Thermodynamic and kinetic aspects of fat crystallization

Advances in Colloid and Interface Science 122 (2006) 3 – 33 www.elsevier.com/locate/cis

Thermodynamic and kinetic aspects of fat crystallization C. Himawan, V.M. Starov, A.G.F. Stapley ⁎ Department of Chemical Engineering, Loughborough University, Ashby Road, Loughborough, Leicestershire, LE11 3TU, United Kingdom Available online 14 August 2006

Abstract Naturally occurring fats are multi-component mixtures of triacylglycerols (TAGs), which are triesters of fatty acids with glycerol, and of which there are many chemically distinct compounds. Due to the importance of fats to the food and consumer products industries, fat crystallization has been studied for many years and many intricate features of TAG interactions, complicated by polymorphism, have been identified. The melting and crystallization properties of triacylglycerols are very sensitive to even small differences in fatty acid composition and position within the TAG molecule which cause steric hindrance. Differences of fatty acid chain length within a TAG lead to packing imperfections, and differences in chain lengths between different TAG molecules lead to a loss of intersolubility in the solid phase. The degree of saturation is hugely important as the presence of a double bond in a fatty acid chain causes rigid kinks in the fatty acid chains that produce huge disruption to packing structures with the result that TAGs containing double bonds have much lower melting points than completely saturated TAGs. All of these effects are more pronounced in the most stable polymorphic forms, which require the most efficient molecular packing. The crystallization of fats is complicated not just by polymorphism, but also because it usually occurs from a multi-component melt rather than from a solvent which is more common in other industrial crystallizations. This renders the conventional treatment of crystallization as a result of supersaturation somewhat meaningless. Most studies in the literature consequently quantify crystallization driving forces using the concept of supercooling below a distinct melting point. However whilst this is theoretically valid for a single component system, it can only at best represent a rough approximation for natural fat systems, which display a range of melting points. This paper reviews the latest attempts to describe the sometimes complex phase equilibria of fats using fundamental relationships for chemical potential that have so far been applied to individual species in melts of unary, binary and ternary systems. These can then be used to provide a framework for quantifying the true crystallization driving forces of individual components within a multi-component melt. These are directly related to nucleation and growth rates, and are also important in the prediction of polymorphic occurrence, crystal morphology and surface roughness. The methods currently used to evaluate induction time, nucleation rate and overall crystallization rate data are also briefly described. However, mechanistic explanations for much of the observed crystallization behaviour of TAG mixtures remain unresolved. © 2006 Elsevier B.V. All rights reserved. Keywords: Nucleation; Crystal growth; Triacylglycerol; Melts; Polymorphism; Crystal morphology

Contents 1.

2.

Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1. Molecular structure and composition of fats. . . . . . . . . . . . . . . . . . . 1.2. Basic polymorphism of TAGs . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3. Scope of this review. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Thermodynamic aspects of the melt crystallization of fats . . . . . . . . . . . . . . . 2.1. Free energy diagrams and polymorph stability . . . . . . . . . . . . . . . . . 2.2. Correlating and predicting the melting temperature and enthalpy of pure TAGs 2.3. The polymorphic behaviour of pure TAGs . . . . . . . . . . . . . . . . . . . 2.3.1. Monoacid saturated TAGs . . . . . . . . . . . . . . . . . . . . . . .

⁎ Corresponding author. Tel.: +44 1509 222525; fax: +44 1509 223923. E-mail address: [email protected] (A.G.F. Stapley). 0001-8686/$ - see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.cis.2006.06.016

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2.3.2. Mixed-acid saturated TAGs . . . . . . . . . . . . . . 2.3.3. Mixed-acid saturated/unsaturated TAGs . . . . . . . . 2.4. Phase behaviour of binary mixtures of TAGs . . . . . . . . . 2.4.1. Phase diagrams . . . . . . . . . . . . . . . . . . . . 2.4.2. Modelling the solid–liquid equilibria of TAGs . . . . 3. Kinetic aspects of the melt crystallization of fats . . . . . . . . . . . 3.1. Nucleation and crystal growth rates — theoretical aspects. . . 3.1.1. Thermodynamic driving force. . . . . . . . . . . . . 3.1.2. Nucleation thermodynamics, kinetics and mechanisms 3.1.3. Polymorphic-dependent nucleation . . . . . . . . . . 3.1.4. Induction time. . . . . . . . . . . . . . . . . . . . . 3.1.5. Growth rate and mechanisms . . . . . . . . . . . . . 3.1.6. Morphology of TAG crystals . . . . . . . . . . . . . 3.1.7. Spherulitic growth. . . . . . . . . . . . . . . . . . . 3.1.8. Polymorphic transformation . . . . . . . . . . . . . . 3.2. Measurement of fat crystallization kinetics. . . . . . . . . . . 3.2.1. Induction time and nucleation rate . . . . . . . . . . 3.2.2. Overall crystallization rates . . . . . . . . . . . . . . 4. Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1. Introduction Fats are highly abundant compounds in nature and are widely used in food and other consumer products [1]. Their behaviour heavily influences the microstructure and physical properties of these products. The development of solid fat

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microstructure from a liquid melt to create commercial fat products such as margarine or chocolate is schematically presented in Fig. 1 [2], which illustrates how both the initial processing (within the factory) and subsequent storage conditions (in the warehouse, shop or home) ultimately affect final product structure, texture and quality.

Fig. 1. Schematic presentation of processes involved in crystallization and storage of fats (adapted from [2]).

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A number of factors, including crystallization conditions, are important. Crystallization occurs in the initial processing stage, and the relative rates of nucleation and growth determine the initial crystal size distribution. This is a key parameter for texture as crystals greater than a few tens of microns in size are detectable on the tongue, and are thus undesirable in products which require a smooth texture. As the solid fraction increases, individual crystals begin to touch each other which slows crystal growth (growth impingement). Interactions between crystals then start to dominate the process. Depending on the nature of the fat substances, gel formation may also occur [3]. During storage, a number of post crystallization processes occur, which can affect properties such as hardness, which often noticeably increases [4]. This is due to sintering, i.e. the formation of solid bridges between crystals to form a network [2,4,5]. Polymorphic transformation (see Section 1.2) towards more stable phases and changes in size distribution via Ostwald ripening may occur [6]. The above events are not necessarily chronological once nucleation occurs. It is possible, even usual, in processing fats, that after primary nucleation and subsequent growth that secondary nucleation, defined as nucleation occurring due to the presence of the growing crystals [7], can take place simultaneously along with crystal growth and ripening. Furthermore, polymorphic transformations may occur in the processing stage. Transformation into the desirable polymorphic forms that deliver favourable properties is often forced via manipulating conditions. For example, shearing and tempering have been applied in cocoa butter crystallization for controlling its polymorphism [8–12]. The characterization of microstructure and the relation to the mechanical properties of the final product is a difficult (and still largely unresolved) field of study in its own right, and readers are suggested to consult the reviews by Walstra et al. [2],

