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Consistent definitions of “the interface” in surfactant-free micellar aggregates
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Colloids and Surfaces A: Physicochem. Eng. Aspects xxx (2014) xxx–xxx
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Colloids and Surfaces A: Physicochemical and Engineering Aspects journal homepage: www.elsevier.com/locate/colsurfa
Consistent definitions of “the interface” in surfactant-free micellar aggregates Sebastian Schöttl a , Didier Touraud a , Werner Kunz a , Thomas Zemb b,∗ , Dominik Horinek a,∗∗ a b
Institut für Physikalische und Theoretische Chemie, Universität Regensburg, Universitätsstraße 31, 93053 Regensburg, Germany Institut de Chimie Séparative de Marcoule (ICSM) UMR 5257 CEA/CNRS/UM2/ENSCM, F-30207 Bagnols sur Céze, France
h i g h l i g h t s
g r a p h i c a l
a b s t r a c t
• Surfactant-free micelle aggregates are studied with molecular dynamics simulations. • Aggregates of the different preferred aggregation numbers are analyzed. • Different criteria for an interface definition are tested. • The location of the interface can be defined in a consistent way.
a r t i c l e
i n f o
Article history: Received 31 August 2014 Received in revised form 12 November 2014 Accepted 13 November 2014 Available online xxx Keywords: Molecular dynamics Simulations Clusters Microemulsion Ternary solution Ouzo effect
a b s t r a c t The theoretical description of systems with soft interfaces requires a precise definition of the interfaces that are present. Such definitions are common for interfaces of simple liquids and for surfactant assemblies like micelles, membranes, and microemulsions. Here, we focus on several possible rigorous definitions in a recently described surfactant-free microemulsion that consists of octanol, ethanol, and water, in which micelle-like aggregates are in dynamic equilibrium with the surrounding pseudo phase. We test different definitions that are based on the radial distribution functions of the components with respect to the center of these surfactant-free micelles. All definitions result in experimentally indistinguishable locations of the interface within the limits of resolution of light, X-ray, and neutron scattering experiments, since their locations differ by at most 0.4 nm, which corresponds to roughly 2-3 bond lengths. © 2014 Published by Elsevier B.V.
1. Introduction The thermodynamic discussion of systems with two phases or pseudo-phases involves the definition of the interface as one of the essential steps [1]. Based on this definition, the excess properties of
∗ Corresponding author. ∗∗ Corresponding author. Tel.: +49 9419434745. E-mail addresses: [email protected] (T. Zemb), [email protected] (D. Horinek).
this interface and their thermodynamic consequences can be analyzed [2]. The exact definition of the interface is however subject to some ambiguity and it depends on the type of system, which definitions are possible and meaningful. At a vapor/liquid interface, the common definition of the interfacial plane is the Gibbs dividing surface (GDS), which is the plane for which the liquid has zero interfacial excess. Other definitions like the plane of maximum density gradient (inflection point of the density profile) or the point at which the density is half of the bulk value usually coincide with the location of the GDS at vapor/liquid interfaces, which have a symmetrically shaped density profile.
http://dx.doi.org/10.1016/j.colsurfa.2014.11.029 0927-7757/© 2014 Published by Elsevier B.V.
Please cite this article in press as: S. Schöttl, et al., Consistent definitions of “the interface” in surfactant-free micellar aggregates, Colloids Surf. A: Physicochem. Eng. Aspects (2014), http://dx.doi.org/10.1016/j.colsurfa.2014.11.029
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These definitions are extendable to liquid/liquid interfaces, in which case the density can be split into the partial densities of the two phases [3]. In this case, a typical definition for the interface is the plane at which the two densities – normalized to their bulk values – are equal [4]. This however can lead to an inconsistent interface location, because a definition using the density profiles can lead to a different position than a definition based on the position of the neutral plane of bending [5]. Moreover, for these interfaces the density profiles are not necessarily symmetric and even different density-based definitions of the interface need not to coincide. This is problematic, because for the prediction of phase diagrams a lateral equation of state is needed, which requires an operative definition of the interface [6]. The consistency of the interface location becomes especially crucial for interdigitated or weakly associated surface layers such as those formed by carboranes, since water molecules are present on both sides of the interface [7]. When interfaces of solutions are studied, the Gibbs dividing surface is especially useful for the determination of surface tension increments [8]. When surfactants are present at the interface, their preferred position is naturally close to the interface in any of the senses mentioned so far. Therefore, the surfactants themselves can be employed for determining the interface. For charged surfactants, e.g., a plane of fixed charge has been employed. [9] In the special case of surfactant solutions, micellar systems, the nature of the interface depends strongly on the type of micelle under consideration. Precise definitions of the location of the “interface” have been proposed for ionic [10] and nonionic micelles [11] as well as for oil soluble surfactants [12], allowing for precise evaluations of the free energy per unit surface as a function of the area per molecule. Recently, scattering experiments with light, X-rays, and neutrons [13–15] demonstrated the existence of micelle-like aggregates of 20–100 molecules in thermodynamically stable ternary solutions, in which an ethanol-soluble but water-insoluble compound like octanol is dissolved in an ethanol/water mixture. Micelle-like aggregates form in these systems even if these systems do not match the classical picture of self-aggregating surfactant molecules. It was argued that in these systems the ethanol molecules play the role of a hydrotrope that mediates the solubility of the ‘nonpolar component’. Classical force-field molecular dynamics simulations were applied for a detailed molecular analysis of these systems [16]. These simulations showed an aggregate size distribution that is distinctly different to simple aggregation in binary solutions [17] or to percolation of water in a binary solution [18]. In this work we closer analyze the structure of the differently sized preferred aggregates that were observed in simulations and compare this to the properties of a flat water/octanol interface in the presence of a small amount of ethanol. We derive and compare different definitions for the interface between the two pseudophases and discuss their implications.
