Multiscale Modeling of Mass Transfer and Adsorption in Liquid–Liquid

Article pubs.acs.org/IECR

Multiscale Modeling of Mass Transfer and Adsorption in Liquid− Liquid Dispersions. 1. Molecular Dynamics Simulations and Interfacial Tension Prediction for a Mixed Monolayer of Mono- and Tetracarboxylic Acids K. Kovalchuk, E. Riccardi,† and B. A. Grimes* Ugelstad Laboratory, Department of Chemical Engineering, Norwegian University of Science and Technology, Sem Sælands vei 4, 7491 Trondheim, Norway ABSTRACT: Phase space trajectories of protonated and deprotonated mono- and tetracarboxylic acid surfactants at an oil/ water interface are obtained from molecular dynamics (MD) simulations and are employed to calculate the interfacial area of the molecules as a parameter for a molecular mixed monolayer model. Three simple methods, based on the available volume at the interface and the solvent accessible area, are applied to calculate the interfacial molecular areas from the MD trajectories. Experimental equilibrium interfacial tension (IFT) data for single component systems are employed with the calculated interfacial molecular areas to fully parametrize the model in order to predict the equilibrium IFT data for mixed solutions of mono- and tetraacids. The agreement between experiment and theory is found to be good. The methodology demonstrated here provides a tool to evaluate the effect of the bulk concentration and composition of various model crude oil surfactant compounds on their interfacial concentrations and composition upon the initial formation of a monolayer at a water/oil interface. In part 2 (Kovalchuk, K.; Riccardi, E.; Grimes, B. A. Multiscale Modeling of Mass Transfer and Adsorption in Liquid−Liquid Dispersions. 2. Application to Calcium Naphthenate Precipitation in Oils Containing Mono- and Tetracarboxylic Acids. Ind. Eng. Chem. Res. 2014, DOI: 10.1021/ie501296t), the parametrized monolayer model is employed to predict phase partitioning of compounds in liquid−liquid dispersions with a continuum model and the parameters determined in this work.

1. INTRODUCTION The dynamic and equilibrium behavior of the interfacial tension due to surfactant adsorption is a significant factor that influences the behavior and stability of liquid−liquid dispersions in a myriad of practical applications.1−8 In many applications, particularly those involving naturally occurring surfactants, surfactant mixtures are encountered where the interfacial tension is determined by competitive adsorption on the interface.1−8 The competitive adsorption of natural surfactant molecules on oil−water interfaces can have a significant impact on the performance of transport and separation systems in the crude oil processing chain.8−15 Specifically, in acidic crude oil systems,8−15 indigenous naphthenic acids act as natural surfactant molecules which can act to stabilize crude oil−water emulsions as well as react with calcium ions to form calcium naphthenate precipitates that foul transport and separation equipment. In particular, a class of C80−82 isoprenoid tetraacids, commonly referred to as ARN acids, has been identified12−17 as a significant constituent of calcium naphthenate precipitates. The ARN acids are nonlinear in structure and multifunctional with each molecule having four carboxylic acid moieties.16,17 These tetraacids have a very high interfacial activity and are hypothesized to form cross-linked networks with calcium ions when adsorbed on an oil−water interface; the cross-linked network is insoluble in both the oil and water phases and consequently deposits on process equipment which can cause frequent process shutdowns.12−17 Observations10,14,15 suggest that the calcium naphthenate precipitation behavior is most likely influenced by competitive © 2014 American Chemical Society

adsorption between the tetraacids and other interfacially active components in crude oil such as monoacids and asphaltenes, while the mechanism of inhibiting naphthenic acid precipitates with petrochemical additives may also be competitive adsorption. In effect, the hypothesis is that the extent of calcium naphthenate precipitation is directly linked to the interfacial concentration of the ARN acids. Consequently, application of a theory for surfactant adsorption1−8 capable of predicting the interfacial concentration of tetraacids in the presence of other pseudo crude oil surfactants and petrochemical inhibitors in well-defined model systems would be valuable for evaluating the risk of naphthenic acid precipitation based on the general oil composition and process variables as well as for developing new naphthenic acid inhibitor chemicals. Crude oil is composed of thousands of compounds, and the composition can vary significantly with time and location. Thus, any attempt to apply a mathematical model for surfacant adsorption1−8 to a real crude oil system would be futile. However, the synthetic asphaltene and tetranaphthenic acid compounds developed by Nordgård and Sjöblom,18 Nordgård et al.,19 and Nordgård,20 together with commercially available surfactants similar to those found in crude oil,21 represent a promising library of compounds that could be used to Received: Revised: Accepted: Published: 11691

March 27, 2014 May 28, 2014 June 2, 2014 June 2, 2014 dx.doi.org/10.1021/ie501295k | Ind. Eng. Chem. Res. 2014, 53, 11691−11703

Industrial & Engineering Chemistry Research

Article

Figure 1. Chemical structures of BP10 and decanoic acid (DA).

obtained from equilibrium interfacial tension measurements on single surfactant solutions of that compound.5−7 Consequently, there is a 1:1 correspondence between the number of equilibrium IFT experiments needed to parametrize the model and the number of surfactants in the solution; as the number of surfactants in the system increases, considerable time and cost could be saved by adopting this approach.5−7 Since the molecular structure of the tetraacid molecules is complex relative to typical surfactant molecules and its orientation at the interface must be determined, molecular dynamics simulation of such molecules at the oil/water interface would be the preferred25−34 computational approach to determine the parameters of the molecular mixed monolayer model.5−7 The molecular dynamics computational approach for water/air interfaces27−29 and liquid/liquid interfaces25,30−34 is a field of active research including simulations of naphthenic (mono) acids31,33 and asphaltene-like compounds.34 In this work, molecular dynamics (MD) simulations of the model tetraacid compound BP10 and decanoic acid (DA), whose structure mimics the arm groups of BP10, are performed to determine their interfacial molecular areas under different conditions at low pH and high pH (below and above the pKa) when the acid molecules are uncharged and charged, respectively. Three basic methods are applied to calculate an interfacial molecular area to represent the hard disk area in the molecular mixed monolayer model for liquid−liquid systems developed by Mulqueen and Blankschtein.27 The standard state chemical potential difference between the interface and the bulk oil or water phases for each surfactant is then obtained from equilibrium interfacial tension (IFT) data for single component systems. Then, the molecular mixed model25−27 is used to predict the equilibrium IFT data for the mixed (binary) surfactant system. Since the interfacial activity of the acid molecules should depend on their dissociation state,23 the monolayer model is parametrized for protonated and unprotonated acids. The parameters determined for the molecular mixed monolayer adsorption model in this work are used in Part 235 to simulate multicomponent interfacial mass transfer in a liquid−liquid dispersion.

