Electrostatic screening and charge correlation effects in micellization of ionic surfactants.

Abstract

We have used atomistic simulations to study the role of electrostatic screening and charge correlation effects in self-assembly processes of ionic surfactants into micelles. Specifically, we employed grand canonical Monte Carlo simulations to investigate the critical micelle concentration (cmc), aggregation number, and micellar shape in the presence of explicit sodium chloride (NaCl). The two systems investigated are cationic dodecyltrimethylammonium chloride (DTAC) and anionic sodium dodecyl sulfate (SDS) surfactants. Our explicit-salt results, obtained from a previously developed potential model with no further adjustment of its parameters, are in good agreement with experimental data for structural and thermodynamic micellar properties. We illustrate the importance of ion correlation effects by comparing these results with a Yukawa-type surfactant model that incorporates electrostatic screening implicitly. While the effect of salt on the cmc is well-reproduced even with the implicit Yukawa model, the aggregate size predictions deviate significantly from experimental observations at low salt concentrations. We attribute this discrepancy to the neglect of ion correlations in the implicit-salt model. At higher salt concentrations, we find reasonable agreement of the Yukawa model with experimental data. The crossover from low to high salt concentrations is reached when the electrostatic screening length becomes comparable to the headgroup size.

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Article
Electrostatic Screening and Charge Correlation
Effects in Micellization of Ionic Surfactants
Arben Jusufi, Antti-Pekka Hynninen, Mikko Haataja, and Athanassios Z. Panagiotopoulos
J. Phys. Chem. B, 2009, 113 (18), 6314-6320• DOI: 10.1021/jp901032g • Publication Date (Web): 10 April 2009
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Electrostatic Screening and Charge Correlation Effects in Micellization of Ionic Surfactants
Arben Jusufi,†Antti-Pekka Hynninen,†,‡Mikko Haataja,§,‡and
Athanassios Z. Panagiotopoulos*,†,‡
Department of Chemical Engineering, Department of Mechanical and Aerospace Engineering, and Institute for
the Science and Technology of Materials, Princeton UniVersity, Princeton New Jersey 08544
ReceiVed: February 4, 2009; ReVised Manuscript ReceiVed: March 12, 2009
We have used atomistic simulations to study the role of electrostatic screening and charge correlation effects
in self-assembly processes of ionic surfactants into micelles. Specifically, we employed grand canonical Monte
Carlo simulations to investigate the critical micelle concentration (cmc), aggregation number, and micellar
shape in the presence of explicit sodium chloride (NaCl). The two systems investigated are cationic
dodecyltrimethylammonium chloride (DTAC) and anionic sodium dodecyl sulfate (SDS) surfactants. Our
explicit-salt results, obtained from a previously developed potential model with no further adjustment of its
parameters, are in good agreement with experimental data for structural and thermodynamic micellar properties.
We illustrate the importance of ion correlation effects by comparing these results with a Yukawa-type surfactant
model that incorporates electrostatic screening implicitly. While the effect of salt on the cmc is well-reproduced
even with the implicit Yukawa model, the aggregate size predictions deviate significantly from experimental
observations at low salt concentrations. We attribute this discrepancy to the neglect of ion correlations in the
implicit-salt model. At higher salt concentrations, we find reasonable agreement of the Yukawa model with
experimental data. The crossover from low to high salt concentrations is reached when the electrostatic screening
length becomes comparable to the headgroup size.
1. Introduction
Self-assembly of amphiphilic molecules is based on the
interplay of solvophilic and solvophobic interactions. Prominent
examples are surfactants and lipid molecules, where the head-
group is typically hydrophilic and the tail consists of hydro-
phobic segments. Above a certain concentration, the amphiphiles
aggregate into micelles or bilayer structures,1,2when the
energetic gain exceeds the entropic cost of aggregation. The
self-assembly process depends on the amphiphile and solvent
molecular architectures, but also on temperature and the presence
of other solution components such as dissolved salts.1-6Am-
phihiles are used in various applications, from cleaning products3,4
to nanotechnology.7,8In nature, lipids are the most important
components of biological cell membranes.1,6,9
The general mechanism of self-assembly is well-understood.
