Article pubs.acs.org/Langmuir
Effect of Hydrostatic Pressure on Gas Solubilization in Micelles Bin Meng and Henry S. Ashbaugh* Department of Chemical and Biomolecular Engineering, Tulane University, New Orleans, Louisiana 70118, United States ABSTRACT: Molecular dynamics simulations of anionic sodium decylsulfate and nonionic pentaethylene glycol monodecyl ether micelles in water have been performed to examine the impact of hydrostatic pressure on argon solubilization as a function of pressure. The potential-of-mean force between the micelles and argon demonstrates that nonpolar gases are attracted to the interiors of both micelles. The affinity of argon for micelle interiors, however, decreases with increasing pressure as a result of the comparatively higher molar volume of argon inside assemblies. We evaluate solubility enhancement coefficients, which describe the drop in the solute chemical potential as a function of the micellized surfactant concentration, to quantify the impact of micellization on gas solubilization. While argon is similarly attracted to the hydrophobic cores of both micelles, the gas is more effectively sequestered within nonionic micelles compared with anionic micelles as a result of salting out by charged head groups and accompanying counterions. The solubility enhancement coefficients of both micelles decrease with increasing pressure, reflecting the changing forces observed in the potentials-of-mean force. An analytical liquid drop model is proposed to describe the pressure dependence of argon solubilization within micelles that captures the simulation solubility enhancement coefficients after fitting an effective micelle radius for each surfactant.
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INTRODUCTION The transfer thermodynamics of nonpolar solutes between aqueous and organic solvents exhibit solubility minima in water with increasing temperature, which is recognized as a signature of the hydrophobic effect that drives self-assembly phenomena in water.1−7 When pressure is increased to values on the order of a thousand atmospheres, however, maxima in the relative solubility of nonpolar solutes in water in equilibrium with aqueous and organic solvents are observed.8−10 Le Chatelier’s quid pro quo dictates that solubility maxima with increasing pressure result from changes in the sign of the solute’s molar volume of transfer from an aqueous to organic phase, from positive at low pressure to negative at high pressure.11 The pressure dependence of hydrophobic species transfer thermodynamics is mirrored as maxima in the critical micelle concentrations of nonionic,12 ionic,13,14 and polymeric surfactants,15 with accompanying changes in the sign of the surfactant volume of assembly from positive to negative with increasing pressure. The analogy between hydrophobic transfer thermodynamics and self-assembly fails for proteins, however, with protein structures becoming destabilized at high pressure as a result of positive folding volumes.16 The differences between surfactant and protein assembly reflect differences in the assembled structures formed. Micelles behave more like oily liquid drops, while proteins are more rigid with internal cavities that add to the biomolecular assembly’s size.17−19 Surfactant micelles potentially face extreme pressures in a range of oil production processes. Dispersant formulations may be applied to remediate deep-sea oil spills several thousand feet below the surface under hundreds of atmospheres of pressure.20 In enhanced oil recovery and fracking by surfactant formulations,21−23 pressures can grow to a thousand atmospheres or more within reservoir rock formations. The changes in surfactant assembly concentrations with increasing pressure described above can subsequently diminish surfactant efficacy © 2015 American Chemical Society
by depriving nonpolar solutes of micelle refuges. Pressure can be anticipated to have an even more significant impact on the micellar solubilization of hydrophobic species when the decrease in the relative solubility of hydrophobic species in organic solvents for pressures below a thousand atmospheres is coupled with micelle assembly changes. Understanding the interplay between pressure, surfactant micellization, and solubilization capacity is therefore essential to designing surfactant formulations for oil field applications. In a recent molecular simulation study, we reported on the effects of pressure on the volumes of assembly for ionic and nonionic surfactant micelles.24 Those simulations verified that changes in the sign of the volume of micellization associated with pressure