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Narine and Marangoni [13,14], and Marangoni [15]. It can be seen, however, that control of the initial crystallization of the fat is crucially important to the final quality of any fat based product. The crystallization of fats also determines the behaviour of fractionation processes in which fat fractions with different melting ranges are separated by crystallizing the higher melting fats and filtering the slurry that is formed. The resulting fractions are used as ingredients in food formulations and the main reason for fractionation is to tailor these fats to improve their functionality. The crystallization conditions in fractionation are different to those in other food processes as growth impingement generally does not occur and larger crystals are required to promote easy filtering [16,17]. The study of fat crystallization is thus a valuable activity as a greater understanding of fat crystallization enables fractionation and food processes to operate more efficiently and the functional effectiveness of fats in food products to be optimised. However, before reviewing fat crystallization in detail, it is necessary to first cover two complicating aspects of fats — their multi-component nature and polymorphism. 1.1. Molecular structure and composition of fats Edible oils and fats mainly consist of a multi-component mix of triacylglycerols (TAGs) with a small amount of other minor components. An edible oil or fat can typically contain more than a hundred different TAGs. A TAG is a triester of glycerol with three fatty acid molecules, and the general chemical structure is depicted in Fig. 2. Fatty acids consist of a hydrocarbon chain terminated by a carboxylic acid group. The hydrocarbon chain length ranges from 4 to 30 carbons (between 12 and 24 are the most common). The chain usually has an even number of carbons and is linear unless double bonds are present in which

Fig. 2. (a) A general molecular structure of triacylglycerol (R1, R2, and R3 are individual fatty acid moieties). (b) The chemical structures of a saturated and a nonsaturated fatty acid [5].

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Table 1 Nomenclature of commonly occurring fatty acids Code Fatty acid 2 4 6 8 C L M

Chain length Double bonds Code Fatty acid

acetic acid (ethanoic acid) 2 butyric acid (butanoic acid) 4 caproic acid (hexanoic acid) 6 caprilic acid (octanoic acid) 8 capric acid (decanoic acid) 10 lauric acid (dodecanoic acid) 12 myristic acid (tetradecanoic acid) 14

none none none none none none none

P S O E l R A B

case the chain becomes kinked. The carbon atoms of these “linear” chains are arranged in a zigzag fashion, which has implications for crystal packing (see next section). The physical properties of TAGs heavily depend upon the fatty acid composition [18]. For convenience, TAGs are usually identified by a 3-letter code. Each of the characters in the code represents a fatty acid with the middle character always indicating the fatty acid that is on the 2-position of the glycerol. For example, PSP represents glycerol-1,3-dipalmitate-2-stearate. If the three fatty acids are the same, the TAG is monoacid; otherwise it is called mixedacid. A TAG is unsaturated if a CfC double bond is present in at least one of the fatty acid moieties, otherwise it is referred to as saturated. The characters used to represent fatty acids are given in Table 1 and will be used throughout this paper. 1.2. Basic polymorphism of TAGs TAG molecules are inherently able to pack in different crystalline arrangements or polymorphs, which exhibit significantly different melting temperatures [19,20]. The polymorphism of most fats is based around three main forms: α, β′, and β; the nomenclature scheme following Larsson [21] as reviewed in Hagemann [20], Hernqvist [22], Wesdorp [23], Sato [24], and Gothra [5]. However, some fats display more polymorphs than this. TAG molecules are “three legged” molecules that can pack with the acyl chains (“legs”) in one of two configurations, neither of which involves all three chains packing alongside each other. They can pack in a “chair” configuration where the acyl chain in the 2 position is alongside the chain on either the 1 or 3 positions. Alternatively, a “tuning fork” configuration can

palmitic acid (hexadecanoic acid) stearic acid (octadecanoic acid) oleic acid (cis-9-octadecanoic acid) elaidic acid (trans-9-octadecanoic acid) linoleic acid (cis-cis-9,12-octadecadienoic acid) ricinoleic acid (12-hydroxy-9-octadecenoic acid) arachidic acid (eicosanoic acid) behenic acid (docosanoic acid)

Chain length Double bonds 16 18 18 18 18 18 20 22

none none 1 1 2 1 none none

be adopted where the acyl chain in the 2 position is alone and the chains in the 1 and 3 positions pack alongside each other. Either configuration naturally packs in a chair-like manner. The stacking of these chairs can be in either a double of triple chain length structure (see Fig. 3a), and these stack side by side in crystal planes, sometimes at an angle. The differences between polymorphs are most apparent from a top view of these planes which shows the subcell structure (Fig. 3b). These structures can be identified by powder X-ray diffraction patterns [22,24], where long spacings give information on the repeat distance between crystal planes (chain length packing) and short spacings give information on subcell structure (interchain distances). These interchain distances depend on how the chains pack together and this is complicated by the “zigzag” arrangement of successive carbon atoms in aliphatic chains. Closer packing is achieved when the zigzags of adjacent chains are in step with each other (“parallel”) as opposed to out of step (“perpendicular”). • The α-form is characterized by one strong short spacing line in the XRD pattern near 0.42 nm. The chains are arranged in a hexagonal structure (H), with no angle of tilt and are far enough apart for the zigzag nature of the chains to not influence packing. • The β′-form is characterized by two strong short spacing lines at 0.37–0.40 nm and at 0.42–0.43 nm. The chain packing is orthorhombic and perpendicular (O⊥), that is adjacent chains are out of step with each other so they cannot pack closely. The chains have an angle of tilt between 50° and 70°. • The β-form is characterized by a strong lattice spacing line at near 0.46 nm and a number of other strong lines around

Fig. 3. (a) Chain-length packing structures in TAGs, and (b) the subcell structures of the three most common polymorphs in TAGs (viewed from above the crystal planes) [24]. Reprinted with permission.

C. Himawan et al. / Advances in Colloid and Interface Science 122 (2006) 3–33

0.36–0.39 nm. This is the densest polymorphic form having a triclinic chain packing, in which adjacent chains are in step (“parallel”), and thus pack snugly together. The chains also have an angle of tilt between 50° and 70°. The β and β′ polymorphs can exist as either double chainlength or triple chain length structures. A double chain length structure normally occurs when the chemical nature of the three fatty acid moieties are the same or very similar. Conversely, if the moieties are quite different to each other (for instance in a mixed saturated-unsaturated TAG), a triple chain-length structure is formed. The α form is normally only found to exist in a double chain length structure. 1.3. Scope of this review Many efforts have been performed to unravel the complex behaviour of fat systems. Crystallization studies are regularly carried out for natural fats and these are classified by their origins, e.g. palm oil and related oils [11,12,25–33], milk fats [11,34–43] and cocoa butter [11,12,44–49]; just to mention a few of the most recent contributions. Further reviews can be

