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Determining Critical Micelle Concentrations of Surfactants Based on Viscosity Calculations from Coarse-Grained Molecular Simulations Hassan Alasiri Energy Fuels, Just Accepted Manuscript • DOI: 10.1021/acs.energyfuels.8b04228 • Publication Date (Web): 30 Jan 2019 Downloaded from http://pubs.acs.org on January 31, 2019
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Determining Critical Micelle Concentrations of Surfactants Based on Viscosity Calculations from Coarse-Grained Molecular Simulations Hassan Alasiri* Chemical Engineering Department/Center for Refining and Petrochemicals, King Fahd University of Petroleum & Minerals, Dhahran, Saudi Arabia ABSTRACT: Alternative methods for determining the critical micelle concentration (CMC) are proposed using coarse-grained molecular dynamics simulations. In the literature, the CMC value is commonly taken to be the “free” (unassociated) surfactant concentration in the presence of micellar aggregates given by molecular dynamics simulations. In the present study, two surfactants with different head groups and the same alkyl tail, C8TAB and S8S, were used to calculate the CMC value. The CMC of surfactants can be calculated from the variation in the zero-shear-rate viscosity as a function of surfactant concentration in both the premicellar and micellar regions. Linear slopes were obtained in both regions with increasing surfactant concentrations. The CMC values for C8TAB and S8S were determined from the intersection of the two lines, and are in good agreement with experimental values.
Introduction Surfactants are attracting increased attention because they are key components in numerous industrial materials such as detergents, cleaners, cosmetics, personal care products, textiles, fibers, paints, plastics, medicine, oilfield chemicals, pharmaceuticals, food, pest control, and plant protection [1–7]. One of the most interesting properties of surfactants in solution is their ability to self-aggregate to form micelles at certain concentrations, known as the critical micelle concentration (CMC). The CMC of a surfactant is a physical value that causes a transition in many of the solution properties, such as the conductivity, surface tension, detergency, interfacial tension, turbidity, selfdiffusion, solubility, and osmotic pressure. Surfactants consists of hydrophilic (polar) and hydrophobic (nonpolar) groups. The arrangement of the surfactants in their surroundings depends on the medium that surrounds them. In a polar solution, the micelles form as the hydrophilic head of the surfactant is facing outwards and the hydrophobic tail is in the micelle core. In a nonpolar solution, the micelles form in the inverse manner. Several experimental techniques have been proposed for measuring the CMC value of surfactants, such as spectroscopic methods, surface tension, viscosity, light scattering, and conductivity [8–11]. Molecular modelling tools can be used to predict surfactant properties such as the CMC. In many simulation studies, the CMC value is obtained from the concentration of free surfactants. The free surfactant concentration attains
a maximum near the CMC and then decreases at higher concentrations of surfactant inside the solution. Desplat and Care [12] investigated the CMC and a cluster size distribution in the micellar region using Monte Carlo simulations of a binary mixture of solvent and amphiphile chains in which free self-assembly of the chains was allowed. The dependence of the weightaverage aggregation number on the total amphiphile concentration confirmed theoretical predictions. The aggregation of a lattice model for amphiphiles using Monte Carlo simulations was studied by Mackie and coworkers [12]. Their research found that the free monomer concentration decreased above the CMC. Lisal et al. [13] have studied the self-assembly of ethylene oxide surfactants in supercritical carbon dioxide as a solvent using large-scale Monte Carlo simulations. They performed different simulations to study the effect of using head and tail groups for the surfactant. The CMC was considered to be the concentration at which the number of surfactants aggregated in micelles was equal to the number of free surfactants. They found that increasing the head group of surfactants decreased the CMC value, whereas increasing the head group made no significant difference to the CMC value. These results agree with experimental observations for nonionic surfactants in water. In 2006, Cheong et al. [14] used Monte Carlo simulations to study the micellization behavior of sodium dodecyl sulfate ionic surfactants. The effects of temperature on CMC, average aggregation number, and the degree of counterion binding were evaluated. The model produced the correct
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shear rates to determine the CMC value. For this purpose, two types of surfactants have been selected: noctyltrimethylammonium bromide (C8TAB) and sodium octyl sulfate (S8S).
