Application of Computer Simulation Free-Energy Methods to Compute

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J. Phys. Chem. B 2008, 112, 1634-1640

Application of Computer Simulation Free-Energy Methods to Compute the Free Energy of Micellization as a Function of Micelle Composition. 1. Theory Brian C. Stephenson, Kate A. Stafford, Kenneth J. Beers, and Daniel Blankschtein* Department of Chemical Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 ReceiVed: April 9, 2007; In Final Form: September 12, 2007

The widespread use of surfactant mixtures and surfactant/solubilizate mixtures in practical applications motivates the development of predictive theoretical approaches to improve fundamental understanding of the behavior of these complex self-assembling systems and to facilitate the design and optimization of new surfactant and surfactant/solubilizate mixtures. This paper is the first of two articles introducing a new computer simulationfree-energy/molecular thermodynamic (CS-FE/MT) model. The two articles explore the application of computer simulation free-energy methods to quantify the thermodynamics associated with mixed surfactant/ cosurfactant and surfactant/solubilizate micelle formation in aqueous solution. In this paper (article 1 of the series), a theoretical approach is introduced to use computer simulation free-energy methods to compute the free-energy change associated with changing micelle composition (referred to as ∆∆Gi). In this approach, experimental critical micelle concentration (CMC) data, or a molecular thermodynamic model of micelle formation, is first used to evaluate the free energy associated with single (pure) surfactant micelle formation, gform,single, in which the single surfactant micelle contains only surfactant A molecules. An iterative approach is proposed to combine the estimated value of gform,single with free-energy estimates of ∆∆Gi based on computer simulation to determine the optimal free energy of mixed micelle formation, the optimal micelle aggregation number and composition, and the optimal bulk solution composition. After introducing the CS-FE/MT modeling framework, a variety of free-energy methods are briefly reviewed, and the selection of the thermodynamic integration free-energy method is justified and selected to implement the CS-FE/MT model. An alchemical free-energy pathway is proposed to allow evaluation of the free-energy change associated with exchanging a surfactant A molecule with a surfactant/solubilizate B molecule through thermodynamic integration. In article 2 of this series, the implementation of the CS-FE/MT model to make ∆∆Gi freeenergy predictions for several surfactant/solubilizate systems is discussed, and the predictions of the CSFE/MT model are compared with the ∆∆Gi predictions of a molecular thermodynamic model fitted to relevant experimental data.

1. Introduction Surfactants are used in many pharmaceutical, industrial, and environmental applications because of their unique solution properties. When dissolved in water, surfactant molecules selfassemble into aggregates (micelles), with their hydrophobic tails shielded from water in the aggregate interior and their hydrophilic heads exposed to water at the aggregate surface. This self-assembly is driven by hydrophobic, van der Waals, hydrogen-bonding, and (in the case of charged surfactants and solubilizates) screened electrostatic interactions and occurs at a surfactant concentration above what is referred to as the critical micelle concentration (CMC).1,2 Surfactant mixtures are useful in many applications because they often perform better than a single surfactant and because purification of a single compound may be costly or difficult.3 Solubilizates are entirely or partially hydrophobic substances with poor water solubility that do not self-assemble to form micelles in aqueous solution but whose solubility can be increased by the addition of surfactants through a process that is referred to as micellar solubilization.4 * Corresponding author address: Department of Chemical Engineering, Room 66-444, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139. Tel: (617) 253-4594. Fax: (617) 2521651. E-mail: [email protected].

Because mixed micellization and micellar solubilization in aqueous solution are such broadly applicable phenomena, gaining a fundamental understanding of the factors that affect mixed micellization and micellar solubilization is of great academic as well as practical interest. Frequently, a highly specific set of micellar solution characteristics is required for a given application. These characteristics typically include the CMC and the shape and size of the micellar aggregates that form in solution. A number of theoretical approaches have been developed to enable the prediction of the free energy of micelle formation (gform) or the free-energy change associated with transferring the surfactant monomers, the solubilizates, and any bound counterions (in the case of ionic surfactants) from their standard states in aqueous solution to a micellar aggregate in its standard state, based on the chemical structures of the solution components and the solution conditions (such as the total surfactant concentration and composition, the temperature, the pressure, and the ionic strength).3,5-13 Determination of gform enables prediction of important equilibrium properties of selfassembled surfactant/solubilizate systems in aqueous solution, including the CMC, the micelle size distribution, the micelle shape and average size, the extent of micellar solubilization, and the locus (or location) of solubilization within a micelle.