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found in Smith [50] for palm oil, in Hartel and Kaylegian [51] for milk fat, and in Sato and Koyano [52] for cocoa butter. Many other studies have investigated the blending of natural fats as means of tailoring the physical and thermal properties of fats [53–58]. The disadvantage of the above approach is the empirical and case by case nature of the information obtained. This can cause difficulties when coping with the compositional variations in natural fats that originate from geographical, climatic, or seasonal factors. A more fundamental approach is to study the crystallization of fats by considering them as multi-component systems. This is a huge challenge but has already given extended insights on the behaviour observed in natural fats as excellently reviewed by Sato [24,59]. This is necessarily a bottom-up exercise, whereby an understanding of pure TAG and binary systems must first be obtained. This review seeks to provide an overview of the current fundamental understanding of fat crystallization approached from the thermodynamic and kinetic behaviour of pure TAGs and binary mixtures of pure TAGs. Fat crystallization differs from most industrial crystallization processes in that crystallization is seldom from a “solvent”, and thus traditional

Table 2 Literature on polymorphic and phase behaviour of pure and binary mixtures of TAGs Author

Systems

Measurement techniques

Remarks

(A) Polymorphic occurrence and transformation of pure TAGs Miura et al. [168] PPP, SSS, POP, SOS, POS, POS/SOS mixtures Ueno et al. [167] PPP, LLL Higaki et al. [48] Pure and impure PPP Smith et al. [213] Different TAGs Sprunt et al. [214] SOS Boubekri et al. [111] SRS Ueno et al. [110] SOS Dibildox-Alvarado et al. [215] PPP in sesame oil Toro-Vazquez et al. [216] PPP in sesame oil Ueno et al. [66] SOS Rousset et al. [197] POP, POS, SOS Yano et al. [109] SOS, POP, POS Kellens et al. [95] PPP Arishima et al. [107] POS Kellens et al. [94] PPP, SSS Kellens et al. [93] PPP Arishima et al. [97] POP, SOS Koyano et al. [105] POP, SOS

DSC, XRD DSC, SR XRD DSC, XRD Light microscopy, DSC, XRD FT Raman spectroscopy, DSC FTIR, SR XRD SR XRD DSC, light microscopy, XRD DSC, light microscopy, XRD DSC, SR XRD Light microscopy, DSC FTIR Light microscopy, DSC DSC, XRD SR XRD SRXRD DSC, XRD DSC, light microscopy, XRD

Effect of Effect of Effect of Effect of

(B) Phase behaviour and polymorphic transformation of binary TAG mixtures Miura et al. [168] POS/SOS Takeuchi et al. [125] LLL/MMM, LLL/PPP, LLL/SSS Takeuchi et al. [124] SOS/SLS Takeuchi et al. [123] SOS/SSO Rousset et al. [146] SOS/POS Minato et al. [121] POP/PPO Minato et al. [122] POP/OPO Minato et al. [120] PPP/POP Engstrom et al. [128] SOS/SSO Kellens et al. [181] PPP/SSS Koyano et al. [119] SOS/OSO Kellens et al. [192] PPP/SSS Wesdorp [23] Binary TAGs Cebula and Smith [194] PPP/SSS

DSC, XRD SR XRD DSC, SR XRD DSC, SR XRD DSC, SR XRD DSC, SR XRD DSC, SR XRD DSC, SR XRD DSC, XRD DSC, XRD DSC, XRD DSC, SR XRD DSC SR XRD

Effect of ultrasound Effect of the difference of molecule length

DSC = Differential scanning calorimetry. SR XRD = synchrotron radiation X-ray diffraction.

ultrasound ultrasound magnetic fields phospholipids additives

Intermediate structured liquids Molecular structure and interactions Variability of morphology

Existence of molecular compounds Phase diagram of metastable phases Existence of molecular compounds Existence of molecular compounds Immiscibility of the least unstable polymorph Existence of molecular compounds Existence of molecular compounds Mixing properties Confirmation of the intermediate phase (β′)

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Table 3 Literature on crystallization kinetics of pure and binary mixtures of TAGs Reference

Systems

(A) Crystallization kinetics of pure TAGs Hollander et al. [178] Different TAGs Meekes et al. [217] Different TAGs Hollander et al. [149] Different TAGs Higaki et al. [48] Pure PPP, impure PPP Smith et al. [213] Different TAGs Dibildox-Alvarado et al. [215] PPP in sesame oil Toro-Vazquez et al. [216] PPP in sesame oil Rousset et al. [146] POP, POS, SOS Kellens et al. [95] PPP Kellens et al. [218] SSS Koyano et al. [199] POS Koyano et al. [106] POP, SOS Sato and Kuroda [92] PPP Zhao et al. [219] PPP, LLL, SSS (B) Crystallization kinetics of binary TAG mixtures Rousset et al. [146] SOS/POS MacNaughtan et al. [127] PPP/SSS Himawan et al. [150,182,193] PPP/SSS

Measurement techniques

Kinetic aspects

Light microscopy Light microscopy DSC, XRD Light microscopy, SEM and DSC DSC, XRD DSC, XRD DSC, light microscopy DSC, light microscopy, XRD DSC, light microscopy, XRD Light microscopy Light microscopy DSC, light microscopy DSC

Crystal growth rate and morphology Simulation of morphology Crystal growth rate and morphology Induction time, effect of ultrasound Crystal growth rate and morphology (effect of additives) Using Avrami model for kinetic analysis Using Avrami model for kinetic analysis Nucleation and growth rates. Mapping of crystal morphology Induction time, nucleation, and growth rate Induction time and nucleation Induction time. Direct melt and melt mediated crystallization Induction time. Direct melt and melt mediated crystallization Induction time Bulk and emulsified samples

DSC, light microscopy DSC DSC, light microscopy

Nucleation and growth rates. Mapping of crystal morphology Induction time and half time of crystallization Nucleation and growth rates. Spherulite morphology

concepts of supersaturation are not helpful. A more detailed examination of thermodynamic driving forces based upon chemical potential relationships is needed. The thermodynamics of fats systems are thus first discussed in Section 2, and subsequently extended in Section 3 to quantify crystallization driving forces and to examine the kinetic aspects of fat crystallization. Tables 2 and 3 list the current literature on the polymorphic and kinetic behaviour of pure and binary mixtures of TAGs, on which much of this review is based. 2. Thermodynamic aspects of the melt crystallization of fats Traditionally, a solid fat mixture is characterized by its solid fraction content (SFC), i.e. the mass fraction of solid present at a certain temperature. The SFC is then normally used as a basis to predict and determine the many physical properties of the material [60]. The typical melting temperature (i.e. normally defined as the temperature at which the SFC is zero) and SFC characteristics of some natural fats are shown in Table 4 [61–63]. These are determined most importantly by the composition of the fat. For instance, the main TAGs in palm oil are POP (22%), POO (22%), PPO (5%), PPP (5%), POS (5%), PlP (7%), PlO (7%),

