Dynamics of Surfactant Clustering at Interfaces and Its Influence on the

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Dynamics of surfactant clustering at interfaces and its influence on the interfacial tension. Atomistic simulation of a sodium hexadecane-benzen sulfonate-tetradecane-water system. Ricardo Paredes, Ana Isabel Fariñas-Sánchez, Bryan Medina-Rodríguez, Samantha Samaniego, Yosslen Rafael Aray, and Luis Javier Alvarez Langmuir, Just Accepted Manuscript • DOI: 10.1021/acs.langmuir.7b03719 • Publication Date (Web): 07 Feb 2018 Downloaded from http://pubs.acs.org on February 7, 2018

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Dynamics of surfactant clustering at interfaces and its influence on the interfacial tension. Atomistic simulation of a sodium hexadecane-benzen sulfonate-tetradecane-water system Ricardo Paredes,∗,†,‡ Ana Isabel Fariñas-Sánchez,¶,# Bryan Medina-Rodríguez,§ Samantha Samaniego,k Yosslen Aray,⊥ and Luis Javier Álvarez∗,¶ †Facultad de Ingeniería en Ciencias de la Tierra, Escuela Superior Politécnica del Litoral, ESPOL, Campus Gustavo Galindo Km 30.5 Vía, Perimetral, P.O. Box 09-01-5863, Guayaquil, Ecuador. ‡Departamento de Física y Matemáticas, Universidad Iberoamericana, Prolongación Paseo de la Reforma, 880, Lomas de Santa Fe, C. P. 01219,Ciudad de México, México. ¶Laboratorio de Simulación, Unidad Cuernavaca, Instituto de Matemáticas, Universidad Nacional Autónoma de México, A.P. 273-3 Admon. 3, Cuernavaca, Morelos, 62251, México. §Facultad de Ingeniería en Ciencias de la Tierra, Escuela Superior Politécnica del Litoral, ESPOL, Campus Gustavo Galindo Km 30.5 Vía, Perimetral, P.O. Box 09-01-5863, Guayaquil, Ecuador. kFacultad de Ciencias Naturales y Matemáticas, Escuela Superior Politécnica del Litoral, ESPOL, Campus Gustavo Galindo Km 30.5 Vía, Perimetral, P.O. Box 09-01-5863, Guayaquil, Ecuador. ⊥Facultad de Ciencias, Universidad de Ciencias Aplicadas y Ambientales, UDCA, Campus Universitario Norte, Calle 222 No. 55-37, Bogotá, Colombia. #Current address: Departamento de Física, Universidad Simón Bolívar, Apartado Postal 89000, Caracas 1080- A, Venezuela.

E-mail: [email protected]; [email protected]

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Abstract The process of equilibration of the tetradecane-water interface in the presence of sodium hexadecane-benzen sulfonate is studied using intensive atomistic molecular dynamics simulations. Starting as an initial point with all surfactants at the interface, it is obtained that the equilibration times of the interface (several µs) are orders of magnitude higher than previously reported simulated times. There is strong evidence that this slow equilibration process is due to the aggregation of surfactants molecules on the interface. In order to determine this fact, temporal evolution of interfacial tension and interfacial formation energy are studied and their temporal variations are correlated with cluster formation. To study cluster evolution, the mean cluster size and the probability that a molecule of surfactant chosen at random is free are obtained as a function of time. Cluster size distribution is estimated and it is observed that some of the molecules remain free while the rest agglomerate. Additionally, the temporal evolution of the interfacial thickness and the structure of the surfactant molecules on the interface are studied. It is observed how this structure depends on whether the molecules agglomerate or not.

