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Water activity in liquid food systems: a molecular
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scale interpretation
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Andy J. Maneffa,1 Richard Stenner,2† Avtar S. Matharu,1 James H. Clark,1 Nobuyuki
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Matubayasi,3,4 and Seishi Shimizu2*
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Heslington, York YO10 5DD, United Kingdom
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2
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York YO10 5DD, United Kingdom
Green Chemistry Centre of Excellence, Department of Chemistry, University of York,
York Structural Biology Laboratory, Department of Chemistry, University of York, Heslington,
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Toyonaka, Osaka 560-8531, Japan
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8520, Japan
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†Present
Division of Chemical Engineering, Graduate School of Engineering Science, Osaka University,
Elements Strategy Initiative for Catalysts and Batteries, Kyoto University, Katsura, Kyoto 615-
Address: School of Physics, University of Bristol, Bristol, BS8 1TL. United Kingdom
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KEYWORDS:
water activity; water structure; statistical thermodynamics; Kirkwood-Buff
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theory; sugar; polyol
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Corresponding Author:
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Seishi Shimizu
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York Structural Biology Laboratory, Department of Chemistry, University of York, Heslington,
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York YO10 5DD, United Kingdom
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Tel: +44 1904 328281, Fax: +44 1904 328281, Email: [email protected]
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ABBREVIATIONS
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KB, Kirkwood-Buff.
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ABSTRACT
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Water activity has historically been and continues to be recognised as a key concept in the area of
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food science. Despite its ubiquitous utilisation, it still appears as though there is confusion
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concerning its molecular basis, even within simple, single component solutions. Here, by close
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examination of the well-known Norrish equation and subsequent application of a rigorous
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statistical theory, we are able to shed light on such an origin. Our findings highlight the importance
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of solute-solute interactions thus questioning traditional, empirically based “free water” and “water
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structure” hypotheses. Conversely, they support the theory of “solute hydration and clustering”
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which advocates the interplay of solute-solute and solute-water interactions but crucially, they do
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so in a manner which is free of any estimations and approximations.
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1. Introduction
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Following its introduction over 60 years ago (Scott, 1953), the application of “water activity”
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has become omnipresent within food science and related disciplines. It serves as a useful indicator
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of the microbiological stability of foodstuffs and food-related systems, much better than mere
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water content (Scott, 1957). Water activity also plays an important role on the sensorial properties
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of foodstuffs such as aroma, taste and texture as well as on their chemical and biological reactivity
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(e.g. lipid oxidation and non-/enzymatic activity) (Labuza & Rahman, 2007). Despite serious
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criticisms on the utility of water activity as a fundamental descriptor of water-related phenomena
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in food and related systems, it continues to be widely used to this day as a tool for product
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development and quality control across multiple areas of the food industry (Slade & Levine, 1991).
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However, what "water activity" really is on a microscopic scale is still a matter of controversy,
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which we focus on exclusively in this paper. There are the following three different views co-
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existing in the literature:
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1. the fraction of "free or available water" due to the presence of the "bound water" around
solutes (sugars or polyols);
2. the measure of "water structure" formation or the “ordering” of water in the presence of the
solutes (sugars or polyols), and;
3. the measure of “hydration water” which comes from the stoichiometric clustering/bindings
models of water and solutes.
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The “Free water” hypothesis (Scott, 1953) advocates the use of water activity “as a measure
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of the availability of water” in an aqueous medium (i.e., solution). The popular view, water activity
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as water “availability”, may have been originated from this seminal paper, giving rise to the
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interpretation that water activity is a measure of water freedom (i.e., boundness). In any case, this
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view suggests that water activity of an aqueous solution is fundamentally reflected by only the
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nature of the water interactions occurring within it. Despite the best attempts to remedy such over-
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simplification (Altunakar & Labuza, 2008), the interpretation of water activity in terms of “free
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water” still persists (Carareto, Monteiro Filho, Pessôa Filho, & Meirelles, 2010; Frosch, Bilde, &
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Nielsen, 2010; Guine, Almeida, Correia, & Mendes, 2015).
