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B: Biomaterials, Surfactants, and Membranes

Coarse-Grain Molecular Dynamics Simulations to Investigate the Bulk Viscosity and Critical Micelle Concentration of the Ionic Surfactant Sodium Dodecyl Sulfate (SDS) in Aqueous Solution Yosadara Ruiz-Morales, and Ascencion Romero-Martínez J. Phys. Chem. B, Just Accepted Manuscript • DOI: 10.1021/acs.jpcb.7b10770 • Publication Date (Web): 13 Mar 2018 Downloaded from http://pubs.acs.org on March 14, 2018

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The Journal of Physical Chemistry

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Coarse-Grain Molecular Dynamics Simulations to Investigate the Bulk Viscosity and Critical Micelle Concentration of the Ionic Surfactant Sodium Dodecyl Sulfate (SDS) in Aqueous Solution

Yosadara Ruiz-Morales* and Ascención Romero-Martínez Instituto Mexicano del Petróleo, Eje Central Lázaro Cárdenas Norte 152, Mexico City 07730, Mexico

ABSTRACT

The first critical micelle concentration (CMC) of the ionic surfactant sodium dodecyl sulfate (SDS), in diluted aqueous solution, has been determined at room temperature from the investigation of the bulk viscosity, at several concentrations of SDS, by means of coarse grain molecular dynamics simulations. The coarse-grained model molecules at the mesoscale level are adopted. The bulk viscosity of SDS was calculated at several millimolar concentrations of SDS in water using the MARTINI force field by means of NVT shear Mesocite molecular dynamics. The definition of each bead in the MARTINI force field is established, as well as their radius, volume, and mass. The effect of the size of the simulation box on the obtained CMC has been investigated as well as the effect of the number of SDS molecules, in the simulations, on the

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formation of aggregates. The CMC, which was obtained from a graph of the calculated viscosities versus concentration, is in good agreement with reported experimental data, and do not depend on the size of the box used in the simulation. The formation of a spherical micellelike aggregate is observed, where the dodecyl sulfate tails point inwards and the heads point outwards the aggregation micelle, in accordance with experimental observations. The advantage of using coarse grain molecular dynamics is the possibility of treating explicitly charged beads, applying a shear flow for viscosity calculation, as well as to process much larger spatial and temporal scales than atomistic molecular dynamics can. Furthermore, the CMC of SDS obtained with the coarse-grained model is in much better agreement with the experimental value than the value obtained with atomistic simulations.

1.

INTRODUCTION

Surfactants are present in a vast number of industrial applications to control interfacial properties ranging from detergents, oil recovery, catalysis, to nanotechnology.1,2 The reduction of surface and interfacial tension in aqueous solutions is one of the key functions of surfactants. Surfactants present the ability to aggregate, in aqueous solution, into structures such as monolayers, bilayers, spherical micelles, cylindrical micelles, and other complex structures.1,2

The formation of

micelles takes place in dilute aqueous solutions and the critical micelle concentration (CMC), above of which aggregates are observed, depends on the molecular structure of the surfactant, ionic character, the concentration, and the thermodynamic conditions.3


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Sodium dodecyl sulfate (SDS), which is an ionic surfactant, aggregates or self-assembles into spherical micelles above its first CMC, at around 8 mM to 8.5 mM in water,4 at 25°C without added salt. After the first CMC the solution rheology remains Newtonian. Also, SDS selfassembles into rod-like micelles above its second CMC, at around 70 mM in water.4-7 It is reported that the average aggregation number in SDS spherical micelles at the first CMC is around 60.8,9 The SDS micelles and their morphology are affected by counterions, salt concentration, and temperature.10-14 The critical micelle concentration is observed as a change in the slope of a plot of a property of the system (for example conductivity, viscosity, velocity of sound, cyclic voltammetry, polarography) versus concentration. 4,15-17

Because of its importance, computer simulations have been used to study the formation of micelles of SDS in aqueous solution. All-atom molecular dynamics (all-atom MD) simulations, containing up to millions of atoms and time scales of microseconds, have been performed for SDS using in many cases a preassembled spherical or elongated micelle with an aggregation number around 60 or larger.18-27 Also the molecular description of micellization in dilute surfactant solutions has been studied with long all-atom MD from the spontaneous self-assembly at elevated concentrations, with subsequent extrapolation to conditions relevant to the CMC.28

Many atomic force fields have been used to study SDS with all-atom molecular dynamics. The most popular are CHARMM, OPLS, AMBER, GROMOS. Tang et al have published a paper, with an extensive comparison, on the performance of the atomistic force fields to study the aggregation of SDS.29 Most of the all-atom MD studies are concentrated on studying the shape of the SDS micelles and the number of dodecyl sulfate molecules that compose the micelles



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rather than on calculating the CMC. Numbers of 60 or even higher than 100 SDS in micelles have been calculated depending on the atomic force field used in the all-atom MD.29

Jusufi and Panagiotopoulos28 have studied the simulation methodologies to obtain the CMC and the equilibrium distribution of aggregate sizes, in dilute surfactant solutions, and have concluded that in general the atomistic force fields underpredict the experimental CMC for zwitterionic and nonionic surfactants.