OOO (5%), and POl (3%) [50]; meanwhile those in coconut butter are POS (46%), SOS (29%), POP (13%), PlS (3%), SOO (2%), and SlS (2%) [52]. In this section the thermodynamic aspects of fat systems are addressed. This begins with a general outline of polymorphism, before focussing on individual systems. The inherently complex nature of fats dictates that the discussion of phase equilibria is best tackled starting with the simplest systems first, namely pure TAGs of a single saturated fatty acid moiety (e.g. PPP, see Table 1 for the nomenclature). Increasing complexity can then be added by the presence of double bonds and mixing different fatty acid moieties within a TAG molecule whilst still maintaining a single component system. Finally, the phase behaviour of binary mixtures of different TAG molecules is introduced. 2.1. Free energy diagrams and polymorph stability Two types of polymorphism generally exist in lipids and organic compounds [20,23,64]. Enantiotropic polymorphism occurs when each polymorphic form is thermodynamically the most stable in a particular range of temperature and pressure. Changing the temperature or pressure to outside this range will

Table 4 Melting temperatures and SFC values of natural fats in their most stable polymorph Fat

Butter Cocoa butter Lard Palm oil Palm kernel oil Tallow

Melting temperature (°C)

SFC (%) at temperature 10 °C

15 °C

20 °C

25 °C

30 °C

35 °C

Data sources

36 34 42 40 28 50

55 – 27 54 68 58

37 – – 40 56 –

19 76 20 26 40 45

11 70 – 16 17 –

5 45 3 11 – 25

1 1 – 8 – 15

Bockisch [61] Gunstone [62] Bockisch [61] Gunstone [63] Gunstone [63] Bockisch [61]

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favour the transformation into a different polymorph (that which is most stable under the new conditions) [6,65]. Long chain odd carbon number alkanes exhibit such behaviour [23]. In monotropic polymorphism, on the other hand, one polymorphic form is always the most thermodynamically stable. Transformations occur from the less stable polymorphs to the more stable ones given sufficient time [6,65]. The relative stability of two polymorphs and the driving force for transformations between them at constant temperature and pressure are determined by their respective Gibbs free energies (G) — the polymorph which has the lowest Gibbs free energy is the most stable. Gibbs free energy–temperature diagrams are utilised to map the thermodynamic stability of the polymorphs. Fig. 4a shows the G–T diagram for the three basic polymorphs in TAGs from which ΔG values between phases can be deduced. The form of the plots follows the defining equation for Gibbs free energy as a function of enthalpy (H), entropy (S) and temperature (T) which is: G ¼ H−TS

ð1Þ

Due to its monotropic nature, the Gibbs free energy values are largest for the α-form (least dense crystal packing), intermediate for the β′-form, and smallest for the β-form (most dense crystal packing). This is mainly a consequence of the higher heats of fusion of polymorphs with higher melting temperature. Each polymorphic form has its own melting temperature, Tm, shown as the intersection points of the G–T curves of the polymorphs and the liquid phase (Fig. 4a). The transformation pathways among the three main polymorphs are shown in Fig. 4b and can be summarised as follows: • The three polymorphic forms can all be directly crystallized from the melt. • Although any polymorph can be returned to the liquid phase by raising the temperature above the melting point, interpolymorphic transformations are always irreversible (i.e. β cannot transform to β′ and β′ cannot transform to α). • Two different modes of transformation are possible: (i) transformations within the solid state, and (ii) a recrystallization of the more stable forms after the less stable forms have

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melted. The latter is normally called “melt-mediated transformation”. • It has been found in some fat systems that a thermotropic liquid crystalline phase exists (not shown in the G–T diagram) as a mesophase or intermediate phase which occurs before the crystallization of the polymorphic crystals or during melt-mediated transformation [66–68]. In such cases, the transformation pathway diagram becomes more complicated (Fig. 4b). The transformations between liquid and crystalline states and between crystalline states are all first order transitions where there is a discontinuity in the first derivative of the free energy [69]. 2.2. Correlating and predicting the melting temperature and enthalpy of pure TAGs The melting temperature and the melting enthalpy of pure TAGs are central to a thermodynamic description of solid liquid phase equilibria in multi-component fat systems as they can be accurately measured and can be used to construct basic free energy diagrams assuming constant ΔH and ΔS. Here correlations between these thermal properties and the chemical structure of the compounds are described. Fig. 4a shows that each polymorph in a pure TAG has its own distinct melting temperature. As at equilibrium ΔG = 0, the melting temperature can be written as the ratio of the enthalpy to the entropy of melting (ΔHm and ΔSm) given by: Tm ¼

DHm DSm

ð2Þ

Thus one strategy for correlating melting points is to combine separate correlations for melting enthalpy and entropy. However, enthalpy and entropy are also difficult to correlate. The values of ΔHm and ΔSm are governed by several factors such as hydrogen bonding, the molecular packing in crystals (influenced by molecular shape, size and symmetry), and other intermolecular interactions such as charge transfer and dipoledipole interactions in the solid phase [70]. These interactions are

Fig. 4. (a) The relation between Gibbs free energy and temperature for the three main polymorphic forms of TAGs (monotropic polymorphism). (b) The polymorphic transformation pathways in fats involving liquid crystals. Adapted from [59].

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complex and it is difficult to predict them (and thus ΔH and ΔS) with confidence. Due to such complex interactions, only limited guidelines exist for describing the relationship between the melting temperature of an organic compound and its chemical structure despite the enormous amount of available melting temperature data. Several recent studies on the estimation of the melting temperature and melting enthalpy of organic compounds have been reported covering a wide variety of classes of organic compounds. A review on this subject was given by Katritzky [70] who classified existing correlations into three categories: • Models utilising physicochemical and structural parameters, such as bulkiness, cohesiveness, hydrogen-bonding parameters, and geometric factors [71–73]. • Group contribution methods in which a molecular breakdown scheme is generally employed and multiple regression analysis is performed to determine the contribution of a large number of molecular groups to the melting temperature [74– 78]. Usually, melting enthalpy is calculated from group contribution methods while melting entropy consists of a group contribution value as well as non-additive molecular parameters. The latter represents rotational and conformational entropies [77,78]. • Estimations from Monte Carlo or molecular dynamics computer simulations for the phase transitions and related properties of compounds including the melting temperature [79–82]. In the case of TAGs, saturated fatty acids are relatively linear molecules (Fig. 2b) and thus TAGs containing only saturated fatty acids can easily align themselves to form a compact mass. On the other hand, unsaturated fatty acids in TAGs have kinks in their aliphatic chains (Fig. 2b). The disrupted packing of the unsaturated TAGs hinders the formation of crystals and causes unsaturated TAGs to have a lower melting temperature than saturated TAGs with the same chain length. Molecular symmetry [83,84] and crystal packing [70,74] are considered to be the most influential factors governing the thermal properties of TAGs. The many different combinations of arranging fatty acid moieties in TAGs, along with polymorphism, means that the estimation of melting temperature of TAGs is more difficult compared to that of most organic compounds. The methods used for general organic compounds can, nevertheless, be applied to TAGs. Normally, the melting enthalpy and entropy are expressed as the sum of a contribution of the hydrocarbon chains (depending linearly on the chain length) and a contribution of the end and head groups (independent of chain length) [23]. DHm ¼ hn þ h0