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The “Water structure” hypothesis advocates that the addition of certain species into solution
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either has: i. the effect of increasing or decreasing the activity of water on account of weakening,
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or; ii. strengthening the ‘structure’ or ‘ordering’ of the water network surrounding the added solute.
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This hypothesis appears to be an extension of the classical view that solutes can either promote or
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interfere with the hydrogen bonding network of water in solution (i.e. act as “structure makers” or
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“structure breakers”) (Frank & Evans, 1945; Frank & Franks, 1968). Such an interpretation
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suggests that water activity should therefore predominantly be a function of water-water
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interaction, with increased interaction (i.e. more order or structure) leading to a reduction in water
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activity (Caurie, 2005; Dutkiewicz & Jakubowska, 2002). Even though the contribution from
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solute-solute interaction is acknowledged to be present in water activity this contribution is
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considered to be small and negligible (Sato & Miyawaki, 2008; Sone, Omote, & Miyawaki, 2015).
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The Hydration number and stoichiometric clustering hypothesis is based upon a
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stoichiometric binding reaction model of solute hydration, Scatchard related water activity to
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‘hydration number’ i.e. number of ‘bound’ water molecules (Scatchard, 1921b). This relationship
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was employed later by Stokes and Robinson to understand the origin of water activity in aqueous
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non-electrolyte solutions (Stokes & Robinson, 1966). Further extension of this approach to
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aqueous sucrose solutions accounts for both hydration and solute clustering modelled in terms of
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a series of stepwise stoichiometric reactions (Gharsallaoui, Rogé, Génotelle, & Mathlouthi, 2008;
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Starzak & Mathlouthi, 2006; VanHook, 1987). This approach has revealed competition between
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hydration and clustering, i.e., i. sucrose hydration lowers water activity, and; ii. sucrose clustering
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raises water activity by increasing the effective mole fraction of water.
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Thus, we see that there have been three different hypotheses on the origin of the water activity.
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Are these hypotheses equivalent or contradictory? To the best of our knowledge this question has
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not been suitably answered. Instead of constructing another thermodynamic model of water
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activity, the aim of the current work is to establish what water activity really means on a molecular
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scale. To do so, we employ the first principles of statistical thermodynamics without any models
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or approximations (Shimizu, 2004; Shimizu & Matubayasi, 2014b) unlike the stoichiometric
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approach to clustering and association which depends on a number of model assumptions (such as
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the size of clusters and equilibrium constants) (Funke, Wetzel, & Heintz, 1989; Gharsallaoui,
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Rogé, Génotelle, & Mathlouthi, 2008; Guggenheim, 1952; Scatchard, 1921a; Schönert, 1986a,
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1986b; Starzak & Mathlouthi, 2006; Starzak, Peacock, & Mathlouthi, 2000; Stokes & Robinson,
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1966; VanHook, 1987), and cannot describe interactions within solutions and mixtures, which are
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weak, non-specific and dynamic in nature, in a realistic manner (Shimizu, 2004, 2013; Shimizu
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& Matubayasi, 2014b; Shimizu, Stenner, & Matubayasi, 2017). The rigorous statistical
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thermodynamic approach, on the contrary, has revealed the molecular picture at odds with most of
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the previous hypotheses on the role of solutes on solution thermodynamics and solubility (Shimizu,
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Stenner, & Matubayasi, 2017).
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Indeed, Shimizu has demonstrated that gradient of water activity with respect to solute mole
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fraction is determined by the compensation between solute-solute and solute-water interactions
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(Shimizu, 2013). This is contradictory to the free water hypothesis and the water structure
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hypothesis, but is consistent with the solute clustering models on the molecular origin of "water
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activity". However, this study focused on the gradient of water activity instead of the water activity
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itself. Hence, herein, the first full statistical thermodynamic clarification of how solute-water and
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solute-solute interactions contribute to the water activity itself is reported.