formulate mixed surfactant oil solutions in order to study their adsorption and interfacial tension behavior under well-defined concentrations and compositions. The synthetic asphaltene and tetranaphthenic acid molecules18−20 mimicked the interfacial behavior of their counterparts indigenous to crude oil reasonably well.18−20 The synthetic tetraacid molecule BP10, which consists of a benzophenone core with four carboxylic acid side chains containing 10 carbon atoms, demonstrated interfacial behavior most consistent with indigenous ARN tetraacids,18−20 and has been successfully employed to experimentally study and quantify the precipitation of calcium naphthenate.22−24 Furthermore, the benzophenone core in BP10 facilitates the measurement of its concentration with UV light. Consequently, the BP10 molecule is appropriate for use in fundamental mechanistic studies of the interfacial behavior of tetraacid molecules involving experiments, molecular simulation, and continuum modeling; the poor solubility of BP10 in many nonpolar solvents is a limitation. Considering the library of synthetic asphaltenes and tetranaphthenic acids with well-defined chemical structures and high purities,18−20 an appropriate thermodynamic model for the adsorption of mixed surfactant solutions could be applied to estimate the interfacial concentration of tetraacid molecules over a wide range of solution compositions and concentrations. The molecular mixed monolayer model developed by Nikas et al.5 employed a generalized twodimensional gas approach to describe mixed nonionic surfactant monolayers as two-dimensional mixtures of hard disks that interact through van der Waals interactions at air− water interfaces. Mulqueen and Blankschtein6,7 extended this approach to ionic surfactants6 and oil−water systems.7 The equilibrium interfacial tension (IFT) predictions for mixed surfactant solutions of nonionic and ionic surfactants were very good.5−7 One significant advantage of this approach is that the area of the hard disks and van der Waals attractions can be calculated from molecular simulation, while the standard state chemical potential difference of the surfactant between the bulk phase and the interface is the only adjustable parameter of the model.5−7 In practical terms, this implies that, for a given interfacially active compound, the adjustable parameter can be 11692

dx.doi.org/10.1021/ie501295k | Ind. Eng. Chem. Res. 2014, 53, 11691−11703


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2. METHODS AND MODELS 2.1. Model Molecules. As mentioned above, the model C80 isoprenoid tetraacid compound developed by Nordgård and Sjöblom,18 Nordgård et al.,19 and Nordgård20 designated as BP10 is employed as the tetraacid in the present study. Unlike the indigenous ARN tetraacids, BP10 adsorbs UV light which facilitates the direct measurement of its concentration in the bulk oil and water phases. Furthermore, the molecular structure is well-defined and has been isolated to at least 97% purity,18−20 which makes it a suitable candidate for the combined simulation and experimental approach presented in this work. It should be noted that BP10 has closely mimicked ARN acids in terms of interfacial activity, interfacial rheology, and critical micelle concentration (cmc);18−20,23 however, the solubility of BP10 in nonpolar solvents is much lower than those of the indigenous ARN tetraacids and much higher in water.20,24 Decanoic acid (DA) was chosen in this work to represent the monoacid primarily because it is similar in molecular structure to the four 10-carbon-chain carboxylic acid groups in the BP10 molecule. The molecular structures of BP10 and DA are given in Figure 1. 2.2. Experimental Methods. Single and mixed surfactant solutions of BP10 and DA were prepared for the measurements of interfacial tension (IFT) at high and low pHs (pH 11 and pH 2, respectively). A solution of 9:1 (v/v) xylene/CHCl3 was employed as the oil phase in the experiments. The aqueous phase was prepared with deionized water, and sodium chloride was added to achieve a 0.6 M NaCl concentration. The selection of the 9:1 (v/v) xylene/CHCl3 oil phase was necessary due to the solubility of BP10;18−20 BP10 is insoluble in xylene without CHCl3. The density of the 9:1 (v/v) xylene/ CHCl3 solution was measured with the density meter DMA 5000 (Anton Paar, Germany). Xylene was obtained from AnalaR NORMAPUR and is composed mainly of the p-xylene isomer. CHCl3 was obtained from Sigma-Aldrich. BP10 was synthesized in-house to at least 97% purity.18−20 The details of the synthesis are given by Nordgård and Sjöblom18 and Nordgård.20 Decanoic acid (DA) was obtained from SigmaAldrich. Solutions were prepared as follows: 1. For low pH (pH 2) experiments BP10 and DA were first dissolved in CHCl3, sonicated for 1 h at 30 °C, and diluted to a 9:1 (v/v) ratio with xylene; the pH of the aqueous phase was lowered to 2 with the addition of HCl (AnalaR NORMAPUR). The mixed DA−BP10 surfactant solution was studied at a molar ratio of DA:BP10 = 5000:1. 2. For high pH (pH 11) experiments, the sodium salt BP10Na4 and DA were dissolved in Milli-Q water and the pH was adjusted to 11 with NaOH (AnalaR NORMAPUR). The mixed DA−BP10 solution was studied at a molar ratio of DA:BP10 = 2000:1. Measurements of the interfacial tension (IFT) for the two described sets of samples were obtained using the Sigma 70 ring tensiometer (KSV Instruments, Finland). The measurements were performed using a platinum du Noüy ring that was washed with toluene, rinsed with Milli-Q water, and heated until glowing before each measurement. The ring was immersed in the aqueous phase before addition of the oil phase on the top. The surface tension of the water/air interface was determined (72.53 mN/m) before the measurements to check the calibration of the instrument. The duration of each experiment was 3−20 h in order to ensure that equilibrium had