ð3Þ

DSm ¼ sn þ s0

ð4Þ

Here, n is the length of hydrocarbon chains, h and s are constants that do not depend on the nature of the compound but

only on the way hydrocarbon chains are packed, thus they are universal constants that only depend on the polymorphic form. The other constants h0 and s0 that account for the end-group contributions (the structure of fatty acid moieties) are specific to each class of lipid. Combining Eqs. (2)–(4), gives:
 DHm hn þ h0 A Tm ¼ ¼ ¼ Tl 1 þ ð5aÞ nþB DSm sn þ s0 with: h Tl ¼ ; s

A¼

h0 s 0 − ; h s

B¼

s0 s

ð5bÞ

This implies that if the melting temperatures of a class of lipids have been correlated, only one data point for the enthalpy of fusion is in principle sufficient to obtain a correlation for the enthalpy of fusion of the complete class of lipids. However, this is an oversimplification, as differences in chain lengths of individual moieties need to be accounted for. Timms [85] compiled Tm and ΔHm data of β′- and β-forms of selected TAGs and gave regressed correlations for each polymorphic form. Zacharis [86] used Eq. (3) to represent the thermal data of monoacid TAGs. Perron [87,88] updated the work of Timms [85] and published correlations for the three polymorphic forms for saturated TAGs. Furthermore, Perron modelled the lower melting enthalpy of unsaturated TAGs (ΔHm,unsat) by comparing them with the corresponding saturated TAG (ΔHm,sat) and then making an adjustment according to the following equation: DHm;unsat ¼ DHm;sat −115ð1−e−0:706d Þ

ð6Þ

where d is the number of double bonds in the unsaturated TAG. Won [89] followed the approach of Zacharis [86] but applied the equations to saturated TAGs with mono and mixed acyl groups. However, data were only correlated with the total number of carbon atoms and the effects of position were not considered. Thus the fitted values were identical for different TAGs with the same total number of carbon atoms. Zeberg-Mikkelsen and Stenby [90] developed empirical correlations based upon a group-contribution method which took into account the position of the acyl groups. The correlations were only valid for saturated TAGs which had an even number of carbon atoms (between 10 and 22) in each acyl group. Chickos and Nichols [74] developed simple relationships for homologous series and showed that they were applicable to the three polymorphic forms of symmetrically substituted TAGs. Anomalous behaviour, which was revealed in some cases, was argued to be caused by different packing between members of a series. Molecular modelling has also recently been applied to estimate the thermal and transport properties of TAGs with reasonable predictive capability [91]. Wesdorp [23] developed a model to estimate Tm and ΔHm for different polymorphic forms of saturated and unsaturated TAGs from a large database. He improved the method of Eqs. (5a) and (5b) to account for the effect of position and chain length of the three acyl groups in TAGs (symbolised by pqr).

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Two parameters were introduced x = q − p and y = r − p, where q is the chain length of the acyl group in position 2 of the TAG and p is the shortest chain length of the acyl group in positions 1 or 3. From many regression trials, Wesdorp [23] identified several factors to be important in order to successfully estimate Tm and ΔHm values of TAGs. These were (1) the length of each chain, (2) whether the chain has an even or odd number of carbon atoms, (3) whether the chain is saturated or unsaturated, and (4) the molecular symmetry. It was also found that the melting enthalpy of the β-form depended on whether it was double chain length or triple chain length packed. Correlations obtained for unsaturated TAGs in the study were found to be less reliable due to the limited data available compared to those for saturated TAGs. Although aimed at the development of an empirical model, the work of Wesdorp [23] indicated that the thermal behaviour of TAGs directly follows from their molecular structure. 2.3. The polymorphic behaviour of pure TAGs The polymorphic nature of TAGs is well established. It is also well known that mixing different fatty acid moieties in a TAG produces more complex polymorphic behaviour (principally the number of observable polymorphs). Thus saturated monoacid TAGs are simplest, followed by mixed acid saturated, with mixed acid saturated/unsaturated being the most complex [18,59]. 2.3.1. Monoacid saturated TAGs This group of TAGs has been examined by thermal techniques (such as DTA and DSC) more than any other group and shows the basic α, β′, and β polymorphic forms [20]. Melting temperature and enthalpy data for the three polymorphic forms with fatty acid chain lengths ranging from 8 to

11

30 have been compiled by Hagemann [20], Wesdorp [23], and by Zelberg-Mikkelsen and Stenby [90]. Generally, the polymorphic behaviour of TAGs with an even carbon number are well represented by the behaviour of PPP [67,92–95] and SSS [20,94,96] and summarised as follows (see Fig. 5 for the SSS thermal behaviour and the structural model of the molecular packing of each polymorph): • The α-form is crystallized upon cooling from the melt at moderate to high cooling rates. Remelting the α-form induces an endotherm at a slightly higher temperature than the cooling exotherm, but this is soon followed by an exotherm associated with the formation of the stable β-form [20,94]. • The β′-form crystallizes if the temperature is maintained slightly above the melting temperature of the α-form (about 30 min induction time for SSS). Several endotherms may be observed upon remelting caused by submodifications of the β′-form [20,94]. • The β-form can be crystallized directly using a solvent [20,97] or by tempering/holding (about 60 min induction time for SSS) slightly above the melting temperature of β′form [94]. Only CCC (tricaprin) was reported to reveal multiple β-forms [98]. The chain length of fatty acid moieties has a significant influence on the polymorphic behaviour. Of particular note is that the crystal packing of β′ and β forms also depends on whether the number of carbons in the chain is even or odd [22]. • For TAGs of C22 and longer, rapid cooling exhibits a single exotherm associated with the formation of the α-form. However, Hagemann [20] showed that tempering can lead to

Fig. 5. (a) Typical thermograms of monoacid saturated TAGs represented by tristearin. Adapted from [20]: cooling from the melt at 20 °C/min (dashed line), followed by heating at 2.5 °C/min (solid line). Intermediate forms (β1′ and β2′) are observed after holding 30 min slightly above the melting point of the α-form. (b) Side-view structural model of molecular packing of the α, β′ and β; the different between the structure of the β′- and the β-form is in their subcell structure (see Fig. 2). Adapted from [22].

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structure due to steric hindrance of the molecular structure of odd number TAGs and the more precise packing of the β polymorph.