10.1021/jp0727603 CCC: $40.75 © 2008 American Chemical Society Published on Web 01/17/2008

Free Energy of Micellization. 1. Theory The most predictive and accurate theoretical models of surfactant micellization and micellar solubilization implement what is known as the molecular thermodynamic (MT) modeling approach.3,9,14 In the MT modeling approach, the free-energy change associated with the formation of the surfactant aggregate is expressed as the sum of several free-energy contributions, all of which can be computed molecularly given the chemical structures of the various micellar components and the solution conditions. The MT model introduced by Nagarajan and Ruckenstein allows prediction of the CMC and of the shape and size of micellar aggregates composed of nonionic, zwitterionic, and ionic surfactants.9 In recent years, our group has also contributed to the development of MT models to predict surfactant behavior in aqueous solution.3,14-22 Molecular thermodynamic models are most successful at modeling the selfassembly of relatively simple surfactant systems, including surfactants that have linear hydrocarbon (or fluorocarbon) tails and a single, rigid head; for example, sodium dodecyl sulfate and cetyltrimethylammonium bromide (CTAB). Some progress has also been made in modeling surfactants with long polymeric heads, including those of the alkyl poly(ethylene oxide) variety.9 For such systems, MT predictions for the CMC are typically within a factor of 5, corresponding to a typical error in the predicted value of gform of less than 1.6 kBT. Recently, our group has shown that by combining molecular thermodynamic models with computer simulations, it is possible to model the micellization behavior of increasingly complex surfactants and solubilizates.5-8,23 We have explored the use of computer simulations to identify which atoms are part of the surfactant head and which atoms are part of the surfactant tail,8,23 as well as the use of computer simulations to determine hydration information in order to accurately quantify the hydrophobic driving force for micelle self-assembly.5-7 Both of these hybrid computer simulation/molecular thermodynamic modeling approaches have made use of computer simulations to determine input parameters for molecular thermodynamic modeling. To our knowledge, only one hybrid computer simulationfree-energy/molecular thermodynamic approach to model micellization has been reported in the literature. Specifically, Mohanty et al. have developed a hybrid modeling approach in which Monte Carlo (MC) simulations are used to evaluate the free-energy contribution associated with forming the head-shell region of the micelle, and a molecular thermodynamic (MT) theory developed by our research group14 is used to calculate the free-energy contribution associated with forming the micelle core. These authors refer to their modeling approach as the Complementary Model. The authors used their model to predict the CMC and the sphere-to-wormlike micelle shape transition for the cationic surfactant CTAB and the partially hydrophobic counterion sodium salicylate (Sal-).24 The extent of penetration of Sal- into the micelle core was determined by performing isobaric (constant NPT) MC simulations of an entire spherical micelle, or of a section of a cylindrical micelle, having various CTAB and Sal- compositions. The core region of the micelle was defined as the spherical, or the cylindrical, region around the aggregate center that had zero concentration of salicylate counterions. The head-shell region of the micelle was defined as the remaining portion of the micelle, which contained a high concentration of CTA + heads, Sal- counterions, and two CH2 groups in the cetyl alkyl chain. To reduce computational expense, Mohanty et al. used a mean-field approach to model water and the counterions. The free-energy contribution associated with the head-shell region was calculated with respect to