Introduction Surfactants in the industry are of great utility in modifying interfacial properties of various systems. 1,2 In the oil industry, the injection of surfactants dissolved in brine has proven to be one of the most used oil recovery methods. 3 It is well known that the modification of interfacial properties is not due solely to amphiphilic molecules added to the brine. It has been shown that modifying the properties of injection water, generates the migration of hydrocarbon macromolecules to the oil-water interfaces, causing a modification of the interfacial properties and thus increasing crude production. 4–6 It has been experimentally observed that the introduction of surfactants into the system yields a reduction of some orders of magnitude in the interfacial tension of the water hydro-

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carbon system 7,8 and this reduction is substantially increased for optimum values of alkalinity or water salinity, and by combining different surfactants. 7,9,10 The atomistic and coarse grained molecular dynamics simulations have been the most popular numerical methods to study surfactants dissolved in water and in vacuum-water and oil-water interfaces. In particular, the most widely used coarse grained models include Martini’s 11,12 and the Dissipative Particle Dynamics (DPD). 13,14 For water-surfactant systems these methods have been very useful to study micellization processes of surfactants and to characterize micelle structures. DPD has been very useful given that one can simulate times of the order of milliseconds and system sizes of some tens of nanometers. Some authors even claim the achievement of quantitative results with this method, for example the estimate of the critical micell concentration (CMC), for some of these systems. 15,16 Sanders and Panagiotopoulos 17 performed mesoscopic simulations using Martini and a Shinoda model 18 to obtain the CMC of several surfactants with simulated times of the order of microseconds. The use of MD simulations is very expensive if one wants to simulate very large systems or long simulated times. Therefore it is very difficult to avoid the effects of size and to obtain a representative sample in order to predict macroscopic properties. On the other hand, it is often difficult to achieve system equilibration times. The equilibration times of micellization processes are found to be of hundreds of nanoseconds or larger. 19 For this reason, authors wishing to study the structure of a single micelle initially start from a spherical array of surfactants. 20 There are many simulation studies of the effect of surfactants on the interfacial properties of vacuum-water, vacuum-oil and water-oil systems. 21 For example, using mesoscopic models, they have studied the dependence of interfacial tension with the concentration of surfactants on the interface, 22,23 the influence of salinity on these values, 24,25 the effect of combining different surfactants on interfacial tension, 26 among others. The behavior of the interfacial tensions obtained in relation to the studied parameters has been qualitatively correct. On the other hand, using atomistic molecular dynamics, several authors have tried to understand

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the effect of amphiphilic molecules on several types of interfaces. 27–35 In particular, they have estimated the interfacial or surface tension for several types of surfactants and have studied their value as a function of the concentration of surfactants at the interface 33 or the type of surfactants present. 27 Some of them have studied various types of interfacial energies such as interfacial formation energy 27 or binding energies of surfactants to the interface. 34 Many of them have found an inverse correlation between the width of the interface estimated in different ways, and the value of interfacial tension. 27,33 In addition to this, some authors have tried to explain what makes one surfactant better than another as an interfacial tension reducer. They have studied the structures of surfactant molecules at the interface in various ways. For example, estimating the length of the hydrophobic tails 27 and comparing it with that of the alkane used, or the inclination of surfactant molecules at the interface via the principal axis of inertia, 27 or the values of the dihedral angles for surfactants with more than one tail, 31 or the deuterium order parameter, 29,32 etc. In general, the authors have achieved good correlations between these quantities and interfacial tension values. A typical problem reported, is that for systems with ultra-low interfacial tensions, the values obtained in the simulations have always been much higher than experimental data. Values obtained through simulations are of the order of dynes/cm whereas experimental values can be up to several thousand times smaller. This could be related to the presence of salts in the aqueous solution. However, there have been simulations including salts in the water and the values previously obtained are not substantially lowered. 31 In order to solve any of the already mentioned problems one must be certain that the system is at equilibrium. The purpose of the following work is to understand how is the equilibration process for the particular system that we are studying. Both in applications and experimental studies, if one wants to address the problem of the influence of surfactants on the interfacial properties of oil-water or air-water systems, a solution of surfactants in water is placed in contact with the hydrocarbon at lower concentrations than CMC. In this way one avoids the process of micellization of surfactants in the water and therefore the vast