Fig. 6. Melting temperatures plotted against fatty acid chain lengths of α-, β′-, and β-forms of monoacid saturated TAGs [99]. Reprinted with permission from the American Oil Chemists' Society.

two submodifications of the α-form with greater separation between the two peaks as the chain length increases. • Three different submodifications of the β′-form were reported in even carbon numbers shorter than C16. The third modification melted close to the β-form, the difference in melting points decreasing wth increasing chain length [20]. • The β′-form of odd carbon number monoacid TAGs is more stable compared to even number TAGs [20]. X-ray diffraction analysis indicates this is due to a closer similarity of the crystal structure of the β′- and β-forms with odd TAGs than is the case with even TAGs [98]. • The melting points of the α-form increase monotonically with fatty acid chain length but those of the β′- and β-forms show fluctuations due to the odd–even chain length effect (see Fig. 6) as reported in hydrocarbon type materials [20,23,99]. The trend of melting temperature versus chain length for odd numbered TAGs is generally lower than that for even numbered TAGs. The effect is most pronounced at lower chain lengths and is maintained for the β polymorph at higher chain lengths. This reflects the less packed crystal

2.3.2. Mixed-acid saturated TAGs Mixed-acid saturated TAGs, mainly those with acids with even carbon number chain lengths in the range 12–20, are widely prevalent in natural fats. Modifications of polymorphic behaviour from that of monoacid saturated TAGs result from differences in chain length between the fatty acid moieties, and this is also influenced by their relative positions [20,59]. This was best described by Sato [59] when analysing the polymorphic and thermal behaviour of the asymmetric PPn TAGs [24,100–102] the symmetric CnCn + 2Cn TAGs [103,104]. Here n represents even chain lengths varying from 0 to 16 in PPn and from 10 to 16 in CnCn+2Cn. Sato and Ueno [59] observed that heterogeneity in the chain lengths of the three acyl groups tends to reduce the gap in stability of the β′-form and β-form such that the β-form is not observed. This is illustrated by the behaviour of asymmetric PPn TAGs, where β′ was the most stable form of PP6, PP8, and PPM, while β was most stable in PP2, PP4, and PPC. The chain-length structure of the most stable forms also varied with increasing n from double (PP2, PP4) to triple (PP6, PP8, PPC) and back to double again (PPL, PPM). The irregular trend of the melting temperatures of the PPn, shown in Fig. 7a, reflects the variation in the chain length structures. In CnCn+2Cn TAGs, β′ was always found to be the most stable form as no β form was observed [103]. The melting temperatures and long spacings of the CnCn+2Cn series increased monotonically with increasing n (Fig. 7b) as would be expected. The complexity of polymorphs of mixed acid TAGs is illustrated by Fig. 8 which shows the polymorph structures of PPC [101]. The most notable aspect is that there are various submodifications of the β′-form of this molecule. The α-form occurs by rapid cooling from the melt which further transforms to β3′ (O⊥ subcell). Upon remelting, the β3′-form transforms to the β2′-form with the same subcell type. All α-, β2′- and β3′forms are double chain length structures. A transformation from

Fig. 7. Long spacing values (open squares) and melting temperatures (closed circles) of (a) PPn TAGs [100] and (b) CnCn+2Cn TAGs [103]. Adapted from Sato and Ueno [59].

C. Himawan et al. / Advances in Colloid and Interface Science 122 (2006) 3–33

Fig. 8. Polymorphic transformations in PP10 [59,101,220].

the β2′-form to the triple chain length β-form proceeds at higher temperatures. Additionally, rapid melting of the α-form induces another β′-form showing a hexa-layered structure (β1′-6). Many issues regarding the polymorphic behaviour in asymmetric mixed-acid saturated TAGs remain unresolved [59], due to the various interchain interactions of the methyl end groups, aliphatic chains and glycerol groups [24]. 2.3.3. Mixed-acid saturated/unsaturated TAGs TAGs with unsaturated fatty acids at the sn-2 position and saturated acids at the other positions (Sat-U-Sat) are the main components of a number of widely used vegetable fats such as palm oil and cocoa butter. These will be considered here to illustrate the complexities of unsaturated systems. Particularly commonplace are those containing oleic acid at the sn-2 position. The presence of the double bond (with the inflexible “kink”) gives greater steric hindrance than found in completely saturated TAGs, which forces specific structures to be formed to enable the saturated and unsaturated fatty acid moieties to be

13

packed together in the same lamella leaflet. Consequently, this TAG group exhibits still more complicated polymorphic behaviour as observed in the systems of SOS, POP, POS, SRS, and SlS [66,105–112]. Kaneko et al. [113] and Sato [24] expressed this complexity by highlighting the importance of olefinic conformations (see Fig. 9) in addition to the molecular chain packing (subcell packing) and the chain-length structure. These relate to how the aliphatic chains on either side of the double bond are twisted with respect to the plane of the double bond. Information on these structures can be obtained from XRD, Fourier Transform Infra Red (FTIR) [109,114,115] and Nuclear Magnetic Resonance (NMR) [116,117]. The polymorphic structures of all Sat-O-Sat TAGs (with Sat being saturated fatty acid and O being oleic acid) are similar, with the exception of POP [24,59]. Fig. 10 shows the structures of both POP and SOS (which can be taken to be representative of the other Sat-O-Sat TAGs) [109]. Particularly noteworthy for this TAG group are: • Another intermediate phase, γ can occur which has a triple chain-length structure. The saturated and oleic acid chains of this form are disordered with oleic acid chains packing in a hexagonal subcell (as in the α-form) whilst the saturated chain leaflet shows a parallel packing. • The β′-form is a triple chain-length structure, whereby the saturated chain leaflets form an ordered O⊥ subcell whilst the oleic acid chain leaflets remain in a disordered hexagonal subcell. • In the case of the two β-forms the saturated and oleic acid leaflets both pack in an ordered manner. There is a slight difference in the length of the triple chain-length structure of these two forms, and a small difference in melting temperature of 1.5–2.0 °C. The presence of a double bond in Sat-O-Sat TAGs generally forces the β′- and β-forms to adopt a triple chain-length

Fig. 9. Representation of the olefinic conformations of fatty acids in TAGs containing oleic acid moieties; S–C–S′ when ω- and Δ-chains are placed in the same plane and S–C–S when the two chains are normal to each other [113].

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Fig. 10. A structural model of the polymorphic behaviour in Sat-O-Sat TAGs represented by the behaviour of POP and SOS [109]. Reprinted with permission from The Journal of Physical Chemistry. Copyright (1993) American Chemical Society.

structure so that the oleic acid chains are packed together and separately from the saturated chains. The exception is the β′form of POP which forms a double chain-length structure (Fig. 10)). This is probably because the palmitic and oleic acid chains pack to a similar length once the kink in the longer oleic acid chain is taken into account. This would result in a weaker steric hindrance to the formation of a double chain length structure than would be the case with the other Sat-OSat TAGs. The long spacings (representing the chain-length structure) and melting temperatures of Sat-O-Sat TAGs are presented in Fig. 11. In general, a smooth increase of the long spacing and melting temperature with increasing length of the saturated acid chains is observed except for the more stable polymorphs which show rather jagged profiles. The long spacing of the β′-form of POP is much shorter than for the other TAGs as it forms a double rather than triple chain-length structure (Fig. 10). An exception to the general pattern is POS, which does not show a