All-atom MD involves a high computational cost. This cost can be mitigated by accelerating the all-atom MD by grouping each carbon, in the hydrocarbon tail, with its bonded hydrogen atoms, into a united-atom (UA) and applying a larger time step size.29-31 Another way to mitigate the high computational cost of all-atoms molecular dynamics is to perform mesoscale coarse grain molecular dynamics. Mesoscale modeling allows for key chemical interaction to be represented without requiring excessive book keeping at the atomic level. Bead−spring-type particle models, where particles represent a group of atoms or a liquid volume, are used in these calculations. Coarse grain molecular dynamics (CG MD) is a widely used methodology at the mesoscale and can process much larger spatial and temporal scales than all-atom molecular dynamics can.32-38

The micelle formation and micelle structure of SDS have been studied with coarse grain molecular dynamics. Shinoda et al have mapped SDS into a coarse grained structure that involves three heavy atoms represented by one bead (3:1 mapping), and the water has been mapped as three water molecules per bead. The coarse grain force field SDK33,34 has been used in the simulations. It has been found that the CG molecular dynamics simulation of sodium


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dodecyl sulfate micellar solution produced a reasonable distribution of monomers between micelles and the bulk solution.33,34

Jalili and Akhavan have mapped SDS into a coarse grained structure that involves four heavy atoms represented by one bead (4:1 mapping), and the water has been mapped into four water molecules per coarse grain bead. The coarse grain force field MARTINI was used in the simulations35,37 as well as a preassembled SDS micelle. Properties such as micelle size and shape, and water penetration have been examined. The obtained results closely reproduced the results obtained with all-atom molecular dynamics.37

Jusufi et al39have reported an implicit water model for SDS, where removing explicit water reduces the degrees of freedom, within the coarse grain framework. The sulfate head is treated as one bead, while the CH2 and CH3 of the tail are treated using a unite atom model. Using modified potentials and grand canonical Monte Carlo simulations a CMC of 9.3 mM has been obtained with an aggregate size of 57 at 298 K.39

Wang and Larson have performed coarse-grained molecular dynamics simulations for SDS using a modification of the Dry MARTINI force field with implicit water. These authors have obtained structural and thermodynamic properties of SDS micelles that are close to those obtained from the standard MARTINI force field, with explicit water, and in good agreement with the all-atoms molecular dynamics, in terms of size and shape of the micelles.40



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Coarse grain dissipative particle dynamics of SDS using the smeared charge approximation has been carried out by Mao et al.41 Dissipative particle dynamics (DPD) is a coarse-grained simulation technique introduced by Koelman and Hoogerbrugge to simulate the Navier−Stokes hydrodynamics of complex fluid systems over long length and time scales. Bead−spring-type soft particle models are used in the DPD calculations.42 Dynamic evolution of the beads is governed by Newton’s second law. In Mao’s et al work41 the electrostatic interactions of charged beads have been treated using the smeared charge approximation of Groot43 where the charge is distributed around the bead center in an implicit dielectric medium. The coarse-grained models of the surfactants have been parameterized using a combination of atomistic molecular simulation and infinite dilution activity coefficient calibration. The CMC has obtained from a curve of the dependence of the free surfactant concentration on the total surfactant concentration for SDS. The calculated CMC was 19 mM, which exceeds the experimental value of 8.4-7 The authors have considered that the discrepancy may be related to the short-range repulsion parameters. Also, the model underestimates the micelle size.

None of the published all-atom MD or CG-MD theoretical simulations have been able to reproduce the first CMC of SDS. The published studies have mainly been focused on the equilibrium aggregate size distributions, shape of micelles, counter ion distribution, and surface structure. In the cases where the CMC was obtained it was overestimated. In most cases the CMC is considered as the concentration in which there is a change in the slope of a plot of the number of free molecules in the simulation versus total surfactant concentration during micellization. However, the definition of “free” molecules is not clear and it is not operationally defined.28


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Here we propose to calculate the first CMC of SDS by simulating a bulk property of the system, as it is done experimentally, in this case the viscosity, at different SDS millimolar (mM) concentrations, using the coarse grain molecular dynamics framework (CG-MD), and the force field MARTINI. The CMC is obtained as the concentration in which there is a change in the slope of the plot of viscosity versus concentration. This concentration is obtained at the crossing of two linear extrapolations (two linear regimes) obtained from the viscosity versus concentration before and after the slope change (inflexion point).

In here we are more interested in establishing a coarse grain simulation methodology to calculate the first CMC of SDS in aqueous solution, at 298.15 K, that agrees with the experimental value, rather than in the equilibrium aggregate size distributions. Even with CG models, equilibrium can be difficult to achieve for dilute solutions when a sufficiently large system is studied to allow the formation of multiple micellar aggregates.36 Therefore, we consider the simulation a small system, ensuring the millimolar (mM) concentration, with the lowest possible number of SDS. The purpose of studying a small system was to not having too many degrees of freedom, and to be able to equilibrate the system in a reasonable computing time.

Our goal in the near future is to be able to simulate with coarse grain molecular dynamics the promotion and/or inhibition of methane hydrates by SDS. Surfactants have shown the ability to control gas hydrate formation and sodium dodecyl sulfate has been proven to promote the formation of methane hydrate by increasing the rate of gas hydrate formation and decreasing the induction times.44-48 At the same time, SDS has also been shown to inhibit the formation of gas



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hydrate at low concentrations, below the CMC.

44,49

The mechanisms of the promotion and/or

inhibition effect of SDS remain unclear.50 To understand the inhibiting and promotion effects of SDS on methane hydrates, at the molecular level, the use of coarse grain molecular simulation would be a promising tool; however, first we require to be able to reproduce with coarse grain molecular dynamics simulations the CMC of SDS and to observe the formation of micelles, which is reported in this study. The advantage of using coarse grain molecular dynamics, as it is shown in here, is the possibility of treating explicitly charged beads, applying a shear flow for viscosity calculation, as well as to process much larger spatial and temporal scales than all-atom molecular dynamics can.

2.

SIMULATION METHOD.