2. Results and discussion The simulations of a macroscopically stable one-phase system comprising octanol (NOct = 224), ethanol (NEth = 6366), and water (NWat = 21 839) reveal that the histogram of aggregate sizes shows several broad peaks. These peaks correspond to preferred aggregates that occur for octanol aggregation numbers N approx. equal 22, 33, 55, and 100. In Fig. 1, we show exemplary aggregates with these aggregation numbers. We demonstrate the generality of the interface definition by showing the results of a planar water/octanol interface in the presence of ethanol with a concentration similar to the aggregate systems. The aggregates of size 22, 33, and 55 are
Fig. 1. Snapshots of the different aggregates of size 22, 33, 55, and 100, and of the planar interface with 5% ethanol in the aqueous phase. Shown are only the octanol molecules. The three smaller aggregates clearly correspond to one concise object. To the contrary, the aggregate with 100 octanol molecules comprises more than one object that are grouped together as an artifact of the algorithm used for aggregate determination.
Please cite this article in press as: S. Schöttl, et al., Consistent definitions of “the interface” in surfactant-free micellar aggregates, Colloids Surf. A: Physicochem. Eng. Aspects (2014), http://dx.doi.org/10.1016/j.colsurfa.2014.11.029
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Fig. 3. Histogram of the van der Waals (top) and Coulomb (bottom) interactions of ethanol molecules with an octanol aggregate of size 22, 33, and 55. The histograms are normalized such that the minimum has a value of one. For both interactions, the distributions are independent of the cluster sizes. In all cases, the majority of ethanol molecules have a very small interaction, and there is a peak for zero interaction energy. All plots show a second population of ‘bound’ ethanol molecules with more favorable interaction energies, which is separated from the first population by a local minimum. Fig. 2. Radial distributions of water, octanol, and the octanol head group oxygens as a function of the center of mass of the octanol aggregate for the aggregates of size 22, 33, 55, and of the planar phase boundary. Different definitions of the interface are based on the radius at which the water and octanol curves cross, the radius at which the octanol oxygen density is maximal, and the radius at which the octanol density has decayed to 0.5.
well described by an aggregate with a core and an – albeit rather diffuse – interfacial region. For the N = 100 aggregate, the situation is different: the snapshot shows a sparsely packed aggregate of octanol molecules that possesses no core region but appears to be loose agglomeration of more than one aggregate. Therefore we interpret these large aggregates as an artifact of the algorithm by which we determine the aggregates. The observations in Ref. [16] also give hints that such large aggregates are very short lived, which also is a sign that two aggregate come close to each other by diffusion without fusing into one single “object”. We conclude that only the smaller objects with aggregations size 55 or smaller are real aggregates showing up in the simulations and we restrict all further analysis to these aggregates. We now treat all aggregates as on average spherical objects. This allows for the easy definition of radial distribution functions. Before we have a closer look at the role of the ethanol molecules, we discuss the behavior of octanol and water in the pre-Ouzo micelles. Fig. 2 shows the radial distribution functions of water, octanol, and of the octanol head group oxygens. The expected behavior is observed: the octanol RDF decays with increasing radius, the water
density decays with decreasing radius. The interface can now be defined by different criteria: first, we determine the radius at which the octanol RDF decays to 0.5. Second, we take the radius at which the octanol and water RDFs cross each other. A third commonly used definition is the point at which the RDF of the solvent (in our case the water RDF) has an inflection point. Finally, we also use a criterion based on the octanol head groups, which show a distinct peak in the RDF. The position of this peak can be taken as another interface definition. The resulting interface locations are shown in Table 1. The first Table 1 Location of the interface for the clusters with aggregation number 22, 33, and 55 and for the slab system obtained with different estimates: the point, where the RDFs of water and octanol cross (rc ), the point at which the octanol RDF decays to 0.5 (rd ), the inflection point of the octanol RDF (ri ), the maximum of the octanol head group oxygen RDF (rox ), and rEW , the crossing point of the bound ethanol and water densities. The last row shows the maximum of the RDF of the ‘bound’ ethanol molecules. For the planar slab system, r corresponds to the coordinate normal to the interface. The interface positions of the slab system are given relative to rc (located at 6.6 nm in the simulation box), which is arbitrarily defined as zero point. Aggregate size
22
33
55
Slab
rc (nm) rd (nm) ri (nm) rox (nm) rEth (nm) rEW (nm)
1.1 1.0 1.0 1.0 1.2 1.4
1.2 1.2 1.1 1.0 1.2 1.5
1.4 1.4 1.3 0.9 1.3 1.6
0.0 0.0 0.0 −0.2 −0.1 0.2
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Fig. 4. Snapshots of the different aggregates of size 22, 33 and 55, and of the planar interface. Left: The vdW bound ethanol molecules are shown in addition to the octanol molecules. Right: The Coulomb bound ethanol molecules are shown in addition to the octanol molecules.