J. Phys. Chem. B, Vol. 112, No. 6, 2008 1635 a pure CTA+ head shell through the use of semi-grand-canonical (constant Ntot, P, T, and µSal-) MC simulation. The free energy of the head-shell region was added to that of the core region to obtain gmic, or the free-energy change associated with transferring the surfactant monomers and the Sal- counterions from bulk aqueous solution to the micelle head-shell region. The computed value of gmic was then combined with a thermodynamic description of self-assembly to predict the CMC, the micelle shape, the micelle aggregation number, and the degree of counterion binding. In this article, we propose a hybrid computer simulation/ molecular thermodynamic modeling approach in which computer simulation free-energy methods are used to determine the free-energy change associated with mixed (multicomponent) micelle formation. This hybrid modeling approach will be referred to as the computer simulation-free-energy/molecular thermodynamic (CS-FE/MT) model, reflecting the fact that computer simulation free-energy methods are used to estimate the free energy of mixed micelle formation. By relying on computer simulations to evaluate free energies, the CS-FE/ MT model avoids many of the approximations and assumptions inherent in molecular thermodynamic modeling; for example, the decomposition of gform into a number of free-energy contributions that are assumed to be functionally independent of each other (although they are coupled through minimization of gform with respect to the geometry and composition of the micellar aggregate).3,14 In contrast to the approach described by Mohanty et al., the computer simulations performed in implementing the CS-FE/MT model will be conducted at an atomistic level of detail and will be used to model an entire micelle rather than only a portion (the head-shell region) of the micelle. In article 2 of this series, ∆∆Gi free-energy predictions made using the CS-FE/MT model are discussed for several surfactant/solubilizate systems. The remainder of this article is organized as follows: In Section 2, an overview of the CS-FE/MT modeling approach is presented, including a description of the inputs and outputs of the model. In Section 3, a thermodynamic framework is developed to enable predictions of micellar solution properties using the free-energy values obtained from the CS-FE/MT model. Note that although presented in the context of modeling two-component micelles (binary surfactant micellization and micellar solubilization), the theoretical framework presented in Section 3 may be generalized in a straightforward manner to describe the micellization and micellar solubilization behavior of n-component systems. In Section 4, an introduction to computer simulation free-energy methods is presented. In Section 5, a description of the free-energy calculations made in the context of the CS-FE/MT model is presented. Conclusions are presented in Section 6. 2. Overview of the CS-FE/MT Model The proposed CS-FE/MT model is a hybrid approach in which computer simulations are used to find the free-energy change associated with changing micelle composition from a single (pure) surfactant micelle reference state. A flowchart outlining the computational approach applied in the CS-FE/ MT model is shown in Figure 1. As shown in Figure 1, the CMC of a single surfactant system is taken as the starting point for the CS-FE/MT model. This CMC may be determined experimentally or predicted using the MT model for single surfactant micellization.14 In the next step shown in Figure 1, the CMC is converted to the free energy associated with single surfactant micelle formation using the

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Figure 1. Computational strategy used in the CS-FE/MT model.