reentrant assembly result from the enhanced compressibility of surfactants in assemblies compared with monomers, which can be largely attributed to the hydrocarbon core of the micelles. Here we extend our previous study to examine the impact of pressure on the uptake of argon, a model nonpolar gas, by anionic and nonionic surfactant micelles. We quantify the micelle/solute interactions via molecularly-detailed potentials-of-mean force and solubility enhancement coefficients, which describe the response of gas solubility to increasing micelle concentration. Solubilization differences between nonionic and anionic surfactants are traced to distinctly different interactions between the headgroup and solute. Our simulations are compared to available experimental results and the solubilization properties of argon in water and octane, a model organic liquid. Based on this comparison an analytical liquid drop model is proposed to describe the variation in argon solubility within micelles with increasing pressure. Received: December 2, 2014 Revised: February 26, 2015 Published: March 2, 2015 3318
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THEORY Here we describe the thermodynamic theory of nonpolar gas hydration and impact of micellization on enhancing solubilization. From statistical thermodynamics, the chemical potential of a monatomic solute (a) in a solvent (s) medium is25 μas = kT ln(Cas Λa 3) + μaex,s
mic exp(−βμaex,wat (Csurf )) ≈
Caig
= exp( −βμaex,s )
(1)
(2)
mic ⎡ ⎤ Csurf mic mic λCsurf = kT ln⎢1 + mic (exp(−β Δμaex,wat (Csurf,sim )) − 1)⎥ ⎢⎣ ⎥⎦ Csurf,sim
(6)
ex,wat ex,wat where Δμex,wat (Cmic (Cmic (Csurf = 0). a surf,sim) = μa surf,sim) − μa Expanding the right-hand side of this expression to first order in mic Cmic surf and dividing by Csurf, we find the solubility enhancement coefficient to be
(3)
where the angular brackets, ⟨...⟩0, indicate averages performed over simulation configurations with no added solutes and φas is the interaction energy between the solute and solvent species for a solute randomly inserted anywhere within the solvent. Experimentally, gas solubility in aqueous surfactant solutions is independent of the surfactant concentration in water, Csurf, below the critical micelle concentration, CMC or CCMC surf , and linearly increases with surfactant concentration above the CMC.28 This observation suggests that the excess chemical potential of a gas molecule in aqueous solution (wat) can be expanded as mic mic mic μaex,wat (Csurf ) ≈ μaex,wat (Csurf = 0) − λCsurf
(5)
In this expression Vsim is the volume of the micellar simulation, and V = Vsim + Vadd is the total volume of the composite micellar and added water system. Assuming all the surfactant molecules, Nsurf, in the simulation are assembled into a single micelle, the micellized surfactant concentration in the simulation box is Cmic surf,sim = Nsurf/Vsim, while the effective micelle concentration upon dilution is Cmic surf = Nsurf/V. The excess chemical potential of the solute in bulk water, μex,wat (Csurf a = 0), can be evaluated from independent simulations of the pure solvent or by adopting an effective value from micelle simulations beyond the solute/micelle correlation length where the excess chemical potential is constant. Combining eqs 4 and 5 and rearranging, we determine the change in the excess chemical potential of the solute upon micellization to be
where β = 1/kT. Gas solubility can be interrogated from simulations using Widom’s test particle insertion formula26,27 μaex,s = −kT ln⟨exp( −βφas)⟩0
Vadd exp(−βμaex,wat (Csurf = 0)) V
+
where kT is the product of Boltzmann’s constant and the temperature, Csa is the solute concentration in the solvent, Λa is the solute’s thermal de Broglie wavelength, and μex,s is the a solute’s excess chemical potential resulting from interactions with the solvent. The excess chemical potential in an ideal gas is zero since there are no interactions. In distribution of the solute between an ideal gas (ig) and liquid solvent, thermodynamic equilibrium is reached when the solute concentrations in both phases satisfy the Ostwald solubility relationship25 Cas
Vsim mic exp(−βμaex,wat (Csurf,sim )) V
λ=
kT mic (exp( −β Δμaex,wat (Csurf,sim )) mic Csurf,sim
− 1) (7)
μex,wat (Cmic a surf,sim)