However, the role of specific interactions on the molecular scale
and their influence on the critical association concentration and
the structure of aggregates in solution are still major points of
interest. A convenient approach to investigate such problems
is through molecular simulations.10Atomistic molecular dynam-
ics (MD) simulations have been used to study surfactant
micelles11-24and lipid membranes.25-31With fully atomistic
simulations the self-assembly process itself is difficult to model,
due to the large computational effort required to reach time
scales of microseconds relevant for micellization.32Implicit-
solvent models are helpful in reducing the computational cost
of simulations.33-42In comparison with atomistic simulations,
one can save 2 orders of magnitude in computation time using
implicit-solvent models,42thus enabling self-assembly studies
on longer time and length scales. We recently developed such
an implicit-solvent model that is capable of reproducing the
experimental critical micelle concentration (cmc) and the
aggregate structures of ionic surfactants, such as sodium dodecyl
sulfate (SDS) and dodecyltrimethylammonium chloride (DTAC).
The approach was found to be transferable to related ionic
surfactants of different headgroups and tail lengths.42
In many implicit-solvent models of amphiphilic molecules,
electrostatic interactions between charged species have been
treated with a Debye-Hu ¨ckel or Yukawa potential that accounts
for Coulomb screening.43-49This approach reduces the com-
putational cost, allowing large-scale investigations of structural
properties. Similar methods have also been used in protein
simulations.50-52Yukawa models in ionic systems are justified
at low Coulomb couplings where the underlying linear
Poisson-Boltzmann (PB) theory is valid and charge correlation
effects are not important. The question arises, however, whether
simple Yukawa models of salt-mediated interactions between
amphiphiles can describe aggregate formation, thermodynamics,
and micellar structure in simulations. In other words, how
important are charge correlation effects for the self-assembly
process? In this work we address this question by using the
micellization of ionic surfactants as representative of self-
assembly processes of building blocks with ionic interactions.
We do this by comparing a Yukawa-type model to an explicit-
salt model with respect to their ability to predict the cmc and
the aggregation number of the micelles as a function of added
salt. In both cases, we use as starting point a surfactant model
developed for the salt-free case in ref 42 and study added-salt
cases with no adjustment to any of the model parameters. We
show that electrostatic correlation effects become important at
low ionic strengths, where the Yukawa model fails to describe
realistically the micellization process. Furthermore, we dem-
onstrate for the first time that our explicit-ion model is capable
* To whom correspondence should be addressed. E-mail: azp@
princeton.edu.
†Department of Chemical Engineering.
‡Institute for the Science and Technology of Materials.
§Department of Mechanical and Aerospace Engineering.
J. Phys. Chem. B 2009, 113, 6314–6320
6314
10.1021/jp901032g CCC: $40.75
 2009 American Chemical Society
Published on Web 04/10/2009

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of predicting an accurate salt dependence of the cmc, aggregate
size, and micellar shapes. At very high salt and surfactant
concentration we observe the occurrence of cylindrical micelles,
again in agreement with experiment.
The structure of this paper is as follows. In the next section
we give a brief description of the methods that were used in
the simulations and the main features of the implicit-solvent
model. The results are summarized in section 3, where we
discuss the micellization properties of SDS and DTAC in the
presence of explicit NaCl. The results are compared to the
Yukawa-type potential with implicit salt. Finally, in section 4,
we conclude and give a short outlook for further applications
of the proposed explicit-ion model.
2. Simulation Model and Methods
Thermodynamic properties were determined using grand
canonical Monte Carlo simulations, in which both the number
of surfactants and the number of salt ions were allowed to
fluctuate, while the volume and the temperature were fixed. The
cmc was obtained as the point of maximum change in slope of
the pressure-surfactant concentration equation of state.39,42
Unless explicitly specified, simulation runs were performed in
cubic boxes of length L ) 6.1 nm. We used histogram
reweighting techniques53-55to circumvent the strong hysteresis
effects56at the target temperature of 298 K. Specifically, we
performed a series of runs at a series of chemical potentials
and temperatures up to 402 K. At each run the number of
surfactants, the number of co-ions, and the total energy were
recorded in a histogram. Using the histogram reweighting
technique, we combined the histograms and determined the
equation of state at the target temperature of 298 K.