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2. A statistical thermodynamic basis of water activity and the Norrish equation
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To reveal the molecular basis of water activity, we combine insights from food chemistry with
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respect to statistical thermodynamics. In recent years, the application of an exact, model-free
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approach, i.e., the Kirkwood-Buff (KB) theory of solutions (Ben-Naim, 2006; Chitra & Smith,
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2002; Kirkwood & Buff, 1951; Shimizu, 2004), has proven to be a powerful tool for improving
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the microscopic understanding of various liquid food systems including gelatin (Shimizu &
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Matubayasi, 2014a), tofu (Shimizu, Stenner, & Matubayasi, 2017) and aqueous sucrose solutions
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(Shimizu, 2013). Here we clarify the molecular basis of water activity within this rigorous
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theoretical framework.
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In food science, water activity of liquid food has been modelled successfully by the Norrish
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equation (Norrish, 1966)
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𝑎𝑤 = 𝑥𝑤 𝑒 𝐾𝑥𝑠
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where 𝐾 is referred to as the Norrish constant and the system consists of two components (water
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and solute). Eq. (1) is reported to fit the water activity in high to medium water content and has
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been used in the context of food science as a useful fitting equation (Baeza, Pérez, Sánchez,
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Zamora, & Chirife, 2010; Fysun, Stoeckel, Thienel, Waschle, Palzer, & Hinrichs, 2015). Note that
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Eq. (1) is identical to one-parameter Margules equation where K is often represented as α.
2
(1)
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In statistical thermodynamics, Eq. (1) can be derived rigorously from first principles (see Appendix
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A). In the framework of the Kirkwood-Buff theory (Chitra & Smith, 2002; Kirkwood & Buff,
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1951; Shimizu, 2004; Shimizu & Matubayasi, 2014b), the following three contributions to the
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Norrish constant can be identified
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𝐾=
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where 𝐺𝑤𝑤 , 𝐺𝑠𝑠 and 𝐺𝑠𝑤 respectively signify the water-water, solute-solute and solute-water KB
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integral (KBI), 𝑉𝑤 is the partial molar volume of water, and the superscript ∞ signifies at the
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infinite dilution of solute. KBI is defined as
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𝐺𝑖𝑗 = 4𝜋∫ 𝑑𝑟 𝑟 2 [𝑔𝑖𝑗 (𝑟) − 1]
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where 𝑔𝑖𝑗 (𝑟) refers to the radial distribution function between the species 𝑖 and 𝑗. KBIs are the
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quantitative measure of affinity between species in solution. Note that Eq. (2) has been derived at
1
∞
2𝑉𝑤
∞
∞
∞)
(𝐺𝑤𝑤
+ 𝐺𝑠𝑠
− 2𝐺𝑠𝑤
(2)
(3)
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∞
∞
the infinite dilution of solute. Here the difference between self- ( 𝐺𝑤𝑤
and 𝐺𝑠𝑠
) and mutual
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∞
interaction (𝐺𝑠𝑤
) has been identified, for the first time, as the molecular-level interpretation of the
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∞
∞
∞
Norrish constant. Note that 𝐺𝑤𝑤
+ 𝐺𝑠𝑠
− 2𝐺𝑠𝑤
has been shown previously to be the key for the
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gradient of water activity with respect to solute concentration (Shimizu, 2013). Previous works on
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the application of the Kirkwood-Buff theory to binary mixtures (Ben-Naim, 2006; Chitra & Smith,
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2002; Shimizu, 2013) have employed the general and rigorous theoretical expressions applicable
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to all concentration ranges, while the present work, aiming at clarifying the meaning of the Norrish
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constant, focuses on the infinite dilution limit of the solute. The theoretical expressions used in the
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present paper can be derived directly from the previous, more general theory, as has been
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demonstrated in Appendices A and B.