trend for the temperature, but the CMC values were underestimated compared with experimental results. A combination of coarse-grained molecular modeling and graphical processing unit (GPU) acceleration has been used to study different nonionic polyethylene glycol (PEG) surfactants [15]. The CMC values for more hydrophilic surfactant systems were found to be in quantitative agreement with experimental values, but the CMC values for more hydrophobic surfactants were underestimated. Sanders et al. [16] studied the micellization behavior of sodium alkyl sulfate surfactants in water through molecular dynamics simulations. They found a strong dependence between the free surfactant concentration and the overall surfactant concentration. The CMC values were obtained for alkyl tail lengths of 6–9 carbon atoms at different temperatures. The dependence of CMC on temperature and tail length qualitatively agreed with experimental trends. Dissipative particle dynamics (DPD) is a mesoscale simulation model that has become a powerful tool for modelling surfactant systems. The advantage of DPD over atomistic methods such as molecular dynamics (MD) is its lower computational cost for complex fluids. DPD has been used to study the CMC [17,18], with the results indicating a decrease in the free surfactant concentration with an increase in the total surfactant loading. The DPD simulations obtained qualitative agreement with the experimental CMC and aggregated number of surfactants. There are alternative methods for determining the CMC that do not rely on the free surfactant concentration. The equilibrium micelle size distribution can determine the CMC value, where the lowest concentration in the distribution with a micellar peak provides the CMC [19,20]. The CMC can also be obtained from the osmotic pressure through histogram-reweighting grand canonical Monte Carlo simulations [21–23]. In this paper, an alternative methodology is presented to determine the CMC from the viscosity using coarsegrained molecular simulations. This method was used experimentally to determine the CMC values from variations in the relative viscosity of various surfactants in regions below and above the CMC [9]. To the best of our knowledge, no previous molecular simulations have calculated the CMC value using the viscosity method. Molecular simulations provide a molecular interpretation of experimental results, which can often be difficult to obtain experimentally. We can determine the CMC value by applying different shear rates to the surfactant–water system to find the viscosity as a function of concentration. The aim of this work is to study the behavior of surfactants in water at different
Methodology Coarse-grained molecular simulations were used to calculate the CMC of surfactants. The coarse-grained molecular model simulates molecules as chains of spherical beads. The motion of the beads is described according to Newton’s equations of motion: 𝑑𝑟𝑖 𝑑𝑝𝑖 (1) = 𝑣𝑖; = 𝑓𝑖 𝑑𝑡 𝑑𝑡
where ri is the position of bead i, vi pi / mi is the
velocity of bead i, f i is the force on bead i, and mi is the mass of bead i. To model the behavior of complex fluids in coarse-grained molecular simulation, the forcefield between the beads must be determined. This forcefield describes the interaction between beads representing groups of atoms. This study uses one of the most popular forcefields in coarse-grained molecular simulations, known as the Martini forcefield. The Martini forcefield was developed for coarse grained simulations of lipids, surfactants, and polymers [24,25], and has been extended to simulate proteins [26]. The Martini forcefield employs a four-to-one mapping to build up the coarse-grained beads. In this manner, each bead represents a group containing, on average, four non-hydrogen atoms. For example, one bead may compose a butylene group (C4H9) or a lump of four water molecules. The Martini forcefield considers four main types of interaction sites: polar (P), nonpolar (N), apolar (C), and charged (Q). These sites are formulated in terms of the van der Waals parameters, particularly the energy parameter in the Lennard–Jones 12-6 function. To allow for a more accurate representation of the chemical nature of the underlying atomic structure, each Martini forcefield type is divided into subtypes. Further details about the classification of the Martini forcefield can be found in [24,25]. Figure 1 and Table 1 depict the mapping of the system from an atomistic model to a mesomolecular model. The molecular simulations were performed in an NPT ensemble (constant number of particles N, pressure P, and temperature T). A cubic box was created for each system and all simulations began with a random distribution of the beads under periodic boundary conditions. The total length of each simulation run was 2×106 steps, with a time step set to 1 fs to ensure that the equilibrium stage was reached. Different shear rates were then applied to each box. The viscosity can be