been reached in the system. Room temperature was maintained at 25 °C for the duration of each experiment. 2.3. Computational Methods. MD simulations were employed to calculate the interfacial area per molecule at the oil/water interface. Simulations were performed with the GROMACS 4.5 simulation package36 employing the OPLSAA force field.37 The surfactant molecules (BP10 and DA as well as mixtures of those) have been simulated at the interface between water and an oil phase represented by a mixture of pxylene/CHCl3 9:1 v/v. The acids were studied in the protonated (uncharged) and deprotonated (charged) states. Electroneutrality in the system was maintained by introducing sodium counterions (Na+) with a corresponding charge equal to +1. Consistent with the tetraacids, p-xylene and CHCl3 have been modeled in full atomistic details with the OPLS-AA force field37 while water has been modeled with the TIP4P/2005 force field.38 The interfacial simulations have been constructed by first equilibrating the different solvents in independent simulations. Next, the equilibrated water slab was placed in the middle of the multiphase simulation box, followed by surfactant slabs on either side of the water box along the z-axis, and subsequently, an equilibrated oil slab was placed on each open side of the surfactant slab along the z direction. Therefore, for each multiphase simulation box, there are two interfaces. The slab dimensions have been chosen to avoid finite size effects of the surfactant molecules which are especially significant for increasing numbers of the large tetraacid molecules. Specifically, the length of the x and y coordinate directions of the slabs were fixed at 8 nm in order to maintain a constant value for the area, Axy, of the oil−water interface in the x−y plane. The MD simulations were performed in a NAxyPzT ensemble, and the dimension of the simulation box along the direction, z, normal to the interfaces was controlled by a barostat in order to guarantee the desired pressure in the system; the resulting dimension of the simulation box along the direction, z, was in the range ∼13 to ∼15 nm depending on the type and number of surfactant molecules in the system. The oil solvent is composed of 2256 xylene molecules and 315 chloroform molecules, while the water phase is composed of 9983 molecules. Interfacial systems containing 2, 4, 16, 32, and 48 BP10 molecules per interface and interfacial systems containing 4, 8, 16, 64, and 128 DA molecules per interface have been constructed to simulate the results presented below. The choice of this wide range of numbers of molecules at the interface was to determine the interfacial molecular areas for increasing levels of monolayer saturation.5 The initial temperature for the simulation has been taken to be equal to 298 K and has been obtained by assigning the molecular velocities according to a Maxwell−Boltzmann distribution. The temperature has then been maintained at 298 K via the Berendsen temperature coupling method, while the Berendsen bath coupling method has been imposed to keep a constant normal pressure of 1 bar. The cutoff distance of short-range nonbonded interactions (van der Waals and real space Coulomb) was chosen to be 12 Å. Particle mesh Ewald (PME) summation was used for computing the electrostatic interactions. Periodic boundary conditions (PBC) have been applied along all three dimensions, and the leapfrog algorithm has been selected to integrate Newton’s equation of motion. Each system has been independently equilibrated with a temperature and pressure annealing procedure. The systems have been kept at 400 K and 100 bar for 1 ns and then 11693



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progressively relaxed to ambient conditions with a transition time 5 ns long. An additional 10 ns has been simulated at the ambient conditions before initiation of the system sampling. The sampling has been performed for a period of 10 ns in 2.5 ps intervals. 2.4. Molecular Mixed Monolayer Model. As stated above, the molecular mixed monolayer model5−7 represents the interfacial layer as a mixture of two-dimensional hard disks. The assumptions of this model5−7 are reasonable provided the interface can be considered to behave as a surfactant monolayer and the interactions are not strong enough to induce longrange ordering and segregation of species in the monolayer. The surfactant molecules can be considered to be charged or uncharged based on the value of the pH of the aqueous phase relative to the pKa of the acids. Provided that the pH of the aqueous phase is significantly below the pKa of the acidic surfactant compound, the acid molecules can be considered to be uncharged (treated as a nonionic surfactant). On the other hand, if the pH of the aqueous phase is significantly greater than the pKa of an acidic surfactant, all the molecules of the acidic surfactant can be considered to be charged (treated as an ionic surfactant) and partitioning of the charged acids into the oil phase can considered to be negligible. It should also be noted that Mulqueen and Blankschtein7 indicated that the magnitude of the attractive van der Waals interactions between the surfactant alkyl tails is negligible when they interact through an intervening linear alkane oil phase. Consequently, they7 neglected the van der Waals interaction terms in their molecular mixed monolayer model for oil−water systems. Within the scope of this work, the van der Waals attractions are also neglected for the acidic surfactant tails interacting through the 9:1 (v/v) p-xylene/CHCl3 oil phase solvent considered in this work due to the fact that, for the interfacial molecular areas calculated and presented below, the molecular mixed monolayer model (eqs 1 and 2) predicted a negligible effect on the interfacial tension and surface concentrations even for very large hypothetical values of the second surface virial coefficient. Therefore, the virial expansion in surface concentration is not employed in the equation of state considered here. While this oil phase is composed of aromatic and polar compounds, for the molecular cross-sectional areas calculated above and the standard state chemical potential differences determined from the experimental data below, the effect of the second virial coefficient on the interfacial tension was insignificant. Therefore, the equation of state for the model system defined above can be written in terms of the molar interfacial concentration as follows:5−7 N

ψ=

+

⎛ ⎞ Nav Γi ⎟ + ln⎜⎜ ⎟ N ⎝ 1 + Nav ∑k = 1 Γkak ⎠ N

Nav(ai ∑k = 1 Γk + 2πri ∑k = 1 Γkrk) N

1 + Nav ∑k = 1 Γkak πai(Nav ∑ Γkrk)2 N

(1 + Nav ∑k = 1 Γkak )2

for i = 1, 2, ..., N

⎛ λ ⎞2 ⎫ ⎪ d ε 1 + ⎜ zi Γi⎟ ⎬ + s zi Γi ⎪ 2ψ εs ⎝ 4ψ ⎠ ⎭ (2)

In eq 2, xi denotes the mole fraction of component i in the bulk oil phase, and Δμ̂ °i,σ/b is the modified7 standard state chemical potential difference of component i between the interface and the bulk oil (b → o) or water (b → w) phase at infinite dilution. The standard state chemical potential of component i on the interface is defined at a reference surface pressure of 1 dyn/cm and temperature T.5−7 The term Δμ̂ °i,σ/b is called “modified” because it accounts for phase partitioning.7 In the high pH systems considered in this work, partitioning of the charged acids to the oil phase can be considered to be negligible. Thus, for the water phase Δμ̂ i,σ/w ° should be equivalent to the actual standard state chemical potential difference, Δμi,σ/w ° , of component i between the interface and ° ≈ Δμi,σ/w ° ). Given known values the bulk water phase (Δμ̂i,σ/w of the mole fraction, xi, of each component in the bulk oil or water phase as well as the parameters Δμ̂°i,σ/b/kBT and ai, the values of the interfacial concentration, Γi, can be determined from the simultaneous solution of the set of N equations given in eq 2 . It should be noted here that the only unknown, adjustable parameter in the model is the term containing the modified standard state chemical potential difference Δμ̂ i°,σ/b/kBT. This