relationship gform,single ) kBT ln( CMC ).5 Next, computer simulations are used to find the free-energy change associated with changing the micelle composition. When added to gform,single, the computer simulation result enables determination of the free energy associated with mixed micelle formation (gform,mixed). The computed value of gform,mixed is then used in a thermodynamic description of micelle self-assembly to make predictions of all relevant micellization properties, including the CMC, the micelle shape, the micelle aggregation number, the locus and the extent of solubilization, and the degree of counterion binding. Note that direct computer simulation determination of gform using free-energy methods is very computationally expensive (requiring months of simulation time on today’s computers) 25 In contrast, using MT modeling, it is possible to predict gform with very little computational expense (requiring only seconds using today’s computers).35 However, the traditional MT model is capable of making quantitatively or semiquantitatively accurate predictions of gform only for relatively simple surfactants and solubilizates. Ideally, the CS-FE/MT modeling approach, by combining an experimental or an MT value of gform,single with computer simulation free-energy methods to find the free-energy change associated with changing micelle composition, would enable evaluation of gform,mixed in a relatively computationally efficient way. Furthermore, it was hoped that successful implementation of the CS-FE/MT model would (i) improve our fundamental understanding of multicomponent surfactant micellization and micellar solubilization in aqueous solution by providing additional insight into the thermodynamics associated with changes in micelle composition and (ii) advance the current state-of-the-art in computer simulation free-energy methods. To the best of our knowledge, the CS-FE/MT modeling approach presented here represents the first attempt to evaluate the free energy associated with mixed micelle formation using atomisticlevel computer simulations. 3. Formulation of the Thermodynamic Framework Used in the CS-FE/MT Model A thermodynamic framework to describe single and mixed surfactant micellization in aqueous solution in the context of traditional MT modeling has been described in detail elsewhere.3,14 Nevertheless, the essential elements of this framework are briefly reviewed here to provide the proper context to understand the CS-FE/MT model. The thermodynamic framework described in this section is formulated in the context of two-component surfactant micellization and two-component micellar solubilization. To simplify notation, the thermodynamic framework is formulated only for nonionic surfactants and nonionic solubilizates. However, this theoretical framework can be reformulated in a straightforward manner to describe ncomponent surfactant micellization and micellar solubilization, as well as the micellization and micellar solubilization of ionic

surfactants and solubilizates. As discussed in Section 2, in the CS-FE/MT modeling approach, the free-energy change associated with changing the composition of a single (pure) surfactant micelle is evaluated using computer simulations. Throughout the remainder of this section, the surfactant present in the single surfactant micelle is denoted as component A, and the surfactant or solubilizate that is added through computer simulation is denoted as component B. At thermodynamic equilibrium, the chemical potential of a micelle of aggregation number n and composition R (µnR) can be related to the chemical potentials of the monomers of component A (µA) and component B (µB) in bulk aqueous solution as follows,

nAµA + nBµB ) µnR

(1)

where nA is the number of component A molecules in the micelle, nB is the number of component B molecules in the micelle, n is equal to nA + nB, and R ) nA/n. Each chemical potential appearing in eq 1 can be expressed as follows,

µi ) µoi + kBT ln(ai)

(2)

where µoi is the standard-state chemical potential of component i, where the standard state is infinite dilution in the bulk aqueous solution, kB is the Boltzmann constant, T is the absolute temperature, and ai is the activity of component i in the aqueous solution. Because the CS-FE/MT model will only be used to model micellization and micellar solubilization at low monomer and micelle concentrations, the term ln(ai) in eq 2 may be replaced by ln(Xi), where Xi is the mole fraction of component i in the aqueous solution. Substituting the expression for µi given in eq 2 for each component i in eq 1 yields the following expression,

nA[µoA + kBT ln(XA)] + nB[µoB + kBT ln(XB)] ) n[µonR + kBT ln(XnR)] (3) which can be rearranged as follows,

[

XnR ) X n1AA X n1BB exp -

]

Gform kBT

(4)

where Gform ) µonR - nµoA - nµoB and represents the total freeenergy change associated with transferring nA component A and nB component B monomers from their standard states in bulk aqueous solution to form a micelle of aggregation number n ) nA + nB and composition R ) nA/n. Equation 4 can be reformulated in terms of R as follows,

Free Energy of Micellization. 1. Theory

XnR )

( )( ) Rn

X1X1A X1 Rn

Rn

X1X1B X1

) X1 R1 X1

(1-R)n

(1-R)n

[

exp -

(1 - R1)

(1-R)n

J. Phys. Chem. B, Vol. 112, No. 6, 2008 1637

]

Gform kBT

[

(5)

]

Gform exp kBT

(6)

where Rn ) nA, (1 - R)n ) nB, R1 ) X1A/X1, and (1 - R1) ) X1B/X1, where R1 is the monomer composition. Equation 6 can be rearranged to move the entropic free-energy contribution associated with the composition of the monomeric phase (R1Rn(1 - R1)(1-R)n) into the Boltzmann factor as follows,

XnR )

[

X1n exp -

]

Gform + Rn ln R1 + (1 - R)n ln(1 - R1) (7) kBT

[ ]