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calculated once the system has reached a steady state under the influence of shear flow. The concentration of surfactant inside the box varied below and above the CMC values. To calculate the concentration in physical units (mol/l), the following equation was used to convert the amount of surfactant in the system to a concentration (C) in real units (mol/l), enabling a comparison with experimental values:
𝐶=
𝑛
viscosity of water is determined in the plateau regain by taking the constant value for the zero-shear-rate viscosity (𝜂0). For water, 𝜂0 is 1.21 cP. The simulation value is larger than the experimental value (0.89 cP). This is expected in coarse-grained models, which tend to move on faster times scales as some atomistic details are neglected and groups of molecules, rather than every individual molecule, are tracked [27]. As the goal of this study is to determine the CMC value, the benefits of coarse-grained models can be used to reach the equilibrium faster. Figures 3(a) and 3(b) show how the viscosity varies with shear rate at different concentrations for our model surfactant, as obtained from NVT simulations. In these figures, for clarity, only three different concentrations are shown for each surfactant system (below, near, and above the CMC value). The values obtained from the simulation of water surfactant systems show the same behavior as for pure water. In general, the viscosity increases with decreasing shear rate until reaching some constant value, regardless of surfactant concentration. The power law index (n) is plotted in Figure 4 for C8TAB and S8S as a function of surfactant concentration. Each system has a power law index that describes its general viscosity behavior with respect to the concentration. This is important in attempting to understand what can be expected as the concentration of surfactant in water changes. For systems with a low concentration, the power law index has a constant value. As the concentration of surfactant increases, the power law index decreases. We can conclude that n depends on the concentration of surfactant. The reason for the constant value of n at low concentrations is that the surfactants are free in systems without aggregation. However, when the concentration is near to and above the CMC value, n decreases because the system begins forming aggregates. Comparing the power law index for both surfactants, S8S exhibits higher values than C8TAB because it has a lower CMC value. In general, n ranges from 0.5–0.55 in this study. The power law index for worm-like micelles (cylinder-shaped) of cationic surfactants/water systems was found to be 0.89 [28]. The n values in our results are lower because the micelles in this study are spherical near to and above the CMC value. We may conclude that, under high shear rates, spherical micelles can orient themselves faster than cylindrical aggregate shapes.
(2)
3
𝐿 𝑁𝐴
where n is the number of surfactant molecules in the box, L is the side length of the simulation box (L = 150 Å3), and NA is Avogadro’s number. Table 1. Coarse-grained particle types and building blocks. Note: Further details about the interaction matrix of the Martini forcefield can be found in [24,25]. Name
Type
Building Block
P4
polar
4 H2O
Qa for S8S
charged
SO4-
Qa for C8TAB
charged
Br-
Qd
charged
Na+
Q0
charged
C3N+
C1
apolar
C4
Results and discussion The viscosity of a fluid can be measured experimentally or from simulations to obtain the CMC value. The rheological properties of simulation systems were analyzed under different shear rates (𝛾) to determine the viscometric flow 𝜂(𝛾) for each shear rate. This study considered a system for pure water to determine the viscosity at 298 K and 1 atm. Figure 2 shows the dependence of the viscosity on the applied shear rate. As can be seen from the figure, the viscosity increases as the shear rate decreases, eventually reaching a constant value. At high shear rates, linear behavior can be observed—this linear region corresponds to the power law: (3) 𝜂 = 𝐴𝛾𝑛 ― 1 where A is a constant of proportionality and n is a power law index. The value of n for pure water systems is 0.553. The power law index is a convenient way of comparing different materials. We will use this index to study two different surfactants as well as the concentration of surfactant. At lower shear rates, viscosity is expected to approach a constant value. The
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(a) Atomistic Model
(b) Mesomoleculer Model
S8S
Qd
Qa
C1
C8TAB
Qa
Q0
C1
Water
C1
C1
P4
Figure 1. (a) Atomistic model of surfactants (S8S and C8TAB) and water. (b) Mesomolecular model: the coarse-grained model of S8S is represented as two tail (C1) beads and two head beads Qa and Qd. C8TAB is represented by two tail (C1) beads and two head bead Qa and Q0. Water (P4) bead has four water molecules.