three definitions agree quite well for all three aggregate sizes and for the planar interface. The interface of the aggregates increases with increasing aggregation number. This shift in interface by 0.3 nm between the N = 22 and N = 55 aggregate reflects the increase in volume required to accommodate the higher number of octanol molecules within the aggregate. However, the peak position of the octanol head groups gives different results: for the N = 22 aggregate, the interfaces position r = 1.0 nm is only slightly smaller than what is predicted with the other criteria, but contrary to the other definitions, this number does not increase for the larges cluster. For the N = 55 cluster, the interface position is even slightly decreased by 0.1 nm. A possible explanation for this is that the N = 55 aggregate
is in fact too large to form a perfectly spherical aggregate (similar to classical micelles formed by surfactants with octyl tails). Therefore, octanol head groups are increasingly observed inside the aggregate’s core, as shown in Fig. 2. We conclude that the head group position is not a good estimate for the interface location. Another remarkable observation is that the interface of the octanol/water planar slab is distinctly organized: the head group density profile shows signatures of a double layer of octanol at the interface. Without further elaborating this point in this work, we focus on the location of the interface, and stick with the three definitions that do not involve the octanol head groups. These definitions agree very well and give rise to a consistent definition of the interface.
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the surfactant-like behavior of these bound ethanol molecules. The snapshots also demonstrate that – as expected – the Coulombbound ethanol molecules are predominantly solvating the polar alcohol group, while the van der Waals bound ethanol also solvates the nonpolar tails. The RDFs of all bound ethanol molecules, which are shown in Fig. 5, show a clear peak close to the interfacial locations that were determined before. Clearly the ethanol has the role of the surfactant-like component in this ternary system, since the ethanol molecules accumulate near “the interface”. However, ethanol does not form a well-defined monolayer based on a fixed value of the area per head-group. This ethanol-rich interface mediates the solubility of the otherwise water-insoluble octanol aggregates in the water-rich pseudo phase. There is a significant amount of ethanol inside the aggregates, which shows that they are best describe as ethanol-swollen octanol aggregates. The concentration of ethanol inside the aggregates increases with increasing aggregation number. This can be explained by geometric constraints: due to the finite length of the hydrophobic tail of octanol, there is an upper limit on the size of a spherical aggregate (similar to classical surfactant micelles [19]). Larger aggregates can only exist as a spherical object with an increasing amount of ethanol in the core of the aggregate. However, we also note that the aggregates that appear in the studied system have no clearly defined spherical shape, and the deviations from spherical shape – another way to avoid the geometrical constraints – are more pronounced for the larger aggregates. In most aggregate systems described up to now, the connected cylindrical shape is linked to bicontinuity coming from minor domain coalescence, and the sample studied here is not relevant to that regime [20]. Owing to the surfactant-like behavior of the bound ethanol, their concentration profile allows for another definition of the interfacial plane. In Table 1, we show the interface locations defined by the peak of their RDF. These interfaces are in good agreement with the three definitions of the interface as determined above.
3. Conclusions Fig. 5. Radial distribution functions of the octanol head groups and of the bound ethanol molecules (van der Waals bound ethanol shown). Two interface definitions are based on the position of the bound ethanol molecules and on the octanol head group positions, another one is based on the crossing point of the bound ethanol curve with the water density, which is also shown.
In a previous paper [16], we have shown that the role of ethanol in the aggregates can be analyzed based on an interaction energy classification. Adopting a self-consistent definition of an interface requires as a first preparation step a rigorous partition of ethanol inside the two pseudo-phases present, the octanol-rich droplets and the water-rich “solvent”. A close examination of the histograms of van der Waals or Coulomb interaction energies between individual ethanol molecules and the studied aggregate provides a natural classification: both histograms show a minimum that separates two populations, as is demonstrated in Fig. 3. The vdW energy histograms are very similar for all studied systems. Also, the different Coulomb energy histograms are very similar. Using the energy values of these minima as a cutoff criterion, the ethanol molecules are divided into a group of molecules that does not interact significantly with the considered aggregate (the ethanol not bound to the octanol-rich aggregates) and into molecules that are either electrostatically ‘bound’, or bound by van der Waals forces. With these definitions we represent all bound ethanol molecules around an aggregate in snapshots in Fig. 4. There, we show the same core of octanol aggregate as the one shown in Fig. 1. These snapshots show
Different criteria for the definition of the interface in a surfactant-free octanol/ethanol/water mixture are tested. A consistent definition of the micellar interface based on different criteria emerges: with the exception of the head group peak position, the different criteria tested all lead to results that agree within the limits of the resolution of 2/qmax that is achieved in scattering experiments. Therefore, a self-consistent concept of an interface is meaningful even in the absence of classical surfactants: in the case of SFME, one can choose either the crossing of octanol and water or the maximum of ethanol radial distribution function. The key structural properties are clearly defined and a reliable estimate of the size and composition of pre-Ouzo micelle aggregates observed in simulations is available. The results are summarized in Fig. 6, where we show the range of different interface locations obtained with the different criteria. This also paves the way toward a further analysis of aggregate stability by analytical models that rely on an appropriate tool for interface definition.