We used a combination of MC moves: 25% of the moves
were deletions/insertions of surfactants and salt ions, 20% were
displacements/rotations of surfactants, 10% were used to regrow
the surfactants, 15% were ion displacements, and 5% were
cluster moves. In the insertion/deletion moves charge neutrality
was ensured by always pairing the insertion/deletion of a
surfactant or co-ion with that of a counterion. Regrowth and
chain insertion/deletion were carried out using configurational
bias techniques.57Electrostatics were treated with the Ewald
method58using 518 Fourier-space vectors and a real-space cutoff
of 3.05 nm. Typical runs comprised 10-30 million MC moves
for equilibration, and, depending on the system, 50-100 million
MC moves for production. The longer runs were carried out
for systems with larger aggregate sizes.
We used the implicit-solvent model developed in our previous
work42for the salt-free case. In ref 42 we parametrized the
hydrophobic potential between the tail beads by matching
the cmc and the aggregate size of SDS to experimental values.
The simultaneous matching of the cmc and the aggregate size
was made possible by concentrating the hydrophobic interaction
between the terminal (i.e., furthest from the headgroup) tail
beads only. This also makes it favorable for the surfactants to
aggregate into concentric objects; i.e., the chains are buried
inside the micelles, whereas the headgroups are mainly placed
on the surface. We note that the model reproduces the liquid-
like characteristics of the micelle interior, in agreement with
atomistic simulation19,22and experimental results.59
The interactions between all other tail beads (i.e., except
terminal beads) were taken from the OPLS force field.60We
used the atomistic settings for the bond potentials13,22and did
not coarse-grain the tail beads. The headgroup and counterion
interactions were described by effective potentials that account
for various force contributions: electrostatics, hydration, disper-
sion, and short-range repulsion. In the headgroup-counterion
interaction a Coulomb correction was employed to account for
close distance dependence of the dielectric permittivity. The
parameter values for these interactions are given in ref 42. We
note, however, that the prefactor of the electrostatic potential
contributions, the Bjerrum length λB) e2/(4π?kT) (dielectric
constant ?, elementary charge e, and thermal energy kT), was
set to 0.71 nm instead of the experimental value of 0.72 nm at
the temperature of 298 K. In the present study, we used the
experimental value and recalculated the quantities shown in ref
42, particularly SDS and DTAC in the salt-free case. Apart from
this minor correction, no other modification of the model
parameters shown in ref 42 was made. For the interaction
between the NaCl salt ions we used the parametrization given
in ref 61. In ref 42 we also demonstrated the transferability of
the SDS surfactant model to cationic surfactants (DTAB, DTAC)
and sodium alkyl sulfate with alkyl chain lengths between 6
and 14.
In our Yukawa model, the surfactant tails interact as just
described, with potential parameters obtained from ref 42. The
headgroups interact with each other via a short-range repulsive
Weeks-Chandler-Anderson (WCA) potential62and the screened
Coulomb potential. The WCA interaction accounts for the finite
size of the headgroup. The energy scale is 2.5 kJ/mol at the
target temperature of 298 K and is roughly equal to the thermal
energy. Headgroup-tail WCA interactions are obtained using
the Lorentz-Berthelot combining rule. The Coulomb screening
is modeled through a Yukawa/Debye-Hu ¨ckel potential
Vyuk(r))e2
4π?
exp(-κ(r-σWCA))
r
,
(1)
where κ ) [2e2c/(?kT)]1/2is the inverse screening length and c
is the total counterion concentration. The counterion concentra-
tion c is given by the sum of the salt and surfactant concentra-
tions as
c)cmc0+cs
(2)
The only unspecified parameter is the headgroup diameter σWCA.
The value was obtained by fitting the cmc to the salt-free case.
In eq 2 we used cmc0) 8 mM for SDS and cmc0) 20 mM for
DTAC (cs) 0). These values are in the experimental range,
namely, 7.763to 9.3 mM64for SDS and 17.265to 21.3 mM66
for DTAC. Best fits for the cmc were obtained with σWCA)
0.44 nm for SDS which corresponds to the smallest headgroup-
headgroup distance that we measured in separate atomistic MD
simulations of headgroups and counterions.42For DTAC we
used σWCA) 0.65 nm, a value that is somewhat larger than the
minimum separation between the headgroups (around 0.55 nm).