10
Viscosity (cP)
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1
0.1 1E9
1E10
1E11
Shear Rate (1/s) Figure 2. Logarithm of viscosity vs logarithm of shear rate for water.
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1E12
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10
1
0.0492 M 0.295 M
Viscosity (cP)
Viscosity (cP)
10
1
0.0492 M 0.295 M
0.492 M
0.492 M
0.1
0.1 1E9
1E10
1E11
1E12
1E13
1E9
1E10
Shear Rate (1/s)
1E11
1E12
1E13
Shear Rate (1/s)
(a)
(b)
Figure 3. Logarithm of viscosity vs logarithm of shear rate for water/surfactant system. (a) C8TAB surfactant (b) S8S surfactant.
n
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0.555 0.55 0.545 0.54 0.535 0.53 0.525 0.52 0.515 0.51 0.505 0.5 0.495 0.49
C8TAB S8S
0
0.1
0.2
0.3
0.4
0.5
0.6
concentration (M) Figure 4. Power law index vs surfactant concentration for C8TAB and S8S.
Figure 5 depicts the zero-shear-rate viscosity as a function of concentration for C8TAB and S8S. The zero-shear-rate viscosity of surfactant solutions with respect to concentration across the micellization region was calculated by varying the shear rate. 𝜂0 increased with increasing surfactant concentration both below and above the micellar region, albeit at a different rate, which made it possible to differentiate between the two regions. The slope and intercept for the two linear
regions of the zero-shear-rate viscosity as a function of surfactant concentration for each of the surfactants are presented in Table 2. The slope of the region below the CMC value is shallower than in the micellar region because of the spherical shape that appears in the micellar state. We defined the CMC of the surfactant from the intersection of the two straight lines. Table 2 gives the CMC values determined in this study. The CMC values for C8TAB and S8S are 0.272 M and 0.176
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M, respectively. The MD results are in good agreement with the experimental values.
Zero-shear rate viscosity (cP)
1.6
Zero-shear rate viscosity (cP)
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1.5 1.4 1.3 1.2 1.1
1.5 1.45 1.4 1.35 1.3 1.25 1.2 1.15 1.1 1.05 1
1 0
0.1
0.2
0.3
0.4
0.5
0
0.6
0.1
0.2
0.3
0.4
0.5
Concentration (M)
Concentration (M)
(a)
(b)
Figure 5. Zero-shear-rate viscosity vs concentration for water/surfactant system. (a) C8TAB (b) S8S (blue color: below the micellar region and red color: upper the micellar region).
Table 2. Slope and intercept below and above the micellar region with CMC values from MD and experiments.
Surfactant C8TAB S8S
Below
Upper
Slope
Intercept
Slope
Intercept
CMC value MD (M)
CMC value Exp. (M)
0.2288
1.2151
1.2010
0.9508
0.272
0.293 [29]
0.1180
1.2151
0.7343
1.1066
0.176
0.155 [30]
model will provide new insights into the molecular-scale behavior of systems, especially for the viscosity of Newtonian and non-Newtonian fluids.
Conclusion Studying self-assembly in surfactant solutions by experimental methods or molecular modelling is important for many industrial processes. The molecular modelling can provide valuable information on the behavior of surfactants in solution and the predictions of surfactant properties such as CMC. In this paper, a new methodology has been proposed for determining the CMC of surfactants in water systems using coarsegrained MD simulations. The good agreement between the simulation results and the experimental values validates the use of this approach for CMC determination. We believe that the current method could be employed to study the rheological behavior of surfactants in different phases of its micellization, such as cylindrical and lamellar phases. This would aid researchers who are interested in synthesizing new surfactants. For specific applications, the proposed
AUTHOR INFORMATION Corresponding Author * E-mail:
[email protected]..