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The statistical thermodynamic derivation of the Norrish equation shows that, strictly speaking, it
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is accurate only at the infinite dilution of solutes. However, in practice, the Norrish equation can
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be used over much wider range of solute concentrations with relatively good accuracy. This means
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that the water-water, solute-water and solute-solute KBIs at infinite solute dilution are crucial
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factors that determine the water activity up to moderate solute concentrations (ca. 60 wt.%, ca. 5-
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10 mol dm-3).
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To quantify the relative contributions of water-water, solute-water and solute-solute KBIs to the
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Norrish constant, additional experimental data are indispensable. Volumetric data will indeed
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complement the Norrish constant and lead to the determination of individual KBIs through the use
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of the following well-known relationships (Shimizu, 2004; Shimizu & Matubayasi, 2014b):
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∞
𝐺𝑠𝑤
= −𝑉𝑠∞ + 𝑅𝑇 𝜅𝑇∞
(4)
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∞
𝐺𝑤𝑤
= −𝑉𝑤∞ + 𝑅𝑇 𝜅𝑇∞
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where 𝑉𝑠∞ , 𝑉𝑤∞ and 𝜅𝑇∞ respectively signify the partial molar volume of solute, water (at infinite
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dilution of solutes) and isothermal compressibility of pure water. Hence the three KBIs can be
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determined via three experimental data (Eqs. (2), (4) and (5)).
(5)
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3. Molecular basis of water activity: both solute-solute and solute-water
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interactions contribute to the Norrish constant
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3.1. The Norrish constant as a competition between solute-water and solute-solute
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interactions
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The combination of the Norrish equation with rigorous thermodynamic theory reveals an entirely
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different molecular-based make-up of the Norrish constant, thereby leading to a reconsideration
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of the molecular basis of water activity.
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The Norrish constant is made up of water-water, solute-water and solute-solute interactions,
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expressed via the KBIs. In the following, their relative contributions to the Norrish constant are
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quantified. Fig. 1. and 2. highlight the contribution of each of the KBI terms for binary aqueous
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solutions of various sugars and polyols respectively, to the overall value of the Norrish constant
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(data is summarised in Table 1). In the case of the solutes studied in the present work, it can be
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∞
∞
∞
seen that both 𝐺𝑠𝑠
and 𝐺𝑠𝑤
make significant contributions to the Norrish constant with 𝐺𝑠𝑤
the
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more dominant of the pair. As has been mentioned previously, this conclusion is markedly
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different to the “free water” and “water structure” hypotheses which were outlined in Section 1.
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The following has emerged from our analyses:
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1. Solute-solute interaction drives up the Norrish constant
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2. Solute-water interaction drives down the Norrish constant
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3. The Norrish constant, much smaller than 1 and 2, is the result of compensation between 1
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and 2.
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These insights have been obtained directly from experimental data and the principles of statistical
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thermodynamics without any model assumptions. Because of the fundamental nature of the above
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insights, we are now in the position to examine the accuracy and validity of the previous
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hypotheses on the molecular basis of water activity, summarized in Introduction.
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With respect to the “Free water” hypothesis, if the “bound water” (i.e. water which is not
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∞
∞
“free”) in solution can be interpreted as 𝐺𝑠𝑤
in the context of our theory, then 𝐺𝑠𝑤
is, in almost all
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∞
cases, a less dominant contributor to the Norrish constant than 𝐺𝑠𝑠
. Similarly, if it is simply “free”
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or “bound” water which are the primary origins of water activity, we may also expect an interplay
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∞
∞
between 𝐺𝑤𝑤
and 𝐺𝑠𝑤
to have a substantial influence on K. This however, is not observed.
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∞
Furthermore, the signs of 𝐾 and 𝐺𝑠𝑤
are opposite.