While depends on the simulated system size, eq 7 is independent of size assuming the original simulated system is large enough so that dissolution in the corners of the system box furthest away from the micelle is representative of the bulk solvent. While the discussion above adopts an average value of the excess chemical potential across the simulation box, in a micellar solution the excess chemical potential can be expressed as a function of distance from the micelle center-of-mass, that is, μex,wat (r) where r is the solute/micelle center-of-mass a separation. This position dependent free energy, or potentialof-mean force, is related to the radial distribution function between the solute and a single micelle in solution as
(4)
where Cmic surf is the total micellized surfactant concentration, which is equal to zero below the CMC and Csurf − CCMC surf above the CMC. The solubility enhancement coefficient, λ, captures the impact of micelle formation on gas solubilization. As defined in eq 4, a positive value of λ has the result of micelles lowering a solute’s solvation free energy and increasing solubility, while a negative value results in micelles increasing the solvation free energy and lowering solubility. The more positive a value of λ the greater the efficiency of a micelle at solubilizing a solute. While in principal λ can be evaluated from excess chemical potentials determined from molecular simulations performed over a range of surfactant concentrations, in practice this is computationally challenging. In particular, since nonpolar gas interactions are relatively short ranged, a significant number of gas particle insertions will be dominated by solute interactions with water alone. As a result, the excess chemical potential for a gas molecule evaluated over an entire simulation box volume is significantly weighted by its value in bulk solvent, which is exacerbated with increasing simulation box size for a single micelle in water. Assuming simulations are conducted with a box volume large enough so that any volume of liquid water added, Vadd, to dilute the micelle will behave like bulk water, the excess chemical potential in the diluted volume can be accurately approximated from the average
−kT ln ga/mic(r ) = μaex,wat (r ) − μaex,wat (∞)
(8)
μex,wat (∞) a
where is the excess chemical potential infinitely far away from the solute in bulk solution. Assuming μex,wat (∞) is a equal to that in pure water, the expression for the solubility enhancement coefficient in eq 7 can be re-expressed as λ=
kT Nsurf
∫ (ga/mic(r) − 1)4πr 2 dr
(9)
A similar expression has been recently been used to model solute and drug molecule partitioning into micelles.29,30 Depending on the circumstances, we will show below that eq 9 can be more advantageous to use than eq 7 when evaluating λ.
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COMPUTATIONAL METHODS
Isothermal−isobaric molecular dynamics simulations31 were performed using GROMACS 4.0.32 The temperature and pressure were maintained using the Nosé−Hoover thermostat33,34 and Parrinello− 3319
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Rahman barostat,35 respectively. Pressures of 1, 500, 1000, 1500, 2000, and 2500 bar were simulated at 300 K. Water was modeled using the TIP4P-Ew potential,36 while surfactants and n-octane were modeled using the Generalized Amber Force Field.37 The surfactants simulated were the anionic sodium decylsulfate (SDeS) and nonionic pentaethylene glycol monodecyl ether (C10E5). Tail lengths were set to 10 carbon units while micelle aggregation numbers were fixed at 40, the experimental aggregation numbers of SDeS,38 to facilitate comparison between surfactants. Simulations of C10E5 and SDeS micelles included 13176 and 4278 water molecules, respectively. Simulations of 500 water molecules and 100 n-octane molecules were performed to characterize the pure solvents. Nonbonded Lennard-Jones interactions were truncated beyond 10 Å, while particle mesh Ewald summation was used to evaluate long-range electrostatic interactions.39 Cross interactions between Lennard-Jones sites were evaluated using Lorentz−Bethelot mixing rules.31 Bonds involving hydrogen were constrained using the LINCS algorithm.40 A 2 fs time step was used to integrate the equations of motion. After equilibration for at least 10 ns, thermodynamic averages were evaluated over 40 ns. Test particle averages evaluated using eq 3 were performed using argon as a model nonpolar gaseous solute. Simulation configurations for interrogation of test particle averages were saved every picosecond. Test particle calculations were performed two different ways (Figure 1) to evaluate
RESULTS AND DISCUSSION Below we discuss the impact of pressure on argon solubilization within micelles. For the interested reader, the structure and thermodynamic response of micelles to increasing pressure is reported in ref 24. Argon/Micelle Interactions. The excess chemical potentials of argon as a function of distance from the center-of-mass of C10E5 and SDeS micelles are reported in Figure 2. The excess
