Origin of nonlinearity in phase solubility:
Solubilisation by cyclodextrin beyond stoichiometric
complexation
Thomas W. J. Nicol,1 Nobuyuki Matubayasi2,3, and Seishi Shimizu1*
1
York Structural Biology Laboratory, Department of Chemistry, University of York, Heslington,
York YO10 5DD, United Kingdom
2
Division of Chemical Engineering, Graduate School of Engineering Science, Osaka University,
Toyonaka, Osaka 560-8531, Japan
3
Elements Strategy Initiative for Catalysts and Batteries, Kyoto University, Katsura, Kyoto 615-
8520, Japan
KEYWORDS: Hydrotropy; McMillan-Mayer Theory; Water; Cosolvent; Cooperativity;
Minimum CD Concentration
AUTHOR INFORMATION
Corresponding Author:
Seishi Shimizu
1
York Structural Biology Laboratory, Department of Chemistry, University of York, Heslington,
York YO10 5DD, United Kingdom
Tel: +44 1904 328281, Fax: +44 1904 328281, Email: [email protected]
ABSTRACT
Low solubility of drugs, which poses a serious problem in drug development, can in part be
overcome by the use of cyclodextrins (CD) and their derivatives. Here, the key to solubilisation is
identified as the formation of inclusion complexes with the drug molecule. If inclusion
complexation were the only contribution to drug solubility, it would increase linearly with CD
concentration (as per the Higuchi-Connors model); this is because inclusion complexation is a 1:1
stoichiometric process. However, solubility curves often deviate from this linearity, whose
mechanism is yet to be understood. Here we aim to clarify the origin of such non-linearity, based
upon the Kirkwood-Buff and the McMillan-Mayer theories of solutions. The rigorous statistical
thermodynamic theory shows that solubilisation non-linearity can be rationalised by two
contributions: CD-drug interaction and the drug-induced change of CD-CD interaction.
1. Introduction
Cyclodextrin (CD) complexes have been used in pharmacy to improve the aqueous solubility of
drug molecules, as well as targeted drug delivery in terms of controlled release.1–5 The ability of
CDs to enhance solubility has been identified, within the literature, to stem from its formation of
inclusion complexes with the drug molecule.6–8 However, accumulating evidence indicates that
2
CD-drug complexation may not be the only driving force of CD-induced solubilisation.9–14 Despite
this, elucidating quantitatively the origin and relative importance of non-complexation
contributions has not yet been successful due to the lack of a rigorous theoretical framework. The
aim of this paper is to clarify the origin and the magnitudes of such contributions, based upon a
rigorous statistical thermodynamic theory.
The classical model for drug solubilisation by CDs was established by Higuchi and Connors
(HC), which attributed the cause of solubility increase to stoichiometric 1:1 inclusion
complexation.9 The HC model in its predominant form predicts a linear increase of solubility with
CD concentration, in agreement with many experimental data.10,11 However, deviations from this
linear solubility increase have also been reported in the literature,11–14 and standard nomenclature
has been established to classify the different ways in which the so called phase solubility curves
deviates from linearity (Figure 1).9,15 In addition, to address such deviation from linearity, the HC
model was extended to incorporate different stoichiometries;12,16–18 yet these treatments cannot
accurately model systems with large numbers of weak interactions.19,20 Hence, prominent
questions still remain: what is the origin of non-linearity in phase solubility diagrams? What is the
origin of the non-stoichiometric driving forces?
To give a clear answer to this historical question, a link between solubility behaviour and the
molecular interactions causing it is indispensable. Such a link, provided by the rigorous KirkwoodBuff (KB) theory of solutions,21–27 has recently made significant progress in clarifying the
mechanism of drug solubilisation by the addition of hydrotropes.28–33 Drug solubility commonly
exhibits non-linear dependence on hydrotrope concentration, often with a sudden offset of
3
solubility increase at a threshold hydrotrope concentration referred to as the minimum hydrotrope
concentration (MHC).28,29 The origin of MHC has been attributed by our rigorous KB-based theory
to the solute-induced enhancement of the interactions among hydrotropes, which gives rise to a
cooperative increase in solubility, showing that many-body interactions indeed play a crucial role
in solubilisation.29–36 Our strategy, drawing from the theory of hydrotropy, is to construct a
rigorous statistical thermodynamic theory of drug solubilisation by CDs to replace the purely
phenomenological HC model.
In this vein, how drug solubility depends on CD concentration has previously been modelled by
the McMillan-Mayer (MM) theory of solutions,37–40 another rigorous statistical thermodynamic
theory;41–45 and though the MM theory has successfully produced regression equations for CDdrug systems,37–40 the molecular basis of non-linearity has remained unresolved.46–49 Here a clear
understanding of the relationship between non-linear solubility and many-body interactions is
indispensable.
The goal of this paper is to characterise drug solubility in ternary CD-drug aqueous systems,
utilising statistical thermodynamics, through a rigorous theory superior to stoichiometric models.
The relationship between non-linear solubility and many-body interactions will be clarified from
both KB and MM theoretical bases. Such a theory will help extract molecular scenarios from the
wealth of phase solubility data in the presence of CDs and provide a long-awaited understanding
of the origin of the non-linear phase solubility curve.
2. Theory of solubilisation in the presence of cyclodextrin
4
2.1. The goal and the formalism
The goal of this paper is to rationalise how the solubility enhancement of drugs depends on the
CD concentration and deviate from linearity.
To this end, consider a three component solution consisting of a drug (solute) (𝑖 = 𝑢), water (𝑖 =
1), and CD (𝑖 = 2) molecules. Let 𝜇𝑖 and 𝑁𝑖 respectively be the chemical potential and the number
of species 𝑖, 𝑇 and 𝑉 be the temperature and the pressure of the system. Let 𝜌𝑖 = 𝑁𝑖 /𝑉 be the
number density or concentration of species 𝑖; in the grand canonical ensemble, the ensemble
1
average of 𝑁𝑖 is used instead to define 𝜌𝑖 , such that 𝜌𝑖 = 〈𝑁𝑖 〉/𝑉. The convention 𝛽 = 𝑘𝑇 (where
𝑘 is the Boltzmann constant) is used throughout. Since 𝑇 is kept constant throughout this paper, it
is often omitted in the subsequent discussions.
Solubility enhancement is quantified by the change of solute’s hydration free energy 𝜇𝑢∗ in the
presence of CD from its value in pure water 𝜇𝑢0∗ .24–26,28–32 Let us express this hydration free energy
change, Δ𝜇𝑢∗ = 𝜇𝑢∗ − 𝜇𝑢0∗ in terms of CD concentration 𝜌2 . To achieve this goal, let us first expand
𝜇𝑢∗ in terms of CD fugacity, 𝜆2 = exp(𝛽𝜇2 ) for convenience, and thereafter express 𝜆2 in terms of
𝜌2 . To this end, we need to calculate the following partial derivatives
𝜕𝜇 ∗
Δ𝜇𝑢∗ = ( 𝜕𝜆𝑢)
2
1 𝜕2 𝜇∗
𝑇,𝑃;𝑁𝑢 →0,𝜆2 →0
𝜆2 + 2 ( 𝜕𝜆2𝑢)
2
𝑇,𝑃;𝑁𝑢 →0.𝜆2 →0
𝜆22 + ⋯
(1)
5
We emphasise here that the insertion of the solute, as well as the addition of CD, are carried out
under constant 𝑁, 𝑃 and 𝑇. This is different from the insertion condition of the MM theory, which
is normally carried out under constant 𝜇, 𝑉 and T.41,50,51
The understanding of CD-induced solubility enhancement has thus been reduced to the
calculation of partial derivatives in Eq. (1).
2.2. Basic concepts of the inhomogeneous Kirkwood-Buff theory
The calculation of partial derivatives in Eq. (1) will be facilitated greatly by our recent
development in the inhomogeneous KB theory.30 In this section we review the basic formulae
necessary to achieve this goal.
The chemical potential of the solute fixed at an arbitrarily-chosen origin 𝜇𝑢∗ can be expressed in
terms of the grand partition functions Ξ in the following manner:30
𝜇𝑢∗ = 𝐽𝑢 − 𝐽0 = −𝑘𝑇 ln
Ξu (𝑉𝑢 ,𝜇1 ,𝜇2 )
Ξ0 (𝑉,𝜇1 ,𝜇2 )
(2)
where the subscripts u and 0 denote the systems with and without the solute, respectively. As
shown in our previous paper,30 the solution volume with the solute 𝑉𝑢 is treated as equal to that
without solute V to assure that 𝜇𝑢∗ is an intensive property. In the following, 𝑉𝑢 is thus written as 𝑉
and the volume is the same whether the solute is present or absent.
Successive differentiation of Eq. (2) with respect to 𝜇𝑖 yields the following key relationships:30
−𝑑𝜇𝑢∗ = Σ𝑖 (〈𝑁𝑖 〉𝑢 − 〈𝑁𝑖 〉)𝑑𝜇𝑖
(3)
𝑑(〈𝑁𝑖 〉𝑢 − 〈𝑁𝑖 〉) = 𝛽Σ𝑘 [(〈𝑁𝑖 𝑁𝑘 〉𝑢 − 〈𝑁𝑖 〉𝑢 〈𝑁𝑖 〉𝑢 ) − (〈𝑁𝑖 𝑁𝑘 〉 − 〈𝑁𝑖 〉〈𝑁𝑖 〉)]𝑑𝜇𝑘
(4)
6
where 〈 〉 denotes ensemble averaging. The number averages and fluctuations in Eqs. (3) and (4)
can be transformed to KB integrals through the following relationships:30
𝐺𝑖𝑗 = 𝑉 [
〈𝑁𝑖 𝑁𝑗 〉−〈𝑁𝑖 〉〈𝑁𝑗 〉
G𝑢,𝑖𝑗 = V [
𝛿
− 〈𝑁𝑖𝑗〉]
〈𝑁𝑖 〉〈𝑁𝑗 〉
(5)
𝑖
〈𝑁𝑖 𝑁𝑗 〉𝑢 −〈𝑁𝑖 〉𝑢 〈𝑁𝑗 〉𝑢
〈𝑁𝑖 〉𝑢 〈𝑁𝑗 〉𝑢
𝛿
− 〈𝑁𝑖𝑗〉 ]
(6)
𝑖 𝑢
where 𝛿𝑖𝑗 is Kronecker’s delta.