2.1. Coarse Grained Model Molecules.

The molecular dynamics of SDS in water at several concentrations, below and above the CMC, was described using the MARTINI 2.0 force field.35 This force field is based on a four-to-one mapping, i.e., on average four heavy atoms are represented by a single interaction center (bead). Four main types of interaction sites are considered in the MARTINI 2.0 force field: polar (P), nonpolar (N), apolar (C), and charged (Q). Each type of interaction particle has subtypes, which are either distinguished by letter, denoting the hydrogen-bonding capabilities (d= donor, a= acceptor, da=both, 0=none), or by numbers, which indicate the degree of polarity (from 1, low polarity, to 5, high polarity).35 The nonbonded interactions among the different bead types are described by a simple 6-12 Lennard-Jones potential and Coulombic interaction for the charged


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beads. Different levels of Lennard-Jones potentials are assigned to each bead type pair to match thermodynamic experimental data. The bonded interactions are modeled by weak harmonic stretching and bending potentials.35

In Figure 1 the coarse grained mapping of sodium dodecyl sulfate is presented. Each dodecyl sulfate molecule has been mapped into two types of beads. The sulfate group , hydrophilic head, has been coarse grained into one bead (yellow bead in Figure 1). The MARTINI type Qa with a charge of -1 has been assigned to this bead. The aliphatic chain, hydrophobic tail, has been coarse grained into three beads of the same MARTINI type C1, pink beads in Figure 1. Each C1 bead represents four CH2 group, while the outermost bead represents three CH2 groups and one CH3 group. See Figure 1. The hydrated sodium cation has been coarse grained into a bead of the type MARTINI Qd with charge +1 (orange bead in Figure 1). Four water molecules represent one water bead (blue bead in Figure 1). The water beads have been described as MARTINI type P4.



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Figure 1. Coarse grain mapping into beads, MARTINI bead type assignment, and charge of dodecyl sulfate, hydrated sodium cation, and water bead. For explanation see text.

2.2. Simulation Details.

All the calculations have been set up using the mesostructure tool in the Materials Studio package.51 In the simulation all beads have the same volume, radius, and mass. These bead parameters have been obtained as explained in the following text.

The size of a bead corresponds to water molecules, in this case 4. is the degree of coarse graining. The molar volume of water is 18 (cm3/mol),52 therefore one water molecule has a volume of 30 Å3 following:


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/

. ×

!/

3 × 10 %& 30Å

(1)

and the volume of one water bead, with 4 is 120 Å3, as follows: (

(2)

Where ( is the volume of a bead; is the volume of one water molecule, i.e. 30 Å3 (Eq. 1). The mass, in mass units, can be defined in terms of the coarse graining as: & &

(3)

Where & is the mass of a bead; & is the mass of one water molecule (18 amu). Therefore, each bead have a mass of 72 amu.

To define the diameter and radius of each bead, which are parameters required in the mesostructure properties of beads in Materials Studio, we used the equations 4 and 5.53,54 A bead corresponds to water molecules. Hence, a cube of volume ) represents * water molecules, where * is the number of beads per cubic )+ . The volume of this cube would be * multiply by the volume of one water molecule ( ). Therefore: )+ *

(4)

) * / *30 / 3.107* /

[Å]

(5)

Where ) is the diameter of a bead; is the volume of one water molecule, i.e. 30 Å3 (see Eq. 1); , is the degree of coarse graining used, 4 in this case; *, is defined as the number of beads, or bead density, in a cubic volume of length ) . We use * 3. By choosing * 3, a liquid with the compressibility of water is simulated.53,54

According to Eq. 5, ) 7.11 Å, which corresponds to the diameter of each bead, and half of it is the radius of a bead, 3.555 Å. In the simulation all beads have the same volume, mass, and



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12 radius. The volume of each bead is 120 Å3 (Eq. 2), the mass of each bead is 72 amu (Eq. 3), and the radius of each bead is 3.555 Å (Eq. 5). In the simulations the volume, size and mass of the beads are maintained constant.

All the calculations are performed in a cubic simulation box with size 150 Å × 75 Å × 75 Å , with periodic boundary conditions applied in all directions. Therefore, the total volume of the simulation box, .(/ ,is 843750 Å3. The enclosed system is composed of beads of water, dodecyl sufate (DS-) and hydrated sodium ions (Na+). The total number of beads in the system is given by: (

01234 5

(6)

Where ( is the total number of beads in the system, .(/ is the total volume of the simulation box, *, is defined as the number of beads, or bead density, in a cubic volume of length ) (see above), ) is the diameter of a bead. Therefore, the total number of beads in the system, ( = 7042 beads.

Several SDS concentrations, below and above the CMC, have been calculated. To establish the number of SDS coarse grained molecules required to ensure the desire concentration we apply: 6

7898 :; 1234

= 6? .(/

(7) (8)

Where 6 is the target concentration in mol/liter (M, molar), = is the number of coarse grained molecules of SDS, ? is the Avogadro’s number (6.022140 ∗ 10 molecules mol-1, .(/ is the volume of the box in liters.


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In Table 1 the number of SDS molecules required to ensure each millimolar concentration, number of tail beads, number of head beads, number of sodium beads, number of water beads, and total number of beads in each calculated system are presented.

Table 1. Number of SDS molecules required to ensure each millimolar concentration, number of tail beads, number of head beads, number of sodium beads, number of water beads, total number of beads in each calculated system, and experimental density for each SDS concentrations. SDS Concentration (mM)

Number of Total Total Total Total Total Exptl. SDS Tail Head beads Water number Density coarse beads beads of Na+ b Beadsc of beadsd (g/cm3)e grained moleculesa 2 1 3 1 1 7032 7037 0.997207 4 2 6 2 2 7028 7038 0.997297 6 3 9 3 3 7024 7039 0.997406 8 4 12 4 4 7020 7040 0.997499 10 5 15 5 5 7015 7040 0.997592 12 6 18 6 6 7010 7040 0.997673 14 7 21 7 7 7005 7040 0.997752 22 11 33 11 11 6989 7044 0.9981248f 32 16 48 16 16 6967 7047 0.9985814f a Values obtained with Eq. 8. These values were rounded. b Hydrated sodium cation. c Total number of water beads for each concentration obtained by subtracting the sum of columns 3-5 to the total number of beads in each system (column 7). d Total number of beads in each system (sum of columns 3-6) e Experimental (Exptl.) density measured at 25 °C for aqueous solutions of SDS at a given concentration.55 f Values obtained by interpolation of the experimental data in the column.