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In contrast to the “Water structure” hypothesis, it is the compensation between solute-solute
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and solute-water interaction rather than solely the properties of solvents which drives up the
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Norrish constant. In the context of the Kirkwood-Buff theory, if “water structure” or “ordering” is
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∞
the origin of water activity then we should expect the water-water interaction term, 𝐺𝑤𝑤
to have a
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∞
∞
significant influence on K. In fact, what we actually observe is that 𝐺𝑤𝑤
is small relative to 𝐺𝑠𝑤
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∞
and especially 𝐺𝑠𝑠
and thus, has negligible effect on the Norrish constant. This conclusion is
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consistent with previous criticisms of ‘water structure’ by KB theory (Shimizu, Stenner, &
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∞
Matubayasi, 2017). Note that our results also highlight that inter-solute interactions (𝐺𝑠𝑠
) are a
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key contributor to water activity in contrast to their supposedly negligible role as advocated in both
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“free water” and “water structure” hypotheses.
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With respect to the “Hydration” and clustering hypothesis, our results have identified that the
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∞
activity coefficient of water is a result of a compensation of two large contributions; i. 𝐺𝑠𝑤
which
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∞
drives down the water activity, and; ii. 𝐺𝑠𝑠
which drives up the water activity. This is contradictory
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to the earlier theories that attempted to explain water activity solely from hydration (i.e., solute-
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water interaction) but is consistent with the later development of the models that incorporated the
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effect of solute (sucrose) clustering. Intuitively speaking, water activity is driven up by solute
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clustering as it increases the effective mole fraction of water (Gharsallaoui, Rogé, Génotelle, &
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Mathlouthi, 2008; Starzak & Mathlouthi, 2006; Starzak, Peacock, & Mathlouthi, 2000; VanHook,
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1987). Previous models have reached such a conclusion through a series of equilibrium constant
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estimations that was crucial for modelling the water activity over a wide range of concentrations.
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Our interpretation, unlike previous models, is based only on the first principles of statistical
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thermodynamics, which supports the insights from the clustering models.
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3.2. Why the Norrish equation describes water activity beyond infinite dilution
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The Norrish equation (eq. 1), according to rigorous statistical thermodynamics, has been shown to
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be valid only at infinite solute dilution. In contrast to this theoretical foundation, Norrish equation
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has been used to fit the water activity far beyond infinite dilution. It has been reported that the
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Norrish equations holds well until ca. 60% w/w% of various nonelectrolytes, including sucrose
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(Baeza, Pérez, Sánchez, Zamora, & Chirife, 2010). Why is the Norrish equation applicable beyond
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infinite dilution? This can be understood by a comparison between the “Norrish approximation”
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and the rigorous KB theory. To do so, we need to employ the general formalism of the KB theory
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for binary solution mixture, applicable to the entire concentration range, which has been well-
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established (Ben-Naim, 2006; Chitra & Smith, 2002) and has been applied for the analysis of
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aqueous sucrose solutions (Shimizu 2013). We have shown in Appendix B that the Norrish
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equation is accurate when
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2𝐾 = 𝑥
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behaves virtually as a constant over a wide concentration range, where 𝑥𝑤 is the mole fraction of
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water, 𝑥𝑐 is for solute, 𝑐𝑤 is the molarity of water, and the KBIs in the r. h. s. are in principle
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dependent on the concentration.