⎤ ⎞2 ⎥ ∑ zi Γi⎟⎟ − 1⎥ ⎠ i=1 ⎥⎦ N

(1)

εεokBT 2e 2NavI

kBT

⎧ ⎪ λ + 2zi ln⎨ zi Γi + ⎪ 4ψ ⎩

where

λ=

Δμî °, σ /b

N

+

N

ds ε (∑ zi Γi)2 4ψ εs i = 1

(1b)

(1 + |zi|) ln(xi) =

N

+

2e 2Nav

In eq 1, Π is the surface pressure, R denotes the ideal gas constant, T is the absolute temperature (T = 298 K), Nav represents Avogadro’s number, Γi denotes the interfacial molar concentration of component i, ai represents the hard disk area of molecule i, ri is the hard disk radius of molecule i (ai = πri2), λ denotes the Debye−Hückel screening length and is given in eq 1a, ψ represents a constant term given by eq 1b, zi denotes the charge number of molecule i, ds is the Stern layer thickness (ds = 2.32 Å based on the ionic diameter of a sodium ion), ε represents the dielectric constant in the bulk aqueous phase (ε = 80.1), εs denotes the dielectric constant of the Stern layer (εs = 42 was employed6,7), εo is the permittivity of free space, e represents the elementary charge, and I denotes the ionic strength of the aqueous phase (I = 0.6 M). Equation 1 can be used to obtain the adsorption isotherm of a given component i5−7 since the chemical potential of molecule i on the oil−water interface can be obtained from eq 1. By equating the interfacial chemical potential obtained from eq 1 to the bulk oil chemical potential (see refs 5−7 for details of the formulation), one can obtain an adsorption isotherm for each component i in terms of the molar interfacial concentrations.

∑i = 1 Γi π ( Nav ∑i = 1 Γiri)2 Π = + N N RT 1 − Nav ∑i = 1 Γiai (1 − Nav ∑i = 1 Γiai)2 ⎡ ⎛ λ 8ψ ⎢ + 1 + ⎜⎜ ⎢ λ ⎢ ⎝ 4ψ ⎣

εεokBT

(1a) 11694



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Figure 2. Normalized number density distribution of water, xylene, CHCl3, and surfactant (DA and BP10/charged and uncharged) molecules along the z coordinate of the simulation box.

parameter can be determined by fitting the model to equilibrium IFT curves of single surfactant solutions.5−7 Mulqueen and Blankschtien7 indicate that one only needs one experimental IFT measurement at a carefully selected bulk concentration to determine the value of Δμ̂°i,σ/b/kBT. However, the model is regressed to experimental IFT data at several different bulk concentrations in order to determine confidence intervals in the Δμ̂i°,σ/b/kBT parameter.

the points where the values of the number density of oil or water molecules become less than 0.1% of their bulk number density in systems with no surfactant molecules present. It can be noted from Figure 2 that, as the number of tetraacid molecules in the system increases, the surfactant density increases with increasing distance from the water phase. Moreover, all number density profiles demonstrate that increasing the number of surfactant molecules (DA and BP10) at the interface significantly offsets the xylene density distribution. This result suggests a more condensed interfacial layer as solvent molecules are excluded. The offset in the xylene density distribution is more significant for the charged acid molecules relative to the uncharged states, implying more tight packing of charged molecules. Since the acid molecules appear to pack tighter, exclude more solvent, and increase in number density at distances further from the interfacial region as their total number increases, the density profiles in Figure 2 could suggest where the condition for a monolayer is violated. For BP10 in the charged and uncharged state (parts a and b, respectively, of Figure 2), the 32 and 48 molecule cases could potentially violate the condition for a monolayer due to the significant amount of surfactant density and solvent exclusion that occurs outside of the interfacial region, while for DA in the

3. RESULTS AND DISCUSSION 3.1. Monolayer Formation. Normalized molecular density profiles (Figure 2) of water, xylene, CHCl3, and the carboxylic acid molecules along the direction normal to the oil/water interface (z direction) as well as trajectory snapshots (Figure 3) are presented for the purpose of selecting the simulations where the acid compounds form a saturated monolayer at the interface in order to calculate the interfacial molecular area for each acid near the point of interfacial saturation. The molecular density profiles along the coordinate (z) direction normal to the interface were plotted up to the centerline of the oil phase due to symmetry along the z dimension and periodic boundary conditions. The interfacial region is indicated with vertical lines in the density profile plots of Figure 2 that are defined based on 11695



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Figure 3. Trajectory snapshots of BP10 at the oil−water interface: (a) 32 molecules of charged BP10, (b) 48 molecules of charged BP10, (c) 32 molecules of uncharged BP10, and (d) 48 molecules of uncharged BP10.

Figure 4. Trajectory snapshots of decanoic acid (DA) at the oil−water interface: (a) 64 molecules of charged DA, (b) 128 molecules of charged DA, (c) 64 molecules of uncharged DA, and (d) 128 molecules of uncharged DA.