4. Methods The simulations were carried out using the GROMACS 4.5 software package [21]. The simulation box of the monophasic system is cubic with an edge length of 11.2 nm. The phase separated system is of cuboid shape with edge lengths 6.1, 6.1 and 11.6 nm. This system contains 912 octanol molecules, 5496 water molecules and 288 ethanol molecules.
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and this value was chosen as the cut-off distance. It should be emphasized that deviations from this exact value within reasonable boundaries (i.e., 0.8 nm) do not notably influence the results. A self-written program was used to evaluate the radial density distribution of a selected group around the geometric center of a cluster of given aggregation number. Acknowledgments T.Z. thanks the Alexander von Humboldt Foundation for a Humboldt Award and the CNRS Max Planck French-German L.E.A. “RECYCLING” as well as the ERC project “Ree-cycle”.W. K. acknowledges support from the Excellence Laboratory in Systems Chemistry (Chemisyst) from the University of Montpellier. References Fig. 6. A summary of the results for the interface location. The polar and nonpolar pseudo phases are marked together with the interface region, which spans the range of different interface locations obtained with the different criteria. The data obtained for the aggregate with 22 particles are shown.
4.1. Simulation box preparation The initial configuration of the one-phase system was generated by subsequently inserting octanol, ethanol and water molecules into the simulation box, ensuring a random distribution of the components throughout space. The slab system is constructed by joining two separate simulation boxes, one containing only octanol, and the other containing a homogeneous solution of ethanol in water. The initial configurations were minimized by applying a steepest descent algorithm with a maximum force of 10.0 kJ/(mol nm). Further equilibration was carried out for 100 000 steps of 1 fs using the same parameters as for the production run (see next paragraph) except for employing a Berendsen barostat. 4.2. Simulation parameters For the organic compounds ethanol and octanol, we used the OPLS all-atom force field. While the original parameters [22] were adopted for ethanol, we employed a modified torsional potential for the backbone of octanol as described by the L-OPLS [23] force field, an optimized version of OPLS/AA for long hydrocarbons. The TIP4P/2005 water model [24] was used. A 1.0 nm long-range cutoff was used for Lennard–Jones interactions and the particle mesh Ewald algorithm was employed to quantify electrostatic interactions. The temperature was adjusted to 300 K by using the velocity rescaling algorithm and a pressure of 1.0 bar was applied by means of a Parinello–Rahman barostat, using a compressibility of 4.5 × 105 . For both methods, the time constant was set to 1.0 ps. Periodic boundary conditions were applied in all directions and the LINCS algorithm was used to constrain all bond-lengths to their equilibrium values. The two-phase system was simulated for 200 ns and the monophasic system for 550 ns, using a leapfrog integrator with a time step of 2 fs. 4.3. Trajectory analysis Octanol molecules are assigned to the same cluster when any of their atoms are closer than a cut-off distance. For this analysis, we considered only the carbon atoms of octanol as well as the oxygen and its hydrogen. The first distinct minimum in the radial distribution function of all atoms in this subset is observed at 0.478 nm,
[1] J.W. Gibbs, The Scientific Papers of J. Willard Gibbs, Vol. I, Thermodynamics, Dover Publications, New York, 1961, pp. 219. [2] R. Evans, The nature of the liquid–vapour interface and other topics in the statistical mechanics of non-uniform, classical fluids, Adv. Phys. 28 (1979) 143–200. [3] I. Benjamin, Molecular structure and dynamics at liquid–liquid interfaces, Annu. Rev. Phys. Chem. 48 (1997) 407–451. [4] C. Tanford, The Hydrophobic Effect. Formation of Micelles and Biological Membranes, Wiley-Interscience, New York, 1973. [5] M.M. Kozlov, M. Winterhalter, Elastic moduli and neutral surface for strongly curved monolayers – analysis of experimental results, J. Phys. II 1 (1991) 1085–1100. [6] M. Kanduc, E. Schneck, R.R. Netz, Hydration interaction between phospholipid membranes: insight into different measurement ensembles from atomistic molecular dynamics simulations, Langmuir 29 (2013) 9126–9137. [7] P. Bauduin, T. Zemb, Perpendicular and lateral equations of state in layered systems of amphiphiles, Curr. Opin. Colloid Interface Sci. 19 (2014) 9–16. [8] K. Lunkenheimer, K.D. Wantke, Determination of the surface-tension of surfactant solutions applying the method of Lecomte-Du-Nouy (ring tensiometer), Colloid Polym. Sci. 259 (1981) 354–366. [9] M.M. Kozlov, M. Winterhalter, D. Lerche, Elastic