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majority of surfactant molecules migrate to the interface. The equilibration times of this type of processes are of the order of minutes. 9 Simulating this, through atomistic or mesoscopic simulations is very difficult, not only because of large equilibration times but because of the fact that if one wants to reach the typical concentrations of surfactants at the interface, the sizes of systems to be simulated would have to be too large to be under the CMC. There are Monte Carlo hybrids with DPD to avoid this problem. 36 For this reason, in both mesoscopic and atomistic simulations people prefer to work with an initial condition where surfactants are already at the interface. Therefore simulation times have been relatively short, maximum some tens of nanoseconds. 30,35 In 2010, Shi et al. 30 were also concerned with structural changes of surfactant molecules at vacuum-water interfaces. They noted that for certain relatively high concentrations of surfactants at the interface, for both non-polar, as the hexaethylene glycol monododecyl ether, C12 E6 , and anionic as the sodium dodecyl sulfate, SDS, the phenomenon of aggregation of surfactants was observed. They emphasized the importance of the dynamics of surfactants at the interface by performing preliminary studies of diffusive processes. If the surfactant aggregation process occurs it is to be expected that the equilibration times are much longer than those previously simulated. If this is true, results obtained so far by atomistic simulations would have to be reconsidered because the systems may not be at equilibrium. The question we address is whether there is a process of equilibration of molecules at the interface, fast in relation to the migration of molecules from the aqueous phase to the interface but slow in relation to the typical simulation times. To do so we performed intensive molecular dynamics simulations of a relatively low concentration of the 8-phenyl hexadecane sulfonate, 8C16, in tetradecane(C14)-water(SOL) interfaces. In this first work, we used low concentrations of 8C16 (0.79 molecules/nm2 ) to avoid additional problems in the interpretation of results. The concentration of maximum coverage to form this surfactant mono layer is around twice this value. 34,37 We study the evolution of interfacial tension and interfacial energy as a function of time and we found that these quantities vary in times up to microsec-

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onds or more. It was observed that these variations are due to a process of agglomeration of 8C16 molecules at the interface. We use elements of percolation theory 38 to analyze these agglomerates. The remainder of the paper is organized as follows: In section 2 we explain the computational model used. In section 3 we describe the results and correlate the interfacial values with the structural changes of the set of surfactants at the interface. In section 4 we summarize the most important conclusions of our work.

Simulations Details In this work a type of alkyl benzene sulfonate molecules were simulated at tetradecanewater interfaces. Sodium linear alkylbenzene sulfonates (LAS) belong to the kind of very popular surfactants which are widely used in industry and daily life. In fact, linear alkyl benzene sulfonates form the most important synthetic surfactant family in the world. 39 This compound has a sulfonate group and a linear alkyl chain which is usually attached to the para-positions of a benzene ring (Figure 1). The alkyl chain may have m + n carbon atoms with the benzene ring attached to the (m +1) carbon (m < n). 39 Figure 1 shows the specific molecule used in this work. It has a chain of 16 carbon atoms, and the benzene ring is attached to the 8th carbon. This is the O-(1-hetylnonyl) benzene sulfonate, more briefly denoted as m-phenyl hexadecane sulfonate, with m= 8, 8C16. In this way, this molecule has two hydrophobic tails one with 8 and the other one with 7 carbon atoms. In this work we use an alkylbenzene sulfonate due to the presence of the benzene ring that can favour the attractive interaction between surfactants and in this way induces its agglomeration 40 (See Supporting Information Figures S5 and S6). In particular, we use 8C16 because it induces ultra-low interfacial tensions when placed in alkane-water interfaces (see the dependence on m of the interfacial tension in ref. 9 ). To model this molecule, the united atom method approximation was used. The same approximation was used for tetradecane molecules. The force field used was gromos53a6. 41 Molecules were obtained from the Automated Topology

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Builder (ATB) and Repository. 42 Water molecules were modeled using the extended single point charge (SPC/E) potential.