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hydrophobic and hydrophilic solvent accessible areas of the acid molecules were also estimated by the Connolly surface method,40 which is widely accepted for calculating the surfaces of biomolecules and is implemented in the GROMACS software package.36 Due to the tight packing and solvent exclusion in the interfacial layers demonstrated in Figure 2, the solvent molecules primarily have access to the surfactant molecules in the direction normal to the interface. Consequently, the hydrophobic or hydrophilic solvent accessible areas could be a reasonable representation of the interfacial molecular area. The results for the calculated interfacial molecular areas are given in Table 1. Previous experimental work19,20 has indicated

charged and uncharged states (parts c and d, respectively, of Figure 2), the 128 and 192 molecule cases could potentially violate the condition for a monolayer for the same reasons. The snapshots presented in Figures 3 and 4 should help elucidate these observations. In parts a and b of Figure 3, trajectory snapshots of BP10 in the charged state are given for 32 and 48 molecules, respectively, while for parts c and d of Figure 3 trajectory snapshots of BP10 in the charged state are given for 32 and 48 molecules, respectively. The trajectory snapshot in Figure 3a seems to suggest that BP10 is packed in a tight interfacial monolayer, while in Figure 3b it is clear that there is a secondary layer of BP10 molecules forming above the interfacial layer. The same observation is apparent for BP10 in the uncharged state (Figure 3c,d). In parts a and b of Figure 4 trajectory snapshots of DA in the charged state are given for 64 and 128 molecules, respectively, while for parts c and d of Figure 4 trajectory snapshots of DA in the charged state are given for 64 and 128 molecules, respectively. For the 64 DA molecule case in both the charged and uncharged states (parts a and c, respectively, of Figure 4), tight packing of the DA molecules in a single layer is observed, while for the 128 molecule case secondary DA layers appear to be forming above the interfacial layer. It is clear from Figures 2−4 that interfaces with 48 BP10 molecules and 128 or more DA molecules per interface tend to form secondary layers above the monolayer and cannot be considered for the further calculations. Therefore, the results presented below include only simulation cases with up to 32 BP10 and 64 DA molecules. 3.2. Calculation of the Hard Disk Areas. The crosssectional area of a molecule on the x−y plane at the interface is referred to here as the interfacial molecular area. The calculations of the interfacial molecular areas were performed at the various concentrations of mono- and tetraacid molecules to represent the hard disk area in the mixed monolayer adsorption model.5−7 Three different methods were applied to compute the interfacial molecular area with the following algorithms adopted for calculation. The first two methods (M1 and M2) are based on calculating the available volume of a surfactant on the interface39 by wrapping the molecule in a cylinder whose axial direction is normal to the interfacial plane. The cross-sectional area of this cylinder would then be considered as a hard disk area for the surfactant on the interface. For methods M1 and M2, the x and y coordinates of atoms of each surfactant molecule were extracted from simulated trajectory files and used for calculations. For method 1 (M1), the geometric center of a molecule was computed first, by averaging the x and y coordinates of all atoms in each surfactant molecule. The lateral distances in the x−y plane between the geometric center and every atom within each molecule were estimated and compared to determine the maximum lateral distance from the geometric center of the molecule. The maximum lateral distance determined is then taken to be the hard disk radius of the molecule on the interface. For method 2 (M2), the lateral distances between all atoms of an acid molecule in the x−y plane were determined and the maximum distance was employed as the diameter of the hard disk. For both methods 1 and 2, the selection of maximum distances ensures that each atom in the molecular structure was taken into account for calculations. The calculations were performed by utilizing the atoms of each acid molecule and averaged by the number of molecules in each simulation. In the third method (M3), the

Table 1. Interfacial Molecular Areas for BP10 and DA in the Charged and Uncharged State Determined from the Three Calculation Methods interfacial area, nm2 no. molecules per interface

M1

2 4 16 32

6.92 6.73 6.79 7.21

2 4 16 32

7.49 7.13 6.96 6.67

4 8 16 64

0.63 0.61 0.62 0.57

4 8 16 64

0.81 0.81 0.82 0.78

M2

solvent accessible area (M3) hydrophobic area, nm2

BP10 Charged 5.84 9.13 5.58 9.26 5.64 7.44 5.83 5.27 BP10 Uncharged 6.20 10.27 5.78 10.03 5.59 8.29 5.30 7.22 DA Charged 0.54 2.38 0.52 2.37 0.53 2.28 0.49 1.84 DA Uncharged 0.70 2.67 0.70 2.60 0.70 2.58 0.68 2.27

hydrophilic area, nm2

average area, nm2

4.54 4.57 4.04 2.71

6.83 6.91 5.74 3.99

3.51 3.45 3.06 2.68

6.88 6.71 5.68 4.95

1.06 1.06 1.04 0.90

1.74 1.71 1.66 1.37

0.82 0.80 0.79 0.72

1.74 1.70 1.68 1.49

that the estimation of the area per molecule for the tetraacids could be difficult due to a complex adsorption mechanism and the interval for the correct value could be as wide as 2.61/2.32 to 13.05/11.62 nm2; all areas calculated here and presented in Table 1 fall in this interval. There are two different types of information that can be extracted from the summary of the results in Table 1. First, the Connolly method40 demonstrates significant reduction in the values of solvent accessible areas of the acid molecules with increasing concentration of molecules between 16 and 64 molecules at the interface for DA and between 4 and 16 molecules for BP10 in both charged and uncharged states. This probably shows that solvent has less access to surfactant molecules due to their much more tight packing at the water−oil interface as clearly demonstrated by the molecular density distributions in Figure 2. For further increases in the concentration of acid molecules, a change in the area occupied by the surfactant is observed, suggesting that the interface is reaching its saturation point. This observation is supported by the trajectory snapshots in Figures 3 and 4. Second, methods 1 and 2 show almost no change in the 11697



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Figure 5. Interfacial orientation angle distributions for the arm groups of BP1025 or individual decanoic acid (DA) molecules: (a) BP10 charged, (b) BP10 uncharged, (c) DA charged, and (d) DA uncharged.

projected areas of surfactants with the increase of the number of molecules at the interface for all studied cases indicating probably only minor conformational changes of DA and BP10 molecules at the highest concentrations. To elucidate this, it is interesting to look at the interfacial orientation angle distribution plots reported in Figure 5. The interfacial orientation angles are determined by assigning a representative vector that quantifies the distance and direction between the acidic head groups and the terminal atoms of DA and the four individual arm groups of BP10.25 The angle of this vector relative to the x−y plane is taken to be the interfacial orientation angle of the DA molecule or arm group of BP10. Thus, an orientation angle of +90° implies that the DA molecule or BP10 arm group is perpendicular to the oil/ water interface and the carboxylic acid group is in the water phase. If the interfacial orientation angle is 0°, the DA molecule or BP10 arm group is lying flat on the interface, while if the interfacial orientation angle is −90°, the DA molecule or BP10 acidic arm group is perpendicular to the interface, but the carboxylic acid group is pointed away from the water phase. Figure 5a suggests that the acidic arm groups for BP10 in the charged state are predominantly oriented toward the water phase; in the case of 32 molecules, the arms have a small