In column 7 of Table 1 the total number of beads in each system are presented. These are the actual numbers in the simulations that have been calculated and that resulted from the setting up of the systems using the mesostructured tool in the Materials Studio package.51 The number of



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total beads calculated with Eq. 6 (7042) differs from the numbers shown in column 7 by ± 5 beads. The percentage error corresponds to ±0.071 %, which is negligible.

As mentioned in the introduction, in the present study we are interested in establishing a coarse grain simulation methodology to calculate the first CMC of SDS in aqueous solution, at 298.15 K, that agrees with the experimental value, rather than in the equilibrium aggregate micelle size distributions. Therefore, we simulate a small system with the desire millimolar (mM) concentration, but with the lowest possible number of SDS, to be able to equilibrate the system in a reasonable computing time. This is done because even with CG models, equilibrium can be difficult to achieve for dilute solutions when a sufficiently large system is studied to allow the formation of multiple micellar aggregates.36

2.3. Simulation Methodology.

All the calculated systems are charged neutral. Instead of carrying out for each system a NPT simulation for the density adjustment, the experimental55 density measured at 25 °C for aqueous solutions of SDS, at the given concentration, has been introduced during the setting up of each systems with the mesostructure tool in the Materials Studio package.51 In the last column of Table 1 the experimental reported densities are provided for each SDS concentration. When setting up the systems with the mesostructured tool in the Materials Studio package51 the final density agree with the input density up to the fourth digit after the decimal point.


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All the calculations are carried out using the Mesocite module embedded in the Materials Studio package.51 Initially the systems are randomly distributed. The dodecyl sulfate anions and the sodium cations are dispersed in the simulation box. Each system is initially geometry optimized using the geometry optimization task in the Mesocite module, with the algorithm smart and a fine quality, with the purpose of removing bad contacts. The default values of the convergence thresholds between optimization cycles for the fine quality in the Mesocite module, embedded in the Materials Studio package,51 are: energy 1x104 kcal mol-1, force 0.005 kcal mol-1 Å-1, stress 0.005 GPa, and displacement 5x10-5 Å.51

After the corresponding system is geometry optimized, an NVT Mesocite Dynamics is carried out at a temperature of 298.15 K to bring the system near to equilibrium. A small time step is used, 15.08 fs, for 5x106 steps; therefore, a total simulation of 75.4 ns is carried out. The NoséHoover thermostat is used with a Q ratio of 1.0.

After the mesocite dynamics is completed, a NVT Shear Mesocite production simulation is carried out to calculate the bulk viscosity of each system, where the dynamics simulation is ran in the presence of a shear flow, and the components of the stress tensor are calculated. The production simulation is carried out at a temperature of 298.15 K, with a fine quality. The quality determines the parameters that control the simulation speed and accuracy. The quality level sets the non-bond cutoff distance in the case of the bead-bead method to 15.5 Å.

The applied shear rate is 0.007869 ps-1, and the shear direction (plane) is B(BC). Each system is simulated as long as 150.8 ns. The time step used in the calculations is 15.08 fs and the number



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16 of steps is 1x107. Once every 2000 steps, the configurations are saved. The Nosé-Hoover thermostat is used with a Q ratio of 1.0.

Periodic boundary conditions are applied in all directions with an electrostatic/van der Waals nonbonded cut off distance of 15.5 Å (fine quality value). The summation method chosen is bead based with a cubic spline truncation of the non-bond energy terms, with a spline width of 1 Å, a buffer width of 0.5 Å, and with application of long range correction to take into account truncation of the van der Waals term.

The MARTINI coarse grain potential is smoother than an atomic potential, and based on the comparison of the diffusion constant for atomic and coarse grain it is found that the effective time sampled is four times larger than the actual simulation time.56 Therefore, the production run has been sampled an effective time of 0.6 µs. 3.

RESULTS AND DISCUSSION

The final conformations of the systems obtained in the NVT Mesocite equilibration are the initial configurations for the production NVT Shear Mesocite run to calculate the bulk viscosity. In Figures 2-4 the configuration obtained from the NVT Mesocite equilibration, or initial configuration in the production run, (a), and the lowest energy conformation obtained in the production run (b) are shown, for all the concentrations presented in Table 1. The water beads are not shown for clarity. During the production NVT Shear Mesocite run the energy of the total system oscillates. We looked at the lowest total energy obtained during this run and associated it


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with the corresponding frame. The thus obtained configuration for each concentration of SDS are presented in Figures 2b-4b.



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Figure 2. Initial configurations (a), obtained from the NVT equilibration run, and lowest energy configurations (b) obtained in the NVT Shear Mesocite run for the 2 mM, 4 mM and 6 mM SDS concentrations. The configurations shown in (b) were obtained at 98.1 ns, 102 ns, and 72 ns respectively.


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The initial conformation for the case of the concentrations 4mM and 6mM (see a in Figure 2) show no aggregation of the dodecyl sulfate anions after the equilibration run. For the 6 mM concentration some aggregation of the dodecyl sulfate anions is observed at end of the equilibration run (see a in Figure 2 for 6mM). However, during the production Shear run the lowest energy conformation obtained show no aggregation (see b in Figure 2 for 6 mM). Also at the end of the production run no aggregation is observed. The reported CMC for SDS is 8 mM4; therefore, the fact that no aggregation of the dodecyl sulfate ions is observed for 4mM and 6 mM concentrations is in agreement with the experimental data. However, it could be also an effect of the box size and the number of dodecyl sulfate units present in the box, which will be discussed later on.