2.3. Cyclodextrin fugacity expansion of the hydration free energy
2.3.1. First order in 𝝀𝟐
𝜕𝜇 ∗
Here we evaluate 𝛽 ( 𝜕𝜆𝑢)
2
in Eq. (1). To do so, we need to bridge the 𝜇VT ensemble in
𝑇,𝑃;𝑁𝑢 →0
which the KB theory is constructed and the NPT ensemble in which solubilisation experiments are
carried out.30 To this end, we employ the following thermodynamic relationship:
𝜕𝜇 ∗
𝛽 ( 𝜕𝜆𝑢)
2
𝜕𝜇 ∗
𝑇,𝑃
= 𝛽 ( 𝜕𝜆𝑢 )
2
𝜕𝜇 ∗
𝑇,𝜇1
+ 𝛽 (𝜕𝜇𝑢)
1
𝜕𝜇
𝑇,𝜆2
(𝜕𝜆1 )
2
(7)
𝑇,𝑃
The evaluation of each term in Eq. (7) straightforwardly follows our previous paper.30 The first
term of Eq. (7), using Eqs. (3) and (4) becomes
𝜕𝜇 ∗
𝛽 ( 𝜕𝜆𝑢)
2
1
𝑇,𝜇1 ;𝑁𝑢 →0
𝜌
= − 𝜆 [〈𝑁2 〉𝑢 − 〈𝑁2 〉] = − 𝜆2 𝐺𝑢2
2
2
(8)
The second term of Eq. (7) can be evaluated by a combination of Eqs. (3) and (4), as well as the
Gibbs-Duhem equation23,24,28–31,52 as
𝜕𝜇∗
𝛽 ( 𝜕𝜇𝑢)
1
𝜕𝜇
𝑇,𝜆2 ;𝑁𝑢 →0
(𝜕𝜆1 )
2
𝜌
𝑇,𝑃;𝑁𝑢 →0
= 𝜆2 𝐺𝑢1
2
(9)
Combining Eqs. (7)-(9), we obtain
7
𝜕𝜇 ∗
𝛽 ( 𝜕𝜆𝑢)
2
𝜌
𝑇,𝑃;𝑁𝑢 →0
= − 𝜆2 (𝐺𝑢2 − 𝐺𝑢1 )
(10)
2
2.3.2. Second order in 𝝀𝟐
𝜕2 𝜇∗
Overall strategy. Here we evaluate ( 𝜕𝜆2𝑢)
2
in Eq. (1). To do so, as in 2.3.1., we need
𝑇,𝑃;𝑁𝑢 →0,𝜆2 →0
a relationship between the 𝜇VT ensemble (the KB theory) and the NPT ensemble (solubilisation
𝜕2 𝜇∗
measurements).30 To this end, we express ( 𝜕𝜆2𝑢)
2
𝜕2 𝜇∗
𝑇,𝑃;𝑁𝑢 →0
in terms of ( 𝜕𝜆2𝑢)
2
by virtue of
𝑇,𝜇1 ;𝑁𝑢 →0
the following thermodynamic relationship:
𝜕2 𝜇∗
𝛽 ( 𝜕𝜆2𝑢)
2
𝜕2 𝜇∗
𝑇,𝑃;𝑁𝑢 →0
= 𝛽 ( 𝜕𝜆2𝑢)
2
𝜕
𝜌
+ [𝜕𝜆 (𝜆2 𝐺𝑢1 )]
2
𝑇,𝜇1 ;𝑁𝑢 →0
2
𝜌
𝑇,𝜇1 ;𝑁𝑢 →0
𝜌2 (𝐺𝑢2 −𝐺𝑢1 )
𝜕
+ 𝛽𝜌 2𝜆 [𝜕𝜇 (
1 2
𝜆2
1
(11)
)]
𝑇,𝜆2 ;𝑁𝑢 →0
Derivation of the thermodynamic relationship, Eq. (11). We start from the following
thermodynamic relationship:
𝜕𝜇 ∗
𝜕
[𝜕𝜆 𝛽 ( 𝜕𝜆𝑢)
2
2
]
𝑇,𝑃 𝑇,𝑃
𝜕𝜇 ∗
𝜕
= [𝜕𝜆 𝛽 ( 𝜕𝜆𝑢)
2
2
𝜕𝜇 ∗
𝜕
]
𝑇,𝑃 𝑇,𝜇
1
+ [𝜕𝜇 𝛽 ( 𝜕𝜆𝑢)
1
2
]
𝑇,𝑃 𝑇,𝜆
2
𝜕𝜇
[𝜕𝜆1 ]
2
(12)
𝑇,𝑃
Using Eqs. (7) and (9) for the first r.h.s. term, and Eq. (10) for the second r.h.s. term, Eq. (12) can
be rewritten as
𝜕𝜇 ∗
𝜕
[𝜕𝜆 𝛽 ( 𝜕𝜆𝑢)
2
2
]
𝑇,𝑃 𝑇,𝑃
𝜕𝜇 ∗
𝜕
= [𝜕𝜆 (𝛽 ( 𝜕𝜆𝑢 )
2
2
𝜌
𝑇,𝜇1
+ 𝜆2 𝐺𝑢1 )]
2
𝑇,𝜇1
𝜌2 (𝐺𝑢2 −𝐺𝑢1 )
𝜕
− [𝜕𝜇 (
1
𝜆2
𝜌
)]
𝑇,𝜆2
[− 𝛽𝜌 2𝜆 ]
1 2
(13)
from which Eq. (11) can be obtained easily.
8
𝝏𝟐 𝝁∗
Evaluation of ( 𝝏𝝀𝟐𝒖 )
𝟐
𝜕2 𝜇∗
−𝛽 ( 𝜕𝜆2𝑢)
2
in Eq. (11). Differentiating Eq. (8) with respect to 𝜆2 we obtain
𝑻,𝝁𝟏 ;𝑵𝒖 →𝟎
1
1
= − 𝜆2 [〈𝑁2 〉𝑢 − 〈𝑁2 〉] + 𝛽𝜆2 (
2
𝑇,𝜇1 ;𝑁𝑢 →0
𝜕[〈𝑁2 〉𝑢 −〈𝑁2 〉]
𝜕𝜇2
2
(14)
)
𝑇,𝜇1 ;𝑁𝑢 →0
The second term on the r.h.s. of Eq. (14) can be rewritten with the help of Eq. (4) as
1
𝛽𝜆22
𝜕[〈𝑁2 〉𝑢 −〈𝑁2 〉]
(
𝜕𝜇2
1
)
𝑇,𝜇1 ;𝑁𝑢 →0
= 𝜆2 [〈𝑁22 〉𝑢 − 〈𝑁2 〉2𝑢 − (〈𝑁22 〉 − 〈𝑁2 〉2 )]
(15)
2
Eqs. (14) and (15) can be rewritten in terms of KB integrals, with the help of Eq. (5) and (6), as
𝜕2 𝜇∗
−𝛽 ( 𝜕𝜆2𝑢)
2
2
𝜌
𝑇,𝜇1 ;𝑁𝑢 →0
〈𝑁2 〉2𝑢 𝐺𝑢,22 −〈𝑁2 〉2 𝐺22
= (𝜆 2 ) 𝑉 {
𝝏
𝝆
Evaluation of [𝝏𝝀 (𝝀𝟐 𝑮𝒖𝟏 )]
𝟐
𝜕
2
+
𝟐
𝜌
[𝜕𝜆 (𝜆2 𝐺𝑢1 )]
2
〈𝑁2 〉2
2
in Eq. (11). Using Eqs. (3) and (4), we can show that
𝑻,𝝁𝟏 ;𝑵𝒖 →𝟎
〈𝑁2 〉2 〈𝑁1 〉𝑢 −〈𝑁1 〉 〈𝑁1 𝑁2 〉−〈𝑁1 〉〈𝑁2 〉
𝜌
𝑇,𝜇1 ;𝑁𝑢 →0
(16)
}
= − 𝜆22 𝐺𝑢1 −
𝜆22
2
〈𝑁1 〉
〈𝑁1 〉〈𝑁2 〉
〈𝑁2 〉2 〈𝑁22 〉−〈𝑁2 〉2 〈𝑁1 〉𝑢 −〈𝑁1 〉
〈𝑁2 〉2 〈𝑁1 〉𝑢 〈𝑁2 〉𝑢 〈𝑁1 𝑁2 〉𝑢 −〈𝑁1 〉𝑢 〈𝑁2 〉𝑢
+
[ 〈𝑁 〉〈𝑁 〉
2
2
〈𝑁2 〉
〈𝑁1 〉
〈𝑁1 〉𝑢 〈𝑁2 〉𝑢
𝜆2
𝜆22
1
2
−
〈𝑁1 𝑁2 〉−〈𝑁1 〉〈𝑁2 〉
〈𝑁1 〉〈𝑁2 〉
(17)
]
Eq. (17) can be rewritten in terms of KB integrals by the help of Eqs. (5) and (6) as
𝜕
𝜌2 2
𝜌2
[𝜕𝜆 (𝜆 𝐺𝑢1 )]
2
2
𝑇,𝜇1 ;𝑁𝑢 →0
𝝆
2
〈𝑁1 〉𝑢 〈𝑁2 〉𝑢 𝐺𝑢,12 −〈𝑁1 〉〈𝑁2 〉𝐺12
= (𝜆 ) 𝐺𝑢1 (𝐺22 − 𝐺21 ) + (𝜆 ) 𝑉 {
2
2
𝝆𝟐 (𝑮𝒖𝟐 −𝑮𝒖𝟏 )
𝝏
Evaluation of 𝜷𝝆 𝟐𝝀 [𝝏𝝁 (
𝟏 𝟐
𝜌2
𝟏
} (18)
in Eq. (11). Again, using Eqs. (3) and (4), we
)]
𝝀𝟐
〈𝑁1 〉〈𝑁2 〉
𝑻,𝝀𝟐 ;𝑵𝒖 →𝟎
can show that
𝜌2
𝛽𝜌1 𝜆2
−
𝜌2 (𝐺𝑢2 −𝐺𝑢1 )
𝜕
[𝜕𝜇 (
𝜆2
1