probability to orient slightly away from the interface suggesting a slight oversaturation at this molecular concentration, but the arm groups are still predominantly oriented toward the water phase. The presence of symmetry and rigidity of the core in its chemical structure could suggest the adsorption of the molecule with all four, three, and only two carboxylic groups at the interface. Therefore, it is interesting to highlight that a large fraction of BP10 in the charged state remains in relatively the same orientation at all molecular concentrations studied up to the saturation point and a large fraction of molecules in the monolayer are absorbed with four carboxylic groups pointing toward the water phase. This behavior confirms the high surface activity of this tetraacid and explains the large area occupied at the interface by BP10 in the charged state. On the other hand, BP10 in the uncharged state has some probability to orient its acidic arm groups in all directions relative to the interface as shown in Figure 5b, which suggests that uncharged BP10 molecules could adsorb with one, two, or three carboxylic groups at the interface. Figure 5c suggests that the monoacid DA in the charged state is oriented predominantly with the carboxyl head pointed toward the water phase, while the decanoic acid in the uncharged state seems to preferentially lie flat on the interface as suggested by Figure 5d. 11698



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Figure 6. Equilibrium interfacial tension (IFT) data and model predictions for uncharged decanoic acid (DA) and uncharged BP10 at pH 2: (a) experimental and theoretical IFT for single surfactant solutions of DA, (b) experimental and theoretical IFT for single surfactant solutions of BP10, (c) experimental and theoretical IFT for mixed surfactant solutions of DA and BP10 at a DA:BP10 molar ratio of 5000:1, and (d) corresponding interfacial concentrations for the 5000:1 DA:BP10 mixed surfactant solution.

3.3. Application of the Molecular Mixed Monolayer Model for Prediction of Interfacial Tension Curves. The interfacial molecular areas or solvent accessible areas calculated in section 3.2 for each molecule can be taken to represent the hard disk area ai, of that molecule and employed in the molecular mixed monolayer model5−7 presented in section 2.4. The molecular areas for the regression of experimental interfacial tension (IFT) data were selected from the simulations at the highest molecular concentrations where the monolayer formation condition was satisfied such that the hard disk areas represent the interfacial areas in a saturated monolayer.5−7 Therefore, the hard disk areas of the BP10 and DA molecules were taken from the results of the simulations with 64 DA molecules at the interface and the simulations with 32 molecules of BP10 at the interface. For each case, there are five calculated hard disk areas, and the area that provided the best fit to the data was taken to be the representative hard disk area. The model regression to the experimental data was obtained using least-squares employing a modified Levenberg−Marquardt method to generate a sequence of approximations to a minimum point;41 the

confidence intervals in the parameters were determined by calculating the variance−covariance matrix of the estimated nonlinear regression parameters. The equilibrium IFT curves for single surfactant and mixed surfactant solutions of uncharged BP10 and DA are given in Figure 6 (low pH case). The filled circle symbols represent the experimental data, and the solid lines represent the model simulation with the best-fit parameters. In Figure 6a the equilibrium IFT data and model simulation are given for single surfactant solutions of uncharged DA, while the experimental IFT data and model simulations for single surfactant solutions of uncharged BP10 are given in Figure 6b. Examination of Figure 6a,b indicates that there is a good agreement between theory and experiment. The values of Δμ̂°i,σ/o/kBT determined for BP10 and DA were −66.312 ± 1.764 and −50.070 ± 0.532, respectively (Table 2). This suggests that the tetraacid (BP10) is significantly more interfacially active than the monoacid (DA) at acidic conditions below the pKa. This fact is reflected in the IFT curves by observing that the IFT starts to decease at very low concentrations and reaches approximately the same final IFT value (∼25 mN/m) as DA at a much lower 11699



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histograms (Figure 5b) suggested that uncharged BP10 molecules could adsorb with one, two, or three carboxylic groups at the interface. Consequently, the interfacial molecular area for uncharged BP10 could be smaller compared to the charged BP10 since not all four carboxyl groups might be oriented toward the interface and the cross-sectional area of a cylinder representing the accessible molecular volume at the interface may not be equivalent to the hard disk area. Therefore, the hydrophobic solvent accessible area, which accounts only for the carboxyl groups accessible to water at the interface, may indicate the area occupied by the uncharged BP10 in the interfacial region and be a good representation of the hard disk area as suggested by the data regression. The model prediction for the mixed DA−BP10 surfactant solution at the low pH condition for the water phase is presented and compared to the experimental data in Figure 6c. The molar ratio of DA: BP10 is 5000:1 for all values of the BP10 mole fraction; the 5000:1 molar ratio of these concentrations was employed to facilitate competition for interfacial area in the mixed surfactant solution. Figure 6c

Table 2. Best Fit Parameters of the Molecular Mixed Monolayer Model Determined from MD Simulation and Regression of Experimental IFT Data molecular mixed monolayer model params system name

Δμ̂i,σ/b ° /kBT

ai, nm2

BP10 charged BP10 uncharged DA charged DA uncharged

−184.705 ± 5.124 −66.312 ± 1.764 −75.787 ± 0.319 −50.070 ± 0.532

7.21 2.68 1.37 0.78

concentration. The best fit hard disk area for uncharged DA was 0.78 nm2 determined by method 1, while the hydrophilic solvent accessible area of 2.68 nm2 provided the best fit of the experimental IFT data for uncharged BP10. The good fit for the uncharged DA with method 1 (and method 2 as well) is somewhat surprising since the trajectory snapshots (Figure 4c) and orientation angle histogram suggest that uncharged DA has a tendency to lie flat on the interface. The trajectory snapshots for uncharged BP10 (Figure 3c) and orientation angle

Figure 7. Equilibrium interfacial tension (IFT) data and model predictions for charged decanoic acid (DA) and charged BP10 at pH 11: (a) experimental and theoretical IFT for single surfactant solutions of DA, (b) experimental and theoretical IFT for single surfactant solutions of BP10, (c) experimental and theoretical IFT for mixed surfactant solutions of DA and BP10 at a DA:BP10 molar ratio of 2000:1, and (d) corresponding interfacial concentrations for the 2000:1 DA:BP10 mixed surfactant solution. 11700