These diameter values were used in the added-salt cases as well.
3. Results
Using explicit-ion simulations in the salt-free solution, we
obtained cmc0) 8.4 ( 0.3 mM for SDS and cmc0) 16.7 (
0.5 mM for DTAC. The results are close to the ones reported
earlier.42The difference is due to a larger box length used in
the present study, L ) 6.1 nm (compared to L ) 4.6 nm), and
the correction of the Bjerrum length, λB, to the experimental
value of 0.72 nm instead of 0.71 nm used in the previous study.
A larger box was used here since we expect bigger aggregates
at high salt concentrations. While the cmc’s are slightly affected
(12-15% smaller than in ref 42) by the correction of λB, the
aggregation numbers remain unchanged. Our simulated cmc’s
agree with experimental results that lie between 7.763and 9.3
mM64for SDS and between 17.265and 21.3 mM66for DTAC.
Micellization of Ionic Surfactants
J. Phys. Chem. B, Vol. 113, No. 18, 2009 6315

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We note that the DTAC results are pure predictions. The DTAC
tail parameters are gained from the SDS parametrization,
including the hydrophobic energy parameter, while the param-
etrization of the headgroup ion interaction potentials is based
on fitting of the pair correlation functions obtained from separate
atomistic simulations; see ref 42 for further details. The pair
correlation functions of the ion systems are not affected by the
small change in the Bjerrum length.
For the Yukawa-SDS model we obtain cmc0) 13.2 ( 0.5
mM, which is somewhat above the experimental range. A better
result would have been obtained if we had used a headgroup
diameter smaller than σWCA) 0.44 nm, which would weaken
the effective headgroup repulsion further. Smaller values of
σWCA, however, would have been unrealistic. For the Yukawa-
DTAC model we obtain cmc0) 15.9 ( 1.0 mM, which is closer
to the lower experimental range.
Addition of salt leads to a significant drop in the cmc in line
with the empirical power-law salt dependence observed for
many ionic surfactants,2as shown in Figure 1. The explicit-ion
simulation results agree very well with experimental data over
the whole salt concentration range, i.e., over 2 orders of
magnitude. The model is robust with respect to headgroup
change from SDS to DTAC without any adjustment of the tail
bead parameters. As already stated, model parameters were taken
unmodified from ref 42; the excellent agreement between
simulations and experiments has not been forced. In the same
figure we also plot results from the Yukawa model. The overall
agreement with experiments is again reasonable and demon-
strates that even the simple Yukawa model is able to reproduce
the dependence of the cmc on salt concentration, despite some
deviations particularly for SDS.
We computed the aggregate size M by defining the aggregate
with a simple cutoff criterion: two surfactants belong to the same
micelle if the center of mass of one tail is separated by less
than 0.4 nm from that of another tail. This cutoff value was
tested to give the right aggregation number by visual inspection
of representative configurations. Figure 2 shows the salt
dependence of the mean aggregation number 〈M〉. Without added
salt, we obtain values similar to those reported in ref 42. 〈M〉
increases with increasing salt concentration cs, in agreement with
experimental findings. SDS and DTAC explicit-ion results both
exhibit a power law dependence, as seen experimentally for
many ionic surfactants related to the ones studied here.72-75In
particular for SDS explicit-ion results, the agreement with
experimental results is good. For DTAC, the simulation model
systematically underestimates the experimentally observed ag-
gregation number in the added salt cases. The reason for the
underestimation could lie in the overestimation of the DTAC
headgroup size. The headgroup parameters of the implicit model
result from matching of the headgroup counterion pair correla-
tion function obtained from implicit-solvent simulations with
the same correlations obtained from atomistic simulations. There
are many possible error sources: from the quality of atomistic
ion force field parameters to the assumption that the implicit-
solvent interaction parameters in bulk are used in the micellar
system without further adjustments. Despite these uncertainties,
the deviations for the aggregation numbers are always less than
20%.