Figure 2. Excess chemical potential of argon as a function of distance from the micelle center-of-mass for C10E5 (a) and SDeS (b) at 300 K. The filled red circles, open red circles, filled blue squares, open blue squares, filled green triangles, and open green triangles indicate results obtained at pressures of 1, 500, 1000, 1500, 2000, and 2500 bar, respectively. The solid black lines indicate fits of eq 10 to the simulation results. The open black triangles on the right-hand y-axis indicate the values of the excess chemical potential of argon in bulk water (Figure 4). Error bars are neglected for clarity.
Figure 1. Schematic illustration of a C10E5 micelle in a simulation cell defining the coordinates used to evaluate argon’s excess chemical potential via particle insertion. In the first set of test particle calculations, argon was randomly inserted within the simulation box (the black bounding box) to evaluate the mean excess chemical potential to evaluate the solubility enhancement coefficient using eq 7. In the second set of calculations, argon was randomly inserted on the surface of a series of concentric spheres of increasing radii r (blue lines) centered on the micelle’s center-of-mass (red point at micelle center) to determine the excess chemical potential as a function of distance from the micelle center. The resulting distant dependent free energies were used to evaluate solubility enhancement coefficients using eq 9
chemical potential is generally lower in the micelle interiors (r < ∼10 Å), indicating that argon prefers solubilization within micelles over bulk solution. These nonpolar gas/micelle distance dependent interactions are comparable to those reported in previous simulation studies.29,42 As the hydrostatic pressure, P, increases the distance dependent excess chemical potential systematically shifts upward over all separations. The changes in μex,wat (r, P) with increasing pressure are well a described by the expression
the impact of the micelle on gas solubility. In the first set of calculations, test particle averages were evaluated by randomly inserting 5000 test argons within the simulation box to determine the mean excess free energy of argon within the simulation box. In the second set of calculations, we focused on the radial dependence of the excess chemical potential by locating the micelle center-of-mass and then randomly inserting 2500 test argons on successive spherical radial shells 0.1 Å thick out from the micelle center to half the simulation box length. Argon was modeled as a single Lennard-Jones site with a selfinteraction diameter and well depth of 3.415 Å and 1.0393 kJ/mol, respectively.41
ex,wat ex,wat μaex,wat (r , P) = μa,0 (r ) + υa,0 (r )(P − 1 bar)
(10)
as observed by the quality of the fits shown in Figure 2. In this ex,wat expression the functions μex,wat a,0 (r) and υa,0 (r) correspond to the distance dependent excess chemical potential at 1 bar pressure and the average distance dependent excess volume of argon/micelle association. The micelle and bulk solution are 3320
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Langmuir assumed to be incompressible in eq 10 since the compressibility is small and difficult to obtain reliably given the statistical noise in the distance dependent chemical potentials. The fitted results for μex,wat a,0 (r) for argon interacting with C10E5 and SDeS micelles corresponds to the bottom most curves in Figure 2a,b at ambient pressure. The fitted results for υex,wat a,0 (r) are reported in Figure 3. The solute volume within
Figure 4. Excess argon solvation chemical potentials as a function of pressure at 300 K. The solid red squares and solid blue circles indicate chemical potentials in water and octane, respectively. The solid black lines indicate linear fits of the chemical potential, assuming the ex,s ex,s solvation volume is incompressible, that is, μex,s a (P) = μa,0 + υa,0 (P − 1 bar) in analogy to eq 10. The open red and blue triangles on the lefthand y-axis indicate experimental values for the excess chemical potential in water (ref 28) and octane (ref 50). Error bars are smaller than the figure symbols. Fits of the chemical potential to the expression derived from the Tait equation (eq 12b) are not included for clarity. While eq 12b does a comparable job to the linear fit at describing argon’s chemical potential in water, eq 12b does a better job at describing the results in octane capturing the slight bow in the chemical potential that results from the organic solvent’s greater compressibility.