XnR ) X1n exp -

Gf kBT

(8)

where Gf ) Gform - kBT[RnlnR1 + (1 - R)n ln(1 - R1)] and is referred to as the modified free energy of mixed micellization.22 In this article, intensive free-energy contributions are represented with g, and extensive free-energy contributions are represented with G. The free-energies Gf and Gform in eqs 6 and 7, respectively, are given as extensive quantities because it is most intuitive to develop the CS-FE/MT model on an extensive basis. In the traditional MT model, Gform is modeled as a series of reversible steps, each of which is computed molecularly on the basis of the chemical structures of components A and B. In the traditional MT model, Gf is computed using the following expression,

Gf ) n[gtr + gint + gpack + gst + gelec + gent] - kBTn [R ln R1 + (1 - R) ln(1 - R1)] (9) where each of the six free-energy contributions appearing in the square brackets in eq 8, whose sum is equal to gform, is multiplied by n because they are expressed on a per surfactant molecule basis. In the CS-FE/MT model, in contrast, Gf is formulated in terms of variables that can be determined through computer simulation. In the CS-FE/MT modeling approach, computer simulations are used to find the free-energy change associated with exchanging molecules of component A with molecules of component B. This free-energy change will be referred to hereafter as ∆∆Gi, where ∆∆Gi is the free-energy change associated with the ith exchange of component B with component A, and i is an index which ranges from 1 to nB ) n(1 R) (the number of component B molecules in the micelle). Note that the free-energy change is referred to as ∆∆Gi rather than as ∆Gi because, as will be discussed in Section 5, in implementing the CS-FE/MT model, exchanging molecules of component A with molecules of component B in the micelle also requires a corresponding transformation in bulk aqueous solution, and it is the difference between these two transformation free energies that is used in the CS-FE/MT model. Accordingly, in the CS-FE/MT model, Gf is computed using the following expression, n(1-R)

Gf ) Gform,single(R ) 0) + or

∑ i)1

∆∆Gi + Gent1 + Gent

(10)

Gf ) Gform,mixed + Gent1 + Gent

(11)

where Gform,single(R ) 0) is equal to n‚gform,single (see Figure 1) and is determined from the experimental CMC value or is computed using the traditional MT model for single surfactant micellization, Gform,mixed is equal to n‚gform,mixed (see Figure 1), Gent1 ) -kBTn[R ln R1 + (1 - R) ln(1 - R1)] and represents the entropic contribution to Gf associated with the monomer composition, and Gent ) -kBTn[R ln R + (1 - R) ln(1 - R)] and represents the entropic contribution to Gf associated with the micelle composition. Note that the free-energy contribution Gent must be included because all molecules modeled during a MD computer simulation are distinguishable; hence, the entropic effect associated with changing n indistinguishable (identical) molecules of component A into nA molecules of component A and nB molecules of component B, where molecules of type A are distinguishable from molecules of type B, gives rise to an entropy of mixing free-energy contribution that is not accounted for in the computer simulation result for ∆∆Gi. Therefore, by adding Gent to Gform,mixed, this entropy of mixing free-energy contribution is accounted for. A detailed description of the manner in which ∆∆Gi is obtained using computer simulations is presented in Section 4. To use the CS-FE/MT model to make predictions of surfactant solution properties, an iterative procedure is required to determine the optimum (minimum) value of Gf, which we denote hereafter as G/f . From G/f , the surfactant solution properties of interest can be obtained. For example, the size and composition of the micelles that will be observed experimentally are determined as the size and composition of the micelles that yield the optimal value of Gf, or G/f . In addition, the CMC can be estimated using the expression exp(G/f /nkBT). The sequence of steps that are required to implement the CSFE/MT model include: (1) Guess a value of R1. (2) At the specified value of R1, solve for the value of R that yields the minimum value of Gf (or equivalently, solve for the value of R that yields the maximum value of XnR; see eq 7). This value of R represents the most thermodynamically favorable micelle composition at the specified value of R1. (3) After solving for the optimal value of R at the specified value of R1, update R1 by evaluating eq 7 with the current estimate of Gf and by solving a mass balance for the solution.15,26 The implementation of this mass balance has been discussed by other researchers in the context of the traditional MT modeling approach.3,14,15,26 (4) Repeat steps 2 and 3 until the value of R1 becomes constant. (5) Compute G/f using eq 11. For reasons that will be discussed in detail in Section 4, steps 1-5 outlined above are sufficient to determine the optimal aggregation number for a cylindrical or a discoidal micelle because the value of lc (the core-minor radius of a cylindrical micelle or the half width of a bilayer) is free to come to an equilibrium value at any micelle composition given the NγT (constant N, interfacial tension, and T) periodic boundary conditions that are applied during simulation. On the other hand, to determine the optimal value of lc for spherical micelles, the iterative procedure outlined above must be modified slightly to permit changes in the aggregation number (because lc for spherical micelles can change only if the aggregation number changes). In step 1, a value must be guessed for both R1 and n. In step 2, the computer simulation procedure used to compute ∆∆Gi must be modified to evaluate the free-energy change