properties of strongly curved monolayers – effect of electric surface charges, J. Phys. II 2 (1992) 175–185. [10] T. Zemb, P. Charpin, Micellar structure from comparison of X-ray and neutron small-angle scattering, J. Phys. France 46 (1985) 249–256. [11] M. Zulauf, K. Weckstrom, J.B. Hayter, V. Degiorgo, M. Corti, Neutron scattering study of micelle structure in isotropic aqueous solutions of poly(oxyethylene) amphiphiles, J. Phys. Chem. 89 (1985) 3411–3417. [12] P. Guilbaud, T. Zemb, Solute-induced microstructural transition from weak aggregates towards a curved film of surface active extractants, ChemPhysChem 13 (2011) 687–691. [13] M.L. Klossek, D. Touraud, T. Zemb, W. Kunz, Structure and solubility in surfactant-free microemulsions, ChemPhysChem 13 (2012) 4116–4119. [14] O. Diat, M.L. Klossek, D. Touraud, B. Deme, I. Grillo, W. Kunz, T. Zemb, Octanolrich and water-rich domains in dynamic equilibrium in the pre-ouzo region of ternary systems containing a hydrotrope, J. Appl. Crystallogr. 46 (2013) 1665–1669. [15] D. Subramanian, C.T. Boughter, J.B. Klauda, B. Hammouda, M.A. Anisimov, Mesoscale inhomogeneities in aqueous solutions of small amphiphilic molecules, Faraday Discuss. 167 (2014) 217–238. [16] S. Schöttl, J. Marcus, O. Diat, D. Touraud, W. Kunz, T. Zemb, D. Horinek, Emergence of surfactant-free micelles from ternary solutions, Chem. Sci. 5 (2014) 2949–2954. [17] B.M. Rankin, D. Ben-Amotz, Analysis of molecular aggregation equilibria using random mixing statistics, J. Phys. Chem. B 117 (2013) 15667–15674. [18] A. Oleinikova, I. Brovchenko, A. Geiger, B. Guillot, Percolation of water in aqueous solution and liquid–liquid immiscibility, J. Chem. Phys. 117 (2002) 3296. [19] Y. Chevalier, T. Zemb, The structure of micelles and microemulsions, Rep. Prog. Phys. 53 (1990) 279–371. [20] T. Zemb, Flexibility, persistence length and bicontinuous microstructures in microemulsions, C. R. Chim. 12 (2009) 218–224. [21] B. Hess, C. Kutzner, D. van der Spoel, E. Lindahl, GROMACS 4: algorithms for highly efficient, load-balanced, and scalable molecular simulation, J. Chem. Theory Comput. 4 (2008) 435–447. [22] W.L. Jorgensen, D.S. Maxwell, J. Tirado-Rives, Development and testing of the OPLS all-atom force field on conformational energetics and properties of organic liquids, J. Am. Chem. Soc. 118 (1996) 11225–11236. [23] S.W.I. Siu, K. Pluhackova, R.A. Böckmann, Optimization of the OPLS-AA force field for long hydrocarbons, J. Chem. Theory Comput. 8 (2012) 1459–1470. [24] J.L.F. Abascal, C. Vega, A general purpose model for the condensed phases of water: TIP4P/2005, J. Chem. Phys. 123 (2005) 234505.
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Contents lists available at ScienceDirect
Colloids and Surfaces A: Physicochemical and Engineering Aspects journal homepage: www.elsevier.com/locate/colsurfa
Consistent definitions of “the interface” in surfactant-free micellar aggregates Sebastian Schöttl a , Didier Touraud a , Werner Kunz a , Thomas Zemb b,∗ , Dominik Horinek a,∗∗ a b
Institut für Physikalische und Theoretische Chemie, Universität Regensburg, Universitätsstraße 31, 93053 Regensburg, Germany Institut de Chimie Séparative de Marcoule (ICSM) UMR 5257 CEA/CNRS/UM2/ENSCM, F-30207 Bagnols sur Céze, France
h i g h l i g h t s
g r a p h i c a l
a b s t r a c t
• Surfactant-free micelle aggregates are studied with molecular dynamics simulations. • Aggregates of the different preferred aggregation numbers are analyzed. • Different criteria for an interface definition are tested. • The location of the interface can be defined in a consistent way.
a r t i c l e
i n f o
Article history: Received 31 August 2014 Received in revised form 12 November 2014 Accepted 13 November 2014 Available online xxx Keywords: Molecular dynamics Simulations Clusters Microemulsion Ternary solution Ouzo effect
a b s t r a c t The theoretical description of systems with soft interfaces requires a precise definition of the interfaces that are present. Such definitions are common for interfaces of simple liquids and for surfactant assemblies like micelles, membranes, and microemulsions. Here, we focus on several possible rigorous definitions in a recently described surfactant-free microemulsion that consists of octanol, ethanol, and water, in which micelle-like aggregates are in dynamic equilibrium with the surrounding pseudo phase. We test different definitions that are based on the radial distribution functions of the components with respect to the center of these surfactant-free micelles. All definitions result in experimentally indistinguishable locations of the interface within the limits of resolution of light, X-ray, and neutron scattering experiments, since their locations differ by at most 0.4 nm, which corresponds to roughly 2-3 bond lengths. © 2014 Published by Elsevier B.V.