Funding Sources Author thanks the support provided by the Deanship of Scientific Research (DSR) at King Fahd University of Petroleum & Minerals (KFUPM) for funding this work through project No.SR17018.
ACKNOWLEDGMENT Author acknowledge the support for the current research from King Fahd University of Petroleum and Minerals (KFUPM) (SR171018).
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ABBREVIATIONS
Octyl , and Nonyl Sulfates. J Phys Chem B Were 2012;116:2430–7. doi:10.1021/jp209207p. [17] Mao R, Lee M, Vishnyakov A, Neimark A V. Modeling Aggregation of Ionic Surfactants Using a Smeared Charge Approximation in Dissipative Particle Dynamics Simulations. J Phys Chem B 2015;119:11673–83. doi:10.1021/acs.jpcb.5b05630. [18] Vishnyakov A, Lee M, Neimark A V. Prediction of the Critical Micelle Concentration of Nonionic Surfactants by Dissipative Particle Dynamics Simulations. J Phys Chem Lett 2013;4:797–802. doi:10.1021/jz400066k. [19] Wijmans CM, Linse P. Modeling of Nonionic Micelles. Langmuir 1995;11:3748–56. doi:10.1021/la00010a027. [20] Ben-Nalm A, Stllllnge FH. Critical Micelle Concentration and the Size Distribution of Surfactant Aggregates. J Phys Chem 1980;84:2872–6. [21] Floriano MA, Caponetti E. Micellization in Model Surfactant Systems. Langmuir 1999;15:3143–51. doi:10.1021/la9810206. [22] Panagiotopoulos AZ, Floriano MA, Kumar SK. Micellization and Phase Separation of Diblock and Triblock Model Surfactants. Langmuir 2002;18:2940–8. doi:10.1021/la0156513. [23] Gindy ME, Prud’homme RK, Panagiotopoulos AZ. Phase behavior and structure formation in linear multiblock copolymer solutions by Monte Carlo simulation. J Chem Phys 2008;128:164906. doi:10.1063/1.2905231. [24] Marrink SJ, Vries AH De, Mark AE. Coarse Grained Model for Semiquantitative Lipid Simulations. J Phys Chem B 2004;108:750–60. [25] Marrink SJ, Risselada HJ, Yefimov S, Tieleman DP, Vries AH De. The MARTINI Force Field: Coarse Grained Model for Biomolecular Simulations. J Phys Chem B 2007;111:7812–24. [26] Monticelli L, Kandasamy SK, Periole X, Larson RG, Tieleman DP, Marrink S. The MARTINI Coarse-Grained Force Field : Extension to Proteins. J Chem Theory Comput 2008;4:819–34. [27] Gyawali G, Stern S, Kumar R, Rick SW. Coarse-Grained Models of Aqueous and Pure Liquid Alkanes. J Chem Theory Comput 2017;13:3846–53. doi:10.1021/acs.jctc.7b00389. [28] Antunes FE, Coppola L, Gaudio D, Nicotera I, Oliviero C. Shear rheology and phase behaviour of sodium oleate / water mixtures. Colloids Surfaces A Physicochem Eng Asp 2007;297:95– 104. doi:10.1016/j.colsurfa.2006.10.030. [29] Zieliński R. Effect of temperature on micelle formation in aqueous NaBr solutions of octyltrimethylammonium bromide. J Colloid Interface Sci 2001;235:201–9. doi:10.1006/jcis.2000.7364. [30] Klevens H. Critical micelle concentrations as determined by refraction. J Phys Chem 1948;111:130–48. doi:10.1021/j150457a013..
CMC: critical micelle concentration; MD: molecular dynamics; C8TAB: n-octyltrimethylammonium bromide; S8S: sodium octyl sulfate.
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