γ-form and only shows a single β-form. Sato and Ueno [59] have suggested that this might be due to the racemic nature of POS (although the similarly racemic SOA does not show the same behaviour). Boubekri et al. [111] and Takeuchi et al. [112] in turn reported that SRS and SlS exhibit similar polymorphism to the other Sat-U-Sat TAGs, except that their polymorph stability and thermal properties are modified significantly. In SRS, hydrogen bonding in the ricinoleoyl chains of the β′-form is much tighter than that in the case of SOS so that the β′-form is much more stable. Evidence for the greater hydrogen bonding comes from the much higher melting enthalpy and entropy of the β′-form of SRS than in SOS and SSS [59]. In SlS, the γ-form is stabilised due to interactions among the linoleoyl chains at the sn-2 position. Accordingly, the enthalpy and the entropy values for the melting of γ of SlS are much larger than those of SOS and SRS. We have discussed here only the Sat-U-Sat TAGs to give an impression of the complex polymorphism that can occur in fats. Other mixed acid saturated–unsaturated TAGs also exist such as Sat-Sat-U and Sat-U-U. For information on these systems the reader is recommended to consult the review by Sato and Ueno [59]. 2.4. Phase behaviour of binary mixtures of TAGs The next step up in complexity of systems is to consider binary mixtures of TAGs. The equilibrium behaviour of a binary mixture is best illustrated using phase diagrams. 2.4.1. Phase diagrams Timms [118] identified four main types of phase diagram that are commonly observed in binary mixtures of TAGs (Fig. 12): • Monotectic continuous solid solutions, which are formed when the TAGs, are very similar in melting temperature, molecular volume and polymorphism (e.g. SSS/SSE, POS/ SOS). • Eutectic systems, which are the most commonly found, tend to occur when the components differ in molecular volume,

Fig. 11. Long spacing values (left) and melting temperatures (right) of polymorphs of Sat-O-Sat TAGs [108]. Adapted from [59].

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Fig. 12. The four main types of phase diagram in binary mixtures of TAGs (a) monotectic, continuous solid solution, (b) eutectic, (c) monotectic, partial solid solution, (d) peritectic [118].

shape, and polymorph but not greatly in melting temperature (e.g. PPP/SSS, POS/POP, SOS/SSO). • Monotectic partial solid solutions form in preference to a eutectic system if the difference in melting temperature of the TAG components is increased (e.g. PPP/POP). • Peritectic systems (2 solid solutions and 1 liquid) have only been found to occur in mixed saturated/unsaturated systems where at least one TAG has two unsaturated acids (e.g. SOS/ SOO, POP/POO). An extensive compilation of phase diagrams of binary TAG mixtures from the literature has been made by Wesdorp [23] who identified three critical issues when considering such diagrams: (i) the purity of materials used in experiments, (ii) the stabilisation procedure for producing the most stable phase (which must be standardised to reduce error), and (iii) difficulties in the determination of the solidus resulting from kinetic effects (discussed in Section 3). Recently, binary phase diagrams have been constructed via the use of synchrotron radiation (SR) XRD [119–125]. The high intensity of this X-ray technique provides richer information about the polymorphic phases and it is also gained in real time which allows metastable polymorphs to be

characterized distinctly, in contrast to traditional methods [125]. For binary TAG mixtures, the primary factors determining phase behaviour are differences between the TAGs in chain length, the degree of saturation and position of the fatty acid moieties, and which polymorphs are involved. Different phase behaviour is frequently observed for different polymorphs, e.g. PPP/SSS shows complete miscibility of the less stable forms (α and β′) but a eutectic system for the β-form [126,127]. The effect of the differences in chain length is illustrated by the behaviour of mixtures of two monosaturated TAGs. Takeuchi et al. [125] studied the phase diagrams of LLL/ MMM, LLL/PPP, and LLL/SSS and after also considering that of PPP/SSS, came to the following conclusions for binary monosaturated TAG mixtures: • The metastable α- and β′-forms are miscible when the carbon numbers for the fatty acid chains of the three TAGs differ by 2 or less. This is the case, for example, with PPP/ SSS and LLL/MMM (see Fig. 13a). • Immiscibility of the metastable phases appears when differences in carbon chain lengths of 4 or 6 are present

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Fig. 13. The effect of the difference of carbon numbers in binary saturated TAG mixtures on phase behaviour: (a) miscible metastable phases in LLL/MMM, (b) immiscible metastable phases in LLL/SSS [125]. The melting temperatures reported are slightly higher than the onset temperatures of melting. Reprinted with permission from Crystal Growth and Design. Copyright (2003) American Chemical Society.

such as with LLL/PPP and LLL/SSS (see Fig. 13b). Eutectic and monotectic behaviour are observed in the β-form for the LLL/PPP and LLL/SSS systems, respectively, with the α form of SSS co-existing with the β form of LLL under certain conditions. As already mentioned, increasing the difference between the melting temperatures of the pure TAG's shifts the phase behaviour from eutectic to monotectic. The reasons for this are largely unexplored [125]. In mixtures where monosaturated and mixed-acid saturatedunsaturated TAGs are combined, such as the PPP/POP system (see Fig. 14), there is a pronounced steric effect. It is difficult for the oleic acid chain to pack directly with PPP and this results in

Fig. 14. The effect of steric hindrance in the PPP/POP system, an example of a mixture of a monosaturated and a mixed-acid saturated–unsaturated TAG. All three polymorphs show eutectic behaviour [120]. Reprinted with permission from the American Oil Chemists' Society.

limited miscibility and is reflected by eutectic behaviour for all three polymorphic forms α, β′ and β [120]. Combining two TAGs which both contain an unsaturated fatty acid is less problematic as like chains from either TAG can arrange themselves together. Indeed it is sometimes the case that two TAGs can display a synergistic compatibility and pack more easily together than on their own. These form so-called “molecular compounds” with a 50:50 ratio of the two components. This is observed in systems such as SOS/OSO [119], SOS/SSO [123,128], POP/PPO [121], and POP/OPO [122]. As an example, the phase behaviour of the POP/PPO system is presented in Fig. 15. The three polymorphs α, β′ and β form eutectic phases at the 50:50 molar composition. The properties of molecular compounds have been investigated using FT-IR and XRD, and show significant deviations from those of the component molecules [113]. Molecular compounds also consistently form double chain length structures in the metastable and stable phases in contrast to the triple chain length structures that are found in the stable polymorphs of the pure TAG components. These molecular compounds also crystallize faster than the pure components of the same polymorph [59,123]. The formation of molecular compounds impacts upon the performance of fractionation processes, as only limited separation is thus experienced. On the other hand this can be useful for blending purposes [59,119]. 2.4.2. Modelling the solid–liquid equilibria of TAGs With the plethora of binary phase diagrams in existence for TAGs, it is useful to be able to condense this information into a (relatively) small number of parameters by the use of modelling. This also potentially enables extensions to be made to describe ternary and higher systems. The equilibrium condition for a multi-component system with a liquid phase and at least one solid phase can be described as the point where the chemical potential of each component (i)