Figure 3. Argon/micelle association volume as a function of distance from the micelle center-of-mass. The filled blue circles and filled red squares indicate results for C10E5 and SDeS, respectively. The horizontal red and blue dashed curves denote the excess partial molar volumes of argon in water and octane, respectively, assuming the volumes are incompressible (Figure 5). Error bars indicate one standard deviation. The excess partial molar volume describes the nonideal contribution to the volume determined from the pressure derivative of the excess chemical potential. In the bulk fluid at infinite dilution the excess volume is related to partial molar volume through = υsa − kTκs, where κs is the isothermal the relationship υex,s a compressibility of the pure solvent.49
potential of argon determined in pure water is in excellent agreement with the results obtained for argon in bulk water outside the C10E5 and SDeS micelles, that is, r ≫ Rg in Figure 2. We note that the excess chemical potential at r = 25 Å outside the SDeS micelle (Figure 2b) is slightly greater than the value in pure water as a result of interactions with the charged micellar headgroup units. This distance dependent excess chemical potential appears to decay exponentially to the value in bulk water with increasing distance, which we attribute to the exponential decay in the screening sodium countercation cloud enshrouding the SDeS micelle. Indeed an exponential decay of the counterion density is expected from simple primitive model descriptions of charged micelles.43 We make use of this exponential asymptotic decay to the bulk argon excess chemical potential in water in the following section to evaluate argon’s solubility enhancement coefficient for SDeS micelles from the potential-of-mean force using eq 9. Analogous exponential decays have been successfully used to extend simulation correlation function integrals for osmotic virial coefficients beyond the simulation box bounds.44 If we assume that the excess volume of solvation is incompressible, as previously assumed in eq 10, the pressure dependence of the excess chemical potential of argon is well described by a linear fit (Figure 4). The “incompressible” excess volume of argon in octane determined from this linear fit is greater than that in water (Figure 5), consistent with pressure pushing argon from octane into water with increasing pressure. While it is not surprising that the excess volume of argon far away from the center of either micelle is equal to that in bulk water, we more interestingly observe that the excess volume of argon in octane agrees with that obtained inside either micelle (Figure 3), supporting the idea that the micelle interior is similar to liquid hydrocarbon.