In Figures 3 and 4 the initial and lowest total energy conformations for the production run for the concentrations from 8 mM, 10 mM, 12 mM (Figure 3), and 14 mM, 22 mM, and 32 mM (Figure 4) are presented. The water beads are not shown for clarity. For the case of the concentrations 8 mM to 32 mM it is observed that the dodecyl sulfate anions aggregate during the equilibration run, see a in Figures 3 and 4. Once the aggregate is formed during the equilibration run it is not disaggregated. Also, the aggregate is maintained during the whole production Shear Mesocite run for the concentrations 8 mM to 32 mM. In the micelle-like aggregate it is observed that the tails point inwards and the heads point outwards the aggregation micelle, in accordance with experimental observations.57 The shape of the aggregation micelle seems to be spherical but because there is a small number of dodecyl sulfate anion units, it is not possible to see the full



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spherical shape. The spherical shape is more evident in the case of the 32 mM concentration, see Figure 4.


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Figure 3. Initial configurations (a), obtained from the NVT equilibration run, and lowest energy configurations (b) obtained during the NVT Shear Mesocite run for 8 mM, 10 mM and 12 mM concentrations. The configurations shown in (b) were obtained at 95.2 ns, 80.1 ns, and 150 ns respectively.



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Figure 4. Initial configurations (a), obtained from the NVT equilibration run, and lowest energy configurations (b) obtained during the NVT Shear Mesocite run for 14 mM, 22 mM and 32 mM concentrations. The configurations shown in (b) were obtained at 135 ns, 139 ns, and 142 ns respectively.

During the production NVT Shear mesocite run the simulation box is deformed along the direction of the shear flow -shear plane B(BC). The shear rate applied to each system is 0.007869 ps-1. Also, the spherical aggregate is distorted, elongated, during the shear run due to the application of stress in the presence of shear flow. This effect is better seen for the case of the 32 mM concentration (see Figure 4b, for concentration 32 mM).

As mentioned earlier the objective of this study is to calculate the first CMC of SDS in aqueous solution from viscosity calculations using a coarse grain framework. It is not the objective to study the shape of the aggregation micelle, also the number of units of dodecyl sulfate in each simulation is such that the desire concentration is ensured in a small system, where the equilibration could be reached faster. Therefore, an aggregate with 60 units of dodecyl sulfate cannot be observed. The highest number of dodecyl sulfate units in this study is 16 (see Table 1). However, from 8 mM to 32 mM concentrations it is observed the aggregation of all the dodecyl sulfate units (see Figures 3 and 4). The experimental CMC of SDS is reported at 8-8.5 mM,4 depending on the measurement technique. Therefore, the fact that there is aggregation of the dodecyl sulfate units from 8mM to 32 mM is in good agreement with the experimental observation. However, the observed aggregation could be also an effect of the box size and the number of dodecyl sulfate units present, which will be discussed later on.


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In general, the sodium cations are close to the dodecyl sulfate heads, see Figure 2 to 4. In several of the studied concentration the formation of ion pairs between sodium cations and head groups is observed, see Figures 2 to 4.

From the production NVT Shear mesocite run the components of the stress tensor are calculated as well as the bulk viscosity. The bulk viscosity obtained for each calculated SDS concentrations are presented in Table 2.

Table 2. Calculated viscosity for each concentration. SDS Concentration (mM) 2 4 6 8 10 12 14 22 32

Bulk SDS

Calculated Bulk viscosity (cP) 0.806 0.807 0.809 0.810 0.814 0.815 0.818 0.830 0.843

In Table 2 it is observed that as the concentration of SDS is increased the bulk viscosity also increases. Even though there is the formation of dodecyl sulfate aggregate from 8mM to 32 mM the increase in the bulk density is gradual. However, the observed aggregation could be also an effect of the box size and the number of dodecyl sulfate units present, which will be discussed later on.



24

In Figure 5 there is presented a plot of the calculated bulk viscosity (cP) versus the SDS concentration (mM) in aqueous solution. In this plot it is observed that there are two linear regimes. The concentration at which these two lines cross corresponds to the CMC. From the two linear fit equations presented in Figure 5 a CMC of 7.8752 mM is obtained.

0.85 0.845 y = 1.373E-03x + 7.992E-01 R² = 9.972E-01

0.84 0.835

Bulk viscosity (cP)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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0.83 0.825 0.82 0.815 0.81 y = 7.000E-04x + 8.045E-01 R² = 9.800E-01

0.805 0.8 0

5

10

15

20

25

30

35

Concentration (mM)

Figure 5. Calculated bulk viscosity (cP) versus SDS concentration.

The linear correlation coefficients for both linear fits, in Figure 5, corresponds to R2=0.9800 and R2=0.9972, respectively; which can be considered as good correlations. The linear fit with a linear correlation coefficient of R2=0.9800 comprises four data points while the curve with the linear correlation coefficient of R2=0.9972 comprises six data points.


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The obtained CMC of 7.8752 mM is in very good agreement with the experimental data, which is reported to be from 8 to 8.5 mM4 -depending on the measurement technique. The experimental CMC value obtained from viscosity measurements is 8.3.58 Therefore, the proposed method to obtaining the first CMC of SDS in aqueous solution from viscosity coarse grain calculations is capable to reproduce the observed experimental data.

3.2. Effect of the Simulation Box Size on the Obtained CMC.

It can be considered that the box size used in the calculations is small (150 Å×75 Å×75 Å) as well as the number of SDS molecules in the calculations. For example, for the 2 mM concentration there is only one SDS molecule in the box and hence there is no possibility of formation of an aggregate. Also, aggregation was observed when 4 molecules of SDS were present in the box (8 mM concentration system) and it cannot be said if the aggregation is an effect of the concentration or of the number of SDS molecules. Therefore, in this section we investigate the effect of the box size on the calculated CMC, and the effect of the number of SDS molecules in the box on the formation of aggregates.

The size of the simulation box has been increased to a cubic simulation box with size 150 Å×150 Å×150 Å. In Table 3 the number of beads in each calculated system are presented for the different concentrations.