Figure 1: Chemical structure of the 8C16 surfactant. For all simulations shown here, the number of molecules were: 4154 for water, 229 for tetradecane and 32 for 8C16 (16 at each interface). The initial condition of the system was constructed so that the dimensions of the system along the x-axis and the y-axis, both parallel to the interfaces, were maintained in the neighborhood of Lx = Ly = 4.5 nm. Lz will depend on the number of surfactant molecules at the interface. For N = 16 molecules at each side Lz =12.0 nm. All these dimensions were obtained to guarantee a temperature of 300 K and a pressure of 1 bar. Periodic boundary conditions were imposed in all directions. Simulations were performed with the public domain program Gromacs 4.6. 43 Initially, a box containing 229 molecules of tetradecane and a box containing 2077 molecules of water were prepared separately. Both boxes are of 4.5 nm in the x and y directions. The size in the z direction is determined in such a way that the densities of both tetradecane and water correspond to those of temperature and pressure conditions of the simulations. Then, the tetradecane box is placed right at the center of a larger box of dimensions 5 nm × 5 nm × 18 nm. On the other hand, two water layers were placed. One on the left end and the other on the right end of the larger box (see Supporting Information Figure S1). Both are placed in such a way that there is 1 nm of empty space between the ends of the box and water. In the free spaces between water and tetradecane, the two layers of 16 molecules of 8C16 are periodically arranged with the hydrophilic head pointing towards water. Then a minimization process, using steepest descent method, is performed, keeping the positions of 8C16 molecules fixed. Then, also keeping the positions of these molecules 7


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fixed, 1 ns of NVT (T = 300 K) simulations are performed. The 8C16 molecules are held fixed, in these two steps, to prevent them from agglomerating while they are not immersed in the other phases. Just after that, an NPT (P = 1 bar, T = 300 K) simulation is performed. The configuration with approximately Lx = Ly = 4.5 nm is taken as a starting point for an NPT semi-isotropic simulation of 1.5 ns, where only the volume in the z direction could change. All this procedure is done to ensure that the dimensions parallel to the interface remain around 4.5 nm. At the end, Lz is of the order of 12.0 nm. Once this step is completed, the production stage is started with NPT (P = 1 bar, T = 300 K) simulations. The leap frog integration method was used in all simulations with a time step of 1 fs. For long-range electrostatic interactions Ewald lattice sum methods was used. Force cut off of 1.3 nm was used. Nose-Hoover (τT = 0.5 ps) method was used as thermostat and Parrinello-Rahman (τP = 5 ps) as barostat. Supporting Information Figure S1 shows the initial configuration just after the energy minimization process. Figure 2 shows water, tetradecane, 8C16− and Na+ density profiles for three different time intervals. In these profiles it is clearly observed that two water-8C16-tetradecane interfaces are formed. It is observed that the density profile of surfactants has two clearly defined maxima. The z-position of these maxima was averaged and all the profiles were plotted by taking the position of this average as z = 0. It is observed how different profiles coincide independently of the time interval where they are estimated. The density of water in the center of its region, reaches a value of 0.997 g/cm3 and that of tetradecane takes 0.768 g/cm3 in the center of its region. Both quantities differ from experimental values by less than 1%. This indicates that the corresponding phases have been correctly formed, indicating that the simulated z-size is adequate. In principle, it is not necessary to go to larger sizes. The overlap of the curves apparently show that the system has stabilized at short times for average bulk properties. However, as it will be shown, interfacial properties take a long time to equilibrate. To observe more carefully the overlap, we estimate the interfacial thickness, from the