)]
=
𝑇,𝜆2 ;𝑁𝑢 →0
〈𝑁2 〉2 〈𝑁1 〉2𝑢 〈𝑁12 〉𝑢 −〈𝑁1 〉2𝑢
𝜆22
{ 〈𝑁
2
1〉
〈𝑁1 〉2𝑢
−
〈𝑁2 〉2 〈𝑁1 〉𝑢 〈𝑁2 〉𝑢 〈𝑁1 𝑁2 〉𝑢 −〈𝑁1 〉𝑢 〈𝑁2 〉𝑢
{ 〈𝑁 〉〈𝑁 〉
〈𝑁1 〉𝑢 〈𝑁2 〉𝑢
𝜆22
1
2
〈𝑁12 〉−〈𝑁1 〉2
〈𝑁1 〉2
}−
−
〈𝑁2 〉2 〈𝑁1 〉𝑢 −〈𝑁1 〉 〈𝑁1 𝑁2 〉−〈𝑁1 〉〈𝑁2 〉
𝜆22
〈𝑁1 〉
{
〈𝑁1 〉〈𝑁2 〉
〈𝑁1 𝑁2 〉−〈𝑁1 〉〈𝑁2 〉
−
〈𝑁1 〉〈𝑁2 〉
〈𝑁12 〉−〈𝑁1 〉2
〈𝑁1 〉2
} (19)
}
9
Eq. (19) can be rewritten in terms of the KB integrals through the help of Eqs. (5) and (6) as
𝜌2
𝛽𝜌1 𝜆1
𝜕
𝜌
[𝜕𝜇 (𝜆2 (𝐺𝑢2 − 𝐺𝑢1 ))]
1
2
2
𝜌
− ( 𝜆2 ) 𝑉 {
𝜌
2
= (𝜆 2 ) 𝑉 {
𝑇,𝜆2 ;𝑁𝑢 →0
𝑉
𝑉
)−〈𝑁1 〉2 (𝐺11 +〈𝑁 〉)
〈𝑁1 〉𝑢
1
〈𝑁1 〉2
〈𝑁1 〉2𝑢 (𝐺𝑢,11 +
2
〈𝑁1 〉𝑢 〈𝑁2 〉𝑢 𝐺𝑢,12 −〈𝑁1 〉〈𝑁2 〉𝐺12
〈𝑁1 〉〈𝑁2 〉
2
𝜌
}
2
1
} − (𝜆2 ) 𝐺𝑢1 (𝐺12 − 𝐺11 − 𝜌 )
2
(20)
1
Summary. Combining Eqs. (11), (16), (18) and (20) we obtain
𝜕2 𝜇∗
𝛽 ( 𝜕𝜆2𝑢)
2
𝜌
𝑇,𝑃;𝑁𝑢 →0
2
2
𝜌
1
= − (𝜆2 ) 𝑊 + (𝜆2 ) 𝐺𝑢1 (𝐺11 − 2𝐺12 + 𝐺22 + 𝜌 )
2
2
(21)
1
where 𝑊 denotes the contribution from solute-induced change of self-association, defined as
𝑊 ≡ 𝑉{
𝑉
𝑉
)−〈𝑁1 〉2 (𝐺11 +〈𝑁 〉)
〈𝑁1 〉𝑢
1
〈𝑁1 〉2
〈𝑁1 〉2𝑢 (𝐺𝑢,11 +
−2
〈𝑁1 〉𝑢 〈𝑁2 〉𝑢 𝐺𝑢,12 −〈𝑁1 〉〈𝑁2 〉𝐺12
〈𝑁1 〉〈𝑁2 〉
+
〈𝑁2 〉2𝑢 𝐺𝑢,22 −〈𝑁2 〉2 𝐺22
〈𝑁2 〉2
}
(22)
∗
1 𝜕 2 𝜇𝑢
The equivalence between Eq. (21) and our previous expression for (
𝛽
𝜕𝜇22
30
)
can be
𝑇,𝑃;𝑁𝑢 →0
shown straightforwardly in Appendix B.
2.3.3. The CD fugacity expansion of hydration free energy to describe solubility enhancement
Now we can conclude our quest for the 𝜆2 -expansion of Δ𝜇𝑢∗ . Combining Eqs. (1), (10) and (21),
we obtain
1 𝜌
𝜌
2
1
𝛽Δ𝜇𝑢∗ = − (𝜆2 ) (𝐺𝑢2 − 𝐺𝑢1 )0 𝜆2 − 2 (𝜆2 ) {𝑊 − 𝐺𝑢1 (𝐺11 − 2𝐺12 + 𝐺22 + 𝜌 )} 𝜆22 + ⋯
2
0
2
0
1
(23)
0
where the subscript 0 denotes the 𝜆2 → 0 limit, and 𝜆2 → 0 corresponds to 𝜌2 → 0 with finite
𝜌2 /𝜆2 .
10
∗
Solubility increase, namely 𝑒 −𝛽Δ𝜇𝑢 ,28–31 can be expressed in the following manner using Eq.
(23):
𝜌
∗
𝑒 −𝛽Δ𝜇𝑢 = 1 + (𝜆2 ) (𝐺𝑢2 − 𝐺𝑢1 )0 𝜆2
2
0
1 𝜌
2
1
+ 2 (𝜆2 ) {𝑊 − 𝐺𝑢1 (𝐺11 − 2𝐺12 + 𝐺22 + 𝜌 ) + (𝐺𝑢2 − 𝐺𝑢1 )2 } 𝜆22 + ⋯
2
0
1
(24)
0
2.4. CD concentration expansion of solubilisation
The solubility of drugs in the presence of cyclodextrin is commonly plotted against the CD
concentration, 𝜌2 in phase solubility diagrams.9,48,49 Hence we need to change the variable of Eq.
(24) from 𝜆2 to 𝜌2 . Note that the procedure of 𝜆2 to 𝜌2 conversion is different from that of the MM
theory ( 𝜇 VT ensemble),44,45,51,53 since the experiments are conducted entirely in the NPT
ensemble.
Thus our goal in the forthcoming section is to determine a relationship between 𝜆2 and 𝜌2 in the
NPT ensemble. To do so, let us start from a well-known relationship from KB theory, derived in
the NPT ensemble:28–31,44,45
𝛽(
𝜕𝜇2
)
𝜕𝜌2 𝑇,𝑃
=
1
𝜌2
−
𝐺22 −𝐺21
1+𝜌2 (𝐺22 −𝐺21 )
(25)
Eq. (25) indicates that 𝜇2 can be expanded in the following manner
𝛽𝜇2 = 𝛽𝜇20 + ln 𝜌2 + 𝑎𝜌2 + ⋯
(26)
where 𝜇20 is a constant. Note that ln 𝜌2 originates from translational degrees of freedom.
Substituting Eq. (26) into Eq. (25) at the 𝜌2 → 0 limit yields
11
(27)
𝑎 = −(𝐺22 − 𝐺21 )0
Note that 𝜌2 → 0 is equivalent to 𝜆2 → 0. Eq. (27) leads to the following 𝜌2 expansion for 𝜆2
𝜆
0
𝜆2 = 𝑒 𝛽𝜇2 = 𝑒 𝛽𝜇2 +ln 𝜌2 −(𝐺22 −𝐺21 )0 𝜌2 +⋯ = (𝜌2 ) 𝜌2 𝑒 −(𝐺22 −𝐺21 )0 𝜌2
2
0
𝜆
= (𝜌2 ) 𝜌2 [1 − (𝐺22 − 𝐺21 )0 𝜌2 + ⋯ ]
2
(28)
0
𝜆
where the existence of (𝜌2 ) is justified clearly from the limiting behaviour.
2
0
Using Eq. (28), the 𝜆2 expansion given by Eq. (24) can now be rewritten as
∗
𝑒 −𝛽Δ𝜇𝑢 = 1 + (𝐺𝑢2 − 𝐺𝑢1 )0 𝜌2
1
(29)
1
+ 2 {𝑊 − 𝐺𝑢1 (𝐺11 − 2𝐺12 + 𝐺22 + 𝜌 ) + (𝐺𝑢2 − 𝐺𝑢1 )2 − 2(𝐺𝑢2 − 𝐺𝑢1 )(𝐺22 − 𝐺21 )} 𝜌22 + ⋯
1
0
This is the CD concentration dependence of solubilisation that we sought after.