Table 3. Number of SDS molecules required to ensure each millimolar concentration, number of tail beads, number of head beads, number of sodium



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beads, number of water beads, total number of beads in each calculated system. The size of the box is 150 Å×150 Å×150 Å. SDS Concentration (mM)

Number of Total Total Total Total Total SDS Tail Head beads Water number coarse beads beads of Na+ b Beadsc of beadsd grained moleculesa 2 4 12 4 4 28130 28150 4 8 24 8 8 28112 28152 6 12 36 12 12 28096 28156 8 16 48 16 16 28078 28158 10 20 60 20 20 28061 28161 12 24 72 24 24 28043 28163 14 28 84 28 28 28025 28165 22 45 135 45 45 27951 28176 32 65 195 65 65 27864 28189 a Values obtained with Eq. 8. These values were rounded. b Hydrated sodium cation. c Total number of water beads for each concentration obtained by subtracting the sum of columns 3-5 to the total number of beads in each system (column 7). d Total number of beads in each system (sum of columns 3-6)

The number of total beads calculated with Eq. 6 (28170) differs, by up to 20 beads, from the number obtained by setting up the systems using the mesostructured tool in the Materials Studio package51 (see column 7 in Table 3). The highest percentage error corresponds to 0.071 %, which is negligible. After setting up the systems shown in Table 3 it is observed that the final density of each system agree with the experimental density55 (see column 8 in Table 1) up to the fourth digit after the decimal point.

All the systems presented in Table 3 are calculated following the methodology presented in section 2.3. Simulation Methodology. In Figure 6 to Figure 9 the final configuration obtained from the shear Mesocite production simulation for the studied concentrations, in a bigger box (150 Å×150 Å×150 Å), are presented.


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Figure 6. Final configurations that are obtained from the shear Mesocite production simulation for concentrations 2mM, 4mM, 6mM, and 8 mM. In all cases the size of the box is 150 Å×150



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28 Å×150 Å. The simulations are carried out for 150.8 ns. The total number of steps are 1x107 steps, and the time step is 15.08 ps. The water beads are not shown for clarity.

All the systems presented in Figure 6 have been ran the same amount of time as the systems with a smaller box (Figures 2-4), i.e. 1x107 steps of 15.08 ps, each step, for a total of 150.8 ns. It is observed for all the cases, except for the 2 mM concentration, that there is aggregation of the dodecyl sulfate units. The number of dodecyl sulfate units in each system are presented in Table 3. There are 4, 8, 12, and 16 dodecyl units in the 2 mM, 4 mM, 6 mM, and 8 mM concentrations respectively. In the case of the small box (Figures 2-3) the aggregation is observed starting at 8 mM, after 1x107 simulation steps, where there are 4 dodecyl sulfate units. For the case of the bigger box (Figure 5) the aggregation is observed starting at 4mM, after 1x107 simulation steps, where there are 8 dodecyl sulfate units. Therefore, the number of simulation steps have been increased for the case of the 2mM system, which contains 4 dodecyl sulfate units, to a total of 1.3x107 simulations steps, i.e. the calculation has been increased from 150.8 ns to 193 ns. The viscosity did not change but aggregation of SDS is observed (see Figure 7). Therefore, it can be said the aggregation is an effect of the number of SDS molecules in the box more than of the concentration.


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Figure 7. Aggregation of dodecyl sulfate units observed at 193 ns for the 2 mM concentration in a cubic box of size 150 Å×150 Å×150 Å, and 4 units of dodecyl sulfate in the box. The water beads are not shown for clarity.



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Figure 8. Final configurations obtained from the shear Mesocite production simulation for concentrations 10mM, 12mM, 14mM, and 22 mM. In all cases the size of the box is 150 Å×150 Å×150 Å. See text for explanation on the simulation times. The water beads are not shown for clarity.

In Figure 8 the final configurations obtained for the 10mM, 12mM, 14mM, and 22 mM concentrations in a box of size 150 Å×150 Å×150 Å are presented. The simulations have been carried out as long as it was necessary to ensure aggregation of most of the dodecyl sulfate units. All the systems shown in Figure 8 have been ran for 1.02 x107 simulation steps (153.8 ns),


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31 except the 12 mM system that has been ran for a longer time 158.3 ns (1.05 x107 simulation steps). There are 20, 24, 29, and 45 dodecyl sulfate units in the 10 mM, 12 mM, 14 mM, and 22 mM systems respectively. In most of the concentrations presented in Figure 8 all the dodecyl units aggregate. In all cases the tails point inwards the aggregate and the heads point outwards.

Figure 9. Final configurations obtained for the system with a concentration of 32 mM in a box of size 150 Å×150 Å×150 Å. The system shown in a was obtained after 1x107 simulation steps (150.8 ns), and the system shown in b was ran for 1.07x107 simulation steps (161.3 ns). The water beads are not shown for clarity.

In Figure 9 the final configurations obtained for the 32 mM system in a simulation box size of 150 Å×150 Å×150 Å are presented at different simulation times. There are 65 dodecyl sulfate units in the box (see Table 3). The configuration shown in Figure 9a has been obtained after a total of 150.8 ns of simulation. It is observed the formation of a micelle which contains 49



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dodecyl units and a smaller aggregate which contains 16 dodecyl units. The system has been ran for another 10.5 ns for a total of 161.3 ns of simulation but no changes in the aggregation or in the calculated viscosity are observed, see Figure 8b. It is reported that the average aggregation number in SDS micelles at the first CMC is around 60.8,9 With the proposed calculation method here presented it is observed that only 49 dodecyl sulfate units form a micelle (see Figure 9), which would correspond to 22% of error when comparing with the experimental result. This number might change if the calculated concentration is increased; however, as mentioned in the introduction, the interest of this work is to propose a method to calculate the CMC of an ionic surfactant more than the shape and number of units in the micelle formed.

In Table 4 the calculated bulk viscosity obtained for all the systems reported in Table 3 are presented. In Table 4 it can be seen that as the concentration of SDS increases the viscosity also increases.