1. Introduction The thermodynamic discussion of systems with two phases or pseudo-phases involves the definition of the interface as one of the essential steps [1]. Based on this definition, the excess properties of
∗ Corresponding author. ∗∗ Corresponding author. Tel.: +49 9419434745. E-mail addresses: [email protected] (T. Zemb), [email protected] (D. Horinek).
this interface and their thermodynamic consequences can be analyzed [2]. The exact definition of the interface is however subject to some ambiguity and it depends on the type of system, which definitions are possible and meaningful. At a vapor/liquid interface, the common definition of the interfacial plane is the Gibbs dividing surface (GDS), which is the plane for which the liquid has zero interfacial excess. Other definitions like the plane of maximum density gradient (inflection point of the density profile) or the point at which the density is half of the bulk value usually coincide with the location of the GDS at vapor/liquid interfaces, which have a symmetrically shaped density profile.
http://dx.doi.org/10.1016/j.colsurfa.2014.11.029 0927-7757/© 2014 Published by Elsevier B.V.
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These definitions are extendable to liquid/liquid interfaces, in which case the density can be split into the partial densities of the two phases [3]. In this case, a typical definition for the interface is the plane at which the two densities – normalized to their bulk values – are equal [4]. This however can lead to an inconsistent interface location, because a definition using the density profiles can lead to a different position than a definition based on the position of the neutral plane of bending [5]. Moreover, for these interfaces the density profiles are not necessarily symmetric and even different density-based definitions of the interface need not to coincide. This is problematic, because for the prediction of phase diagrams a lateral equation of state is needed, which requires an operative definition of the interface [6]. The consistency of the interface location becomes especially crucial for interdigitated or weakly associated surface layers such as those formed by carboranes, since water molecules are present on both sides of the interface [7]. When interfaces of solutions are studied, the Gibbs dividing surface is especially useful for the determination of surface tension increments [8]. When surfactants are present at the interface, their preferred position is naturally close to the interface in any of the senses mentioned so far. Therefore, the surfactants themselves can be employed for determining the interface. For charged surfactants, e.g., a plane of fixed charge has been employed. [9] In the special case of surfactant solutions, micellar systems, the nature of the interface depends strongly on the type of micelle under consideration. Precise definitions of the location of the “interface” have been proposed for ionic [10] and nonionic micelles [11] as well as for oil soluble surfactants [12], allowing for precise evaluations of the free energy per unit surface as a function of the area per molecule. Recently, scattering experiments with light, X-rays, and neutrons [13–15] demonstrated the existence of micelle-like aggregates of 20–100 molecules in thermodynamically stable ternary solutions, in which an ethanol-soluble but water-insoluble compound like octanol is dissolved in an ethanol/water mixture. Micelle-like aggregates form in these systems even if these systems do not match the classical picture of self-aggregating surfactant molecules. It was argued that in these systems the ethanol molecules play the role of a hydrotrope that mediates the solubility of the ‘nonpolar component’. Classical force-field molecular dynamics simulations were applied for a detailed molecular analysis of these systems [16]. These simulations showed an aggregate size distribution that is distinctly different to simple aggregation in binary solutions [17] or to percolation of water in a binary solution [18]. In this work we closer analyze the structure of the differently sized preferred aggregates that were observed in simulations and compare this to the properties of a flat water/octanol interface in the presence of a small amount of ethanol. We derive and compare different definitions for the interface between the two pseudophases and discuss their implications.
2. Results and discussion The simulations of a macroscopically stable one-phase system comprising octanol (NOct = 224), ethanol (NEth = 6366), and water (NWat = 21 839) reveal that the histogram of aggregate sizes shows several broad peaks. These peaks correspond to preferred aggregates that occur for octanol aggregation numbers N approx. equal 22, 33, 55, and 100. In Fig. 1, we show exemplary aggregates with these aggregation numbers. We demonstrate the generality of the interface definition by showing the results of a planar water/octanol interface in the presence of ethanol with a concentration similar to the aggregate systems. The aggregates of size 22, 33, and 55 are
Fig. 1. Snapshots of the different aggregates of size 22, 33, 55, and 100, and of the planar interface with 5% ethanol in the aqueous phase. Shown are only the octanol molecules. The three smaller aggregates clearly correspond to one concise object. To the contrary, the aggregate with 100 octanol molecules comprises more than one object that are grouped together as an artifact of the algorithm used for aggregate determination.
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Fig. 3. Histogram of the van der Waals (top) and Coulomb (bottom) interactions of ethanol molecules with an octanol aggregate of size 22, 33, and 55. The histograms are normalized such that the minimum has a value of one. For both interactions, the distributions are independent of the cluster sizes. In all cases, the majority of ethanol molecules have a very small interaction, and there is a peak for zero interaction energy. All plots show a second population of ‘bound’ ethanol molecules with more favorable interaction energies, which is separated from the first population by a local minimum. Fig. 2. Radial distributions of water, octanol, and the octanol head group oxygens as a function of the center of mass of the octanol aggregate for the aggregates of size 22, 33, 55, and of the planar phase boundary. Different definitions of the interface are based on the radius at which the water and octanol curves cross, the radius at which the octanol oxygen density is maximal, and the radius at which the octanol density has decayed to 0.5.