2.5. Comparison to the McMillan-Mayer theory of solutions
Part of the 𝜆2 expansion presented above can be derived also from the MM theory. To demonstrate
this, let us first appreciate the difference between the NPT process (Eq. (1)) and the osmotic
equilibrium treated by the MM theory. The latter corresponds to
𝜕𝜇 ∗
∗
Δ𝜇𝑢,𝑀𝑀
= ( 𝜕𝜆𝑢)
2
1 𝜕2 𝜇∗
𝑇,𝜇1 ;𝑁𝑢 →0,𝜆2 →0
𝜆2 + 2 ( 𝜕𝜆2𝑢)
2
𝑇,𝜇1 ;𝑁𝑢 →0,𝜆2 →0
𝜆22 + ⋯
(30)
Note that 𝜇1 , instead of 𝑃, is kept constant in the partial derivatives; such partial derivatives have
already been evaluated in Eqs. (8) and (16). Combining them with Eq. (30) yields
1 𝜌
𝜌
2
〈𝑁2 〉2𝑢 𝐺𝑢,22 −〈𝑁2 〉2 𝐺22
0
〈𝑁2 〉2
∗
βΔ𝜇𝑢,𝑀𝑀
= − (𝜆2 ) (𝐺𝑢2 )0 𝜆2 − 2 (𝜆2 ) {𝑉
2
0
2
} 𝜆22 + ⋯
(31)
0
12
Eq. (31), which has been derived via the KB theory, can also be derived via the MM theory,
which is shown in Appendices C and D. The insight from this comparison is that the solute-induced
CD-CD interaction change term comes from MM theory, whereas those in CD-water and waterwater interactions account for the difference between NPT and 𝜇VT ensembles.
3. Molecular basis of non-linearity in the phase solubility diagram
Here we apply our theory to analyse the experimental solubility of drugs in the presence of CDs.
The aim is to elucidate how non-linearity in phase solubility diagrams change as we modify the
molecular structures of the drug and CD.
3.1. Origin of non-linearity in the phase solubility diagram
According to the classical 1:1 stoichiometric CD-solute complexation model of Higuchi and
Connors,9 solubility is expected to increase linearly with the CD concentration. In reality,
deviations from such linearity have been documented, suggesting that factors other than 1:1 CDsolute complexation are at work.11–14,54 Commonly observed modes for deviations from linearity
in phase solubility data have been classified (Figure 1),
9,15
yet the molecular-based mechanism
behind these different modes has remained a mystery.46,47 This was due to the lack of a true
microscopic theory, which has been overcome by Eq. (29).
13
Interpreting phase solubility diagrams using the MM theory requires us first of all to have a clear
physical meaning of each term in Eq. (29).
The 𝝆 term of Eq. (29), (𝐺𝑢2 − 𝐺𝑢1 )0 𝜌2 , clearly shows its equivalence to the 𝜌2 → 0
expression of preferential solvation within KB theory.27 Thus the linear term of phase diagram,
interpreted in the HC model as representing solute-CD complexation9 has now been generalised
to drug-CD preferential interaction .
The 𝝆𝟐 term of Eq. (29) accounts for deviations from linear phase solubility. Positive and
negative deviations from linearity, commonly referred to as 𝐴𝑃 and 𝐴𝑁 (Figure 1) can be attributed
to its sign, yet it has a very complex expression with many terms. We aim, first of all, to simplify
the 𝜌2 term of Eq. (29) based upon experimental data. As will be demonstrated in Section 3.2, the
following order of magnitude relationships hold true: |𝐺𝑢2 | ≫ |𝐺𝑢1 |, |𝐺𝑢2 | ≫ |𝐺22 | and |𝐺𝑢2 | ≫
1
|𝐺21 |. Note also that, at 𝜌2 → 0 limit, 𝐺11 + =
𝜌
1
𝜅𝑇
𝛽
, where 𝜅𝑇 is the isothermal compressibility,
which makes a negligibly small contribution.45 Considering the above contributions,
1
𝐺𝑢1 (𝐺11 − 2𝐺12 + 𝐺22 + 𝜌 ) can be ignored. Hence
1
2
(𝐺𝑢2 − 𝐺𝑢1 )2 − 2(𝐺𝑢2 − 𝐺𝑢1 )(𝐺22 − 𝐺21 ) ≃ 𝐺𝑢2
(32)
Using Eq. (32), Eq. (29) can be simplified as
∗
1
2 } 2
𝑒 −𝛽Δ𝜇𝑢 = 1 + (𝐺𝑢2 )0 𝜌2 + 2 {𝑊 + 𝐺𝑢2
0 𝜌2 + ⋯
(33)
The following two factors contribute to the non-linear term in Eq. (33):
(i)
𝑊, defined by Eq. (22), represents solute-induced change of the self-association of CD
and water, as has been introduced previously.30,31 Because CD-solute interactions are
14
much stronger than those between CD and water, we expect this term to be dominated
by 𝑉
〈𝑁2 〉2𝑢 𝐺𝑢,22 −〈𝑁2 〉2 𝐺22
〈𝑁2 〉2
, the solute-induced CD-CD interaction change.31 Justification of
this approximation is given in Appendix E. Furthermore, at the thermodynamic limit (i.e.,
𝐺𝑢𝑖
𝑉
𝑉
(ii)
→ 0), we can show
〈𝑁2 〉2𝑢 𝐺𝑢,22 −〈𝑁2 〉2 𝐺22
〈𝑁2 〉2
= 𝑉 [(1 +
𝐺𝑢2 2
𝑉
) 𝐺𝑢,22 − 𝐺22 ] ≃ 𝑉(𝐺𝑢,22 − 𝐺22 )
(34)
𝐺𝑢2 can be interpreted as the CD-solute interaction.24,28–33 Hence the origin of nonlinearity can be attributed to the interplay between CD-solute interaction and soluteinduced increment of CD-CD interaction. The significance of each of the above factors’
contributions will be further discussed below through the use of experimental data.
On the other hand, phase solubility exhibits an apparently linear behavior when the third term
1
2
of Eq. (33) is negligibly small compared to the second, namely |𝐺𝑢2 | ≫ |2 𝑊 + 𝐺𝑢2
| 𝜌2 , where 𝜌2
here should represent the maximum CD concentration for experiment. To clarify the possible
scenarios by which this condition is satisfied or broken would require an extensive analysis of
experimental data, however, solute-induced weakening of CD-CD interaction sufficiently strong
2
(but not too strong) to compensate 𝐺𝑢2
would be required to make the nonlinear term vanish.
3.2. Non-linearity of phase solubility diagram: effect of solute and CD structural changes
When the structures of the solute and CD are modified, what is the consequent change in the
non-linearity of phase solubility? Here we answer this question through the analysis of
experimental data, in order to address the long standing need for a molecular-based interpretation
15
of phase solubility diagrams in the presence of CDs.15,46 Non-linearity in phase solubility diagrams
has now been attributed to the sign of
1
2
1
2
𝑊 + 𝐺𝑢2
, composed of the two factors as clarified in
2
Section 3.1. Here we calculate their contributions from experimental data.
In order to understand the behaviour more clearly we have chosen data, from two drug molecules
of comparable structure and two CD derivatives, to analyse using our method. Two specific
solution comparisons stand out in terms of the insights that they can provide within our literature
data survey (see Figure 2 for molecular structures):
1. Aqueous solubility of naringenin with β-CD versus with 2-hydroxypropyl-β-cyclodextrin
(HP-β-CD).2,55
2. Aqueous solubility of naringin versus naringenin with HP-β-CD.55
This set of examples was chosen for the following reasons, which are beneficial for investigating
the structural cause of the nonlinearity:
1. Phase solubility of naringenin in β-CD exhibits close to linear AL (or a very weak positive
deviation, AP) behaviour in contrast to a weak AP in HP-β-CD, which can be attributed to
the derivatisation of β-CD to HP- β-CD.6
2. Entirely different deviations from phase solubility linearity between naringin (AN) and
naringenin (AP) can be attributed to the additional neohesperidose (disaccharide) present
in naringin.
16
The phase solubility diagrams of these drugs are shown in Figure 3. Naringin’s solubility
behaviour does not change majorly between different CD derivatives, however, for completeness
it will also be analysed.
Phase solubility data in the presence (S) and absence (𝑆0 ) of CD, taken from the literature,55 has
∗
𝑆
been converted to Δ𝜇𝑢∗ through a well-established relationship: 𝑒 −𝛽Δ𝜇𝑢 = 𝑆 . Such data have been
0
fit to the following polynomial (i.e., the second order truncation of Eq. (33)), whose fitting
constants 𝑎 and 𝑏 are tabulated in Table 1.
∗
𝑒 −𝛽Δ𝜇𝑢,𝑁𝑃𝑇 = 1 + 𝑎𝜌2 + 𝑏𝜌22
(35)
The corresponding KB integrals can be determined through a comparison between Eqs. (33) and
(35) as
𝐺𝑢2 = 𝑎
(36)
And therefore
𝑊 ≃ 2𝑏 − 𝑎2
(37)
The approximations made to derive Eq. (37) (and Eq. (33)) can be justified using Table 2.
(a) |𝑎| = |𝐺𝑢2 | ≫ |𝐺22 | can be shown simply in from Table 2. Note that we have used a wellknown relationship between 𝐺22 and the osmotic second virial coefficient 𝐵2 , 𝐺22 =
−2𝐵2,38,45 to obtain 𝐺22 . Combining this with (b) leads to |𝐺𝑢2 | ≫ |𝐺22 |.
(b) |𝐺𝑢2 | ≫ |𝐺𝑢1 | can be justified through an order-of-magnitude analysis. According to the KB
theory, 𝐺𝑢1 ≃ −𝑉𝑢 holds true, ignoring the negligibly small contribution from the isothermal
compressibility; hence 𝐺𝑢1 can be obtained from the partial molar volume of solutes in pure
water 𝑉𝑢 . Yet the experimental values for 𝑉𝑢 do not exist for naringin and naringenin to the
17
best of our knowledge, likely due to their low aqueous solubilities. However, it is wellestablished observation that 𝑉𝑢 is in a similar order-of-magnitude to its crystal volume.56
Table 2 indeed shows that the crystal volumes are much smaller in magnitude than 𝐺𝑢2 .