either surfactant micelle is larger inside the micelle than in bulk water, indicating that argon is expelled into bulk solution with increasing pressure over the pressure range examined. The excess volumes of argon inside both micelles are the same within the simulation errors and drop to the same values in bulk solution as expected. The most significant difference between υex,wat a,0 (r) for the two surfactants is that the volume drops from the interior to exterior value at ∼17 Å for SDeS versus ∼20 Å for C10E5. Assuming that the surfactant chains for both micelles are uniformly packed into a sphere, the radius of a micelle can be obtained from the relationship between the radius of a solid sphere and its radius-of-gyration R sphere = R g / 3/5
(11)
The radii-of-gyration for SDeS and C10E5 micelles are 13.4 and 15.6 Å, respectively, corresponding to micelle radii of 17.3 and 20.1 Å, in excellent agreement with the sizes inferred from the distance dependent excess volumes. For comparison with liquid/liquid transfer thermodynamics, the excess chemical potentials of argon in water and octane as a function of pressure are reported in Figure 4. The excess chemical potential in water is greater than that in octane, corresponding to a greater solubility in oil, while the solvation free energy in both phases is an increasing function of pressure. The simulated chemical potential of argon in water and octane at 1 bar pressure is in good agreement with experimental results at ambient pressures (Figure 4). In addition, the chemical 3321
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volume crossing point we expect the solubility of argon in octane to increase relative to that in water. This in turn could impact the pressure dependence of argon’s solubility enhancement in micelles outside the simulated pressure range as discussed below. Argon Solubility Enhancement. The net impact of the attractive interactions between argon and micelles is to increase argon’s solubility as demonstrated by the positive solubility enhancement coefficients reported in Figure 6. The solubility
Figure 5. Excess argon solvation volumes as a function of pressure at 300 K. The horizontal dashed red and blue lines indicate results in water and octane, respectively, obtained fitting the chemical potentials in Figure 3 to a line assuming the excess volume is incompressible. The solid red and blue curves indiate results for argon in water and octane, respectively, obtained from the Tait equation (eq 12) fitted to the chemical potentials in bulk solvent (Figure 4). The open red and blue triangles on the left-hand y-axis indicate the average experimental excess volumes for argon in water (refs 51 and 52) and octane (refs 53 and 54). Error bars indicate one standard deviation. At infinite dilution, the excess solvation volume is related to partial molar volume s s s through the relationship υex,s a = υa − kTκ , where κ is the isothermal compressibility of the pure solvent.49 Extrapolating the Tait equation fits for water and octane, the excess volumes of these two solvents are expected to cross one another at a pressure of ∼5000 bar.
Figure 6. Argon solubility enhancement coefficients as a function of pressure at 300 K. The open blue circles and open red squares report results for C10E5 and SDeS, respectively, evaluated from direct particle insertion (eq 7). The filled blue circles and filled red squares report results for C10E5 and SDeS, respectively, evaluated by integration of the pair correlation function (eq 9) using the chemical potentials fit to eq 10. The solid blue and red lines indicate fits of eq 14 for C10E5 (Rmic = 11.2 Å) and SDeS (Rmic = 9.0 Å) micelles. Error bars indicate one standard deviation. The long thin error bars are for coefficients determined direct particle insertion (eq 7), while the short thick error bars are for coefficients determined by integration of the pair correlation function (eq 9). For comparison, the experimental solubility enhancement coefficient for argon in sodium dodecyl sulfate micelles is reported as the open black triangle on the left-hand yaxis.28,55
If we assume incompressible solvation volumes, the excess volumes of argon extracted from simulation under predict reported experimental values, especially in octane (Figure 5). While statistical noise in the distance dependent chemical potentials for the micellar solutions is too large to extract reliable compressibility effects (Figures 2 and 3), this is not a problem for argon in the pure solvents. A more accurate description of the pressure dependence of argon solvation is obtained by assuming that the partial molar volume follows the Tait equation45 υaex,s(P) = a + b ln(1 + cP)
(12a)
enhancement coefficients obtained by test particle insertion analysis (eq 7) of simulation configurations show that solubility is more significantly enhanced by the nonionic micelles compared with the anionic micelles and that solubility enhancement decreases with increasing pressure. The simulation results for the argon solubility enhancement coefficient for SDeS are in excellent agreement with that obtained experimentally for argon dissolved in sodium dodecyl sulfate micelles at ambient pressure (Figure 6), giving confidence that our simulation predictions of the impact of micellization on gas solubility compare with experimental expectations. While the reduction in λ with increasing pressure is consistent with the larger volumes of argon inside micelles (Figure 3), the error bars obtained by direct interrogation of simulation configurations using eq 7 are significant and make this trend tenuous at best. These errors largely arise from the fact that λ is determined from small solubility differences between simulations with and without a micelle. This error is especially large for C10E5, which was simulated in a box with approximately three times as many waters as for the SDeS simulations. Those additional waters effectively reduce the difference between the excess chemical potentials in the micelle simulation box and pure water, giving rise to larger statistical uncertainties.