well described by an aggregate with a core and an – albeit rather diffuse – interfacial region. For the N = 100 aggregate, the situation is different: the snapshot shows a sparsely packed aggregate of octanol molecules that possesses no core region but appears to be loose agglomeration of more than one aggregate. Therefore we interpret these large aggregates as an artifact of the algorithm by which we determine the aggregates. The observations in Ref. [16] also give hints that such large aggregates are very short lived, which also is a sign that two aggregate come close to each other by diffusion without fusing into one single “object”. We conclude that only the smaller objects with aggregations size 55 or smaller are real aggregates showing up in the simulations and we restrict all further analysis to these aggregates. We now treat all aggregates as on average spherical objects. This allows for the easy definition of radial distribution functions. Before we have a closer look at the role of the ethanol molecules, we discuss the behavior of octanol and water in the pre-Ouzo micelles. Fig. 2 shows the radial distribution functions of water, octanol, and of the octanol head group oxygens. The expected behavior is observed: the octanol RDF decays with increasing radius, the water
density decays with decreasing radius. The interface can now be defined by different criteria: first, we determine the radius at which the octanol RDF decays to 0.5. Second, we take the radius at which the octanol and water RDFs cross each other. A third commonly used definition is the point at which the RDF of the solvent (in our case the water RDF) has an inflection point. Finally, we also use a criterion based on the octanol head groups, which show a distinct peak in the RDF. The position of this peak can be taken as another interface definition. The resulting interface locations are shown in Table 1. The first Table 1 Location of the interface for the clusters with aggregation number 22, 33, and 55 and for the slab system obtained with different estimates: the point, where the RDFs of water and octanol cross (rc ), the point at which the octanol RDF decays to 0.5 (rd ), the inflection point of the octanol RDF (ri ), the maximum of the octanol head group oxygen RDF (rox ), and rEW , the crossing point of the bound ethanol and water densities. The last row shows the maximum of the RDF of the ‘bound’ ethanol molecules. For the planar slab system, r corresponds to the coordinate normal to the interface. The interface positions of the slab system are given relative to rc (located at 6.6 nm in the simulation box), which is arbitrarily defined as zero point. Aggregate size
22
33
55
Slab
rc (nm) rd (nm) ri (nm) rox (nm) rEth (nm) rEW (nm)
1.1 1.0 1.0 1.0 1.2 1.4
1.2 1.2 1.1 1.0 1.2 1.5
1.4 1.4 1.3 0.9 1.3 1.6
0.0 0.0 0.0 −0.2 −0.1 0.2
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Fig. 4. Snapshots of the different aggregates of size 22, 33 and 55, and of the planar interface. Left: The vdW bound ethanol molecules are shown in addition to the octanol molecules. Right: The Coulomb bound ethanol molecules are shown in addition to the octanol molecules.
three definitions agree quite well for all three aggregate sizes and for the planar interface. The interface of the aggregates increases with increasing aggregation number. This shift in interface by 0.3 nm between the N = 22 and N = 55 aggregate reflects the increase in volume required to accommodate the higher number of octanol molecules within the aggregate. However, the peak position of the octanol head groups gives different results: for the N = 22 aggregate, the interfaces position r = 1.0 nm is only slightly smaller than what is predicted with the other criteria, but contrary to the other definitions, this number does not increase for the larges cluster. For the N = 55 cluster, the interface position is even slightly decreased by 0.1 nm. A possible explanation for this is that the N = 55 aggregate
is in fact too large to form a perfectly spherical aggregate (similar to classical micelles formed by surfactants with octyl tails). Therefore, octanol head groups are increasingly observed inside the aggregate’s core, as shown in Fig. 2. We conclude that the head group position is not a good estimate for the interface location. Another remarkable observation is that the interface of the octanol/water planar slab is distinctly organized: the head group density profile shows signatures of a double layer of octanol at the interface. Without further elaborating this point in this work, we focus on the location of the interface, and stick with the three definitions that do not involve the octanol head groups. These definitions agree very well and give rise to a consistent definition of the interface.
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the surfactant-like behavior of these bound ethanol molecules. The snapshots also demonstrate that – as expected – the Coulombbound ethanol molecules are predominantly solvating the polar alcohol group, while the van der Waals bound ethanol also solvates the nonpolar tails. The RDFs of all bound ethanol molecules, which are shown in Fig. 5, show a clear peak close to the interfacial locations that were determined before. Clearly the ethanol has the role of the surfactant-like component in this ternary system, since the ethanol molecules accumulate near “the interface”. However, ethanol does not form a well-defined monolayer based on a fixed value of the area per head-group. This ethanol-rich interface mediates the solubility of the otherwise water-insoluble octanol aggregates in the water-rich pseudo phase. There is a significant amount of ethanol inside the aggregates, which shows that they are best describe as ethanol-swollen octanol aggregates. The concentration of ethanol inside the aggregates increases with increasing aggregation number. This can be explained by geometric constraints: due to the finite length of the hydrophobic tail of octanol, there is an upper limit on the size of a spherical aggregate (similar to classical surfactant micelles [19]). Larger aggregates can only exist as a spherical object with an increasing amount of ethanol in the core of the aggregate. However, we also note that the aggregates that appear in the studied system have no clearly defined spherical shape, and the deviations from spherical shape – another way to avoid the geometrical constraints – are more pronounced for the larger aggregates. In most aggregate systems described up to now, the connected cylindrical shape is linked to bicontinuity coming from minor domain coalescence, and the sample studied here is not relevant to that regime [20]. Owing to the surfactant-like behavior of the bound ethanol, their concentration profile allows for another definition of the interfacial plane. In Table 1, we show the interface locations defined by the peak of their RDF. These interfaces are in good agreement with the three definitions of the interface as determined above.
