Hydrotropy: MonomerâMicelle Equilibrium and Minimum Hydrotrope

Article pubs.acs.org/JPCB

Hydrotropy: Monomer−Micelle Equilibrium and Minimum Hydrotrope Concentration Seishi Shimizu*,† and Nobuyuki Matubayasi*,‡,§ †

York Structural Biology Laboratory, Department of Chemistry, University of York, Heslington, York YO10 5YW, United Kingdom Division of Chemical Engineering, Graduate School of Engineering Science, Osaka University, Toyonaka, Osaka 560-8531, Japan § Elements Strategy Initiative for Catalysts and Batteries, Kyoto University, Katsura, Kyoto 615-8520, Japan ‡

ABSTRACT: Drug molecules with low aqueous solubility can be solubilized by a class of cosolvents, known as hydrotropes. Their action has often been explained by an analogy with micelle formation, which exhibits critical micelle concentration (CMC). Indeed, hydrotropes also exhibit “minimum hydrotrope concentration” (MHC), a threshold concentration for solubilization. However, MHC is observed even for nonaggregating monomeric hydrotropes (such as urea); this raises questions over the validity of this analogy. Here we clarify the effect of micellization on hydrotropy, as well as the origin of MHC when micellization is not accompanied. On the basis of the rigorous Kirkwood-Buff (KB) theory of solutions, we show that (i) micellar hydrotropy is explained also from preferential drug−hydrotrope interaction; (ii) yet micelle formation reduces solubilization effeciency per hydrotrope molecule; (iii) MHC is caused by hydrotrope−hydrotrope self-association induced by the solute (drug) molecule; and (iv) MHC is prevented by hydrotrope self-aggregation in the bulk solution. We thus need a departure from the traditional view; the structure of hydrotrope-water mixture around the drug molecule, not the structure of the aqueous hydrotrope solutions in the bulk phase, is the true key toward understanding the origin of MHC. conclusion is in stark contrast with the previous view,5,8−14 it has been supported by a more recent study.33 The rigorous KB theory has thus challenged the long-held assumptions on the mechanism of hydrotropy. All the above conclusions have been obtained by analyzing small molecule hydrotropes. However, many of the hydrotropes that are in clinical use are micellar, not monomeric.33−36 These micellar hydrotropes (whose typical concentration is 1−10 mM)33−37 are known to be far more efficient than small molecular hydrotropes (1−3 M).38,39 The efficiency of micellar hydrotropes seems to be in apparent contradiction to our previous, KB-based conclusion that the self-aggregation of smallmolecule hydrotropes reduces the per-hydrotrope solubilization efficiency.6,7 Consequently, we ask: is self-aggregation a hindrance also for micellar hydrotropes? This is the first question we shall address, through an extension of the KB theory of hydrotropy to incorporate micellar hydrotropes. This comparison between monomeric and micellar hydrotropes brings us to another unresolved question, the origin of the minimum hydrotrope concentration (MHC), i.e., the concentration below which solubilization does not take place. Its apparent similarity to the critical micelle concentration (CMC)8,12,14,37−39 has inspired the view that the self-aggregation of hydrotrope molecules drives solubilization.8,12,14,37−39 However, such an explanation seems to be in contradiction to our previous study, according to which MHC can occur even in the

1. INTRODUCTION The low solubility of drug molecules poses a serious difficulty for drug delivery.1−5 This problem, however, has been overcome effectively by formulating the drug with a class of molecules called hydrotropes, which are nontoxic and water-soluble.1−5 Through the use of hydrotropes, solubility increases up to several orders of magnitude have been reported.1−5 Despite widespread use of hydrotropes, how they actually work to solubilize hydrophobic drugs has long remained a puzzle.1−5 The quest for a molecular-based understanding has revolved around the following three hypotheses:6,7 (i) selfassembly of hydrotrope molecules;5,8−14 (ii) disruption of water structure by hydrotropes;3,15−18 or (iii) formation of solute− hydrotrope complexes.19−22 On top of these conflicting hypotheses, the fact that the word “hydrotropes” have been used to cover a wide variety of molecules, ranging from small organic molecules to micelles and microemulsions, has made it even more difficult to resolve this mystery.6,7 To clear up this situation, we have recently proposed a novel approach:6,7 the application of the Kirkwood−Buff (KB) theory (or the Fluctuation Solution Theory),6,7,23−32 which is an exact theory of statistical thermodynamics. Two major driving forces for solubilization have been identified for small-molecule hydrotropes:6,7 (i) solute−hydrotrope binding which drives solubilization (major contribution); (ii) self-association of the hydrotropes, which, through reducing the effective number of hydrotrope molecules, reduces the solubilization per hydrotrope molecule (secondary contribution).6,7 Here, what is the most unexpected is (ii); self-aggregation actually makes the hydrotropes less efficient solubilizer per molecule. Although this © 2014 American Chemical Society

Received: June 12, 2014 Revised: July 25, 2014 Published: August 21, 2014 10515

dx.doi.org/10.1021/jp505869m | J. Phys. Chem. B 2014, 118, 10515−10524

The Journal of Physical Chemistry B

Article

absence of hydrotrope self-aggregation and micellization.6,7 What, then, is the true origin of MHC? This is our second question, which we shall address through extending our KB theory of hydrotropy, to provide a molecular-based explanation of the concentration-dependence of solubility. Thus, we attempt, in this paper, to clarify the effect of micellization on hydrotropy (where there is micellization), as well as to dispel some of the false analogies between hydrotropy and micellization (where there is no micellization).

which has the following microscopic interpretation through the solute−water or solute−hydrotrope distribution function, gui(r), as a function of solute−solvent distance r: Nui = niGui , Gui = 4π

= n1Gu1dμ1 + (n2Gu2 + mn3Gu3)dμ2 + = n1Gu1dμ1 + nmGum ̅ dμ2 +

nm = n2 + mn3

s* − s dT nu

s* − s dT nu

(7)

Gum ̅ =

n2Gu2 + mn3Gu3 nm

(8)

Equation 7 is the fundamental KB relationship for solvation free energy μu* represented in terms of μ1,μ2, and T. Note that n2/nm is the fraction of the hydrotrope molecule present at the monomeric state and that mn3/nm is the fraction at the micellar state. Thus, eq 8 is the weighted average of the solute-hydrotrope KB parameters over the two states (monomeric and micellar) of the hydrotrope. The same relationship can also be represented in terms of μ1,P and T or μ2, P and T which can be obtained, via eq 2, by eliminating μ1 or μ2:

(1) (2)

⎛ s* − −dμu* = n1(Gu1 − Gum ̅ )dμ1 − ⎜Gum ̅ s− nu ⎝

s⎞ ⎟dT ⎠

+ Gum ̅ dP

(9)