𝑐𝑤 (𝐺𝑤𝑤 +𝐺𝑠𝑠 −2𝐺𝑠𝑤 )
𝑤 1+𝑥𝑠 𝑐𝑤 (𝐺𝑤𝑤 +𝐺𝑠𝑠 −2𝐺𝑠𝑤 )
1
(6)
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Based upon previous analysis of sucrose (Shimizu, 2013) using the fitting model of Starzak and
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Mathilouthi (Starzak & Mathlouthi, 2006), Fig. 3 shows the change of the r.h.s. of Eq. (6) against
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sucrose w/w%, which demonstrate its increase is very slow, thereby demonstrating the accuracy
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of the Norrish approximation based on its constancy. Note that the r.h.s. of Eq. (6) computed from
266
preceding data used by Shimizu (-16.4) (Shimizu 2013), is closer to the value (-14.8) calculated
267
from two-parameter Margules (Miyawaki, Saito, Matsuo, & Nakamura, 1997) rather than the one
268
from the Norrish constant from literature (2𝐾 = −12.9) (Taoukis & Richardson, 2007), which
12
269
may come from the fact that the Norrish 𝐾 represents an average of the r.h.s. of Eq. (6) over a wide
270
sucrose concentration range. What is important here is that the weak sucrose concentration
271
dependence of 𝐾 comes from 𝐺𝑤𝑤 + 𝐺𝑠𝑠 − 2𝐺𝑠𝑤 , whose sucrose concentration dependence has
272
been shown to be much weaker than the much stronger concentration dependencies of 𝐺𝑠𝑠 and 𝐺𝑠𝑤
273
(Shimizu, 2013). This suggests that the presence of compensation between 𝐺𝑠𝑠 and 𝐺𝑠𝑤 that is
274
responsible for the near-constancy of 𝐾. Whether this mechanism holds true for other solutes will
275
be investigated in our future publications.
276
277
The effective radii of sucrose and water can provide a rough justification of the KBIs and the
278
Norrish constant. Assuming water and sucrose as spheres, the sucrose-sucrose and water-water co-
279
volumes, −𝐺𝑠𝑠 and −𝐺𝑤𝑤 , correspond to the effective radii of 3.2 and 0.94 Å for water and
280
sucrose, which are smaller than their commonly-quoted hard-sphere radii, because the peaks of the
281
correlation functions contribute negatively to co-volumes (Shimizu, 2013). Using these effective
282
radii, −𝐺𝑠𝑤 can be estimated to be 174 cm3 mol-1, which is only ca. 15 % smaller than the value
283
in Table 1. The reasonable success of this rough estimation suggests that there seems to be a simple
284
volumetric mechanism at work, which may determine much of the KBIs and the Norrish constant
285
and may be behind the slow change of 2𝐾 in Fig. 3. A more quantitative treatment is possible
286
only by an explicit treatment of intermolecular interactions via molecular simulations.
287
288
4. Conclusion
289
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290
Due to the lack of a theoretical foundation, the molecular origin of water activity in liquid food
291
systems has long been obscure. There have been three hypotheses in the literature, yet even
292
whether they are consistent or contradictory has remained unanswered.
293
294
To address this historical question, we have combined the wisdom of food science with the
295
rigorous statistical thermodynamics. Based upon the Kirkwood-Buff theory of solutions, we have
296
identified the origin of the Norrish constant, i.e., water-water, solute-water and solute-solute
297
interactions, amongst which water-water is a minor contribution. The Norrish constant is a product
298
of compensating contributions from the solute-solute and solute-water interactions. Solute-solute
299
interaction drives up the Norrish constant while the solute-water interaction is in the opposite
300
direction to the Norrish constant.
301
302
In contrast to the previous work, solute-solute interaction has quantitatively been identified as a
303
crucial contributor to the water activity. This conclusion based on a rigorous theory is inconsistent
304
with two traditional hypotheses of water activity origin; “free water” and “water structure”, both
305
of which are primarily built upon solely empirical data and argue that solute-solute interplay is
306
effectively negligible. The compensation between solute-solute and water-solvent interactions
307
clarified by our rigorous theory corroborates more recent solute hydration and clustering models
308
from the first principles of statistical thermodynamics.
309
Acknowledgements
310
ASM wishes to thank Nestlé and BBSRC for provision of a CASE studentship (AJM). SS is
311
grateful to the Gen Foundation. NM is supported by the Grants-in-Aid for Scientific Research
312
(Nos. 15K13550 and JP26240045) from the Japan Society for the Promotion of Science, by the
14
313
Elements Strategy Initiative for Catalysts and Batteries and the Post-K Supercomputing Project
314
from the Ministry of Education, Culture, Sports, Science, and Technology, and by the HPCI
315
System Research Project (Project IDs: hp170097 and hp170221).