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associated with exchanging one component A molecule with one component B molecule (a process that keeps n constant), one component A molecule with two component B molecules (a process that increases that value of n), and two component A molecules with one component B molecule (a process that decreases the value of n). By evaluating the free-energy change associated with these three exchanges, eq 10 can be used to solve for Gf for the optimal value of R and n associated with a specified value of R1. In step 3, R1 would be updated by solving a mass balance for the solution.14,15 In step 4, steps 2 and 3 would be repeated until the value of R1 becomes constant, and in step 5, G/f would be computed using eq 11. To reduce the computational expense associated with determining the optimal value of R (step 2 of the iterative procedure outlined above), it may be possible to express ∆∆Gi as an analytical function of R. If the dependence of ∆∆Gi on R is found to be smooth and monotonic, it may be necessary to perform only a few exchanges of molecules of component A with molecules of component B to determine the dependence of ∆∆G on R. Once this dependence is determined, identifying the value of R that minimizes Gf in eq 11 may be accomplished by taking the derivative of eq 10 with respect to R and solving for the value of R for which the derivative is equal to zero. Specifically,

(∑

)

n(1-R)

∂ ∂Gf ∂R

)

∆∆Gi(R)

i)1

∂R

+

∂Gent1 ∂R

+

∂Gent

)0

(12)

∂R

4. Introduction to Computer Simulation Free-Energy Methods Calculation of free-energy changes for physical and chemical systems is one of the most important applications of computer simulations. All molecular behavior, from the self-assembly of surfactants to form micelles to protein-ligand binding, can be directly linked to free energy and free-energy changes. Computer simulation estimates of free energy have been made in many different contexts, including protein folding,27 protein stability,28 enzyme reaction paths,29 ligand binding,30 ion transport,31 solvation processes,32,33 and conformational equilibria.34 Two commonly used methods to determine the free-energy difference between two states, I and II, from either experiment or computer simulation are (i) evaluation of the probability of finding the system in state I or state II and (ii) evaluation of the reversible work required to move from state I to state II.35 Free-energy perturbation and thermodynamic integration are two of the most generally applicable and accurate free-energy methods developed to date.36 In free-energy perturbation, the free-energy difference between two states is evaluated with approach (i), whereas in thermodynamic integration, it is evaluated with approach (ii). The equations involved in freeenergy perturbation and thermodynamic integration are exact,36 and the two methods yield the same solution in the limit of infinite sampling of phase space (coordinate and momentum space).35 However, a number of other methods have been developed to estimate free-energy differences, including quantum mechanical methods,37-39 Poisson-Boltzmann-based continuum methods,40,41 integral equation methods,42-44 and linear response theory.45-47 Quantum mechanical methods can be used to determine free-energy changes only for very small systems because of the computational expense associated with solving Schro¨dinger’s equation. To model larger systems, quantum mechanical methods must be combined with molecular dynam-