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Figure 2: Density profiles for tetradecane, water, 8C16− and Na+ estimated at three different time intervals. The curves for each substance coincide for all time intervals shown. This gives the impression that the system has reached its equilibrium at short times. The densities of water and tetradecane in the centers of their respective zones reach values with less than 1% of error with respect to the experimental results corresponding to the same conditions of temperature and pressure. density profiles by using two criteria. The first one is the 90% -90% criterion, ttotal , which is the separation between the z coordinates where the water and tetradecane density have 90% of their bulk value. The other criterion we use is the average of the widths of the surfactant density profile at each interface, tRM S . If we assume that this profile has a gaussian form this is 2σ, as expected. Figure 3(a) shows the value of these two widths as a function of simulation time. For the scale shown in the figure it is observed that these quantities practically do not vary. The average of ttotal is 1.3649 nm with a standard deviation relative to the mean of 0.9%. Previously, 1.883 nm was obtained ttotal for 8C16 at decane-water interfaces but at a much higher concentration of surfactants at the interface, which justifies the difference in the results. On the other hand, the mean of tRM S is 0.7443 nm with a standard deviation relative to the mean also of 1.0%. Figure 3(b) shows the average of the two maxima of the density profile of surfactant as a function of simulation time. The mean of this maximum in the simulation time was 0.534 g/cm3 and the relative standard deviation to the mean was 0.7%. These results, support the overlap. 9


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Figure 3: (a) Interfacial thickness estimated by the 90%-90% criteria, ttotal , and by the width of the surfactant density profile, tRM S as a function of simulation time. (b) Average peak of surfactant density profiles as a function of time. For both figures each point is the average for a time interval of 96 ns. The variability of all these quantities does not exceed 1% over the entire time interval. On the other hand, Figure 4 shows the density profile of Na+ cations exclusively for three different time intervals. It is clearly observed that there are no cations dissolved in tetradecane, given that the density of this ion in this region is zero. However, in the water phase, it is observed that there is a finite fraction of dissolved ions. From these profiles it is estimated that the fraction of Na+ ions not bound to 8C16− is of the order of 20%. This value is at the lower limit of that obtained for counterion dissociation in sodium n-alkyl sulfate micelles. 44

Results and Discussion We shall divide this section in five parts: The first one corresponds to the monitoring of the SOL-8C16-C14 system through macroscopic variables such as the interfacial tension and the interface formation energy. The second part deals with the relation of these quantities with the formation and evolution of 8C16 clusters. The third part is about the interfacial thickness. Finally, the fourth and fifth parts deal with some geometrical characteristics of 10


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Figure 4: Na+ Density profile for three different time intervals. It is observed that the overlap is not as perfect as that observed for the other components (see Figure 2). This component consists of a total of 32 atoms which is much smaller than the total atoms for each one of the other components. An appreciable fraction of this ion is dissolved in the aqueous phase. The shaded area represents the dissociated Na+ fraction. the 8C16 molecule at the interface. All results correspond to the NPT production phase of the simulations.

Interfacial tension and interface formation energy evolution Figure 5 shows the interfacial tension evolution as a function of simulated time. This quantity is estimated using the expression

Z Pxx (z, t) + Pyy (z, t) 1 Lz Pzz (z, t) − dz γ(t) = 2 0 2 Lz Pxx (t) + Pyy (t) = Pzz (t) − 2 2

(1)

where Pzz is the normal pressure, Pyy and Pxx are the tangential pressures with respect to a plane parallel to the interface. Overlines mean spatial averages. The factor 1/2 is due to the presence of two interfaces in the system. The time average of the interfacial tension, hγi,

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is taken each 1 ns and each 72 ns and are shown as circles and continuous lines in Figure 5 respectively. From now on, the time interval taken in each average is represented by the subscripts 1 and 72.