⎛ s* − −dμu* = nm(Gum ̅ − Gu1)dμ2 − ⎜Gu̅ 1s − nu ⎝

Here we have adhered to the classical chemical thermodynamics of chemical equilibria, in which intermolecular interactions are implicitly contained. Now we combine eqs 1−3; by introducing the concentration (number per volume, ni = Ni/V) and the entropy density (s = S/ V), the dP term cancels out:

s⎞ ⎟dT ⎠

+ Gu1dP

(10)

Upon the basis of the above three different representations (eqs 8−10), we can now show that μ1-, μ2-, and P-dependences of μ u* correspond to the following well-known experiments:6,7,23,27,40,41 Preferential Hydration Parameter.

nudμu − RTdnu + (n1* − n1)dμ1 + [(n2* − n2) + m(n3* − n3)]dμ2 + (s* − s)dT = 0 (4)

⎛ ∂μ* ⎞ def νu1 = −⎜⎜ u ⎟⎟ = n1(Gu1 − Gum ̅ ) ⎝ ∂μ1 ⎠T , P , n → 0

Here, in addition, we have used the van’t Hoff equation for the osmotic pressure (Π = RTnu), which holds true at the infinite dilution of the solute. Thus, the Gibbs−Duhem eqs 1 and 2 have now been rewritten explicitly in terms of the concentration change (ni* − ni) in the solute’s vicinity.6,7,23,27,40,41 Now we introduce the excess solvation number of the species i around the solute defined as follows: (ni* − ni) nu

(6)

where we have introduced the total concentration of the hydrotrope nm and the mean KB parameter G̅ um defined as follows:

where Ni and μi are respectively the number and the chemical potential of the species i, S is the entropy, P is the pressure, and Π is the osmotic pressure due to the presence of a solute. The species 2 and 3 are in monomer-micelle equilibrium as in the following: μ3 = mμ2 (3)

Nui =

[gui(r ) − 1]r 2dr

−dμu* ≡ −d(μu − RT ln nu)

Nu*dμu + N1*dμ1 + N2*dμ2 + N3*dμ3

N1dμ1 + N2dμ2 + N3dμ3 − VdP + SdT = 0

∞

Here, G ui is referred to as the Kirkwood−Buff (KB) parameter.6,7,23,27,40,41 From eqs 4−6, we obtain the following:

2. STATISTICAL THERMODYNAMIC THEORY OF SOLUTE−HYDROTROPE INTERACTION UNDER MONOMER−MICELLE EQUILIBRIUM Consider a three component solution consisting of a solute (i = u), water molecules (i = 1), hydrotrope monomers (i = 2) and micelles (i = 3), whose aggregation number is m. In the present work, the monomeric and micellar states of the hydrotrope are treated as distinct species that are in equilibrium and can be connected through the mass-action model. Furthermore, the micelle is assumed to be monodisperse, while the introduction of polydispersity is straightforward. Now, let us divide the solution into two parts; the first part (called the “solute’s vicinity”) contains a solute molecule, the other part (called the “bulk”) is far away from the solute.6,7,23,27,40,41 To explore the thermodynamic consequence of this soluteinduced concentration change of the solvent species, let us first write down the Gibbs−Duhem equation for each part, vicinity (represented by *) and bulk:6,7,23,27,40,41

− V *(dP + d Π) + S*dT = 0

∫0

u

(11)

Preferential Hydrotrope Solvation Parameter. ⎛ ∂μ* ⎞ def = nm(Gum νu2 = −⎜⎜ u ⎟⎟ ̅ − Gu1) ⎝ ∂μ2 ⎠T , P , n → 0 u

(5)

(12)

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⎛ ∂μ* ⎞ ⎛ ∂μ ⎞ ⎛ ∂ln(Nu/V ) ⎞ Vu = ⎜ u ⎟ = ⎜⎜ u ⎟⎟ + RT ⎜ ⎟ ⎝ ⎠T , N ∂P ⎝ ∂p ⎠T , N ⎝ ∂p ⎠ i T ,N i

Now the Gibbs−Duhem relationship under the micellar equilibrium is expressed by the combination of eqs 2 and 3:

i

= −n1V1Gu1 − nmV2Gum ̅ + RTκT

n1dμ1 + nmdμ2 = 0 (13)

Combining eqs 18 and 19, we obtain the following:

where Vi is the partial molar volume of the species i and κT is the isothermal compressibility of the solution, which is usually negligibly small. Thus, Gu1 and G̅ um, which contain crucial information on solute−water and solute−hydrotrope interactions, can be determined from two independent experiments, either eqs 11 and 13 or eqs 12 and 13. Thus, we have established a full KB treatment of micellar hydrotropes on solubilization, following the previous work by one of us.26 Even though KB theory has previously been applied to micelles, the main focus was on micellization equilibrium per se.42,43 One exception is by Hall,44 who made a brief comment on the effect of micellar cosolvents on gas solubility, despite under the assumption that monomers behave ideally.

⎛ ∂μ1 ⎞ ⎜ ⎟ = ⎝ ∂nm ⎠T , P

(14)

⎛ ∂μ* ⎞ ⎛ ∂μ* ⎞ ⎛ ∂μ2 ⎞ −⎜⎜ u ⎟⎟ = −⎜⎜ u ⎟⎟ ⎜ ⎟ ⎝ ∂nm ⎠T , P , n → 0 ⎝ ∂μ2 ⎠T , P , n → 0 ⎝ ∂nm ⎠T , P , nu → 0

(15)

u

u

u

u

2

−

T , μ2

n1 nm

( ) ∂nm ∂μ2

(20)

T , μ1

2

2

1 = [(⟨N1N2⟩ − ⟨N1⟩⟨N2⟩) + m(⟨N1N3⟩ − ⟨N1⟩⟨N3⟩)] V (21)

⎛ ∂n ⎞ 1 ⎛ ∂(N2 + mN3) ⎞ ⎟⎟ = RT ⎜⎜ RT ⎜⎜ m ⎟⎟ ∂μ2 V⎝ ⎝ ∂μ2 ⎠T , μ ⎠T , μ 1

1

1 = [(⟨N2 2⟩ − ⟨N2⟩2 ) + 2m(⟨N2N3⟩ − ⟨N2⟩⟨N3⟩) V + m2(⟨N32⟩ − ⟨N32⟩)]

(22)

where ⟨⟩ signifies the ensemble average. Now we introduce the KB parameter as follows: ⎡ ⟨NN δij ⎤ i j⟩ − ⟨Ni⟩⟨Nj⟩ ⎥ − Gij = V ⎢ ⟨Ni⟩⟨Nj⟩ ⟨Ni⟩ ⎥⎦ ⎢⎣

⎛ ∂n ⎞ = n1nmG̅1m RT ⎜⎜ m ⎟⎟ ⎝ ∂μ1 ⎠T , μ

(24)

⎛ ∂n ⎞ 2 = mn RT ⎜⎜ m ⎟⎟ ̿ ̅ m + nmGmm ⎝ ∂μ2 ⎠T , μ

(25)