ics simulations.48,49 Poisson-Boltzmann based continuum methods are relatively computationally inexpensive, but because they do not model the solvent explicitly, they are approximate and are not well-suited for capturing the free-energy changes involved in hydrophobic interactions.47 Integral equation methods50 and linear response theory45-47 also involve a number of approximations, and linear response theory requires parametrization for specific systems. With the above in mind, we have chosen thermodynamic integration as the free-energy method to implement the CSFE/MT model because we expect it to be well-suited for determining the free-energy changes involved in changing micelle composition. Because surfactant micellization and micellar solubilization are driven by the hydrophobic effect, explicit simulation of water molecules is expected to be necessary to accurately predict G/f . As a result, continuum free-energy methods are not suitable. Similarly, integral equation methods and linear response theory are expected to be too approximate. Implementation of quantum mechanical methods would be too computationally expensive, because explicit simulation of water implies that simulations of several thousand atoms will be required to implement the CS-FE/MT model. 5. Free-Energy Calculations Made in the CS-FE/MT Model As discussed in Section 3, the starting point for the CS-FE/ MT model is a micelle containing only surfactant (component) A. The free-energy change associated with exchanging molecules of surfactant A with molecules of a cosurfactant or solubilizate (component) B, or ∆∆Gi, must be determined through computer simulation, where ∆∆Gi refers to the freeenergy change associated with the ith exchange of component A with component B. As discussed in Section 4, thermodynamic integration will be used to compute ∆∆Gi. To implement thermodynamic integration, a free-energy pathway must be chosen to move between state I and state II. The free-energy pathway that was selected to implement the CS-FE/MT model was inspired by a thermodynamic cycle that has proven useful in performing substrate-enzyme binding calculations in aqueous media. In Figure 2A, we present a typical thermodynamic cycle used to find the difference in binding free energies between two different substrates (represented by the wedge-shaped objects) and an enzyme (represented by the circle), or ∆∆Gbind. Experimentally, the difference in binding free energies for the two substrates can be determined by measuring the free energies Gbind,1 and Gbind,2 associated with the physical binding processes represented by the two horizontal arrows. ∆∆Gbind would then simply be evaluated as Gbind,2 - Gbind,1. However, computer simulation determinations of Gbind,1 and Gbind,2 may be difficult for two reasons: (i) the process of bringing a substrate and an enzyme together may lead to conformational rearrangements of the substrate, the enzyme, or both, which may require extended simulation time to adequately sample; and (ii) choosing an efficient reaction coordinate for binding can be difficult, and the free-energy changes experienced along this coordinate may require significant sampling in order to converge. Even if conformational rearrangements of the substrate and the enzyme do not occur, many of the water molecules associated with the substrate and the enzyme in the bulk aqueous state must be displaced for binding to occur, which may require significant computational expense to sample adequately. Instead of following the physical pathways represented by the two horizontal arrows labeled Gbind,1 and Gbind,2 in Figure

Free Energy of Micellization. 1. Theory

Figure 2. Comparison of alchemical pathways used to determine ∆∆G for substrate-enzyme binding (A), and for a composition change in a micelle (B). For an explanation of the notation used, see the text.

2A, it is much more computationally efficient in practice to determine the free-energy changes associated with the nonphysical processes ∆G1 and ∆G2, as represented by the vertical arrows in Figure 2A.36 Note that ∆G1 represents the free-energy change associated with chemically modifying a substrate in aqueous solution, and ∆G2 represents the free-energy change associated with chemically modifying a substrate that is bound to an enzyme in aqueous solution. In contrast to the two physical processes associated with the free-energy changes Gbind,1 and Gbind,2, the nonphysical processes associated with the free-energy changes ∆G1 and ∆G2 typically (i) do not involve significant conformational rearrangements of the substrate or the enzyme, (ii) do not require the definition of a reaction coordinate, and (iii) do not require displacement of large numbers of water molecules, thus greatly reducing the computational expense associated with making accurate free-energy estimates.36 Since the thermodynamic cycle in Figure 2A satisfies Gbind,1 + ∆G2 - Gbind,2 - ∆G1 ) 0, it follows that ∆∆Gbind ) Gbind,2 - Gbind,1 ) ∆G2 - ∆G1. In the limit of infinite sampling of phase space, the values of ∆∆Gbind computed as Gbind,2 - Gbind,1 or as ∆G2 - ∆G1 are identical because free energy is a state function. Consequently, the computed free-energy difference does not depend on the reversible path taken to move from one state to the other. The type of unphysical path associated with the evaluation of ∆G1 or ∆G2 is referred to as an alchemical path or as computer alchemy because of the chemical transformations involved.51,52