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50

γ [dynes/cm]

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45 720 ns

1887 ns

γ γ γ

40

< > each 1 ns < > each 72 ns < > bare C14-SOL interface

217 ns 0

500

1000

1500

2000

t[ns]

Figure 5: Interfacial tension versus simulation time for 8C16 in a tetradecane-water interface. It is clear that the system has not achieved equilibrium during the first 720 ns. The circles represent the average interfacial tension each nano second and the continuous line represents the average each 72 ns. At least two plateaus are observed along the simulated time. Purple horizontal line represents our simulations result for the bare C14-SOL interface (See Supporting Information section Simulation details). It is clear form the figure that the system has not achieved an equilibrium state during the first 720 ns. This time is much greater than the simulated time reported in previous works. 27–35 For this type of systems, i.e. surfactants at water-alkane interface, the maximum simulation time reported was less than 50 ns.

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In the plot of the evolution of γ three regions

can be distinguished: the first, between 0 and 217 ns, hγi72 where it is approximately constant with an average value of 42.6 ± 0.1 dynes/cm; region two is defined for the interval from 217 to 720 ns where hγi72 suddenly increases its value; the third region is defined for the interval from 720 ns to 1800 ns and has an average value of 47.5±0.1 dynes/cm, which is 12% greater than the value obtained in the first region. There is a change in the behaviour of hγi72 around 1887 ns . Therefore we could define a fourth region for t > 1800 ns. This is due to a change 4

In ref., 35 to study the stability of interfaces, simulations go up to 250 ns but only for extremely high concentration of surfactants at the interface.

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in the molecule aggregates at the interface as will be discussed latter. From this last change there is no certainty that the system has reached equilibrium. At the end of the fourth region hγi72 = 48 dynes/cm, a little bit greater than the value obtained in region 3 (See Supporting Information section Evolution of the 8C16-tetradecane-water-system). It is important to point out that additionally to the problem that the system has not arrived to equilibrium the fluctuations taken for average intervals of just a few nanoseconds introduce multiple fluctuations to the estimates of the interfacial tension. The maximum variation of hγi1 in the first region is about 12 dynes/cm, with a standard deviation of 2.3 dynes/cm. In the third region these values are 16 dynes/cm and 2.6 dynes/cm respectively. The fluctuations for short time averages are too big. The mean value of the total energy of the system, Etot , is −162200 kJ/mol. This quantity varies in the whole interval of simulated time by only 0.01 %. This variation of energy is totally due to changes at the interface. Therefore we calculate the interface formation energy Eif defined by:

Eif =

Etot − (n Ess − Etw ) n

(2)

where n = 32 is the total number of surfactant molecules at the interfaces, Ess is the energy of a single surfactant molecule and Etw the energy of a system with only tetradecane and water. These energies were calculated in separate simulations at the same thermodynamic conditions and Etw with the same number of molecules as the one with all the components. Figure 6 shows Eif as a function of time. Circles and continuous lines represent hEif i1 and hEif i72 respectively. The same four regions observed in Figure 5 for the interfacial tension are observed in this figure for the interface formation energy. The intermediate time variability of hEif i72 is larger than this variability in hγi72 . For example,hEif i72 is less constant in regions 1 and 3 than hγi72 at the corresponding regions. In region 4, it is observed a clear decay of energy, probably showing that the equilibrium state has not been reached in the nearly 2 µs of simulated time. A clear change of behaviour of hEif i72 around 500 ns is observed in region 13


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2. This is not observed at the same region in Figure 5. Changes observed at 500 ns and 1887 ns will be explained later. The average value of Eif is -414.0 kJ/mol in region 1, -423.0 kJ/mol in region 3 and reaches -425 kJ/mol in region 4. The variation between regions 1 and 3 is about 2.1%, much lower that the variation observed for the interfacial tension. The short time variability is less for Eif than for γ.

Figure 6: Interfacial formation energy versus simulation time for 8C16 in a tetradecane-water interface. It is clear the correspondence between this graphic and the one shown in Figure 5.