2

Note that the number of variables here has been reduced by one, by the existence of monomer−micelle equilibrium (eq 3). Now we rewrite eq 16 as follows: ⎛ ∂n ⎞ ⎛ ∂μ ⎞ ⎛ ∂n ⎞ ⎛ ∂μ ⎞ ⎛ ∂μ ⎞ 1 = ⎜⎜ m ⎟⎟ ⎜ 1 ⎟ − ⎜⎜ m ⎟⎟ ⎜⎜ 1 ⎟⎟ ⎜ 2 ⎟ ⎝ ∂μ1 ⎠T , μ ⎝ ∂nm ⎠T , P ⎝ ∂μ1 ⎠T , μ ⎝ ∂μ2 ⎠T , n ⎝ ∂nm ⎠T , P 2

2

1

where G̅ 1m and G̿ mm are the averaged KB parameters of the monomeric and micellar states, and m̅ is the mean aggregation number, defined respectively in the following as

m

(17)

and using the chain rule to simplify as follows: ⎛ ∂n ⎞ ⎛ ∂μ ⎞ ⎛ ∂n ⎞ ⎛ ∂μ ⎞ 1 = ⎜⎜ m ⎟⎟ ⎜ 1 ⎟ + ⎜⎜ m ⎟⎟ ⎜ 2 ⎟ ⎝ ∂μ1 ⎠T , μ ⎝ ∂nm ⎠T , P ⎝ ∂μ2 ⎠T , μ ⎝ ∂nm ⎠T , P 2

1

(23)

where δij is Kronecker’s delta. In our development, the hydrotropes at the monomer and micellar states are treated as distinct species and the KB parameter is introduced as such. Indeed, the KB parameter is written in terms of (co)variance of the fluctuations of numbers of particles, and the number of micellar particles is well-defined in the mass-action model. Using eq 23, we obtain the following:

(16)

m

( )

⎛ ∂n ⎞ 1 ⎛ ∂(N2 + mN3) ⎞ ⎟⎟ = RT ⎜⎜ RT ⎜⎜ m ⎟⎟ ∂μ1 V⎝ ⎝ ∂μ1 ⎠T , μ ⎠T , μ

In order to make this connection, we need to obtain the KB expressions of (∂μ1/∂nm)T,P,nu → 0 and (∂μ2/∂nm)T,P,nu → 0. This can be achieved through a direct use of the grand canonical ensemble as has been done in the inversion of the KB Theory. An alternative derivation based upon the Gibbs−Duhem equation can be found in the Appendix A. The starting point is the following thermodynamic relationship:32 ⎛ ∂μ ⎞ ⎛ ∂μ ⎞ ⎛ ∂μ1 ⎞ ⎛ ∂μ ⎞ + ⎜⎜ 1 ⎟⎟ ⎜ 2 ⎟ ⎜ ⎟ = ⎜ 1⎟ ⎝ ∂nm ⎠T , P ⎝ ∂nm ⎠T , μ ⎝ ∂μ2 ⎠T , n ⎝ ∂nm ⎠T , P

1 ∂nm ∂μ1

Now, we turn to statistical thermodynamics, in order to express (∂nm/∂μ1)T,μ2 and (∂nm/∂μ2)T,μ1 in eq 20 in terms of KB parameters. Using the grandcanonical ensemble,32 and noting that it is N2 + mN3 which is conjugate to μ2, we can easily show the following:

3. MICELLE FORMATION AND SOLUBILIZATION Now we answer the first question: does the formation of hydrotrope micelles promote drug solubilization? To answer this question, let us first connect the preferential solvation theory (section 2) and the experimental solubility curve. 3.1. Theoretical Framework. We first note that, in the experimental literature, it is the nm dependence of μ*u that is of major interest. This connection can be made in the following manner:32 ⎛ ∂μ* ⎞ ⎛ ∂μ* ⎞ ⎛ ∂μ1 ⎞ −⎜⎜ u ⎟⎟ = −⎜⎜ u ⎟⎟ ⎜ ⎟ ⎝ ∂nm ⎠T , P , n → 0 ⎝ ∂μ1 ⎠T , P , n → 0 ⎝ ∂nm ⎠T , P , nu → 0

(19)

G̅1m = G̅3m =

(18) 10517

n2G12 + mn3G13 , nm n2G32 + mn3G33 nm

G̅2m =

n2G22 + mn3G23 , nm

(26)


The Journal of Physical Chemistry B Gmm ̿ ≡

Article

1 (n2G̅2m + mn3G̅3m) nm

1 = 2 (n22G22 + 2mn2n3G23 + m2n32G33) nm m̅ = (n2 + n3m2)/nm

RT m−

(28)

Table 1. Estimation on the Effect of Micellization (Eq 33) at a Typical Experimental Concentration (nm = 10−2 mol dm−3)

⎛ ∂μ* ⎞ ⎤ ⎡ RT ⎥ −⎜⎜ u ⎟⎟ = (Gum ̅ − Gu1)⎢ ⎣ m̅ + nm(Gmm ̿ − G̅1m) ⎦ ⎝ ∂nm ⎠T , P , n → 0

4/3πR3 G33 ≈ − 4/3π(2R)3 nm nmG33 m

u

(29)

The first factor is the preferential solute-hydrotrope interaction with respect to solute-water interaction; the second factor is the nonideality of the bulk solution, due to the formation of micelles, as well as monomer−monomer, monomer−micelle, micelle− micelle, monomer−water, and micelle−water interactions. 3.2. Does Hydrotrope Micellization Promote Solubilization? In the case of small molecule hydrotropes, we have previously shown that it is the self-association of hydrotropes which has reduced solubilization per hydrotrope molecule. In a similar way, is micellization equally a hindrance to solubilization? To answer this question, let us consider the case in which one finds micelles predominantly in the solution, namely, mn3 ≫ n2. Under this condition, the second factor of eq 29 becomes the following:

a

u

Tween 80 55.4 dm3 mol−1a −443 dm3 mol−1 10−2 mol dm−3 −4 124a

From ref 37.

⎛ ∂μ* ⎞ RT (Gum −⎜⎜ u ⎟⎟ ≈ ̅ − Gu1) m̅ ⎝ ∂nm ⎠T , P , n → 0

(30)

u

⎡ n (G − G ) + mn (G − G ) ⎤ u1 3 u3 u1 ⎥ = RT ⎢ 2 u2 n2 + n3m2 ⎣ ⎦

(34)

Since the hydrotrope concentration is low, we can safely assume that Gu1, Gu2, and Gu3 are constants within the relevant concentration region. In this assumption, the nm dependence of μ*u comes only from n2 and n3. When the solubility increases steeply upon the formation of hydrotrope micelle, Gu1 and Gu2 are (relatively) small and Gu3 is large. In this case, eq 34 shows that the increase of solubility is observed when n3 becomes appreciably large. The presence of non-negligible n3 means the formation of micelle; the KB approach leads indeed to conventional notion on the mechanism of micellar hydrotropy. When the hydrotrope is dominantly in the micellar form (namely, mn3 ≫ n2), eq 31 holds, which shows that it is micellesolute interaction that drives solubilization. The aggregation number, however, counteracts the solubilization per hydrotrope molecule.