Table 4. Calculated Bulk viscosity for each SDS concentration obtained using a cubic simulation box of size 150 Å×150 Å×150 Å SDS Concentration (mM) 2 4 6 8 10 12 14 22

Calculated Bulk viscosity (cP) 0.807 0.812 0.813 0.819 0.820 0.824 0.825 0.838


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32

0.848

In Table 5 the calculated viscosities for all the studied concentrations obtained using the small (150 Å×75 Å×75 Å), and the big (150 Å×150 Å×150 Å) simulations boxes are presented for comparison purposes, together with the percentage of error between the obtained viscosities.

Table 5. Comparison of the calculated viscosities obtained for the different concentrations when using different simulation boxes. SDS Calculated Bulk Calculated Bulk %Error Concentration viscosity (cP) viscosity (cP) (mM) Using small box Using large box (150 Å×75 Å×75 Å) (150 Å×150 Å×150 Å) 2 0.806 0.807 0.12 4 0.807 0.812 0.62 6 0.809 0.813 0.49 8 0.810 0.819 1.11 10 0.814 0.820 0.74 12 0.815 0.824 1.10 14 0.818 0.825 0.86 22 0.830 0.838 0.96 32 0.843 0.848 0.59

As it can be seen in Table 5 the viscosity values obtained with a bigger box (150 Å×150 Å×150 Å) are different to the values obtained with a smaller box (150 Å×75 Å×75 Å, Table 2). As it can be seen in Table 5, the smaller percentage error between the calculated viscosities for each concentration is 0.12%, for the 2 mM concentration, and the largest error corresponds to the 8 mM concentration (1.11%). The average error using different box sizes is 0.73%. It can be said that the highest percentage error obtained (1.11%, see Table 5) is actually negligible. The difference in the viscosity when different simulation boxes are used could be because the structure of the solution do affect the calculated viscosity. The number of units aggregated in the



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different boxes for a same concentration are different, but the viscosity values calculated are of similar order of magnitude (see Table 5).

In Figure 10 a plot of the calculated bulk viscosity (cP) versus SDS concentration, for the systems presented in Table 3, and using a bigger simulation box (150 Å×150 Å×150 Å), is presented. As it is the case of the plot presented in Figure 5, in the plot of Figure 10 it is observed that there are two linear regimes, one represented by the blue circles and the other represented by the orange circles (see Figure 10). The difference in the two linear regimes in Figure 10 is not as pronounced as the one obtained when using a smaller simulation box (see Figure 5). The concentration at which these two lines cross corresponds to the CMC, and from the two linear fitted equations (see blue and orange lines), shown in Figure 10, a CMC of 7.8947 mM is obtained. As in the case of Figure 5, the blue line comprises four data points, while the orange line comprises six data points.


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Figure 10. Calculated bulk viscosity (cP) versus SDS concentration for the systems presented in Table 3, calculated with a bigger calculation box (150 Å×150 Å×150 Å).

The CMC of dodecyl sulfate obtained with a small simulation box (150 Å×75 Å×75 Å) is 7.8752 mM (see Figure 5), while the CMC obtained with a bigger box (150 Å×150 Å×150 Å) is 7.8947 mM. The difference is negligible and in both cases the calculated CMC agrees very well with the experimental value of 8.4 The percentage error in the calculation of the CMC for the small simulation box is: %E))F)

GH.HI G G

∗ 100 1.56%

(9)

While the percentage error in the calculation of CMC for the bigger simulation box is: %E))F)

GH.JH G G

∗ 100 1.32%.

(10)

Therefore, it can be said that the error involved in the CMC calculation due to finite size effects, using different simulation box sizes, is between 1.32 percent and 1.56 percent.



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The aggregation observed for concentrations below the CMC (at 2 mM, 4 mM, and 6 mM concentrations), for the bigger simulation box, see Figures 6-9, does not correspond to actual micelles, and therefore does not affect the calculation of the CMC, which is the same (7.9 mM) for the small and big simulation boxes. The aggregates observed below the CMC, in the case of the bigger simulation box, do not resemble the circular shape of micelles. It is from 8mM concentration on, Figures 6-9, that the typical circular micelles are observed.

The interest of the present work is to propose a method to calculate the CMC of SDS, or in general of an ionic surfactant, from viscosity calculations. As shown here the viscosity of each concentration might change depending on the size of the simulation box but the calculated CMC is the same either a small or big box is used in the simulations, and spite of the difference in the calculated viscosities. As mentioned before the difference in the viscosity when different simulation boxes are used could be due to the fact that the structure of the solution do affect the calculated viscosity. The number of units aggregated in the different boxes for a same concentration are different, but the viscosity values calculated are of similar order of magnitude (see Table 5), and the calculated CMC is the same for both cases.

It is not known if increasing the simulation times to values much bigger than the ones used here of 150 ns -160 ns would change the calculated viscosities, and this will be investigated in the near future. However, the obtained CMC, either using a small or a big simulation box, is the same (7.9 mM) and in good agreement with the experimental result. Therefore, it can be said that the calculation of the CMC from viscosity calculations does not depend on the size of the


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simulation box or on the number of dodecyl units in the box. The same CMC result has been obtained, following the proposed method, but using different simulation boxes. The proposed methodology in the present work is useful to calculate the CMC of SDS or of an ionic surfactant.

In Figure 11 a plot which contains the calculated bulk viscosity (cP) versus SDS concentration for both simulation boxes, small and big, is presented. The curve with colors green and red corresponds to the big simulation box (same curve as in Figure 10), while the curve with colors blue and orange corresponds to the small simulation box (same curve as in Figure 5).

Figure 11. Calculated bulk viscosity versus SDS concentration for the small (blue and orange curves) and big (green and red curves) simulation boxes.