(c) |𝐺𝑢2 | ≫ |𝐺21 | can be justified by the help of (a) and |𝐺22 | ≫ |𝐺21 |, which can be justified
through a molecular crowding argument that CD-CD co-volume, which dominates 𝐺22 , is
much larger in magnitude than CD-water covolume, which dominates 𝐺21 .23,24,30 Indeed, the
available data supports this argument: 𝑉2 = −𝐺21 = 0.7 dm3 mol-1, for 𝛽 -CD, which is
much smaller than 𝐺22 = −12.6 dm3 mol-1.57
Based upon the above analysis, now we are in a position to examine changes in phase-solubility
2
non-linearity due to solute or CD structural alterations. The key is the competition between 𝐺𝑢2
and 𝑊in Eq. (33), in which the latter is expected to be dominated by the solute-induced CD-CD
interaction change (Appendix E);31 the former originates from the solute-CD interaction, the latter,
when negative in value, signifies a case where the CD-CD interaction (𝐺22 ) is more attractive than
the solute-induced CD-CD interaction (𝐺𝑢,22 ); such a case with 𝐺𝑢,22 − 𝐺22 < 0 will be referred
to as the solute-induced weakening of CD-CD interaction.
Naringin in β-CD and in HP-β-CD.2,55 Both CD derivatives’ solubility enhancement profiles
exhibit negative deviation from linearity (𝐴𝑁 ), as shown in Figure 3 and Table 3; the deviation is
larger in HP-β-CD compared to β-CD, which can be attributed to the stronger naringin-induced
weakening of the interaction between HP-β-CDs, i.e., larger negative 𝑉(𝐺𝑢,22 − 𝐺22 ) compared to
2
that of β-CD, even though the interaction of naringin 𝐺𝑢2
is stronger with HP-β-CD. The addition
of hydroxypropyl groups, while strengthening the solute-CD interaction, enhances the solute-
18
induced weakening of CD-CD interaction, leading to similar behaviour but enhanced solubility
due to HP-β-CD’s higher solubility in water.
Naringenin in β-CD (AL/P) versus in HP-β-CD (AP). Naringenin-HP-β-CD interaction is
stronger than that of naringenin-β-CD, as indicated by the magnitude of 𝐺𝑢2 ≃ 𝑎 in Tables 1 and
3. (Understanding the detailed molecular basis of this requires simulation; yet the stronger
interaction of HP-β-CD suggests that complexation is not driven by the hydrophobic effect alone;
polar groups contributes significantly to CD-solute interaction). The positive deviation from
linearity, albeit small, is still larger for HP- β-CD than for β-CD. This can again be understood in
terms of the balance between solute-CD interaction and the solute-induced weakening of CD-CD
interaction. Consistent with the previous case, the addition of hydroxypropyl groups increase the
solute-induced weakening of CD-CD interaction, yet this increase is not great enough compared
to the increase of solute-CD interaction contributions, thereby leading to a positive deviation for
HP- β-CD.
Naringenin versus Naringin. An inspection of Tables 1 and 3 reveals a relatively simple
picture. Naringenin interacts more strongly with both β-CD and HP-β-CD (as indicated by a larger
positive 𝑎 or 𝐺𝑢2 , suggesting that the presence of the disaccharide group in naringin weakens
solute-CD interaction probably through steric hindrance. (This would also explain why β-CD to
HP-β-CD increase of 𝐺𝑢2 is less pronounced for naringin).
Consistent with the binding strength, naringenin induces the weakening of both β-CD to β-CD
and HP-β-CD to HP-β-CD interactions more significantly than naringin, as indicated by larger
19
negative 𝑉(𝐺𝑢,22 − 𝐺22 ). However, this naringenin-induced weakening of CD-CD interaction
2
cannot overcome the CD-drug interaction contribution 𝐺𝑢2
, whereas in naringin it can. It is this
distinction which leads to the differences in phase-solubility behaviour. This may be attributed to
the much smaller molecular size of naringenin compared to naringin (Figure 2), which is less
effective for perturbing the CD-CD interaction.
We have shown that the molecular mechanism behind non-linear solubility enhancement can be
attributed to the interplay between solute-CD interaction and solute-induced weakening of CD-CD
interaction. The phase solubility classification (Fig. 1), employed widely in the literature, of an
increase of solubility with CD concentration, in agreement with many experimental data,8–17 has
now been given a molecular interpretation for the first time. The challenge now would be to
2
quantitatively rationalise the competing factors, 𝐺𝑢2
and 𝑉(𝐺𝑢,22 − 𝐺22 ), at a molecular structural
basis. The values of 𝑉(𝐺𝑢,22 − 𝐺22 ), calculated from experimental data,2,55 are indeed helpful and
lead to the description above, yet any interpretation beyond this requires extensive computer
simulation studies.
4. Conclusion
Cyclodextrin (CD), through inclusion complexation, can improve the aqueous solubility of
drugs.1–5 Phase solubility diagrams, determined extensively for drug-CD systems, indicate that
formation of drug-CD inclusion complexes is not the only driving force of solubilisation.15,46 This
has been evidenced by the non-linear dependence of drug solubility on CD concentration, which
20
necessitates a rigorous statistical thermodynamic theory beyond the phenomenological HiguchiConnors model.11–14
The molecular-based origin of such non-complexation contributions has been clarified in this
paper through a rigorous statistical thermodynamic framework, based upon the rigorous
Kirkwood-Buff and McMillan-Mayer theories of solutions.41,44,45,53 The leading contribution to the
deviation from linear phase solubility (i.e., the term proportional to 𝜌22 ) has been given a
microscopic interpretation for the first time.
With the help of published thermodynamic and phase solubility data, we have identified the
origin of the difference between the positive and negative deviation from linearity (AP and AN in
Fig 1): the competition between drug-CD interaction and the drug-induced weakening of CD-CD
interaction. AP behaviour is due to the overriding drug-CD interaction, whereas AN results from
the overriding drug-induced weakening of CD-CD interaction. Our rigorous statistical
thermodynamic approach thus led to a clarification of the factors determining the non-linearity of
phase solubility, and the first guideline for further studies, which should concentrate on drug-CD,
drug-CD-CD, and CD-CD interactions in water.
Acknowledgements
We thank Steven Abbott for his careful and critical reading of our manuscript. This work is
supported by the Grants-in-Aid for Scientific Research (Nos. 15K13550, 23651202, and
26240045) from the Japan Society for the Promotion of Science, and by the Elements Strategy
Initiative for Catalysts and Batteries from the Ministry of Education, Culture, Sports, Science, and
21
Technology, and by Computational Materials Science Initiative, Theoretical and Computational
Chemistry Initiative, and HPCI System Research Project (Project IDs: hp150131, hp150137,
hp150268, hp150294, hp160013, and hp160019) of the Next-Generation Supercomputing Project.
22
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25
Appendix A: Kirkwood-Buff integrals, radial distribution functions, and the potentials of
mean force interactions
The origin of nonlinearity in phase solubility has been attributed, by our theory, to the KB integrals.
Here we emphasise (using well-established statistical thermodynamic results44,45,53) that such KB
integrals, namely 𝐺𝑖𝑗 and 𝐺𝑢,𝑖𝑗 defined by Eqs. (5) and (6), can be linked directly to the two-body
correlation functions (CF) and the orientationally-averaged radial distribution functions (RDF) as
well as to the potentials of mean force (PMF) interactions between the species. Therefore, unlike
activity coefficient models which indeed are useful in modelling and predicting thermodynamic
quantities,58 our theory does not employ any model parameters to characterise interactions between
species; we employ CFs and PMFs instead, that can be accessible directly by molecular dynamics
simulations27,34,35. X-ray and neutron scattering can also yield information on RDFs.27 Hence our
theory does not claim to be a predictable model but has a direct, rigorous link to the structure of
solutions.
Let 𝜌𝑖𝑗 (𝑟⃗1 , 𝑟⃗2 ) be the two-body distribution function between the species 𝑖 and 𝑗 located
respectively at 𝑟⃗1 and 𝑟⃗2 . The corresponding two body CF is defined as
𝑉
𝑉
𝑔𝑖𝑗 (𝑟⃗1 , 𝑟⃗2 ) = 〈𝑁 〉 〈𝑁 〉 𝜌𝑖𝑗 (𝑟⃗1 , 𝑟⃗2 )
𝑖
𝑗
(A1)
which is linked directly to the potential of mean force between the species 𝑖 and 𝑗 as
𝑃𝑀𝐹𝑖𝑗 (𝑟⃗1 , 𝑟⃗2 ) = −𝑅𝑇 ln 𝑔𝑖𝑗 (𝑟⃗1 , 𝑟⃗2 )
(A2)
𝑃𝑀𝐹𝑖𝑗 is the free-energy change required to bring molecule 1 of species i and molecule 2 of species
j from their infinite separation to the positions 𝑟⃗1 and 𝑟⃗2 . It is not the direct interaction between the
26
two molecules, but is determined under the influence of all the other molecules in the system. In
this sense, it is the effective interaction mediated by the molecules in the solution.
