa utility with the default value of
the probe radius (0.14 nm) and increased pre-
cision (parameter -dots 100). The head groups
OSO3
– and CH2N(CH3)3
+ were omitted in the
calculation as not forming the surface but pro-
truding from it. This SAS was considered as the
actual micelle surface, and it set the origin for
measuring distance d towards a bulk solution:
all points of this SAS were attributed with d = 0.
Further, a series of SAS was generated using
larger probe radii of (0.14+x) nm where x
ranged from 0.05 to 1 nm. These surfaces were
larger than the original one and represented an
accurate equivalent to the notion “the place of
points at distance x from micelle”. The princi-
ple is illustrated in Figure 1A.
These surfaces were used then as the regions
for averaging the electrostatic potential distribu-
tion. The average of ψ(x, y, z) evaluated over the
surface generated using probe radius (0.14+x)
nm was considered as the Ψ value at distance
d = x from micelle surface. Because each surface
was intrinsically represented as a set of points,
and the potential distribution was a three-di-
mensional grid, calculation of the average value
was done by enumerating the surface points and
summing up the values of potential there, which
were in turn determined via trilinear interpola-
tion between the neighboring grid points. Im-
portantly, not all surface points were used here,
but only a subset: we excluded the SAS points
located closer than 0.4 nm to any S (N) atom
of head groups and the points within 0.5 nm of
two or more S (N) atoms at the same time. The
justification is provided in the Results section
below. An example of a SAS with the indication
of the excluded regions is shown in Figure 1B.
Table 1
The parameters of micelles and subsequent solutions of OHW equation
surfactant Naggr r, nm s, nm2 |Ψ0|, mV β |Ψδ|, mV
SDS 60 1.81 0.686 139 0.48 94
CTAB 80 2.22 0.774 134 0.61 75
CTAB 95 2.35 0.731 138 0.65 74
Table 2
Electrostatic potential of micelles from numerical solution of PB equation at I = 0.05 M
surfactant Naggr |Ψ0|, mV |Ψ(0.2)|, mV |Ψ(0.4)|, mV |Ψδ|, mV
SDS 60 100 85 68 75
CTAB 80 149 106 76 70
CTAB 95 151 111 85 78
CONTINUUM ELECTROSTATICS INVESTIGATION OF IONIC MICELLES USING ATOMISTIC MODELS
62 ISSN 2708-129X. Укр. хім. журн., 2021
PHYSICAL CHEMISTRY
Firstly, we focus on the theoretical predic-
tions of the OHW equation. The parameters
of micelles obtained from MD simulations (r,
s, β) and the subsequent solutions of Eq. 2 are
collected in Table 1. For Ψδ calculation, the
micelle radius r was increased by 0.1 nm in
order to reflect the incorporation of neighbor-
ing ions (the s value was accordingly changed,
too). We deliberately chose the addition less
than δ because the visual examination of MD
trajectories revealed that most of these coun-
terions are located in between the head groups
or alongside them, therefore, do not advance
far towards a bulk solution. The analysis of
g(d) graphs suggested placing the Stern lay-
er boundary δ at about 0.32 nm for SDS and
0.45 nm for CTAB, Figure 2. The correspond-
ing β values found as N(δ)/Naggr are 0.48 and
0.61, respectively. We note that the experimen-
tally determined β values vary within the range
of 0.5–0.9 for different surfactants and depend
on the method used [3, 15].
There is no significant difference found be-
tween Ψ values of CTAB micelles composed of
80 and 95 monomers indicating the negligible
effect of the micelle size in this case.
Now we turn to the results of PB calcu-
lations. From the qualitative point of view,
visual examination of the electrostatic poten-
tial distribution depicted as equipotential sur-
faces (isosurfaces) reveals its complex shape.
ψ(x, y, z) is quite uniform near the regions
where the hydrocarbon core is exposed to wa-
ter, however, it sharply increases in magnitude
upon approaching the head groups. Its gradi-
ent increases there, too, as is evidenced by a
tighter spacing between isosurfaces: there the
magnitude reaches ≥260 mV/nm, while far
from head groups it drops to ≤80 mV/nm.