(31)

This expression is simply the preferential solute-micelle interaction with respect to solute−water interaction when a micelle is viewed as a single particle. 3.2.2. Micelle−Micelle and Micelle−Water KB Parameters, G33−G13. As examples, we take tween 20 and 80, common polysorbate surfactants used as detergents and emulsifiers.37 Their typical concentration for practical applications is about 10 mM.33−36 As far as we know, there are no published data on the KB parameters or cross virial coefficients, despite some reports on scattering experiments. We therefore perform an order-ofmagnitude estimation based upon the available experimental data.37 Let R be the radius of the micelle. On the basis of the molecular crowding approximation, the micelle−micelle KB parameter should be the micelle−micelle excluded volume, as follows: 4 G33 ≈ − π (2R )3 3

Tween 20 39.6 dm3 mol−1a −317 dm3 mol−1 10−2 mol dm−3 −3 86a

negligibly small compared to the aggregation number m (the first term). This shows that micellization lowers the solubilization efficiency as viewed per hydrotrope concentration, and that the micelle−micelle interaction does not affect this efficiency. 3.3. The Mechanism of Micellar Hydrotropy. We first simplify eq 29 based upon the above analysis. Since we have shown that m̅ ≫ nm(G̅ mm − G̅ 1m), eq 29 can be simplified into the following form:

Equation 30 holds the key toward understanding whether micellization is a hindrance toward solubilization. There are two major factors as below: 3.2.1. The Aggregation Number, m. m works to reduce the solubilization efficiency per hydrotrope molecule. When mn3 ≫ n2 and nm is small enough, eq 29 reduces to the following:26 ⎛ ∂μ* ⎞ −⎜⎜ u ⎟⎟ = RT (Gu3 − Gu1) ⎝ ∂n3 ⎠T , P , n → 0

(33)

Since the hydrotrope concentration is low, we can safely assume that G33 is concentration independent. According to Table 1, the effect of micelle−micelle KB parameter (the second term in the denominator of eq 33) is

(27)

From both routes (eqs 24 or 25), we now have a clear physical interpretation of the solubility curve:

RT RT ≈ m + n ( m̅ + nm(Gmm ̿ − G̅1m) m G33 − G13)

32 πR3nm 3

4. ORIGIN OF THE MINIMUM HYDROTROPE CONCENTRATION Now we turn to the monomeric hydrotropes, which do not form micellar aggregates and are free from the monomer−micelle equilibrium condition (eq 3). Nevertheless such hydrotropes exhibit MHC.8,12,14,38,39 What, then, is the true origin of MHC? In constructing a theory of MHC, let us first note the concentration ranges necessary for the hydrotropes. In the case of micellar hydrotropes, the required hydrotrope concentration is dilute enough (10 mM)33−36 such that we could safely ignore the concentration-dependence of KB parameters. In contrast, the

(32)

since, due to the micellar size, G33 should be much larger than G13. Under this condition, eq 30 can be estimated by the following formula: 10518



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Equation 36 has a clear physical interpretation in terms of number fluctuations. The driving force for MHC is the first term; MHC takes place when (i) number fluctuations of water and hydrotrope molecules in the presence of the solute become greater than those in the bulk, and (ii) the solute-induced increase of number fluctuations exceeds that of number covariance. On the contrary, the bulk phase excess number fluctuation over the covariance (within {}) prevents the occurrence of MHC, considering that ⟨N2⟩u − ⟨N2⟩0 is positive since we are treating the case of preferential solute−hydrotrope interaction. We have thus shown that it is the solute-induced increase of molecular number fluctuations that drive k B T(∂ 2 μ u* / ∂μ21)T,P,Nu → 0 negative. 4.2. Connection to Molecular Distribution Functions. In order to gain a clearer insight into the role of intermolecular interactions on MHC, we rewrite eq 36 in terms of KB parameters, defined respectively for the inhomogeneous and bulk solutions as follows:

required concentration of monomeric hydrotropes is much higher, usually 2−3 M.38,39 This motivates us to undertake an explicit consideration of the concentration-dependence of the KB parameters. 4.1. Fluctuation-Theory-Based Framework. Consider an inhomogeneous solution, which consists of water and hydrotrope molecules, as well as a solute molecule fixed at the origin; the solution is called inhomogeneous when the solute is fixed at the origin, and is called homogeneous when the solute position is not fixed just as water and hydrotrope. We shall start from the following expression for the preferential hydration parameter of a solute in the presence of monomeric hydrotropes in the grand canonical ensemble: ⎛ ∂μ* ⎞ −⎜⎜ u ⎟⎟ = (⟨N1⟩u − ⟨N1⟩0 ) ⎝ ∂μ1 ⎠T , P , N → 0 u

−

⟨N1⟩0 V ⟨N2⟩0 V

(⟨N2⟩u − ⟨N2⟩0 ) (35)

where ⟨⟩u is the ensemble average in the presence of the solute, whereas ⟨⟩0 is that in the bulk solution. Appendix B outlines the derivation of eq 35; Appendix C shows that eq 35 is equivalent to the well-known expression for homogeneous solutions, which can be derived from eqs 11 and 23, as well as the useful formulae in Appendix D. MHC is the point at which a significant decrease of μu* start to take place. Hence we need to calculate the second derivative of μ*u . (Such derivatives have been studied from a KB-based perspective, but under different experimental conditions and thermodynamic constraints.45,46) This can be executed both for inhomogeneous (Appendix B) and homogeneous (Appendix C) solutions, both of which will yield the same result (Appendix C). For the sake of clarity, we present the following expression for the inhomogeneous solution:

⟨NN i j⟩u − ⟨Ni⟩u ⟨Nj⟩u =

⟨Ni⟩u ⟨Nj⟩u ⎛ Vδij ⎞ ⎜Gu , ij + ⎟ V ⟨Ni⟩u ⎠ ⎝

(37)

⟨NN i j⟩0 − ⟨Ni⟩0 ⟨Nj⟩0 =

⟨Ni⟩0 ⟨Nj⟩0 ⎛ Vδij ⎞ ⎜⎜Gij + ⎟ V ⟨Nj⟩0 ⎟⎠ ⎝

(38)

where δij is Kronecker’s delta. Equation 36 can be rewritten through eqs 37 and 38 in the following manner: ⎛ ∂ 2μ* ⎞ ⟨N ⟩2 =− 10 kBT ⎜⎜ u2 ⎟⎟ V ⎝ ∂μ1 ⎠T , P , N → 0 u