Now we link CF to KB integrals. Firstly, it is easy to show that44,45,53
(A3)
∬ 𝑑𝑟⃗1 𝑑𝑟⃗2 𝜌𝑖𝑗 (𝑟⃗1 , 𝑟⃗2 ) = 〈𝑁𝑖 𝑁𝑗 〉 − 〈𝑁𝑖 〉𝛿𝑖𝑗
Combining Eqs. (E1), (E3) and (5), we obtain
〈𝑁𝑖 𝑁𝑗 〉−〈𝑁𝑖 〉〈𝑁𝑗 〉
∬ 𝑑𝑟⃗1 𝑑𝑟⃗2 [𝑔𝑖𝑗 (𝑟⃗1 , 𝑟⃗2 ) − 1] = 𝑉 2 (
〈𝑁𝑖 〉〈𝑁𝑗 〉
𝛿𝑖𝑗
− 〈𝑁 〉) = 𝑉𝐺𝑖𝑗
𝑖
(A4)
Thus we have demonstrated that the KB integral 𝐺𝑖𝑗 has a direct link to the solution structure
characterised by 𝑔𝑖𝑗 (𝑟⃗1 , 𝑟⃗2 ), as well as to the PMF interaction. In the inhomogeneous solution, in
which a solute (drug) molecule is fixed at the origin, Eq. (E4) can straightforwardly be generalised
to yield
〈𝑁𝑖 𝑁𝑗 〉𝑢 −〈𝑁𝑖 〉𝑢 〈𝑁𝑗 〉𝑢
∬ 𝑑𝑟⃗1 𝑑𝑟⃗2 [𝑔𝑢,𝑖𝑗 (𝑟⃗1 , 𝑟⃗2 ) − 1] = 𝑉 2 (
〈𝑁𝑖 〉𝑢 〈𝑁𝑗 〉𝑢
𝛿
− 〈𝑁𝑖𝑗〉 ) = 𝑉𝐺𝑢,𝑖𝑗
𝑖 𝑢
(A5)
which again links the KB integral 𝐺𝑢,𝑖𝑗 to the solution structure in the presence of the solute (drug)
characterised by 𝑔𝑢,𝑖𝑗 (𝑟⃗1 , 𝑟⃗2 ), and to the PMF.
Note the RDF, which is used frequently in molecular simulations, is the orientationally-averaged
CF, which depends only on 𝑟 = |𝑟⃗1 − 𝑟⃗2 |. RDFs are generally oscillatory with the periodicity of
the molecular diameter of water, which possess narrow repulsive regions at short 𝑟. For such
RDFs, Eqs. (A2) and (A4) can be simplified as
𝑃𝑀𝐹𝑖𝑗 (𝑟) = −𝑅𝑇 ln 𝑔𝑖𝑗 (𝑟)
(A5)
𝐺𝑖𝑗 = 4𝜋 ∫ 𝑑𝑟 𝑟 2 [𝑔𝑖𝑗 (𝑟) − 1]
(A6)
which also link the KB integrals to the solution structure and PMF interactions.
27
Appendix B: Equivalence between our theory and our previous paper.
Here we show that our present result (Eq. (21)) is equivalent to a result in our previous paper (Eq.
(39) of Ref 30). To do so, using the symmetry of indexes 1 and 2, Eq. (39) of Ref 30 is cast into
the following form through the use of 𝑊, defined in Eq. (22):
∗
1 𝜕2 𝜇𝑢
(
)
2
𝛽 𝜕𝜇2 𝑇,𝑃;𝑁 →0
𝑢
1
1
= −𝜌2 𝐺𝑢2 − 𝜌22 𝑊 + 𝜌22 𝐺𝑢1 (𝐺11 − 2𝐺12 + 𝐺22 + 𝜌 + 𝜌 )
1
2
(B1)
To derive Eq. (B1) from Eq. (21), we exploit the following thermodynamic relationships which can easily
be derived from the definition of 𝜆2 :
∗
1 𝜕𝜇𝑢
( )
𝛽 𝜕𝜇
2
𝜕𝜇 ∗
𝑇,𝑃;𝑁𝑢 →0
= 𝜆2 ( 𝜕𝜆𝑢 )
∗
1 𝜕 2 𝜇𝑢
(
)
𝛽 𝜕𝜇 2
2
2
(B2)
𝑇,𝑃;𝑁𝑢 →0
𝜕𝜇 ∗
𝑇,𝑃;𝑁𝑢 →0
= 𝛽𝜆2 ( 𝜕𝜆𝑢)
2
𝜕2 𝜇∗
𝑇,𝑃;𝑁𝑢 →0
+ 𝛽𝜆22 ( 𝜕𝜆2𝑢)
2
(B3)
𝑇,𝑃;𝑁𝑢 →0
Substituting Eqs. (10) and (21) into Eq. (B3) yields Eq. (B1).
28
Appendix C: Rederivation of Eq. (31) via the McMillan-Mayer theory
∗
Here we derive Eq. (31) via the MM theory. The goal is to expand 𝑒 −𝛽Δ𝜇𝑢,𝑀𝑀 , defined below, in
terms of the fugacity 𝜆2 and concentration 𝜌2 of CD.
𝑒
∗
−𝛽Δ𝜇𝑢,𝑀𝑀
=
Ξ𝑢 (𝑉,𝜇1 ,𝜇2 )
Ξ𝑢 (𝑉,𝜇1 )
Ξ(𝑉,𝜇1 ,𝜇2 )
Ξ(𝑉,𝜇1 )
(C1)
where the quantities with a subscript 𝑢 refer to those with solute fixed at the origin and the
quantities with the subscript denote those without solute; note that when the solute is present, it is
treated as a source of external field. To this end, let us start from the definition of grand canonical
partition function of the three two component system42–45,53
𝑁
𝑁
(C2)
Ξ(𝑇, 𝑉, 𝜇1 , 𝜇2 ) = ∑𝑁1 ≥0 ∑𝑁2 ≥0 𝜆1 1 𝜆2 2 𝑄(𝑇, 𝑉, 𝑁1 , 𝑁2 )
where Q is the canonical partition function; 𝜆𝑖 is the fugacity of the species i defined as:44,45,53
(C3)
𝜆𝑖 = exp(𝛽𝜇𝑖 )
Using Eq. (C2), the denominator and numerator of Eq. (C1) can be rewritten into the following
manner: 41,43,45,53
Ξ(𝑉,𝜇1 ,𝜇2 )
Ξ(𝑉,𝜇1 )
=
Ξ𝑢 (𝑉,𝜇1 ,𝜇2 )
Ξ𝑢 (𝑉,𝜇1 )
∑𝑁2 ≥0 ∑𝑁1 ≥0 𝑄(𝑉,𝑁1 ,𝑁2 ) 𝑒 𝛽𝜇1 𝑁1 𝑒 𝛽𝜇2 𝑁2
=
∑𝑁1 ≥0 𝑄(𝑉,𝑁1 ) 𝑒 𝛽𝜇1 𝑁1
∑𝑁2 ≥0 ∑𝑁1 ≥0 𝑄𝑢 (𝑉0 ,𝑁1 ,𝑁2 ) 𝑒 𝛽𝜇1 𝑁1 𝑒 𝛽𝜇2 𝑁2
∑𝑁1 ≥0 𝑄𝑢 (𝑉0 ,𝑁1 ) 𝑒 𝛽𝜇1 𝑁1
𝑁
(C4)
= 1 + ∑𝑁2 ≥1 𝑅𝑁2 𝜆2 2
𝑁
= 1 + ∑𝑁2 ≥1 𝑅𝑢,𝑁2 𝜆2 2
(C5)
where the cluster integrals 𝑅𝑁2 and 𝑅𝑢,𝑁2 are defined respectively as
𝑅𝑁2 =
∑𝑁1 ≥0 𝑄(𝑉0 ,𝑁1 ,𝑁2 ) 𝑒 𝛽𝜇1 𝑁1
𝑅𝑢,𝑁2 =
∑𝑁1 ≥0 𝑄(𝑉0 ,𝑁1 ) 𝑒 𝛽𝜇1 𝑁1
∑𝑁1 ≥0 𝑄𝑢 (𝑉0 ,𝑁1 ,𝑁2 ) 𝑒 𝛽𝜇1 𝑁1
∑𝑁1 ≥0 𝑄𝑢 (𝑉0 ,𝑁1 ) 𝑒 𝛽𝜇1 𝑁1
(C6)
(C7)
29
Eq. (C1), with the help of Eqs. (C4) and (C5), can be expanded into the series of 𝜆2 as
∗
𝑒 −𝛽Δ𝜇𝑢,𝑀𝑀 = 1 + (𝑅𝑢,1 − 𝑅1 )0 𝜆2 + (𝑅𝑢,2 − 𝑅2 + 𝑅12 − 𝑅𝑢,1 𝑅1 )0 𝜆22 + ⋯
(C8)
Using the definition of the KB integrals, Eqs. (D14)-(D16), Eq. (C8) can now be expressed as
𝑅
∗
1
𝑒 −𝛽Δ𝜇𝑢,𝑀𝑀 = 1 + (𝐺𝑢2 )0 ( 𝑉1 ) 𝜆2 + 2 {𝑉
〈𝑁2 〉2𝑢 𝐺𝑢,22 −〈𝑁2 〉2 𝐺22
〈𝑁2 〉2
0
𝑅
2
2
+ 𝐺𝑢2
} ( 𝑉1 ) 𝜆22
0
0
(C9)
which can be shown through Eq. (D8) to be equivalent to Eq. (31) derived via the KB theory, since
𝜌
𝑅1 /𝑉 reduces to (𝜆2 ) at the 𝜌2 → 0 limit.
2
0
The variable 𝜆2 can also be converted to 𝜌2 via the MM theory (Eq. (D8)) to yield:
∗
1
𝑒 −𝛽Δ𝜇𝑢,𝑀𝑀 = 1 + (𝐺𝑢2 )0 𝜌2 + 2 {𝑉
〈𝑁2 〉2𝑢 𝐺𝑢,22 −〈𝑁2 〉2 𝐺22
〈𝑁2 〉2
2
+ 𝐺𝑢2
− 2𝐺𝑢2 𝐺22 } 𝜌22 + ⋯
(C10)
0
Note that the 𝜌2 -expansion from the MM theory (Eq. (C10)) is different from the KB theory (Eq.
(29)) in two ways: (1) the lack of CD-water and water-water terms; and (2) the relationship
between 𝜆2 and 𝜌2 are different between the 𝜇 VT ensemble (MM; Eq. (D8)) and the NPT
ensemble (KB; Eq. (28)).