J. Phys. Chem. B, Vol. 112, No. 6, 2008 1639 The thermodynamic cycle proposed for use in the CS-FE/ MT model of micellization or micellar solubilization is shown in Figure 2B. Determining the free-energy changes associated with transferring a surfactant A molecule from aqueous solution to the micellar environment or with transferring a surfactant/ solubilizate B molecule from aqueous solution to the micellar environment is expected to require long simulation times to give accurate free-energy estimates because such transitions involve (i) structural rearrangements within the micelle and (ii) significant rearrangement of the water molecules surrounding the surfactant or solubilizate and the micelle. Therefore, we propose an alternative path involving alchemical transformations, as shown by the two nonphysical processes ∆G1 and ∆G2 in Figure 2B. Note that ∆G1 represents the free-energy change associated with chemically modifying one surfactant A molecule (represented by the brown circular head) into a surfactant/solubilizate B molecule (represented by the brown rectangular head) in aqueous solution, and ∆G2 represents the free-energy change associated with chemically modifying a surfactant A molecule into a surfactant/solubilizate B molecule in a simulated micelle. Since the thermodynamic cycle in Figure 2B satisfies Gform,single + ∆G2 - Gform,mixed - ∆G1 ) 0, it follows that ∆∆G ) Gform,mixed - Gform,single ) ∆G2 - ∆G1. To compute the free-energy differences involved in the thermodynamic cycle shown in Figure 2B, computer simulations of surfactant to cosurfactant or surfactant to solubilizate transformations must be conducted in bulk aqueous solution and in a micellar environment. Because differences in Gibbs free energy are being computed, all bulk aqueous solution simulations should be conducted in the NPT ensemble.53 When implementing the CS-FE/MT model to determine G/f for spherical micelles, simulations in the micellar environment will also be conducted in the NPT ensemble (to correspond with experimental conditions). However, when implementing the CS-FE/MT model to determine G/f for cylindrical or discoidal micelles, simulation in the NPT ensemble in the micellar environment is not appropriate because only a slice of an infinite cylinder or a plug of an infinite bilayer are simulated. The appropriate boundary condition to use parallel to the axis of a cylindrical micelle, or parallel to the surface of a bilayer, during micellar simulation is an interfacial tension that provides a postequilibration value of area per surfactant/solubilizate head that is similar to the area that would be observed experimentally 5. Therefore, for cylindrical and discoidal micelles, simulation in the NγT ensemble is required, where γ is the appropriate interfacial tension. 6. Conclusions In this article, a theoretical approach was introduced to use computer simulation free-energy methods to evaluate the freeenergy change associated with changing micelle composition (as reflected in ∆∆Gi). In this approach, experimental CMC data, or a MT model for single surfactant micellization, is first used to evaluate the free energy associated with single (pure) surfactant micelle formation, gform,single, where the single surfactant micelle contains only surfactant A molecules. An iterative approach was proposed to combine the estimated value of Gform,single with free-energy estimates of ∆∆Gi based on computer simulation to determine the optimal free energy of mixed micelle formation, the optimal micelle aggregation number and composition, and the optimal bulk solution composition. A variety of free-energy methods were briefly reviewed, and the selection of the thermodynamic integration freeenergy method was justified and selected to implement the CS-

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