To get quantitative results we averaged
ψ(x, y, z) following the procedure described in
the Computational procedure section. The pre-
vious observations motivated us to exclude the
vicinities of head groups from averaging: these
regions, while having comparatively small total
area, are located at ψ(x, y, z) values very dif-
ferent from the rest of the surface. Therefore,
keeping them would distort the average.
A B
Fig. 2. A: the number of counterions at given distance from micelle surface; B: radial distribution
function of counterions around the micelle. The origin corresponds to the solvent-accessible surface of
hydrocarbon core.
V. S. Farafonov, A. V. Lebed, N. O. Mchedlov-Petrossyan
63https://ucj.org.ua
UCJ № 6 / Vol. 87
The produced dependences of Ψ on dis-
tance from micelle surface and ionic strength
are presented in Figures 3, 4A and summa-
rized in Table 2. The profiles demonstrate
the expected behavior of gradual monotonic
decrease towards zero with distance and ion-
ic strength. At similar conditions, Ψ around
CTAB micelles have a larger magnitude than
around the SDS ones. It should be emphasized
that the data for high ionic strengths is rather
evaluative because all the computations were
done using the same atomistic model. While
at such conditions the micelles usually change
shape from ellipsoidal to rod-like.
A feature of the numerical computa-
tions is the possibility to handle the case of a
“bare” micelle in the absence of counterions
and background electrolyte. In this case I = 0
and κ–1 tends to infinity preventing the use of
the OHW equation. The estimated Ψ0 values
equaled –535 mV and +630 mV for SDS and
CTAB micelles, respectively.
A B
Fig. 3. Profiles of electrostatic potential fade with distance for SDS (A) and CTAB (B) micelles at dif-
ferent ionic strengths.
A B
Fig. 4. A: Dependence of surface electrostatic potential on ionic strength for SDS and CTAB micelles;
B: Profile of electrostatic potential fade with distance for CTAB micelles of different sizes at I = 0.05 M.
CONTINUUM ELECTROSTATICS INVESTIGATION OF IONIC MICELLES USING ATOMISTIC MODELS
64 ISSN 2708-129X. Укр. хім. журн., 2021
PHYSICAL CHEMISTRY
Table 3
Effect of counterions radius on Ψ0 from the numerical solution of PB equation
at I = 0.05 M. The original values are highlighted in bold
surfactant Naggr
|Ψ0|, mV
Ri = 0.2 nm Ri = 0.1 nm Ri = 0
SDS 60 130 100 71
CTAB 80 149 106 74
To check the robustness of the presented
results, we varied several factors, which can af-
fect the computed ψ(x, y, z) distributions.
The graphs presented above were produced,
basing on a single model for each micelle.
To estimate the effect of the choice of con-
figuration, we repeated the computations at
I = 0.05 M using two other models. The pro-
duced Ψ profiles were within 5 mV at d = 0 and
within 3 mV for larger d indicating that using
a single model is enough to get a correct result.
The possible reason is that the main structur-
al motifs of the micelle surface are observed at
any time during the evolution of its shape. We
note that the values in Table 2 are averages over
three models.
Next, because of the uncertainty with Naggr
of CTAB micelles we repeated the computa-
tions using the models with Naggr = 95 instead of
80. Both results are confronted in Table 2 and
Figure 4B. It is seen that while some difference
is present, the profiles agree well: the Ψ values
almost coincide at the shortest (≤0.15 nm) and
longest (≥0.7 nm) d values. In between the dis-
crepancy is no more than 10 mV that is mo
derate. This is in line with the behavior of the
OHW equation.
A crucial factor is the radius of ions of the
background electrolyte. We tested the values
0.2 nm for SDS (instead of 0.1 nm) and 0.1 nm
for CTAB (instead of 0.2 nm). Also, the point-
like case (Ri = 0) was checked. Unlike the previ-
ous factors, here the influence appeared signif-
icant for both kinds of micelles: each decrease
of Ri by 0.1 nm reduced |Ψ0| by 30–40 mV, Ta-
ble 3. This effect is expected due to the appear-
ance of counterions closer to the surface.