30
Appendix D: Useful formulae of the McMillan-Mayer theory
Virial expansion. The many-body interactions amongst the CD molecules show up most naturally
in terms of the virial expansion of the osmotic pressures.41,44,45,53 Using grand canonical ensembles,
osmotic pressures of the CDs in the bulk phase (Π) and in inhomogeneous solution of aqueous
CDs (Π𝑢 ) can be expressed as:41,44,45,53
𝑒 𝛽Π𝑢𝑉 =
Ξ𝑢 (𝑉,𝜇1 ,𝜇2 )
Ξ𝑢 (𝑉,𝜇1 )
(D1)
Ξ(𝑉,𝜇1 ,𝜇2 )
Ξ(𝑉,𝜇1 )
𝑒 𝛽Π𝑉 =
(D2)
Combining Eqs. (C1), (D1) and (D2), we obtain the following fundamental expression for
solubilisation:
∗
Δ𝜇𝑢,𝑀𝑀
= (Π − Π𝑢 )𝑉
(D3)
Inversion of fugacity expansion. Let us first note the following, which can be derived
straightforwardly from Eq. (D2):
〈𝑁2 〉 =
𝜕 ln Ξ(𝑉,𝜇2 ,𝜇1 )
𝜕(𝛽𝜇2 )
= 𝜆2
𝜕 ln Ξ(𝑉,𝜇2 ,𝜇1 )
𝜕𝜆2
= 𝜆2
𝜕𝛽Π𝑉
𝜕𝜆2
(D4)
where 𝜆2 = 𝑒 𝛽𝜇2 . Combining Eqs. (C4) and (D4), we obtain:41,44,45,53
1
2
𝑅12
𝑅2 − 𝑅1 2
〈𝑁2 〉 = 𝑅1 𝜆2 + 2
(𝑅1 𝜆2 )2 + ⋯
(D5)
Using Eq. (D5), we now express 𝜆2 in terms of 𝜌2 = 〈𝑁2 〉/𝑉. To do so, let us first assume the
following:
𝑅1
𝑉
𝜆2 = 𝑎
〈𝑁2 〉
𝑉
〈𝑁2 〉 2
+𝑏(
𝑉
) +⋯
(D6)
Substituting Eq. (D6) into Eq, (D5) yields
31
〈𝑁2 〉
𝑉
=𝑎
〈𝑁2 〉
𝑉
+ (𝑏 + 2𝑉
1
2
𝑅12
𝑅2 − 𝑅1 2
〈𝑁2 〉 2
𝑎2 ) (
𝑉
Comparing r.h.s and l.h.s., 𝑎 = 1, 𝑏 = −2𝑉
𝑅1
𝑉
𝜆2 = 𝜌2 − 2𝑉
1
2
𝑅1 2
𝑅2 − 𝑅1 2
(D7)
) +⋯
1
2
𝑅12
𝑅2 − 𝑅1 2
. Hence we obtain the following: 41,44,45,53
𝜌22 + ⋯
(D8)
Connection to Kirkwood-Buff integrals. Let us first differentiate Eq. (D5) with respect to
𝜇2 :41,44,45,53
𝑘𝑇
𝜕〈𝑁2 〉
𝜕𝜇2
1
= 〈𝑁22 〉 − 〈𝑁2 〉2 = 𝑅1 𝜆2 + 4 (𝑅2 − 2 𝑅1 2 ) 𝜆22 + ⋯
(D9)
Subtracting Eq. (D5) from Eq. (D9) yields:
1
〈𝑁22 〉 − 〈𝑁2 〉2 − 〈𝑁2 〉 = 2 (𝑅2 − 𝑅1 2 ) 𝜆22
2
(D10)
Combining Eq. (D10) with the first term of Eq. (D8), we obtain
2
1
2
𝑅12
𝑅2 − 𝑅1 2
=
〈𝑁22 〉−〈𝑁2 〉2 −〈𝑁2 〉
〈𝑁2 〉2
(D11)
We can derive the following in a similar manner
2
1
2
2
𝑅𝑢,1
𝑅u,2 − 𝑅u,1 2
Ru,1
𝑅1
−1=
=
〈𝑁22 〉𝑢 −〈𝑁2 〉𝑢 2 −〈𝑁2 〉𝑢
〈𝑁2 〉𝑢 2
〈𝑁2 〉𝑢 −〈𝑁2 〉
(D12)
(D13)
〈𝑁2 〉
Unless otherwise noted, all the ensemble averages are to be evaluated at 𝜌2 → 0.
Combining Eqs. (D11)-(D13) with the definitions of the KB integrals, we obtain the following
relationship between 𝑅s and the KB integrals:
𝐺𝑢2 = 𝑉
〈𝑁2 〉𝑢 −〈𝑁2 〉
〈𝑁2 〉
R
= 𝑉 ( 𝑅u,1 − 1)
1
(D14)
32
𝐺22 = 𝑉
〈𝑁22 〉−〈𝑁2 〉2 −〈𝑁2 〉
𝐺𝑢,22 = 𝑉
〈𝑁2 〉2
= 2V
〈𝑁22 〉𝑢 −〈𝑁2 〉𝑢 2 −〈𝑁2 〉𝑢
〈𝑁2 〉𝑢
2
1
2
𝑅12
𝑅2 − 𝑅1 2
= 2𝑉
(D15)
1
2
2
𝑅𝑢,1
𝑅u,2 − 𝑅u,1 2
(D16)
Note here that all the KB integrals are defined at infinite CD concentration, i.e., 𝑁2 → 0.
33
Appendix E: On the solute-induced change of CD-CD interaction
Justification that 𝑾 ≃ 𝑽(𝑮𝒖,𝟐𝟐 − 𝑮𝟐𝟐 ). Using experimental data, we can show that 𝑊 defined
in Eq. (22) is indeed dominated by the contribution from 𝑉(𝐺𝑢,22 − 𝐺22 ) for AP and AN phase
solubility types. To do so, let us remember that in our theory, solubilisation was divided into two
sub-steps (MM and the rest); each ensemble is expressed in two different, but equivalent (denoted
by ⇔ ) representations using the two sets of variables: (𝑉, 𝜇1 , 𝜇2 ) and (𝑃, 𝑁1 , 𝑁2 ). Under this
setup, the solubilisation process can be expressed as a change from (I) to (III) via (II):
(I) Pure water: (𝑉0 , 𝜇1 , ∞) ⇔ (𝑃, 𝑁1 , 0).
(II) “Intermediate” aqueous CD solution: (𝑉0 , 𝜇1 , 𝜇2 ) ⇔ (𝑃′ , 𝑁1′ , 𝑁2′ ), where 𝑃′ = 𝑃 + Π, where
Π is the osmotic pressure.
(III) Aqueous CD solution of experimental condition: Constant 𝜌2 ensemble (𝑉, 𝜇1′ . 𝜇2′ ) ⇔
(𝑃, 𝑁1 , 𝑁2 ).
where the unprimed quantities correspond explicitly to experimental conditions, and temperature
∗
is kept constant throughout. The first step, (I) → (II), corresponds to Δ𝜇𝑢,𝑀𝑀
. The second step, (II)
∗
→ (III), corresponds to Δ𝜇𝑢∗ − Δ𝜇𝑢,𝑀𝑀
. To ensure this correspondence, μ2 in (II) is determined
under the condition that 𝜌2 (𝑉0 , 𝜇1 , 𝜇2 ) = 𝜌2 (𝑉, 𝜇1′ , 𝜇2′ ) = 𝜌2 .
Using Eqs. (29) and (C10),
∗
𝛽Δ𝜇𝑢∗ − 𝛽Δ𝜇𝑢,𝑀𝑀
1
= (𝐺𝑢1 )0 𝜌2 − 2 {(𝑊 − 𝑉
〈𝑁2 〉2𝑢 𝐺𝑢,22 −〈𝑁2 〉2 𝐺22
〈𝑁2 〉2
) + 2𝐺𝑢2 𝐺21 + 𝐺𝑢1 𝐺22 } 𝜌22 + ⋯
(E1)
0
in which we have ignored the negligibly small contribution coming from the isothermal
1
compressibility 𝜅𝑇 , 𝑖. 𝑒. , −𝐺𝑢1 (𝐺11 + 𝜌 ) = −𝑘𝑇 𝐺𝑢1 𝜅𝑇 . Our aim here is to show that 𝑊 ≃
1
34
𝑉
1
2
〈𝑁2 〉2𝑢 𝐺𝑢,22 −〈𝑁2 〉2 𝐺22
〈𝑁2 〉2
, i.e., the difference between the two is negligible in comparison to
2 )
(𝑊 + 𝐺𝑢2
= 𝑏.
Let us first note that 𝜇𝑢∗ is intensive; this means that 𝜇𝑢∗ cannot be the function of extensive
thermodynamic quantities. Hence 𝜇𝑢∗ (𝑃′ , 𝑁1′ , 𝑁2′ ) = 𝜇𝑢∗ (𝑃′ , 𝑥2′ ) and 𝜇𝑢∗ (𝑃, 𝑁1 , 𝑁2 ) = 𝜇𝑢∗ (𝑃, 𝑥2 ) ,
where 𝑥2′ and 𝑥2 are the mole fractions of the species 2. Consequently,
∗
Δ𝜇𝑢∗ − Δ𝜇𝑢,𝑀𝑀
= 𝜇𝑢∗ (𝑃, 𝑥2 ) − 𝜇𝑢∗ (𝑃′ , 𝑥2′ )
(E2)
Eq. (E2) can be decomposed into the following steps:
𝜇𝑢∗ (𝑃, 𝑥2 ) − 𝜇𝑢∗ (𝑃′ , 𝑥2′ ) = [𝜇𝑢∗ (𝑃, 𝑥2 ) − 𝜇𝑢∗ (𝑃′ , 𝑥2 )] + [𝜇𝑢∗ (𝑃′ , 𝑥2 ) − 𝜇𝑢∗ (𝑃′ , 𝑥2′ )]
(E3)
The first term of Eq. (E3) can be approximated as
∗ (𝑃,𝑥 )
𝜕𝜇𝑢
2
𝜇𝑢∗ (𝑃, 𝑥2 ) − 𝜇𝑢∗ (𝑃′ , 𝑥2 ) ≃ (
)
𝜕𝑃
𝑇,𝑥2
(𝑃 − 𝑃′ ) = −𝑉𝑢 π = 𝐺𝑢1 𝜋
(E4)
where the final step involves a well-known KB relationship, 𝑉𝑢 = −𝐺𝑢1 , by ignoring a negligible
term involving isothermal compressibility. Using Table 2, the magnitude is estimated to be 4-7 J
mol-1 since Π ≃ 0.2 atm and 𝑉𝑢 ≃ 0.2 − 0.35 dm3 mol-1, which is negligible compared to Δ𝜇𝑢∗ .