3. Conclusions Fig. 5. Radial distribution functions of the octanol head groups and of the bound ethanol molecules (van der Waals bound ethanol shown). Two interface definitions are based on the position of the bound ethanol molecules and on the octanol head group positions, another one is based on the crossing point of the bound ethanol curve with the water density, which is also shown.
In a previous paper [16], we have shown that the role of ethanol in the aggregates can be analyzed based on an interaction energy classification. Adopting a self-consistent definition of an interface requires as a first preparation step a rigorous partition of ethanol inside the two pseudo-phases present, the octanol-rich droplets and the water-rich “solvent”. A close examination of the histograms of van der Waals or Coulomb interaction energies between individual ethanol molecules and the studied aggregate provides a natural classification: both histograms show a minimum that separates two populations, as is demonstrated in Fig. 3. The vdW energy histograms are very similar for all studied systems. Also, the different Coulomb energy histograms are very similar. Using the energy values of these minima as a cutoff criterion, the ethanol molecules are divided into a group of molecules that does not interact significantly with the considered aggregate (the ethanol not bound to the octanol-rich aggregates) and into molecules that are either electrostatically ‘bound’, or bound by van der Waals forces. With these definitions we represent all bound ethanol molecules around an aggregate in snapshots in Fig. 4. There, we show the same core of octanol aggregate as the one shown in Fig. 1. These snapshots show
Different criteria for the definition of the interface in a surfactant-free octanol/ethanol/water mixture are tested. A consistent definition of the micellar interface based on different criteria emerges: with the exception of the head group peak position, the different criteria tested all lead to results that agree within the limits of the resolution of 2/qmax that is achieved in scattering experiments. Therefore, a self-consistent concept of an interface is meaningful even in the absence of classical surfactants: in the case of SFME, one can choose either the crossing of octanol and water or the maximum of ethanol radial distribution function. The key structural properties are clearly defined and a reliable estimate of the size and composition of pre-Ouzo micelle aggregates observed in simulations is available. The results are summarized in Fig. 6, where we show the range of different interface locations obtained with the different criteria. This also paves the way toward a further analysis of aggregate stability by analytical models that rely on an appropriate tool for interface definition.
4. Methods The simulations were carried out using the GROMACS 4.5 software package [21]. The simulation box of the monophasic system is cubic with an edge length of 11.2 nm. The phase separated system is of cuboid shape with edge lengths 6.1, 6.1 and 11.6 nm. This system contains 912 octanol molecules, 5496 water molecules and 288 ethanol molecules.
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and this value was chosen as the cut-off distance. It should be emphasized that deviations from this exact value within reasonable boundaries (i.e., 0.8 nm) do not notably influence the results. A self-written program was used to evaluate the radial density distribution of a selected group around the geometric center of a cluster of given aggregation number. Acknowledgments T.Z. thanks the Alexander von Humboldt Foundation for a Humboldt Award and the CNRS Max Planck French-German L.E.A. “RECYCLING” as well as the ERC project “Ree-cycle”.W. K. acknowledges support from the Excellence Laboratory in Systems Chemistry (Chemisyst) from the University of Montpellier. References Fig. 6. A summary of the results for the interface location. The polar and nonpolar pseudo phases are marked together with the interface region, which spans the range of different interface locations obtained with the different criteria. The data obtained for the aggregate with 22 particles are shown.
4.1. Simulation box preparation The initial configuration of the one-phase system was generated by subsequently inserting octanol, ethanol and water molecules into the simulation box, ensuring a random distribution of the components throughout space. The slab system is constructed by joining two separate simulation boxes, one containing only octanol, and the other containing a homogeneous solution of ethanol in water. The initial configurations were minimized by applying a steepest descent algorithm with a maximum force of 10.0 kJ/(mol nm). Further equilibration was carried out for 100 000 steps of 1 fs using the same parameters as for the production run (see next paragraph) except for employing a Berendsen barostat. 4.2. Simulation parameters For the organic compounds ethanol and octanol, we used the OPLS all-atom force field. While the original parameters [22] were adopted for ethanol, we employed a modified torsional potential for the backbone of octanol as described by the L-OPLS [23] force field, an optimized version of OPLS/AA for long hydrocarbons. The TIP4P/2005 water model [24] was used. A 1.0 nm long-range cutoff was used for Lennard–Jones interactions and the particle mesh Ewald algorithm was employed to quantify electrostatic interactions. The temperature was adjusted to 300 K by using the velocity rescaling algorithm and a pressure of 1.0 bar was applied by means of a Parinello–Rahman barostat, using a compressibility of 4.5 × 105 . For both methods, the time constant was set to 1.0 ps. Periodic boundary conditions were applied in all directions and the LINCS algorithm was used to constrain all bond-lengths to their equilibrium values. The two-phase system was simulated for 200 ns and the monophasic system for 550 ns, using a leapfrog integrator with a time step of 2 fs. 4.3. Trajectory analysis Octanol molecules are assigned to the same cluster when any of their atoms are closer than a cut-off distance. For this analysis, we considered only the carbon atoms of octanol as well as the oxygen and its hydrogen. The first distinct minimum in the radial distribution function of all atoms in this subset is observed at 0.478 nm,
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