Finally, we tested the influence of the com-
putation parameters. The box size increase to
12×12×12 nm3 and the grid dimensions de-
crease to 129×129×129 were found to cause
the negligible difference of Ψ0 (<3 mV).
First of all, the comparison of results in Ta-
bles 1 and 2 shows that both OHW equation
and numerical computations provide the mag-
nitude of Ψ0 in the range of 100–150 mV for
both kinds of micelles. The closer look, how-
ever, reveals a discrepancy: while OHW equa-
tion predicts almost equal |Ψ0| for SDS and
CTAB (within 5 mV), their numerical results
are much different (by ~50 mV). This behavior
seems rather surprising because both ways are
based on the same theory (that is Eq. 2) and
requires explanation.
The main distinction between the ways is
the geometry of the model: while the OHW
equation represents the micelle with head
groups as a smooth sphere with the uniformly
distributed charge, in numerical computations
the protrusion of head groups from the hydro-
carbon core and their discrete charge is taken
into account.
V. S. Farafonov, A. V. Lebed, N. O. Mchedlov-Petrossyan
65https://ucj.org.ua
UCJ № 6 / Vol. 87
The second distinction is more subtle but
no less important: in the derivation of the
OHW equation, the ions were treated as point
charges implying they approach infinitely close
to the solute. Oppositely, the used software for
numerical solution allowed tuning the dis-
tance of the closest approach between the ions
and the solute in terms of ions radii Ri. As was
mentioned before, this parameter strongly af-
fects the results. The examination of Table 3
shows that if equal radii are used, the Ψ0 values
of SDS and CTAB become much closer (with-
in 7–20 mV) resembling the behavior of the
OHW equation.
Still, there remains the difference in pre-
dicted Ψ0 magnitude. Surprisingly, the closest
match between OHW and numerical predic-
tions is observed for Ri = 0.2 nm instead of
zero size. We attribute this fact to the geome-
try of the models via the following qualitative
consideration. The discrete head groups have
a high charge density. When their charge is
smeared across the whole surface, the charge
density proportionally decreases. As a result,
such uniformly charged surface exerts pro-
portionally weaker attraction to counterions.
This, however, leads to exponentially smaller
concentration and charge density of the ap-
proached counterions. Therefore the origi-
nal surface charge becomes neutralized to a
lesser extent resulting in higher |Ψ0| values.
To achieve the same magnitude in numerical
computation one has to artificially prevent
counterions from accumulating around high-
ly charged head groups by attributing the ions
with ~0.2 nm radii.
Considering the surface potential of “bare”
micelles without counterions, its large magni-
tude against the |Ψ0| values in Table 2 indicates
the effect exerted by the outer part of EDL.
Turning to the potential of Stern layer Ψδ,
both ways agree that for CTAB it equals ~75 mV.
For SDS the numerical result is –75 mV. Its
direct comparison with the OHW solution is
impossible because of the discrepancies de-
scribed above.
To summarizing, in our opinion the distinc-
tion of results of the used ways is caused by the
different treatment of head groups and coun-
terions. Among the two ways, we have a pre
ference for the numerical solution because it
implies fewer simplifications and assumptions
and, thus, provides a more realistic treatment
of the examined systems. Nevertheless, the
analytical equation produces the values of cor-
rect magnitude with much less computational
effort.
CONCLUSIONS. The surface electrosta
tic potential Ψ0 of micelles of common sur-
factants sodium dodecyl sulfate and cetyltri-
methylammonium bromide was estimated
by two theoretical approaches based on the
Poisson – Boltzmann equation. The analy
tical equation derived for uniformly charged
spherical particles and point-like counterions
was found to provide similar Ψ0 values for both
micelles. Oppositely, the numerical solution
with the geometry of the systems accurate-
ly reproduced and ions attributed with radii
equal to their ionic radii resulted in pronounc-
edly higher Ψ0 of CTAB micelles than of SDS
ones. Namely, at background electrolyte ionic
strength of 0.05 M the values equaled +150 mV
and –100 mV, respectively. By the same ap-
proach, the electrostatic potential on the
boundary of the Stern layer Ψδ was found to
be ca. +74 mV and –75 mV, respectively. The
discrepancy between the approaches was ex-
plained on the basis of the assumptions and
simplifications made in them.