The 𝜌22 term should be smaller than the above, and is expected to be negligible.
The second term of Eq. (E3), let us start from
𝜕𝜇 ∗
𝜇𝑢∗ (𝑃′ , 𝑥2 ) − 𝜇𝑢∗ (𝑃′ , 𝑥2′ ) ≃ ( 𝜕𝑥𝑢 )
2
𝑇,𝑃
(𝑥2 − 𝑥2 ′)
(E5)
35
𝜕𝜇 ∗
In evaluating ( 𝜕𝑥𝑢 )
2
𝑇,𝑃
, we have to bear in mind that 𝑥2 = 𝑥2 (𝑇, 𝑃, 𝜇1 ), since 𝑇 and 𝑃 are fixed in
the partial differentiation in Eq. (E5); here, 𝑥2 is really 𝑥2 (𝜇1 ). Eq. (E5) can therefore be rewritten
using the chain rule into the following form:
𝜕𝜇 ∗
𝜇𝑢∗ (𝑃′ , 𝑥2 ) − 𝜇𝑢∗ (𝑃′ , 𝑥2′ ) ≃ ( 𝜕𝜇𝑢 )
1
𝜕𝜇
𝑇,𝑃 ′
(𝜕𝑥1 )
2
𝑇,𝑃 ′
(𝑥2 − 𝑥2 ′)
(E6)
Since Π is small, we use 𝑃 ≃ 𝑃′ throughout in the following discussion. The first two factors of
Eq. (E6) can be evaluated by the well-known KB relationships44,45
𝜕𝜇 ∗
( 𝜕𝜇𝑢)
1
𝑇,𝑃
𝜕𝜇
(𝜕𝑥1 )
2
𝑇,𝑃
(E7)
= 𝜌1 (𝐺𝑢2 − 𝐺𝑢1 )
𝑅𝑇
= − 1+𝜌
(E8)
2 (𝐺11 +𝐺22 −2𝐺12 )
To evaluate the third factor (𝑥2 − 𝑥2 ′) of Eq. (E6), let 𝜌1 and 𝜌1′ be the density of water
respectively in systems (III) and (II);
𝜌
𝜌
𝜌
𝑥2 − 𝑥2′ = 𝜌2 − 𝜌2′ ≃ 𝜌 22 (𝜌1′ − 𝜌1 )
1
(E9)
1
1
Using system (I) as the reference state, whose density is 𝜌10 , Eq. (E9) can be rewritten as
𝜌1′ − 𝜌1 = (𝜌1′ − 𝜌10 ) − (𝜌1 − 𝜌10 )
(E10)
Since the first and the second terms correspond respectively to constant 𝜇1 and constant 𝑃
processes
𝜕𝜌1
(𝜌1′ − 𝜌1 ) = (
)
𝜕𝜌
𝜕𝜌2 𝜇
1
𝜌2 − (𝜕𝜌1 ) 𝜌2
2
(E11)
𝑃
The first term of Eq. (E11) can be evaluated by noting that 𝜇2 is the function of 𝜌2
𝜕𝜌
(𝜕𝜌1 )
2
𝜇1
𝜕𝜌
= (𝜕𝜇1 )
2
𝜕𝜇
𝜇1
(𝜕𝜌2 )
2
𝜇1
𝜕𝑃
= 𝛽𝜌1 𝐺12 (𝜕𝜌 )
2
𝜇1
≃ 𝜌1 𝐺12 (1 + 𝐵2 𝜌2 + ⋯ )
(E12)
36
𝜕𝜇
1
in which we have used (𝜕𝜌2 )
2
𝜕𝜌
(𝜕𝜇1 )
2
𝜇1
𝜇1
𝜕𝑃
= 𝜌 (𝜕𝜌 ) from the Gibbs-Duhem equation, the KB integral
2
2
𝜇1
𝜕𝜌
𝑉
= 𝛽𝜌1 𝜌2 𝐺12 . The second can be simplified through (𝜕𝜌1 ) = − 𝑉2 ;
2
𝑃
1
44,45
taking all
together, we obtain the following leading term
𝜌2
𝑉
𝑥2 − 𝑥2′ ≃ 𝜌 22 (𝑉2 + 𝜌1 𝐺12 ) =
1
1
𝑅𝑇𝜅𝑇
𝜌1
𝜌22
(E13)
In combination with Eqs. (E7) and (E8), Eq. (E5) makes a negligible contribution because of the
negligibility of 𝑅𝑇𝜅𝑇 .
∗
We have thus shown that Δ𝜇𝑢∗ − Δ𝜇𝑢,𝑀𝑀
, namely the process II → III makes a negligible
∗
contribution; this also means that Δ𝜇𝑢∗ ≃ Δ𝜇𝑢,𝑀𝑀
. In Eq. (E1), 2𝐺𝑢2 𝐺𝑢1 + 𝐺𝑢1 𝐺22 can be
estimated for naringin in β-CD (exhibiting AN non-linearity) to be −7.3 × 102 dm6 mol-2, much
smaller than 𝑏 = −2.0 × 104 dm6 mol-2 (Table 3); considering that 𝐺𝑢1 for HP-β-CD is larger yet
in the same order of magnitude to that of β-CD, 2𝐺𝑢2 𝐺𝑢1 + 𝐺𝑢1 𝐺22 ≪ 𝑏 still holds true also for
AN and Ap types in HP-β-CD. Consequently, 𝑊 − 𝑉(𝐺𝑢,22 − 𝐺22 ) is much smaller than 𝑏, thereby
leading to 𝑊 ≃ 𝑉(𝐺𝑢,22 − 𝐺22 ) for AN and Ap.
An intuitive interpretation on the role of 𝑽(𝑮𝒖,𝟐𝟐 − 𝑮𝟐𝟐 ) on phase solubility nonlinearity. We
∗
have shown above that Δ𝜇𝑢,𝑀𝑀
makes a dominant contribution to Δ𝜇𝑢∗ . According to Eq. (D3), the
∗
sign of Δ𝜇𝑢,𝑀𝑀
is determined by a competition between Π𝑢 and Π: solute-induced lowering of the
∗
osmotic pressure (Π𝑢 < Π) increases Δ𝜇𝑢,𝑀𝑀
and decreases solubility, whereas solute-induced
∗
increase of the osmotic pressure (Π𝑢 > Π) decreases Δ𝜇𝑢,𝑀𝑀
and increases solubility. Note that
1
the second virial coefficients in the presence and absence of the solute are 𝐵𝑢,2 ≡ − 2 𝐺𝑢,22 and
37
1
𝐵2 ≡ − 2 𝐺22 , respectively. This simple setup rationalises how solute-induced changes in CD-CD
interaction occur in non-linear behaviour. For example, AN is caused when (𝐵𝑢,2 − 𝐵2 )𝜌22 > 0,
∗
which contributes to the increase of Δ𝜇𝑢,𝑀𝑀
and the decrease of solubility. From the relationship
between the virial coefficients and the KB integrals, this takes place when 𝐺𝑢,22 < 𝐺22 , i.e., when
2
the solute-induces weakening of CD-CD interaction. (Note that 𝐺𝑢2
comes into 𝜌22 term when
∗
considering solubility, i.e., 𝑒 −𝛽Δ𝜇𝑢,𝑀𝑀 .)
38
Table 1: CD concentration dependence of solubilisation; fitting parameters for Eq. (35). See
Figure 3 for a comparison between experimental data and the fitting curves. Data used was
obtained from the literature.55
Solute
CD
a / dm3 mol-1
b / dm6 mol-2
R2
Naringenin
β-CD
1143
202.2
0.9965
Naringenin
HP-β-CD
2938
47230
0.9969
Naringin
β-CD
518.2
-20240
0.9996
Naringin
HP-β-CD
632.4
-23510
0.9978
39
Table 2: Comparative properties and KB integrals calculated from the published experimental
data.55 B2 values for β-CD and HP-β-CD were taken from the literature.2 The values of 𝑎 in
Table 1 are shown here again for comparison.
Solute
𝐺22
/ dm3 mol-1
Crystal
volume
/dm3 mol-1
CD
𝑎
/ dm3 mol- 1
Naringenin
β-CD
1143
0.1832
12.6
Naringenin
HP-β-CD 2938
0.1832
2.83
Naringin
β-CD
518.2
0.3497
12.6
Naringin
HP-β-CD 632.4
0.3497
2.83
40
Table 3: Results for second order deviations calculated from published experimental data.55 See
Eqs. (35)-(37). The values of 𝑏 in Table 1 are shown here again for comparison.
Solute
CD
𝑏
2 )
(𝐺𝑢2
0
𝑊a
/dm6 mol-2
/ dm6 mol-2
/ dm6 mol-2
Type
Naringenin
β-CD
2.0 × 102
1.3×106
-1.3× 106
AL/P
Naringenin
HP-β-CD
4.7 × 104
8.6×106
-8.5 × 106
AP
Naringin
β-CD
-2.0× 104
2.7×105
-3.1× 105
AN
Naringin
HP-β-CD
-2.3× 104
4.0×105
-4.5 × 105
AN
41