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As it can be seen in Figure 11 the viscosity in these curves shows different behaviors. For the case of the small box, curves blue and orange, before the CMC the viscosity increases slowly with the concentration (blue curve). After the CMC the viscosity increases rapidly with the concentration (orange line). For the case of the big simulation box (curves green and red) before the CMC (green curve) the viscosity increases rapidly, and after the CMC (red curve) the viscosity increases slowly. The orange and red curves in Figure 11 seem to have almost the same slope, therefore it can be said that the viscosity increases in general at the same rate after the CMC for the small and big simulation boxes. The main difference is observed before the CMC for both simulation boxes (green and blue curves). In the green curve (big simulation box) the viscosity increases more rapidly with the concentration than in the case of the blue curve (small simulation box). This is because in the case of the big simulation box (green curve) there is formation of aggregates before the CMC at concentrations 2 mM, 4 mM, and 6 mM (see Figures 6 and 7). These aggregates are not actual micelles, and do not have the characteristic circular shape of micelles, but these aggregates increase the viscosity (green curve). In the case of the small simulation box (blue curve) the viscosity increases slowly because no aggregates are formed before the CMC (see Figure 2).

4.

CONCLUSIONS.

In the present study the first critical micelle concentration, CMC, of the ionic surfactant sodium dodecyl sulfate (SDS) in diluted aqueous solution, at 298.15 K, has been determined from coarse grain NVT shear Mesocite simulations. The bulk viscosity of SDS was calculated for SDS aqueous solutions from 2 mM to 32 mM in concentration using the MARTINI force field. From


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a plot of the calculated bulk viscosity versus SDS concentration the CMC has been obtained. Two different simulation boxes of different size has been used in the calculations, one small box of size 150 Å×75 Å×75 Å and one big box of size 150 Å×150 Å×150 Å. In both cases the CMC is obtained from a plot of the calculated bulk viscosity versus the SDS concentration. The CMC is obtained as the concentration in which there is an inflexion point of two linear regimes.

The same CMC result has been obtained, following the proposed method here presented, but using different simulation boxes. The obtained CMC using the big simulation box was 7.8947 mM, which compares very well with the CMC obtained with the small simulation box of 7.8752 mM. Both calculated CMCs, with a small and big simulation box, agree very well with the experimental result of 8-8.5 mM.4 The error involved in the CMC calculation due to finite size effects, using different simulation box sizes, is between 1.32 percent and 1.56 percent. It can be said that the calculation of the CMC from viscosity calculations does not depends on the size of the simulation box or on the number of dodecyl units in the box but on the concentration of the SDS molecules.

The viscosities obtained for each concentration using the bigger simulation box are of same magnitude as the viscosities obtained with the smaller simulation box, but they are not identical. The difference in the viscosity when different simulation boxes are used could be because the structure of the solution do affect the viscosity calculated. The number of units aggregated in the different boxes, for a same concentration, are different but the calculated viscosity values are of similar magnitude. The highest percentage error in the calculated viscosity when comparing the



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results obtained using different simulation box sizes is as large as 1.11%, which can be considered negligible.

Even with coarse grain models equilibrium can be difficult to achieve for dilute solutions when a sufficiently large system is studied to allow the formation of multiple micellar aggregates. Therefore, here we propose to simulate small systems, but ensuring the millimolar (mM) concentration, to be able to equilibrate the system in a reasonable computing time. Our objective was to estimate the bulk viscosity and the CMC rather than to study the micelle shape and distribution. Even though the number of dodecyl sulfate units consider in the calculations with the small simulation box is small, aggregation of these units is observed, and once the aggregate is formed it does not disaggregate.

For the case of the small simulation box (150 Å × 75 Å × 75 Å) an aggregate with 60 units of dodecyl sulfate cannot be observed -the highest number of dodecyl sulfate units in this study is 16. For the case of the aggregate formed with 16 units of SDS (32 mM) a spherical shape is observed. This shape of the aggregated is elongated during the shear simulation due to the stress applied in the presence of shear flow to calculate the bulk viscosity of the system. The tails point inwards and the heads point outwards the aggregation micelle, in accordance with experimental observations. In general, the sodium cations are closed to the dodecyl sulfate heads. In several of the concentration studied the formation of ion pairs between sodium cations and head groups was observed.


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For the case of the big simulation box, 150 Å × 150 Å × 150 Å, aggregation of all the dodecyl sulfate units is observed for all the concentrations. For the 32 mM concentration there are 65 dodecyl sulfate units in the box. For this concentration it is observed the formation of a micelle which contains 49 dodecyl units and a smaller aggregate which contains 16 dodecyl units. It is reported that the average aggregation number in SDS micelles at the first CMC is around 60.8,9 With the proposed calculation method here presented it is observed that only 49 dodecyl sulfate units form a micelle, which would correspond to 22% of error when comparing with the experimental result. The tails point inwards and the heads point outwards the aggregation micelle, in accordance with experimental observations.

The aggregation observed for concentrations below the CMC (at 2 mM, 4 mM, and 6 mM concentrations), for the bigger simulation box, does not correspond to actual micelles, and therefore does not affect the calculation of the CMC, which is the same (7.9 mM) for the small and big simulation boxes. The aggregates observed below the CMC, in the case of the bigger simulation box, do not resemble the circular shape of micelles. It is from 8 mM concentration on, Figures 6-9, that the typical circular micelles are observed.

To the best of our knowledge, this is the first report of the calculation of the CMC of SDS in aqueous solution from viscosity simulations using the coarse grain framework. The coarse grain framework, permits to process much larger spatial and temporal scales than all-atom molecular dynamics can. Furthermore, the CMC obtained with the coarse-grained model is in much better agreement with the experiment than the value obtained with atomistic simulations, which tend to overestimate the CMC of SDS.



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The results and methodology presented in this work will help in the study, at the coarse grain level, of the promotion/inhibition mechanism of the formation of gas hydrates by SDS below and above the CMC.

AUTHOR INFORMATION Corresponding Author * E-mail: [email protected]

Notes The authors declare no competing financial interest

ACKNOWLEDGEMENT Financial support for this research from Instituto Mexicano del Petróleo under project D.61017 is gratefully acknowledged.

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