CONTINUUM ELECTROSTATICS INVESTIGATION OF IONIC MICELLES USING ATOMISTIC MODELS
66 ISSN 2708-129X. Укр. хім. журн., 2021
PHYSICAL CHEMISTRY
ACKNOWLEDGEMENTS. The authors
thank the Ministry of Education and
Science of Ukraine for financial sup-
port in the framework of the project “Novel
nanomaterials based on the lyophilic self-as-
sembled systems: theoretical prediction, ex-
perimental investigation, and biomedical
applications” (0120U101064).
КОНТИНУАЛЬНО-ЕЛЕКТРОСТАТИЧНЕ ДОСЛІ-
ДЖЕННЯ ЙОННИХ МІЦЕЛ ІЗ ВИКОРИСТАН-
НЯМ АТОМІСТИЧНИХ МОДЕЛЕЙ
В. С. Фарафонов,* О. В. Лебідь,
М. О. Мчедлов-Петросян
Харківський національний університет
імені В. Н. Каразіна, пл. Свободи, 4, Харків,
61022, Україна
*e-mail: farafonov@karazin.ua
Розглянуто питання про ключовий па-
раметр, пов’язаний з будовою подвійно-
го електричного шару міцел йоногенних
ПАР, – електростатичний потенціал. Наве-
дено короткий огляд експериментальних
методів та теоретичних моделей для оцінки
електростатичного потенціалу. Запропоно-
вано метод розрахунку електростатичного
потенціалу, який базується на чисельному
розв’язанні рівняння Пуассона – Больцма-
на, з використанням атомістичної моделі
міцели йоногенної ПАР. Необхідні для по-
будови атомістичних моделей параметри
одержано з молекулярно-динамічного мо-
делювання. Із використанням цього мето-
ду розраховано електростатичний потен-
ціал для міцел додецилсульфату натрію та
броміду цетилтриметиламонію при різних
іонних силах. Результати обговорюють
порівняно зі значеннями, розрахованими
в рамках спрощеної моделі за рівнянням
Ошими – Хілі – Уайта.
Ключові слова: міцела поверхнево-ак-
тивної речовини, шар Штерна, електро-
статичний потенціал, рівняння Пуассо-
на – Больцмана, доступна для розчинника
поверхня.
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Стаття надійшла 12.07.2021.
|
| id | oai:ojs2.1444248.nisspano.web.hosting-test.net:article-321 |
| institution | Ukrainian Chemistry Journal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-07-23T01:06:25Z |
| publishDate | 2021 |
| publisher | V.I.Vernadsky Institute of General and Inorganic Chemistry |
| record_format | ojs |
| resource_txt_mv | ucjorgua/d0/5ab5ee296086090e49532e78679745d0.pdf |
| spelling | oai:ojs2.1444248.nisspano.web.hosting-test.net:article-3212026-07-22T08:23:46Z CONTINUUM ELECTROSTATICS INVESTIGATION OF IONIC MICELLES USING ATOMISTIC MODELS КОНТИНУАЛЬНО-ЕЛЕКТРОСТАТИЧНЕ ДОСЛІДЖЕННЯ ЙОННИХ МІЦЕЛ ІЗ ВИКОРИСТАННЯМ АТОМІСТИЧНИХ МОДЕЛЕЙ Farafonov, Vladimir Lebed, Alexander Mchedlov-Petrossyan, Nikolay surfactant micelle, Stern layer, electrostatic potential, Poisson – Boltzmann equation, solvent-accessible surface. The key parameter related to the structure of the electric double