originally applied to pure liquids. Integrating this expression with respect to pressure, we obtain the solute’s excess chemical potential μaex,s (P) = d + (a − b)P + (b/c + bP)ln(1 + cP) (12b)
The constants a, b, c, and d are obtained by fitting eq 12b to the excess chemical potentials reported in Figure 4 (fit not shown). The excess volumes of argon in water and octane at 1 bar obtained from the Tait equation fit to the simulation free energies are in significantly improved agreement with the experimental volumes (Figure 5). The volume of argon in water decreases only slightly with increasing pressure up to 2500 bar, consistent with the previously reported low compressibility of methane in water.46 The pressure dependence of argon’s volume in octane is much more significant than in water, consistent with the fact that pure octane is three times more compressible than water.47,48 While the volume of argon in octane does not fall below that in water over the pressure range simulated, extrapolation of eq 12a to higher pressures suggests these volumes cross at ∼5000 bar. While this is outside the range for a reliable extrapolation, for pressures beyond the 3322
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Langmuir Alternatively, the errors in λ can be attenuated by focusing on the micelle and integrating the argon/micelle radial distribution function out from the assembly’s center-of-mass using eq 9 (Figure 6). Excellent agreement is obtained between λ values obtained by eqs 7 and 9. More importantly the reduced errors obtained more clearly show that argon’s solubility enhancement decreases with increasing pressure over the pressure range studied. The radii-of-gyration of the hydrophobic domains of the C10E5 and SDeS micelles are 11.7 and 11.5 Å, consistent with the fact that both surfactants have the same alkyl tail lengths and were simulated having the same aggregation number. Since the hydrophobic domains of both micelles effectively have the same volume, a question that arises is why is the solubility of argon greater in the nonionic versus anionic micelles? This can be addressed by examining λ as a function of the upper bound radius used to integrate eq 9. As seen in Figure 7, λ for both
argon affinities, surfactant headgroup interactions strongly influence the ultimate solubility. The significant role of the micelle alkyl center on argon solubilization (Figure 7) and the correspondence between the pressure response of argon/micelle interactions and the transfer of argon between water and octane (Figure 3) suggests a simplified model for describing the solubility enhancement coefficient in which the micelle is treated as a drop of liquid octane in water. Rather than utilizing the detailed distance dependent excess chemical potential (Figure 2) the partitioning of argon between water and the micelle could be treated as a liquid−liquid phase transfer, where argon’s excess chemical potential inside the micelle adopts its bulk octane value and outside it adopts its bulk water value μaex,mic (r )
ex,oct ⎧ r < R mic ⎪ μa ⎨ = ⎪ μex,wat r > R ⎩ a mic
(13)
Here Rmic is the effective micelle radius, which is treated as a fitting parameter encapsulating the size of the hydrophobic domain and the impact of the headgroup interactions on solvation. Substituting eq 13 into eqs 8 and 9, we find the solubility enhancement coefficient to be λ = kT
4πR mic 3 (exp( −β Δμaex,w → o ) − 1) 3Nsurf
Δμex,w→o a
(14)
where = − is the water to octane transfer free energy. The liquid drop model provides an accurate description of the simulation results reported in Figure 6 within the error bars using fitted micelle radii of 11.2 Å for C10E5 and 9.0 Å for SDeS. The smaller radius of the anionic surfactant compared with the nonionic surfactant reflects salting out by the charged headgroup. Both fitted radii are less than the radius of the hydrophobic cores evaluated above (15 Å), which we believe reflects the simplification that the distance dependent excess chemical potential is not a simple step function as presumed in eq 13 but rises to values greater than that in octane even within the micelle interiors (Figure 2) as a result of interactions at the boundary with the surrounding water. While our simulations only indicate that the solubility of argon within micelles decreases with increasing pressure, the liquid drop model suggests that this trend could reverse for pressures beyond the point at which the argon volumes in octane and water cross (see Figure 5 and discussion above) and the net transfer volume changes sign from positive to negative. This minimum in λ lies outside the simulated range of pressures and is anticipated to be quite shallow, making precise determination of this point difficult.