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interface and therefore approximate parameters employed in the model. It should be noted that in Figure 7b,c, the two largest experimental mole fractions of BP10 appear to exceed the cmc value reported by Nordgård20 by approximately 1 order of magnitude. These points were included in the calculation since clear evidence of a cmc was observed beyond the experimental mole fraction of BP10 presented in Figure 7, and thus, it can be assumed that no micelles are present at these points in question.

indicates that the agreement between theory and experiments is good. Moreover, interfacial concentrations of BP10 and DA shown in Figure 6d clearly indicate that, over the majority of the concentrations studied, BP10 dominates the interfacial coverage. The DA only begins to compete for interfacial area and partially replaces BP10 when the total surfactant mole fraction becomes very large. It should also be pointed out here that the interfacial concentrations, Γi, in Figure 6d have been normalized by a term denoted as Γi,max; the normalization term Γi,max is defined for convenience and defined in terms of the hard disk area, ai (Γi,max = 1/(aiNav)) and is used consistently for the remainder of the results presented. In Figure 7, the experimental equilibrium IFT data and model simulations for single surfactant and mixed surfactant solutions of charged BP10 and DA are given (high pH case). For single surfactant solutions of charged DA (Figure 7a) and charged BP10 (Figure 7b), the agreement between experiment and theory was good. A strong increase in interfacial activity of both acids is observed at basic conditions corresponding to pH 11 (above the pKa value). This is demonstrated by a significantly lower interfacial tension reported in the IFT curves of charged DA and BP10 presented in Figure 7 a,b. However, the tetraacid has a higher interfacial activity compared to DA as indicated by the ability of BP10 to reduce the interfacial tension at extremely low concentrations. This observation is supported by the values of Δμ̂ °i,σ/w/kBT = −184.705 ± 5.124 for charged BP10 and Δμ̂ °i,σ/w/kBT = −75.787 ± 0.319 for charged DA (Table 2) that were determined from the data regression (Figure 7a,b). For DA, the calculated interfacial molecular area that provided the best fit to the data was the average solvent accessible area of 1.37 nm2 determined from method 3, while the interfacial molecular area determined by method 1 for BP10, 7.21 nm2, provided the best fit to the data for BP10. The hard disk areas increase for both molecules in the charged state relative to their uncharged state. This may be due to the fact that the distance between the carboxylic acid arms of BP10 or individual DA molecules could be regulated by the repulsive force arising from hydrophilic, steric, and ionic repulsion42,43 which may lead to wider spreading of the molecule at the interface and increase in area per molecule. It should be pointed out that when the tetraacid molecules have a high probability for all four arms to be oriented toward the water phase, the simple accessible volume based methods (methods 1 and 2) might provide a good representation of the hard disk area. However, this is not the case for charged DA, where the representative hard disk area seems to be considerably larger than the cross-sectional area of its accessible volume at the interface. It is worth noting at this point that the selection of the appropriate hard disk areas is ad hoc based on the best fit to the data. This suggests that alternate thermodynamic methods for calculating the appropriate hard disk area could be employed, such as the most probable interfacial molecular area determined from the interfacial formation energy,44 to establish a more theoretically rigorous selection criterion for the representative hard disk area. Figure 7c indicates 82% of the model prediction of experimental interfacial tension data for the mixture of DA: BP10 with the mole ratio of 2000:1. The prediction of the IFT behavior of the mixed system is good considering that the slopes (shape) of the curves match very well, yet the model prediction seems to be shifted to slightly lower mole fractions. The deviation in the prediction and experiments could possibly be attributed to the electrostatic effects accounted for at the

4. CONCLUSIONS In this work, equilibrium interfacial tension (IFT) data and molecular dynamics (MD) simulation of mono- and tetracarboxylic acid compounds at oil/water interfaces are employed together to calculate the adjustable parameters of a molecular mixed monolayer model and predict the equilibrium IFT of mixed mono- and tetraacid monolayers in oil/water systems.5−7 Since the adsorption behavior of acid molecules in crude oil has been observed8 to be different for acids in their protonated (uncharged) and deprotonated (charged) states, the molecular mixed monolayer model was parametrized for both protonation states. BP10, the tetraacid compound studied in this work, is a synthetic molecule developed18−20 to mimic the behavior of ARN tetraacids,12−17 while decanoic acid (DA) was used as the model monoacid due to its structural similarity to the four carboxylic acid groups on the BP10 molecule. The MD simulations provide insight into the distribution and orientation of the tetraacid molecules at oil−water interfaces at successively increasing interfacial concentrations to determine the saturation point of the monolayer. The results indicate that the BP10 tetraacid molecules are very interfacially active and tend to remain exclusively at the interface without significant conformational changes. Methods for computing of interfacial molecular area at the interface were developed and successfully employed in the molecular mixed monolayer model, and the values of the single adjustable parameter were obtained within a narrow error tolerance from equilibrium IFT data in single surfactant systems. Consequently, the a priori predictions of the model5−7 (below the pKa) for the mixed 5000:1 DA:BP10 surfactant system are good and deviation of experimental data points from the model for the system DA:BP10 2000:1 (above pKa) is only 18%. The predictions for the mixed BP10−DA surfactant system indicate that the BP10 dominates the interfacial surface coverage until the total surfactant concentration becomes very large when the DA molecules eventually start to displace the BP10. It is worth noting here that the ability of the molecular mixed monolayer model5−7 to predict the equilibrium IFT behavior of the protonated mono- and tetracarboxylic acids is promising but still needs to be applied to the other components of the crude oil. Furthermore, to fully characterize the system, molecular thermodynamic theory of surfactant solutions45 could be employed to describe the formation of micellar aggregates for the surfactant compounds that partition into the water phase at high pH values. However, since the mixed monolayer model has only one adjustable parameter that can be determined from equilibrium IFT measurements on single surfactant systems, there is a 1:1 correspondence between the number of surfactants present in the system and the number of experiments required to characterize the system. Additionally, since the other parameters of the model5−7 are determined from molecular simulation, the molecular interactions with the solvent molecules in the monolayer are explicitly considered. 11701