layer of ionic surfactant micelles – electrostatic potential – is considered. A brief overview of experimental methods and theoretical models for estimating electrostatic potential- is given. The calculating method for the electrostatic potential based on a numerical solution of the Poisson-Boltzmann equation using an atomistic model of anionic surfactant micelle - is proposed. The parameters necessary for the construction of atomistic models - are obtained from molecular dynamic modeling.&nbsp; The electrostatic potentials for the micelles of sodium dodecyl sulfate and cetyltrimethylammonium bromide at different ionic strengths - were calculated by this method. The results are discussed in comparison with the values calculated in the simplified model, the Ohshima – Healy – White equation. V.I.Vernadsky Institute of General and Inorganic Chemistry 2021-07-26 Article Article Physical chemistry Физическая xимия Фізична xімія application/pdf https://ucj.org.ua/index.php/journal/article/view/321 10.33609/2708-129X.87.06.2021.55-69 Ukrainian Chemistry Journal; Vol. 87 No. 6 (2021): Ukrainian Chemistry Journal; 55-69 Украинский химический журнал; ##issue.vol## 87 ##issue.no## 6 (2021): Ukrainian Chemistry Journal; 55-69 Український хімічний журнал; Том 87 № 6 (2021): Український хімічний журнал; 55-69 2708-129X 2708-1281 en https://ucj.org.ua/index.php/journal/article/view/321/172 Copyright (c) 2021 Vladimir Farafonov, Alexander Lebed, Nikolay Mchedlov-Petrossyan https://creativecommons.org/licenses/by-nc/4.0 |
| spellingShingle | Farafonov, Vladimir Lebed, Alexander Mchedlov-Petrossyan, Nikolay КОНТИНУАЛЬНО-ЕЛЕКТРОСТАТИЧНЕ ДОСЛІДЖЕННЯ ЙОННИХ МІЦЕЛ ІЗ ВИКОРИСТАННЯМ АТОМІСТИЧНИХ МОДЕЛЕЙ |
| title | КОНТИНУАЛЬНО-ЕЛЕКТРОСТАТИЧНЕ ДОСЛІДЖЕННЯ ЙОННИХ МІЦЕЛ ІЗ ВИКОРИСТАННЯМ АТОМІСТИЧНИХ МОДЕЛЕЙ |
| title_alt | CONTINUUM ELECTROSTATICS INVESTIGATION OF IONIC MICELLES USING ATOMISTIC MODELS |
| title_full | КОНТИНУАЛЬНО-ЕЛЕКТРОСТАТИЧНЕ ДОСЛІДЖЕННЯ ЙОННИХ МІЦЕЛ ІЗ ВИКОРИСТАННЯМ АТОМІСТИЧНИХ МОДЕЛЕЙ |
| title_fullStr | КОНТИНУАЛЬНО-ЕЛЕКТРОСТАТИЧНЕ ДОСЛІДЖЕННЯ ЙОННИХ МІЦЕЛ ІЗ ВИКОРИСТАННЯМ АТОМІСТИЧНИХ МОДЕЛЕЙ |
| title_full_unstemmed | КОНТИНУАЛЬНО-ЕЛЕКТРОСТАТИЧНЕ ДОСЛІДЖЕННЯ ЙОННИХ МІЦЕЛ ІЗ ВИКОРИСТАННЯМ АТОМІСТИЧНИХ МОДЕЛЕЙ |
| title_short | КОНТИНУАЛЬНО-ЕЛЕКТРОСТАТИЧНЕ ДОСЛІДЖЕННЯ ЙОННИХ МІЦЕЛ ІЗ ВИКОРИСТАННЯМ АТОМІСТИЧНИХ МОДЕЛЕЙ |
| title_sort | континуально-електростатичне дослідження йонних міцел із використанням атомістичних моделей |
| topic_facet | surfactant micelle Stern layer electrostatic potential Poisson – Boltzmann equation solvent-accessible surface. |
| url | https://ucj.org.ua/index.php/journal/article/view/321 |
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