Cluster formation and its evolution A close view of the configuration of surfactants at the interfaces may be used to explain the changes in the temporal evolution of the interfacial tension and interface formation energy. Figure 7 shows these configurations of 8C16 at 200 ns (a), 924 ns (b) and 1480 ns (c). Figures are views of surfactants at planes perpendicular to the z direction and are viewed from the water phase region. The sulfur (yellow) and oxygen atoms (red) are on the front. The visualisation at 200 ns corresponds to the end of region 1; both visualisations at 924 ns and 1480 ns correspond to region 3. There is a clear difference between the configuration in region 1 and the ones in region 3. In the first one surfactants seem to be randomly located whereas in the other two images it is clear that a clustering process has already taken place. 14


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Figure 7: Images of 8C16 molecules at the interface at three different times. (a) 200 ns, (b) 924 ns and (c) 1480 ns. Two interfaces each time. Sulfures in yellow, oxygens in red and carbon compounds in light blue. In all the images the sulfures are shown toward the front. (a) Molecules seem to be free. (b)-(c) Molecules have formed clusters. (b) and (c) left a cluster of size 11 is observed. Remember that there are periodic boundary conditions. The total number of monomers at the interfaces is 16 at both sides. Considering the periodic boundary conditions imposed to the simulations, one can clearly distinguish clusters of 11 monomers at the interfaces (figure 7(b)-(c) left). Based on visual inspection of configurations at different times, it seems that these clusters of size 11 are stable through time. This sort of aggregation has been reported for both non-polar, C12 E6 , and anionic, SDS, surfactants at interfaces vacuum-water. 30 Assuming that the system is in metastable equilibrium at each of the already mentioned regions, one can use the pair radial distribution function g(r) to characterize the structural features of the surfactant phase by calculating it for the sulfur-sulfur pair. Top of Figure 8 15


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Figure 8: The radial distribution function of sulfur atoms versus distance r. Top: g(r) function is estimated in the interval 84 ns < t < 180 ns. That is, in the first region shown in Figure 5 and Figure 6. It is clear that the head of surfactants are uniformly distributed on the whole interface. Bottom: g(r) function is estimated in two separated intervals belonged to the third region shown in Figure 5 and Figure 6. Both curves have qualitatively the same form. It is clear, that the 8C16 molecules build some nanoscopic structure. shows this function averaged in the time interval [84,180] ns which corresponds to region 1. The form of this function is characteristic of that of a liquid with the main peak at 0.398 nm and it is in agreement with the snapshots shown in Figure 7(a). Radial distribution functions were also calculated in two different subregions of region 3 given by the intervals [878, 924] ns and [1404,1480] ns (see bottom of Figure 8). In these subregions the form of g(r) shows the existence of some structure that extends beyond 1 nm. The g(r) of the two subregions are quite similar indicating that these structures are preserved. The same structure of g(r) is observed in region 4. The first peak of these functions appear at the same distance as in the case of the g(r) of region 1, but there appears a doublet which implies that there are two preferred positions for the nearest sulfur neighbours. These facts suggest that changes observed in both Figure 5 and Figure 6 in region 2 are due to a process of aggregation of the 8C16 molecules that finally form cluster structures. We consider two molecules of 8C16 to belong to the same cluster if their sulfur atoms are separated by a distance less or equal to 0.47 nm. This is the point just to the right to 16


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Figure 9: (a) Cluster size distribution ns averages in two different intervals of time in region 1. All surfactants are free in this region no cluster are formed yet. (b) ns averages in three different intervals in region 3. Two zones are observed in this graphic. A distribution of free surfactants on the left and the distribution of clusters of surfactants on the right. This is completely similar to the distribution of surfactants in a system of surfactant solvated in water. It is clear that a cluster of size 11, observed in Figure 7, is completely stable. On the other hand a transition from a cluster of size 7 to a cluster of size 8 is observed. (c) ns averages in two different intervals of time in region 4. A change in the structure of the cluster is observed. The cluster of size 11 has a tendency of reduce its size. the main maxima where the two curves showed on the bottom of Figure 8 begin to separate. A test was made changing this value in a range of 10% around it and no qualitative change was observed. The size of a cluster, s, is defined as the number of molecules belonging to the same cluster. Then the cluster size distribution, ns , is the time average of number of clusters of sizes s taken on a given time interval. In Figure 9 this distribution function is showed for several time intervals. Figure 9(b) shows ns for three time intervals on region 3, [924, 1020] ns, [1500,1596] ns and [1692,1798] ns. This distribution is analogous to the size distribution function of micelles in a surfactant-water system above the critical micelle concentration. To the left, in the limit s → 1, there is a decaying function that represents 17