In contrast to the explicit-salt results, the Yukawa model does
not reproduce the salt dependence of the aggregation number,
as seen in Figure 2. At low salt concentrations, 〈M〉 predicted
from the Yukawa model is much smaller than experimental
values. Above a certain salt concentration the agreement with
experiments and explicit-ion simulation results becomes better.
The position of the threshold salt concentration is different for
SDS and DTAC. We interpret this as being due to the different
headgroup sizes of the surfactants. The effective (hydrated)
diameter for the SDS headgroup is σ ) 0.54 nm and for DTAC
headgroup σ ) 0.79 nm in our explicit-ion model. These values
were obtained by matching pair correlation functions obtained
from implicit-water simulations of headgroups and counterions
with atomistic explicit-water simulation data.42At low salt
concentrations the screening length κ-1is much larger than σ.
The screening length becomes comparable to the headgroup size;
i.e., κσ ≈ 1 around 150 mM for DTAC and 315 mM for SDS,
as indicated by vertical dotted lines in Figure 2. Interestingly,
at these and higher salt concentrations the Yukawa model seems
to reproduce experimental values for the aggregate size, as can
be particularly seen for DTAC. The Yukawa results of SDS
exhibit scatter at high salt concentration (600 mM) in Figure 2,
but the trend is clearly observable.
The hypothesis put forward based on these results is that the
cmc is mainly determined by the strength of the hydrophobic
Figure 1. cmc as a function of the total counterion concentration cmc0
+ cs. For SDS (black line and symbols): explicit-ion simulations (solid
circles), Yukawa model (open right-pointing triangles), and experimental
data from ref 2 (asterisks), ref 67 (plus signs), and ref 68 (times signs).
For DTAC (red line and symbols): explicit-ion simulations (solid
diamonds), Yukawa model (open boxes), and experimental data from
ref 69 (open downward-pointing triangles) and ref 70 (open upward-
pointing triangles). The lines are linear least-squares fits to the explicit-
ion results. Error bars are comparable to the symbol sizes.
Figure 2.
counterion concentration cmc0+ cs. Symbols are the same as those
for Figure 1. Solid lines are fits of the explicit-ion simulation results,
and dashed lines are guides for the eye for the Yukawa model results.
The dotted vertical lines mark the concentrations at which the screening
length becomes smaller than the headgroup size for the corresponding
surfactant.
Mean aggregation number 〈M〉 as a function of total
6316
J. Phys. Chem. B, Vol. 113, No. 18, 2009
Jusufi et al.

Page 5

interaction and of the effective repulsion between the head-
groups, which contains Coulomb screening. The cmc is the
concentration at which micelles start forming. Up to the cmc,
the system is homogeneous and the screening parameter κ
reasonably accounts for Coulomb screening effects. However,
when micelles are formed, i.e., above the cmc, local headgroup
counterion correlations enhance electrostatic screening. These
effects are not captured by the simple Yukawa and similar mean-
field models. This is particularly important for salt concentra-
tions at which κσ j 1. Above this threshold the screening length
κ is smaller than the effective headgroup size and larger micelles
can form. The Yukawa and explicit-salt surfactants behave
similarly in this regime. Micellar growth in the Yukawa
surfactant system at low salt is limited by the large effective
headgroup size.
To have a measure for the headgroup counterion correlations,
we computed the degree of counterion binding to headgroups
at different salt concentrations. Figure 3 shows the counterion-
headgroup pair correlation function g(r) for SDS and DTAC in
the salt-free case, where only headgroup and counterions are
present. The behavior of g(r) for the added salt cases remains
qualitatively similar, with only the peak heights changing with
respect to the salt-free case. As discussed in detail in ref 42,
the headgroup-counterion correlation for SDS possesses a
double peak profile, accounting for first and second counterion
coordination shells of headgroups due to hydration layers around
the ions.