⎡ V V 2 2 ⎢ ⟨N1⟩u Gu ,11 + ⟨N1⟩u − ⟨N1⟩0 G11 + ⟨N1⟩0 ⎢ ⟨N1⟩20 ⎢ ⎢ ⟨N1⟩u ⟨N2⟩u Gu ,12 − ⟨N1⟩0 ⟨N2⟩0 G12 ⎢ − 2 ⎢ ⟨N1⟩0 ⟨N2⟩0 ⎢ ⎢ V V 2 2 ⎢ ⟨N2⟩u Gu ,22 + ⟨N2⟩u − ⟨N2⟩0 G22 + ⟨N2⟩0 ⎢+ ⟨N2⟩20 ⎣

⎡{(⟨(N )2 ⟩ − (⟨N ⟩ )2 ) ⎤ 1 u 1u ⎢ ⎥ 2 2 ⎢ − (⟨(N1) ⟩0 − (⟨N1⟩0 ) )} ⎥ ⎢ ⎥ ⎢− 2 ⟨N1⟩0 {(⟨N N ⟩ − ⟨N ⟩ ⟨N ⟩ ) ⎥ 1 2 u 1u 2 u ⎥ ⎢ ⟨N ⟩ ∂ 2μu* 2 0 ⎢ ⎥ =− kBT ∂μ12 ⎢ − (⟨N1N2⟩0 − ⟨N1⟩0 ⟨N2⟩0 )} ⎥ ⎢ ⎥ 2 ⎢ ⎛ ⟨N1⟩0 ⎞ 2 2 ⎥ ⎟ {(⟨(N2) ⟩u − (⟨N2⟩u ) )⎥ ⎢+⎜ ⎢ ⎝ ⟨N2⟩0 ⎠ ⎥ 2 2 ⎣⎢ − (⟨(N2) ⟩0 − (⟨N2⟩0 ) )} ⎦⎥

(

)

(

(

)

(

⎛ 1 1⎞ + n12Gu2⎜G11 − 2G12 + G22 + + ⎟ n1 n2 ⎠ ⎝

⎧ ⟨N ⟩ − ⟨N2⟩0 ⎫⎧ ⎪ ⎬⎨(⟨(N1)2 ⟩0 − (⟨N1⟩0 )2 ) +⎨ 2u ⎪ ⟨N2⟩0 ⎩ ⎭⎩ ⟨N ⟩ − 2 1 0 (⟨N1N2⟩0 − ⟨N1⟩0 ⟨N2⟩0 ) ⟨N2⟩0 ⎫ ⎛ ⟨N1⟩0 ⎞2 ⎪ +⎜ ⎟ (⟨(N2)2 ⟩0 − (⟨N2⟩0 )2 )⎬ ⎪ ⎝ ⟨N2⟩0 ⎠ ⎭

)

)

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ (39)

where n1 = ⟨N1⟩0/V and n2 = ⟨N2⟩0/V. Equation 39 is the foundation for understanding MHC at a molecular basis. MHC, as defined above, is the concentration at which kBT(∂2μ*u /∂μ21)T,P,Nu → 0 starts to decrease dramatically from zero. What, then, is the driving force for the dramatic decrease of kBT(∂2μu*/∂μ21)T,P,Nu → 0? Even though eq 39 looks somewhat complicated in the first outlook, we show that there are in fact two competing forces, as explained below: (A). Solute (Drug)-Induced Enhancement of the SelfAssociation of the Water and Hydrotrope Molecules. This is the first term of eq 39. When the solvent-induced self-association Gu,ii becomes much larger than the bulk Gii, kBT(∂2μu*/ ∂μ21)T,P,Nu → 0 is driven down.

(36)

Even though applying the theory requires an explicit connection between kBT(∂2μu*/∂μ21)T,P,Nu → 0 and KB parameters (which will be given in the next subsection), a brief comment on the interpretation of eq 36 will be helpful here. 10519



Article

(B). Bulk-Phase Self-Association. This is the second term of eq 39. Bulk phase self-association of water and hydrotropes (G11 − 2G12 + G22 + 1/n1 + 1/n2) drives kBT(∂2μu/∂μ21)T,P,Nu → 0 up through a large positive Gu2 (preferential solute−hydrotrope interaction). MHC takes place when (A) becomes much greater than (B). 4.3. Solute-Induced Hydrotrope−Hydrotrope Association Is the Cause of MHC. Now we apply eq 39 to clarify the true origin of MHC. Urea is an important example, because it exhibits MHC6,7,38,39 while forming a near-ideal mixture with water.6,24,47 Hence it is a counter-example to the hydrotrope self-aggregation hypothesis on the origin of MHC. Here we show (Figure 1) that MHC is driven instead by the drug-induced solvent self-association which overrides the bulk-phase self-association term.

Figure 2. Origin of MHC for sodium benzoate (sb)-induced solubilization of BA (butyl acetate, solid) and BB (benzyl benzoate, dotted), just as Figure 1. Due to the quality of experimental data used for fitting, there is no clear MHC observed for BA/sb.6,38

Figure 1. Origin of MHC for urea-induced solubilization of BA (butyl acetate, solid) and BB (benzyl benzoate, dotted) based upon eq 39. (a) The solute-induced self-association term, + ((⟨N1⟩20/V)[(⟨N1⟩2u(Gu,11 + (V/⟨N 1 ⟩ u )) − ⟨N 1 ⟩ 02 (G 1 1 + (V/⟨N 1 ⟩ 0 )))/(⟨N 1 ⟩ 02 )) − 2((⟨N1⟩u⟨N2⟩uGu,12 − ⟨N1⟩0⟨N2⟩0G12)/(⟨N1⟩0⟨N2⟩0)) + (⟨N2⟩2u(Gu,22 + (V/⟨N2⟩u)) − ⟨N2⟩20(G22 + (V/⟨N2⟩0))/(⟨N2⟩20)] (thick lines), and bulk self-association in hydrotrope-drug inetraction volume, + n21Gu2(G11 − 2G12 + G22 + 1/n1 + 1/n2) (thin lines). (b) kBT(∂2μ*u /∂μ21)T,P,Nu → 0 is driven negative when the first exceeds the second. The plot is based upon the linear regression of the experimental data (ref 6). As can be seen from eq 39, all the terms considered here are dimensionless.

Figure 3. Origin of MHC for sodium salycilate (ss)-induced solubilization of BA (butyl acetate, solid) and BB (benzyl benzoate, dotted), just as in Figure 1. Note that ss exhibits stronger bulk-phase selfassociation than sb, yet no stronger MHC is seen herein.