Figure 7. Argon solubility enhancement coefficients determined from eq 9 as a function of the upper integration bound r at a temperature of 300 K and pressure of 1 bar. The blue and red solid curves indicate results for C10E5 and SDeS, respectively, while the blue and red open triangles on the right-hand y-axis indicate the ultimate values for the integral. To evaluate the integral for SDeS, the argon chemical potential beyond 25 Å was assumed to exponentially decay to the bulk value in water, capturing the impact of the cloud of sodium counterions.
micelles up to ∼15 Å are nearly identical, although minor differences can be observed reflecting differences in the packing of the alkyl chains in the micelle interior for the anionic and nonionic surfactant. Beyond this radius, however, argon’s solubility enhancement continues to increase to a plateau for C10E5, while for SDeS the integral exhibits a maximum at ∼15 Å before falling to a lower plateau at greater distances. For both micelles, the integral converges to its ultimate value just beyond ∼25 Å. Assuming that the alkyl chains are uniformly packed into a sphere, the spherical hydrophobic domain radius obtained from eq 11 is 15.0 Å for an average radius-of-gyration of the hydrophobic domains of 11.6 Å. This radial size of the hydrophobic domain for both micelles corresponds well with the point at which the λ integrals for both micelles match one another. The continued increase in λ beyond 15 Å for C10E5 indicates that the ethoxy headgroup is a good solvent for argon, which should not be unexpected given the weakly hydrophobic character of the ethoxy carbons. The ionic head groups of the SDeS micelle appear to partially salt out argon, resulting in weak reductions in λ beyond 15 Å. We conclude then that while the hydrophobic domains of both micelles exhibit comparable
μex,oct a
μex,wat a
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CONCLUSIONS Using molecular simulations, we have characterized the impact of micellization on the solubilization of a model nonpolar gas (argon) in aqueous solution as a function of the applied hydrostatic pressure. Based on the potential-of-mean force between the micelle and argon, directly related to the distance dependent excess chemical potentials reported in Figure 2, nonpolar gases are attracted to the micelle interiors. The pressure dependence of the potential-of-mean force indicates that argon is expelled from the micelle interior into bulk solution with increasing pressure, in line with the lower solvation volumes of argon in water compared with octane. 3323
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Langmuir
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Potentials-of-mean force determined from simulation were used to directly evaluate argon’s solubility enhancement coefficient, which describes the lowering of the solute’s excess chemical potential in water as a result of micellization. Argon solubility enhancement coefficients obtained for SDeS micelles match those reported for chemically similar sodium dodecyl sulfate micelles, connecting the molecularly detailed potentials-ofmean force evaluated from simulation and experimental solubility changes. More interestingly, nonionic C10E5 micelles exhibit an enhancement coefficient nearly twice that for anionic SDeS micelles. The greater impact of C10E5 on solubilization can be traced to favorable interactions between argon and the nonionic surfactant head groups, while the anionic surfactant head groups reduce argon solubility due to salting out effects. Assuming that micelle interiors resemble bulk octane, the pressure dependence of the solubility enhancement coefficients of argon in both C10E5 and SDeS micelles is well described using a liquid drop model. While argon is similarly attracted to the hydrophobic cores of both micelles, the larger fitted droplet radius of the nonionic micelles compared with the anionic micelles reflects the interaction differences of argon with the headgroup units of each surfactant.
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AUTHOR INFORMATION
Corresponding Author
*Corresponding author. E-mail:
[email protected]. Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS We gratefully acknowledge financial support from the NSF under a CAREER award (CTS-0746955) and the Gulf of Mexico Research Initiative. We also acknowledge computational support from the Louisiana Optical Network Initiative.
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REFERENCES
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