Those effects have been observed in atomistic simulations
as well, although they differ quantitatively, which in turn lead
to different degrees of counterion binding in the corresponding
shells.19-22The degree of counterion binding ? has been
estimated from the fraction of counterion under the peaks up
to the subsequent minimum. In the DTAC system the hydration
layers of the ions are strong and the cutoff distance is large,
around 1.05 nm. For SDS the first and second minima are at
0.43 and 0.92 nm, yielding a corresponding first and second
degree of counterion binding ?1and ?2, respectively. While the
first peak position agrees well with atomistic results,18,20,22the
second peak position is somewhat bigger than observed in
atomistic studies. This is due to a lack of a third coordination
shell of water molecules around the ions in our implicit-water
model. However, for the degree of counterion binding the area
is of bigger importance than the exact peak position. In Table
1, the degrees of counterion binding ? are given at different
salt concentrations. According to atomistic studies,18,22the vast
majority of ion binding is through its second coordination shell
of the headgroups. The total ion binding degree ? is around
67% in the absence of salt, which lies in the range of atomistic
results which report values of ? between 5018and 80%.22
Experimental values lie in a smaller interval, namely, between
6267and 72%.77Addition of salt generally increases the ion
binding, however, only weakly at modest salt concentration,
i.e., below 100 mM. Within this range our values remain
between 70 and 75%. Experimental groups reported a similar
trend.67,68At high salt concentration experimental measurements
obtained counterion binding of 86% already at 300 mM.67One
can expect a higher degree at 600 mM, where we obtained 92%
binding; i.e., at this salt concentration the micelles are almost
completely neutralized.
DTAC micelles possess similar characteristics. Only at 500
mM salt concentration the counterion binding reaches almost
100%. At lower ionic strengths the degree increases from 65%
in the salt-free case to almost 80% at 150 mM salt concentration.
The first value is in excellent agreement with experimental data
(63%).75Calculated values that were used to match measured
surface potentials report a slight salt dependence, ranging
between 61% in the salt-free case to 82% at cs) 650 mM
NaCl.76Note that our model predicts only a second ion binding
shell. This is a significant difference to SDS. A large headgroup
size and a lack of a first ion binding shell reduce the probability
of ion correlation effects, such as ion bridging observed in SDS
micelles.18In turn, the cmc of DTAC is larger and its micelle
sizes smaller than SDS micelles, irrespective of addition of salt.
A lack of such ion correlations also makes possible the use of
Yukawa models, as long as κσ J 1. In the corresponding salt
regimes we saw that the aggregate size of DTAC micelles is
well-described by a Yukawa model, in contrast to the SDS
micelles, where the Yukawa model predictions of the aggregate
size is rather limited; see Figure 2.
In Table 1 we have given values for micellar radii obtained
as the average distance between the headgroups and the center
of mass of the micelles. For both cases, SDS and DTAC, the
radius R increases very weakly. Information on micellar shape
is obtained from the principal moments of inertia I1g I2g I3.
In Figure 4 we plot the ratio I1/I3as a function of the counterion
concentration for SDS and DTAC. Representative snapshots are
given in Figure 5.
Clearly, the asphericity of SDS micelles increases with
increasing salt concentration. This is in line with experimental
observations2,78,79and indicates a trend toward ellipsoidal shapes
of the micelles. In many experimental studies this trend is
pronounced.71,80,81In contrast to SDS, the DTAC micelles remain
spherical almost over the whole salt concentration range, again
Figure 3. Pair correlation function g(r) between headgroups and
counterions for SDS (black solid line) and DTAC (red dashed line) as
a function of their separation r without added salt.
TABLE 1: Radius R and Counterion Binding Values ? of
SDS and DTAC Micelles for Various NaCl Concentration csa
cs(mM)
R (nm)
?1
?2
?
SDS02.02
2.04
2.09
2.26
2.48
1.84
1.83
1.86
1.93
1.99
0.11
0.11
0.13
0.14
0.19
0.56
0.59
0.60
0.61
0.73
0.67
0.70
0.73
0.75
0.92
0.65
0.69
0.73
0.78
0.95
10
25
100
600
DTAC0
20
70
150
500
aFor the SDS cases the counterion binding is distinguished
between first and second shell binding, ?1 and ?2, respectively.
DTAC micelles posses only second shell counterion binding.
Micellization of Ionic Surfactants
J. Phys. Chem. B, Vol. 113, No. 18, 2009 6317