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15
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Appendix A
322
We derive the statistical thermodynamic expression of the Norrish constant based on a well-known
323
result from the KB theory (Ben-Naim, 2006; Shimizu, 2013), namely
324
( 𝜕𝑥𝑤 )
𝜕𝜇
𝑠
𝑅𝑇
𝑇,𝑃
= −𝑥
1
(A1)
𝑤 1+𝑥𝑠 𝑐𝑤 (𝐺𝑤𝑤 +𝐺𝑠𝑠 −2𝐺𝑠𝑤 )
325
where 𝑐𝑤 is the molar concentration of water. Note that 𝐺𝑤𝑤 + 𝐺𝑠𝑠 − 2𝐺𝑠𝑤 plays a crucial role in
326
Eq. (A1) (Shimizu, 2013). At 𝑥𝑠 → 0, the r.h.s. of Eq. (A1) can be expanded as:
327
( 𝜕𝑥𝑤 )
𝜕𝜇
𝑠
𝑅𝑇
𝑇,𝑃
≈ − 𝑥 [1 − 𝑥𝑠
∞ +𝐺 ∞ −2𝐺 ∞
𝐺𝑤𝑤
𝑠𝑠
𝑠𝑤
𝑤
∞
𝑉𝑤
(A2)
]
328
where 𝑉𝑤∞ is the partial molar volume of pure water. It is useful to rewrite Eq. (A2) in terms of the
329
activity coefficient of water, as
330
(
𝜕 ln 𝛾𝑤
𝜕𝑥𝑠
1
)
𝑇,𝑃
𝜕𝜇
= 𝑅𝑇 ( 𝜕𝑥𝑤 )
𝑠
1
𝑇,𝑃
+ 𝑥 ≈ 𝑥𝑠
𝑤
∞ +𝐺 ∞ −2𝐺 ∞
𝐺𝑤𝑤
𝑠𝑠
𝑠𝑤
∞
𝑉𝑤
(A3)
331
Comparing Eq. (A3) with the Norrish equation, ln 𝛾𝑤 = 𝐾𝑥𝑠2 , we obtain the following final form:
332
𝐾=2
333
∞
Note that the physical meaning of the infinite-dilution KB integral, 𝐺𝑠𝑠
, can be understood directly
334
∞
from its relationship to the second virial coefficient, 𝐵2, via 𝐺𝑠𝑠
= −2𝐵2 (Ben-Naim, 2006).
∞ +𝐺 ∞ −2𝐺 ∞
1 𝐺𝑤𝑤
𝑠𝑠
𝑠𝑤
∞
𝑉𝑤
335
16
336
Appendix B
337
Norrish equation can fit water activity far beyond infinite solute dilution. To explain why this is
338
possible, let us start by rewriting Eq. (A1) as
339
(
𝜕 ln 𝛾𝑤
𝜕𝑥𝑠
1
)
𝑇,𝑃
=𝑥
𝑥𝑠 𝑐𝑤 (𝐺𝑤𝑤 +𝐺𝑠𝑠 −2𝐺𝑠𝑤 )
𝑤 1+𝑥𝑠 𝑐𝑤 (𝐺𝑤𝑤 +𝐺𝑠𝑠 −2𝐺𝑠𝑤 )
(B1)
340
341
The rigorous expression (Eq. (B1)) should be compared with the approximation used in deriving
342
the Norrish equation. Hence the task is to show the accuracy of
343
𝑐𝑤 (𝐺𝑤𝑤 +𝐺𝑠𝑠 −2𝐺𝑠𝑤 )
𝑥𝑤 1+𝑥𝑠 𝑐𝑤 (𝐺𝑤𝑤 +𝐺𝑠𝑠 −2𝐺𝑠𝑤 )
1
≈
∞ +𝐺 ∞ −2𝐺 ∞
𝐺𝑤𝑤
𝑠𝑠
𝑠𝑤
∞
𝑉𝑤
(B2)
344
17
345
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445
446
20
447
Figure captions
448
449
∞
∞
Fig. 1. Molecular origin of the Norrish constant 𝐾. Comparison between KBIs 𝐺𝑤𝑤
, −2𝐺𝑠𝑤
and
450
∞
∞
∞
∞
𝐺𝑠𝑠
and the corresponding 2𝐾𝑉𝑤∞ = 𝐺𝑤𝑤
+ 𝐺𝑠𝑠
− 2𝐺𝑠𝑤
for various binary aqueous sugar
451
solutions.