Self-association in the bulk phase, indeed, is not the driving force of MHC, which can be seen from Figures 2 and 3. We have shown previously that the bulk phase self-association of sodium salicylate (ss) is larger than that of sodium benzoate (sb).6,7 In contrast, MHC for benzyl benzoate (BB) is more pronounced in sb than in ss, while MHC of butyl acetate (BA) is observed clearly only in ss. There is no apparent correlation between bulk-phase self-association and MHC. Hydrotrope−hydrotrope association induced by the drug should therefore be the dominant contribution to the druginduced enhancement of self-association. Even though it is impossible, within our present theory, to compare the individual terms, we know that Gu2 is much larger than Gu1 for hydrotropes.6,7 This implies that the effect of introducing a solute molecule should affect hydrotrope−hydrotrope self-

association more than water−water self-association, for its distribution in solution is affected far more by the solute molecule. The molecular insight into the mechanism of MHC, as obtained here from KB theory, should be examined further by computer simulation, with which each KB parameter can be separately evaluated.

5. CONCLUSIONS Hydrotrope self-aggregation in the bulk phase has been considered to be the cause of hydrotropy. In this work, we have challenged this long-held view, articulated as fol10520



■

lows:8,12,14,37−39 (i) self-aggregation and micelle formation drive the solubilization of the hydrophobic drugs; and (ii) minimum hydrotrope concentration (MHC) is the evidence for the micelle-like behavior of the hydrotropes. Contrary to the above, statistical thermodynamics has shown the following: (i) micelle formation reduces solubilization per hydrotrope molecule; and (ii) MHC is caused by solute-induced increase of solvent−solvent (especially hydrotrope−hydrotrope) interactions f rom the bulk, and not by the hydrotrope self-aggregation in the bulk solution. The above conclusions of ours are in stark contrast to the traditional view. Yet they owe their credibility to the KB theory of solution, which is an exact theory, free of approximations and model assumptions.6,7,23−32,40−46 In order to arrive at the above conclusions, we have (i) extended the KB theory of hydrotropy to incorporate the micellar hydrotropes, subject to monomermicelle equilibrium; (ii) developed the KB theory of hydrotropy for homogeneous and inhomogeneous solutions, which connects MHC to solute-induced changes in solvent−solvent selfassociation. These theoretical developments have laid a solid foundation for the study of hydrotropy at a microscopic scale in the following ways: (i) micellar hydrotropy can be explained by drug-micelle preferential interaction, which is a generalization of the KB theory of hydrotropy for monomeric hydrotropes; (ii) the behavior of hydrotropes around the solutes, not the hydrotrope−hydrotrope interaction in the bulk solution, is a key toward understanding MHC. In particular, resolving the paradox concerning the existence of MHC for nonaggregating hydrotropes has led to a departure from the traditional view. What has emerged is the crucial importance of the structure of aqueous hydrotrope solutions around the drug molecules. On the contrary, the bulk solution structure of such mixtures has little to do with MHC. We thus believe that there is now a unified theory of hydrotropy which incorporates both monomeric and micellar hydrotropes. We have shown that there is no place for an analogy between monomeric hydrotropy and micellization.

APPENDIX B Here we derive eqs 35 and 36 for an inhomogeneous solution in the grand canonical ensemble. Derivation of eq 35

Consider a system which consists of water and hydrotrope molecules. The grand potential J can be expressed in terms of grand partition function Ξ in the following manner: J = −kT ln Ξ(V , μ1 , μ2 )

μu* = Ju − J0 = −kT ln

RTdn3 = n1n3G31dμ1 + n3(nmG̅3m + m)dμ2

(A2)

−

⎛ ∂μ2 ⎞ 1 RT = ⎜ ⎟ nm [m̅ + nm(Gmm ⎝ ∂nm ⎠T , P , n → 0 ̿ − G̅1m)] u

∂Ξ 0(V0 , μ1 , μ2 )/∂μi ⎤ ⎥d μ = Ξ 0(V0 , μ1 , μ2 ) ⎥⎦ i

∑ [⟨Ni⟩u − ⟨Ni⟩0 ]dμi i

Equation B3 is useful in making order of magnitude observations, which will be crucial and indispensable for the subsequent derivations. Here we note that, although ⟨Ni⟩u and ⟨Ni⟩0 are both O(V), their difference is O(1), i.e., an intensive thermodynamic quantity. Since the observable quantities in the present developments are O(1), the quantities used for physical interpretations will also be restricted to O(1) ones. In order to derive eq 35 from eq B3, we need to consider the following two points: (i). Vu and V0. The effect of the solute is localized in a region close to it, and there is no O(1) contribution to thermodynamic quantities from the region far away from the solute. This is satisfied when the following occurs:

Vu − V0 = o(1)

(B4)

where o(1) means a variable which vanishes in the limit of large system size (thermodynamic limit).48 This guarantees that we can set Vu = V0 ≡ V in the following equations. (ii). Pressure. Since Vu = V0 ≡ V, the pressure can be written in the following expression common for both the systems with and without the solute (in the following, equations valid in both systems will be written without the subscript u or 0):

(A3)

By eliminating dμ1 or dμ2 from eq A3 using the Gibbs−Duhem equation (eq 19), we obtain the following:

u

(B2)

(B3)

RTdnm = RTd(n2 + mn3)

⎛ ∂μ1 ⎞ 1 RT =− ⎜ ⎟ n1 [m̅ + nm(Gmm ⎝ ∂nm ⎠T , P , n → 0 ̿ − G̅1m)]

Ξ 0(V0 , μ1 , μ2 )

⎡ ∂Ξu(Vu , μ , μ )/∂μ i 1 2 −dμu* = kT ∑ ⎢ ⎢ ( V , , ) Ξ μ μ ⎣ u u i 1 2

where G̅ 2m and G̅ 3m are introduced by eq 26. From eqs A1 and A2, we obtain the following:

= n1nmG̅1mdμ1 + nm(nmGmm ̿ + m̅ )dμ2

Ξu(Vu , μ1 , μ2 )

where the subscripts u and 0 denote the systems with and without the solute, respectively. From eq B2 follows the Gibbs−Duhem relationship, which will be our staring point:

APPENDIX A Here we present an alternative derivation of eq 20, based upon the Gibbs−Duhem equation.6,7,23,27,40,41 By equating the species u in eq 7 with the species 2 and 3 at dT = 0, respectively, we obtain the following: (A1)

(B1)

in which the temperature T, kept constant throughout the discussion, is omitted. We consider two systems, with and without the solute. When the solute is present, it is fixed at the origin and acts as the source for an external field for the water and hydrotrope molecules. In this case, the solution system is inhomogeneous. When the solute is absent, the system consists only of water and hydrotrope and is homogeneous. The chemical potential of the solute can be expressed in terms of the grand partition functions in the following manner:

■

RTdn2 = n1n2G21dμ1 + n2(nmG̅2m + 1)dμ2

Article

(A4)

p=

kT ln Ξ(V , μ1 , μ2 ) V

+ o(1)

(B5)

This leads to the following expression which will be useful in conjunction with eq B3:

(A5) 10521


The Journal of Physical Chemistry B dp = =

kT V

∑ i

∑ i

∂Ξ(V , μ1 , μ2 )/∂μi Ξ(V , μ1 , μ2 )

Article

⎛ ∂μ* ⎞ ⟨N1⟩0 −⎜⎜ u ⎟⎟ = ⟨N1⟩u − ⟨N2⟩u + o(1) ⟨N2⟩0 ⎝ ∂μ1 ⎠T , P , N → 0

dμi + o(1)

⟨Ni⟩ dμi + o(1) V

The first goal is to show that an expression equivalent to eq C1 can be derived for homogeneous solutions. To do so, let us combine eqs 11 and 23, which yields,

(B6)

which, under a constant pressure condition, yields the following:

⎛ ∂μ* ⎞ ⎜⎜ u ⎟⎟ ⎝ ∂μ1 ⎠

∂μ2

⟨N ⟩/V =− 1 + o(1) ∂μ1 ⟨N2⟩/V

(B7)

⟨Ni⟩u =

This requires the differentiation of ⟨Ni⟩u − ⟨Ni⟩0, as well as ⟨Ni⟩/ V with respect to μ1. We start from the following grand canonical expression:

(C2)

⟨NuNi⟩ ⟨Nu⟩

(C3)

⟨Ni⟩0 = ⟨Ni⟩

(C4)

Our next goal is to derive eq 36 for the homogeneous solution. This is done by differentiating eq C2 with respect to μ1 using eqs D1 and D2.

∑N Ni exp(β ∑k μk Nk)Q (V , N1 , N2) (B8)

⎡ ⎛ ∂ 2μ* ⎞ ⟨NuN2⟩ ⎢ 2 2⟨N1⟩ = ⟨N1 ⟩ − ⟨N1N2⟩ kBT ⎜⎜ u2 ⎟⎟ ⎢ ⟨N2⟩ ⟨ ⟩⟨ ⟩ N N ∂ μ ⎝ 1 ⎠T , P , N → 0 u 2 ⎣

(i). Differentiation of ⟨Ni⟩u − ⟨Ni⟩0. From eq B8, we have the following:

u

⎤ ⎡ ⎛ ⟨N ⟩ ⎞ ⟨N N 2⟩ 2⟨N1⟩ ⟨NuN1N2⟩ + ⎜ 1 ⎟ ⟨N22⟩⎥ − ⎢ u 1 − ⎥⎦ ⎢⎣ ⟨Nu⟩ ⟨N2⟩ ⟨Nu⟩ ⎝ ⟨N2⟩ ⎠ 2

d(⟨Ni⟩u − ⟨Ni⟩0 ) = β ∑ [(⟨NN i k⟩u − ⟨Ni⟩u ⟨Nk⟩u ) k

− (⟨NN i k⟩0 − ⟨Ni⟩0 ⟨Nk⟩0 )]dμk

⟨N1⟩ NuN2 ⟨NuN1⟩ − ⟨Nu⟩⟨N2⟩ ⟨Nu⟩

where refers to the ensemble average in the homogeneous treatment. Equivalence between eqs C1 and C2 is guaranteed by the following correspondence between homogeneous and inhomogeneous number expressions:

Derivation of eq 36

∑N exp(β ∑k μk Nk)Q (V , N1 , N2)

=

T , P , Nu → 0

Equation 35 can be derived by combining eqs B3 and B7. Actually, (⟨N1⟩/V)/(⟨N2⟩/V) in eq B7 can refer to either of (⟨N1⟩u/V)/(⟨N2⟩u/V) and (⟨N1⟩0/V)/(⟨N2⟩0/V); only an o(1) difference is caused by the choice, and the expression with (⟨N1⟩0/V)/(⟨N2⟩0/V) is shown as eq 35.

⟨Ni⟩ =

(C1)

u

2 ⎛ ⟨N1⟩ ⎞2 ⟨NuN22⟩ ⎤ ⎛ ⟨NuN1⟩ ⟨N1⟩ ⟨NuN2⟩ ⎞ ⎥ +⎜ +⎜ − ⎟ ⎟ ⟨N2⟩ ⟨Nu⟩ ⎠ ⎝ ⟨N2⟩ ⎠ ⟨Nu⟩ ⎥⎦ ⎝ ⟨Nu⟩

(B9)

Since the covariance in eq B9 are O(V) for both the systems with and without the solute and the perturbation due to the solute is restricted to the region close it, the [] term of the righthand side is also O(1), i.e., intensive. Thus, we obtain the following:

(C5)

Equation C5 can be rewritten using the correspondence to inhomogeneous number averages (eqs C3 and C4) in the following manner:

⎡ {(⟨NN ⟩ − ⟨N ⟩ ⟨N ⟩ ) ⎤ 1u i 1u i u ⎢ ⎥ − (⟨NN ⎢ i 1⟩0 − ⟨Ni⟩0 ⟨N1⟩0 )} ⎥ ∂(⟨Ni⟩u − ⟨Ni⟩0 ) ⎥ = β ⎢⎢ ⟨N ⟩ ⎥ 1 ∂μ1 i 2⟩u − ⟨Ni⟩u ⟨N2⟩u )⎥ ⎢− ⟨N ⟩ {(⟨NN 2 ⎢ ⎥ ⎣ − (⟨NN i 2⟩0 − ⟨Ni⟩0 ⟨N2⟩0 )} ⎦

⎡ ⎛ ∂ 2μ* ⎞ ⟨N2⟩u ⎢ 2 2⟨N1⟩0 = ⟨N1 ⟩0 − ⟨N1N2⟩0 kBT ⎜⎜ u2 ⎟⎟ ⟨N2⟩0 ⎢⎣ ⟨N2⟩0 ⎝ ∂μ1 ⎠T , P , N → 0 u

⎤ ⎡ ⎛ ⟨N ⟩ ⎞ 2⟨N1⟩0 + ⎜ 1 0 ⎟ ⟨N22⟩0 ⎥ − ⎢⟨N12⟩u − ⟨N1N2⟩u ⎥⎦ ⎢⎣ ⟨N2⟩0 ⎝ ⟨N2⟩0 ⎠ 2

(B10)

⎤ ⎛ ⎛ ⟨N ⟩ ⎞2 ⎞2 ⟨N1⟩0 + ⎜ 1 0 ⎟ ⟨N22⟩u ⎥ + ⎜⟨N1⟩u − ⟨N2⟩u ⎟ ⎥⎦ ⎝ ⟨N2⟩0 ⎝ ⟨N2⟩0 ⎠ ⎠

Note that ⟨N1⟩/⟨N2⟩ without the subscript in eq B10 can be set to either of ⟨N1⟩u/⟨N2⟩u and 0/0; this is because the difference between u/u and 0/0 is o(1), and the {} term is O(1). (ii). Differentiation of ⟨Ni⟩/V. See Appendix D; same as homogeneous solutions. Equation 36 can be derived straightforwardly using eqs B10 and D1.