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the distribution of free surfactants. For larger values of s there is another distribution that represents the surfactants that form genuine clusters, equivalent to micelles in the watersurfactant system. The free surfactants distribution is stable in the whole region 3. In the other case, a stable genuine cluster of size 11, the same that was seen in Figure 7(b)-(c), is kept along the complete interval of time that define region 3. At [924,1020] ns a genuine cluster of size 7 is observed but this cluster is transformed, finally, to a cluster of size 8 in the [1692,1798] ns interval. These three intervals are shown on figures 5 and 6 as small line segments. The values of γ and Eif clearly depend on cluster size distribution and molecular arrangements at the interface. Figure 9(a) shows ns for two different time intervals in region 1, [84,108] ns and [180,204] ns. In this region, ns decreases monotonically in a similar way to the distribution of free surfactants showed in Figure 9(b). For this case, aggregates of 8C16 molecules have not been formed. This completely agrees with the g(r) obtained above for this region. Note that the value of ns for s = 1 is much grater that the values for the same quantity observed in figures 9(b) and (c). Finally, Figure 9(c) shows ns for two different intervals in region 4, [1884,1980] ns and [1980,2076] ns. The distribution is quite similar to the one observed in region 3. The distribution of free surfactants is almost the same as in Figure 9(b). However, there are some changes in the size distribution for genuine clusters. The cluster of size 11 seems to reduce its size to 10 and the cluster of size 8 is divided in some smaller clusters. This change is reflected in a decrease of Eif observed in Figure 6 and in a small increase of γ in Figure 5. It could be that region 3 is not in a real macroscopic equilibrium state and it tends to arrive to the final equilibrium state, or else the system is in a microscopic fluctuation towards the equilibrium observed in region 3. It is clear, that the changes observed in region 2 of figures 5 and 6 are due to the formation of genuine clusters of molecules of 8C16 at interfaces. To describe macroscopically this process of clusterization some definitions of percolation theory are used. 38 The probability that a surfactant molecule chosen at random is part of a cluster of size s and is defined by:

18


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s ns ws = P s s ns

(3)

It is clear that the denominator of this expression is a constant and it is equal to the total number of surfactants at the interface, i.e. 32 for our simulations.

Figure 10: Probability that a molecule of 8C16 does not belong to a genuine cluster wfree versus time (left axis-bold circles). The same four regions observed in the time evolution of γ and Eif . It is clear from this plot that in region 1 almost all molecules are free. In the third region most of them belong to a cluster. Mean cluster size S versus time (right axis - open circles). We can identify the first three regions shown in Figs. 5, 6 and in wfree evolution. In the third region, the system seems to arrive to equilibrium, clusters are formed. The aggregation number, Saggr , is also showed in the figure at the final region (right axis-open squares). In this region the behaviour of this quantity is quite similar to the behaviour of the mean cluster size S. Analogous to the procedures used on the analysis of the distribution of surfactants in a system water-surfactant over the CMC, an intermediate value that separate the two parts of the size distribution cluster, ns , is chosen. The value sc = 4 is used in what follows. In this way the probability that a molecule of surfactant chosen at random is free is defined by:

wfree =

X

ws .

(4)

s

Dynamics of Surfactant Clustering at Interfaces and Its Influence on the

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