452
453
∞
∞
Fig. 1. Molecular origin of the Norrish constant 𝐾. Comparison between KBIs 𝐺𝑤𝑤
, −2𝐺𝑠𝑤
and
454
∞
∞
∞
∞
𝐺𝑠𝑠
and the corresponding 2𝐾𝑉𝑤∞ = 𝐺𝑤𝑤
+ 𝐺𝑠𝑠
− 2𝐺𝑠𝑤
for various binary aqueous polyol
455
solutions.
456
457
Fig. 3. Accuracy of the Norrish equation beyond infinite sucrose dilution, demonstrated by the
458
slow change of the r.h.s. of Eq. (6) with respect to sucrose concentration.
459
21
460
Figure 1
500
∞
𝐺
ABC
𝑤𝑤
∞
−2𝐺
-2Gsw
𝑠𝑤
∞
𝐺
Gss
𝑠𝑠
2KVw
2𝐾𝑉𝑤∞
(cm3/mol)
250
0
-250
-500
-750
461
Fructose
Glucose
Sucrose
Maltose
Xylose
Galactose
462
22
463
Figure 2
300
∞
𝐺𝑤𝑤
Gww
∞
−2𝐺
-2Gsw
𝑠𝑤
∞
𝐺
Gss
𝑠𝑠
2𝐾𝑉𝑤∞
2KVw
200
(cm3/mol)
100
0
-100
-200
-300
464
Glycerol
Xylitol
Arabitol
Mannitol
Sorbitol
465
466
23
467
Figure 3
-2
2K
-6
-10
-14
-18
0
468
20
40
60
80
sucrose w/w %
469
470
24
471
Table 1
472
∞
∞
∞
Values of the Norrish constant, K and individual Kirkwood-Buff integrals (𝐺𝑠𝑤
, 𝐺𝑤𝑤
and 𝐺𝑠𝑠
) for
473
∞
∞
each species used in this study (cm3/mol). 𝐺𝑠𝑤
and 𝐺𝑤𝑤
were calculated according to Eqs. (4) and
474
(5) respectively and using values of 𝑉𝑤∞ = 18.1 cm3 mol-1 and 𝜅𝑇∞ = 4.53 × 10-10 Pa-1. Note that
475
all calculations are strictly only valid at 298K.
K
c
𝑽∞
𝒔
Fructose
-2.25a
Glucose
𝑮∞
𝒔𝒘
𝑮∞
𝒘𝒘
𝑮∞
𝒔𝒔
-110
-109
-16.9
-283
-2.25a
-112
-111
-16.9
-286
Sucrose
-6.47a
-212
-210
-16.9
-638
Maltose
-4.54a
-209
-208
-16.9
-562
Xylose
-1.54a
-95.4
-94.3
-16.9
-227
Galactose
-2.24a
-110
-109
-16.9
-282
Glycerol
-1.16a
-71.0
-69.8
-16.9
-165
Xylitol
-1.66a
-102
-101
-16.9
-246
Arabitol
-1.41b
-103
-102
-16.9
-238
Mannitol
-0.91a
-119
-118
-16.9
-252
Sorbitol
-1.65a
-120
-119
-16.9
-280
Species
Sugars
Polyols
476
477
a
478
b
479
c
Taken from Taoukis and Richardson (Taoukis & Richardson, 2007)
Taken from Rahman (Rahman, 1995)
Taken from Cabani et al. (Cabani, Gianni, Mollica, & Lepori, 1981)
480
481
482
25