(C6)

A straightforward algebra shows that eq C6 is identical to eq 36.

■

APPENDIX D Here we derive the following relationships for homogeneous solutions.

■

⎛ ∂ ⎛ ⟨N ⟩ ⎞⎞ ⟨NN n ⟨NN ⟩ i 1⟩ = − 1 i 2 kBT ⎜⎜ ⎜ i ⎟⎟⎟ ⎝ ⎠ V n2 V ⎝ ∂μ1 V ⎠T , P , N → 0

APPENDIX C Here we show that the results for the inhomogeneous solution (eqs 35 and 36) are equivalent to those for homogeneous solutions; the solution is termed inhomogeneous when the solute is fixed at the origin and is called homogeneous when the solute position is not fixed just as water and hydrotrope. Equation 35 can easily be rewritten in the following form:

u

(D1)

⎛ ∂ ⎛ ⟨N N ⟩ ⎞⎞ ⟨NuNN n ⟨N NN ⟩ i 1⟩ = − 1 u i 2 kBT ⎜⎜ ⎜ u i ⎟⎟⎟ ⎝ ⎠ ∂ μ V V n V ⎝ 1 ⎠T , P , N → 0 2 u

(D2) 10522



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⎛ ∂μ ⎞ n ⎜⎜ 2 ⎟⎟ =− 1 n2 ⎝ ∂μ1 ⎠T , P , N → 0

where n1 = ⟨N1⟩/V and n2 = ⟨N2⟩/V. Here we focus on the derivation of eq D1. Equation D2 can be derived in a similar manner. Step 1:

u

Derivation

⎛ ∂n ⎞ kBT ⎜⎜ i ⎟⎟ ⎝ ∂μ1 ⎠T , P , N → 0

Gibbs−Duhem equation at Nu → 0 leads to eq D9. Combining eqs D3, D5 and D9 yields eq D1. Equation D2 can be derived in a similar manner.

u

⎛ ∂n ⎞ = kBT ⎜⎜ i ⎟⎟ ⎝ ∂μ1 ⎠T , μ

2 , Nu → 0

■

⎛ ∂n ⎞ ⎛ ∂μ ⎞ ⎜⎜ 2 ⎟⎟ + kBT ⎜⎜ i ⎟⎟ μ ∂ ⎝ 2 ⎠T , μ , N → 0 ⎝ ∂μ1 ⎠T , P , N → 0 1

u

u

*Tel: +44 1904 328281; fax: +44 1904 328281; e-mail: seishi. [email protected]. *Tel: +81 6 6850 6565; fax: +81 6 6850 6343; e-mail: [email protected].

Derivation

We express ni in terms of T,μ1,μ2,Nu. Under constant Nu and T, we have the following: ⎛ ∂n ⎞ ⎛ ∂n ⎞ dni = ⎜⎜ i ⎟⎟ dμ1 + ⎜⎜ i ⎟⎟ dμ2 ⎝ ∂μ1 ⎠T , μ , N → 0 ⎝ ∂μ2 ⎠T , μ , N → 0 u

1

u

Notes

The authors declare no competing financial interest.

■

(D4)

ACKNOWLEDGMENTS We are grateful to Eleanor Daly for her help to the precursor of this work, to Steven Abbott for many hours of stimulating discussions and his careful comments, and to Jonathan Booth for his technical assistance. This work is supported by the Grants-inAid for Scientific Research (Nos. 21300111, 23651202, and 26240045) from the Japan Society for the Promotion of Science, by the Grant-in-Aid for Scientific Research on Innovative Areas (No. 20118002) and the Elements Strategy Initiative for Catalysts and Batteries from the Ministry of Education, Culture, Sports, Science, and Technology, and by Computational Materials Science Initiative, Theoretical and Computational Chemistry Initiative, HPCI Strategic Program, HPCI System Research Project (Project IDs:hp120093, hp130022, hp140156, and hp140214), and Strategic Programs for Innovative Research of the Next-Generation Supercomputing Project.

Partial differentiation with respect to μ1 yields eq D3. Step 2: ⎛ ∂n ⎞ kBT ⎜⎜ i ⎟⎟ ⎝ ∂μ1 ⎠T , μ

= 2 , Nu → 0

⟨NN i 1⟩ − ⟨NN i 1⟩ , V

⎛ ∂n ⎞ ⟨NN i 2⟩ − ⟨NN i 2⟩ = kBT ⎜⎜ i ⎟⎟ V ⎝ ∂μ2 ⎠T , μ , N → 0 1

(D5)

u

Derivation

Firstly, we note that μ1 does not depend on V. Hence: ⎛ ∂n ⎞ ⎜⎜ i ⎟⎟ ⎝ ∂μ1 ⎠T , μ

= 2 , Nu → 0

1 ⎛ ∂⟨Ni⟩ ⎞ ⎜⎜ ⎟⎟ V ⎝ ∂μ1 ⎠

1

■

T , μ2 , Nu → 0

⎛ ∂n ⎞ 1 ⎛ ∂⟨Ni⟩ ⎞ ⎜⎜ i ⎟⎟ ⎟⎟ = ⎜⎜ ⎝ ∂μ2 ⎠T , μ , N → 0 V ⎝ ∂μ2 ⎠T , μ , N → 0 u

1

Secondly, we use the grand canonical expression of ⟨Ni⟩: Ni =

1 Ξ(T , V , μ1 , μ2 )

∑ ∑

βN1μ1 βN2μ2 Ne e i

N1≥ 0 N2 ≥ 0

Q (T , V , N1 , N2)

(D7)

where Q(T,V,N1,N2) is the canonical partition function, and β = (kBT)−1. Carrying out the partial differentiation yields the following: ⎛ ∂⟨N ⟩ ⎞ 1 i ⎟⎟ = kBT ⎜⎜ μ ∂ Ξ ( T , V , μ1 , μ2 ) ⎝ 1 ⎠T , μ , N → 0 2

u

∑ ∑ N1≥ 0 N2 ≥ 0

βN1μ1 βN2μ2 NN e Q (T , V , N1 , N2) − ⟨Ni⟩⟨N1⟩ i 1e = ⟨NN ⟩ − ⟨ N i 1 i⟩⟨N1⟩

⎛ ∂⟨N ⟩ ⎞ 1 i ⎟⎟ = kBT ⎜⎜ μ ∂ Ξ ( , , μ1 , μ2 ) T V ⎝ 2 ⎠T , μ , N → 0 2

u

REFERENCES

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2

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∑ ∑ N1≥ 0 N2 ≥ 0

βN1μ1 βN2μ2 NN e Q (T , V , N1 , N2) − ⟨Ni⟩⟨N2⟩ i 2e = ⟨NN i 2⟩ − ⟨Ni⟩⟨N2⟩

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Article

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Hydrotropy: MonomerâMicelle Equilibrium and Minimum Hydrotrope

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