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(10) Hurtevent, C.; Rousseau, G.; Bourrel, M.; Brocart, B. Production Issues of Acidic Petroleum Crude Oils. In Emulsions and Emulsion Stability, 2nd ed.; Sjöblom, J., Ed.; CRC Press: Boca Raton, FL, 2006; pp 477−516. (11) Robbins, W. K. Challenges in the Characterization of Naphthenic Acids in Petroleum. Prepr.Am. Chem. Soc., Div. Pet. Chem. 1998, 43, 137−140. (12) Baugh, T. D.; Wolf, N. O.; Mediaas, H.; Vindstad, J. E.; Grande, K. V. Characterization of a Calcium Naphthenate DepositThe ARN Acid Discovery. Prepr.Am. Chem. Soc., Div. Pet. Chem. 2004, 49, 274−276. (13) Baugh, T. D.; Grande, K. V.; Mediaas, H.; Vindstad, J. E.; Wolf, N. O. The Discovery of High Molecular Weight Naphthenic Acids (ARN Acid) Responsible for Calcium Naphthenate Deposits. SPE Seventh International Symposium on Oilfield Scale, Aberdeen, U.K., May 11−12, 2005; SPE 93011; Curran Associates, Inc.: Red Hook, NY, 2006; pp 9−15. (14) Brocart, B.; Hurtevent, C. Flow Assurance Issues and Control with Naphthenic Oils. J. Dispersion Sci. Technol. 2008, 29, 1496−1504. (15) Baugh, T. Oil Field Fouling: Tales of the Unexpected. Presented at the Thirteenth International Conference on Petroleum Phase Behavior and Fouling, St. Petersburg Beach, FL, June 10−14, 2012; lecture K-3. (16) Lutnaes, F.; Brandal, Ø.; Sjöblom, J.; Krane, J. Archaeal C80 Isoprenoid Tetraacids Responsible for Naphthenate Deposition in Crude Oil Processing. Org. Biomol. Chem. 2006, 4, 616−620. (17) Lutnaes, F.; Krane, J.; Smith, B.; Rowland, S. Structure Elucidation of C80, C81 and C82 Isoprenoid Tetraacids Responsible for Naphthenate Deposition in Crude Oil Production. Org. Biomol. Chem. 2007, 5, 1873−1877. (18) Nordgård, E. L.; Sjöblom, J. Model Compounds for Asphaltenes and C80 Isoprenoid Tetra-Acids. Part I: Synthesis and Interfacial Activities. J. Dispersion Sci. Technol. 2008, 29, 1114−1122. (19) Nordgård, E. L.; Magnusson, H.; Hanneseth, A.-M. D.; Sjöblom, J. Model Compounds for C80 Isoprenoid Tetra-acids: Part II. Interfacial Reactions, Physicochemical Properties and Comparison with Indigenous Tetra-acids. Colloids Surf., A 2009, 340, 99−108. (20) Nordgård, E. L. Model Compounds for Heavy Crude Oil Components and Tetrameric Acids. Ph.D. Dissertation, Norwegian University of Science and Technology, Trondheim, Norway, 2009. (21) Brandal, Ø.; Hanneseth, A.-M. D.; Sjöblom, J. Interactions Between Synthetic and Indigenous Naphthenic Acids and Divalent Cations Across Oil-Water Interfaces: Effects of Addition of Oil-Soluble Non-Ionic Surfactants. Colloid Polym. Sci. 2005, 284, 124−133. (22) Simon, S.; Reisen, C.; Bersås, A.; Sjöblom, J. Reaction between Tetrameric Acids and Ca2+ in Oil/Water System. Ind. Eng. Chem. Res. 2012, 51, 5669−5676. (23) Nordgård, E. L.; Simon, S.; Sjöblom, J. Interfacial Shear Rheology of Calcium Naphthenate at the Oil/Water Interface and the Influence of pH, Calcium, and in Presence of a Model Monoacid. J. Dispersion Sci. Technol. 2012, 33, 1083−1092. (24) Ge, L.; Vernon, M.; Simon, S.; Maham, Y.; Sjöblom, S.; Xu, Z. Interactions of Divalent Cations with Tetrameric Acid Aggregates in Aqueous Solution. Colloids Surf., A 2012, 396, 238−245. (25) Riccardi, E.; Kovalchuk, K.; Mehandzhiyski, A. Y.; Grimes, B. A. Structure and Orientation of Tetracarboxylic Acids at Oil−Water Interfaces. J. Dispersion Sci. Technol. 2014, 35 (7), 1018−1030, DOI: 10.1080/01932691.2013.826584. (26) Kovalchuk, K.; Riccardi, E.; Mehandzhiyski, A. Y.; Grimes, B. A. Aggregates of Poly-functional Amphiphilic Molecules in Water and Oil Phases. Kolloidn. Zh. 2014. (27) Howes, A. J.; Radke, C. J. Monte Carlo Simulation of Mixed Lennard-Jones Nonionic Surfactant Adsorption at the Liquid/Vapor Interface. Langmuir 2007, 23, 11580−11586. (28) Yan, H.; Guo, X.-L.; Yuan, S.-L.; Liu, C.-B. Molecular Dynamics Study of the Effect of Calcium Ions on the Monolayer of SDC and SDSn Surfactants at the Vapor/Liquid Interface. Langmuir 2011, 27, 5762−5771.

It should be stressed here that since crude oils contain a huge number of interfacially active compounds whose molecular structures and compositions are extremely tedious to characterize, the effort to apply the modeling framework discussed in the paper to real crude oil/water systems would be futile. However, interfacially active compounds in crude oil can be assigned to general classes of molecules with certain functional behavior (i.e., resin and asphaltene compounds with acidic, basic, or neutral properties). Therefore, the library of synthetic asphaltene and tetraacid compounds developed by Nordgård et al.18−20 along with the modeling framework for acidic surfactants at an oil−water interface8 and the molecular mixed monolayer model5−7 can be employed to formulate model oils with representative interfacially active molecules with different concentrations and compositions. Such model oils could be used in fundamental studies of the interfacial phenomena encountered in crude oil−water systems.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Tel.: +47 735 90 338. Present Address †

E.R.: Department of Chemistry, Norwegian University of Science and Technology, Høgskoleringen 5, 7491 Trondheim, Norway. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors gratefully acknowledge support for this work by the Norwegian Research Council (PETROMAKS program) and the members of the Ugelstad JIP-2 industrial consortium: Nalco Champion, Conoco Philips, ENI, Petrobras, R.E.P., Statoil, Shell, Talisman, and Total.

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Multiscale Modeling of Mass Transfer and Adsorption in Liquid–Liquid

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