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Fig. 15. Formation of molecular compounds in the mixture of unsaturated TAGs (PPO/POP): (a) the most stable phase and (b) metastable phases; C represents molecular compounds [121]. Reprinted with permission from The Journal of Physical Chemistry B. Copyright (1997) American Chemical Society.

in each phase is equal to that in any other phases present [129], i.e.: lLi ¼ lSj i

ð7Þ

where μiL and μiSj are the chemical potentials of each component i in the liquid and the jth solid phase, respectively. The chemical potential of component i in a mixed phase p (solid or liquid) is given by: lpi ¼ lpi;0 þ RT lnðgpi xpi Þ

ð8Þ

p is the chemical potential of the pure component i in where μi,0 the respective phase, xip is the mole fraction of component i and γip is the activity coefficient for component i. Substitution of Eq. (8) into Eq. (7) results in the equilibrium condition for component i: ! Sj lLi;0 −lSj gSj i;0 i xi ð9Þ ln L L ¼ RT gi xi p p To evaluate the right hand side of Eq. (9), let dμi,0 = − Si,0 dT p p p + Vi,0dP (where Si,0 and Vi,0 are the pure component molar entropy and molar volume of the p phase for component i, respectively, P is pressure) and ΔSi,0 = ΔHi,0/T (where ΔHi,0 is the change of molar enthalpy upon melting of pure component i). Using these definitions we obtain:   d Dli;0 ¼ −DSi;0 dT þ DVi;0 dP DHi;0 dT þ DVi;0 dP ¼− ð10aÞ T or

dðDli;0 Þ DHi;0 DVi;0 ¼− dP dT þ RT RT 2 RT

ð10bÞ

A simplification of Eq. (10b) can be made by assuming the following: • The reference temperature is the melting temperature of the pure component i at the system pressure, Tm,i(P). Thus the

effect of pressure does not need to be considered further (dP = 0). • The change in molar enthalpy can be represented by ΔHi,0 ≅ ΔHm,i,0 + ΔCpi,0(T − Tm,i), where ΔHm,i,0 is the molar enthalpy of melting of pure component i at the reference temperature Tm,i and ΔCpi,0 is the molar heat capacity difference between the liquid and solid for the pure component i (assumed to be independent of temperature). Integration of Eq. (10b) and substitution into Eq. (9) results in [60,130]: ! Sj Dli;0 DHm;i;0 DT DCpi;0 DT gSj i xi ln L L ¼ ¼ − RTm;i;0 T RT RT gi xi
 DCpi;0 Tm;i;0 þ ln ð11Þ R T where ΔT = Tm,i − T. Eq. (11) relates the equilibrium compositions in the two phases (left hand side) to the system temperature (right hand side). These equilibrium compositions are heavily dependent on the activity coefficients, and to describe the equilibrium conditions, the effect of composition and temperature on the activity coefficients (in Eq. (11)) must be appropriately modelled. This is usually only required for the solid phase activity coefficients as the liquid phase can generally be assumed to be ideal. Prausnitz [129] elaborately describes the existing thermodynamic models for such a purpose. The simplest case is where there is a large difference in melting points. The high melting component essentially forms a pure crystal (xiS = 1). Both liquid and solid activity coefficients are unity and Eq. (11) is rearranged and reduced to the so-called Hildebrand equation (where xi in Eq. (12) is the mole fraction of the high melting component in the liquid phase):
 DHm DT DHm 1 1 ¼ − lnxi ¼ ð12Þ RTm T Tm T R Of course, the activity coefficients also dictate the mixing behaviour of the system in both the liquid and solid phases. If it is possible for the overall system Gibbs free energy to be

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Fig. 16. Modelling of the phase diagram of the stable phases in binary TAG mixtures [23]. Examples are shown for PSP/SPS which forms a eutectic (left) and for PPP/ POP without a eutectic point (right) [60].

reduced by splitting the solid into two different solid fractions of different compositions then immiscibility will occur. This generally requires activity coefficients to be greater than unity. The thermodynamic modelling of binary solid liquid equilibria involving solid solutions has been applied to many different areas of application. An example is the long chain hydrocarbons (waxes), which exhibit non-ideal mixing (activity coefficients deviate from unity) in both liquid and solid phases. Investigations have thus focused on finding the appropriate model to describe activity coefficients for liquid and solid phases and then assessing the capability of binary parameters to describe the multicomponent mixtures [131–137]. Equations of state have also been applied in this particular case [138]. The use of thermodynamic models in the food area has also recently been reported [139–142], and Tao [143] has reviewed their application in material science. Despite the usefulness of thermodynamic modelling in many other areas of application, there has been relatively little work on modelling the solid–liquid equilibria of TAG mixtures [60]. Wesdorp [23] studied the mixing behaviour of TAG mixtures in the liquid phase and three different polymorphic forms. He found that melts of TAG mixtures and solid solutions of αpolymorphs behave as ideal mixtures (as long as the difference of chain length does not exceed 15 carbon atoms) while β′- and β-forms exhibit significantly non-ideal behaviour. Based on those findings, a thermodynamic model to describe the phase behaviour of multi-component fats was proposed. The excess Gibbs energy for all solid phases, ΔGES, was successfully fitted using a 3-suffix Margules equation (see Eqs. (13) and (14) for binary systems). A drawback of this equation is the lack of a rational base for its extension to multicomponent systems. It is generally assumed that the contributions of the binary parameters (A12 and A21 in Eqs. (13) and (14)) to the excess Gibbs energy in the multi-component mixture are the same as in the binary mixture at the same relative concentrations. DGE ¼ ðA21 x1 þ A12 x2 Þx1 x2 RT lng1 ¼ x22 ½A12 þ 2ðA21 −A12 Þx1 

ð13Þ

RT lng2 ¼ x21 ½A21 þ 2ðA12 −A21 Þx2 

ð14Þ

All 4 types of binary TAG phase diagram [118] have been well simulated by the 3-suffix Margules equation. Examples are shown in Fig. 16 for eutectic and non-eutectic binary TAG mixtures [60]. Binary interactions parameters of various TAG combinations have been documented [23] and have been used to simulate the SFC of fats containing many TAG components; showing reasonably good agreement with experimental data [23,60,144,145]. A similar approach was employed by Rousset et al. [146] to characterise the equilibrium states of binary mixtures of the POS/SOS system which was then used to define the crystallization driving forces for a kinetic study (see Section 3.1.1). Having demonstrated the ability of the 3-suffix Margules equation to simulate phase diagrams of TAG mixtures, Wesdorp [23] attempted to theoretically estimate the binary interaction coefficients needed in the Margules equation by evaluating the degree of isomorphism [147] and lattice distortion and thus produce a predictive model. However, reliable correlations were not achieved. Ideally, thermodynamics should give a firm foundation for predictive models of SFC provided the compositions of the fat mixture are known. By extracting binary interaction coefficients between the triacylglycerol components in the mixture, it is possible to extrapolate to ternary and more complex mixtures [23,60,146]. In practice, however, kinetics cannot be neglected due to the often slow process of fat crystallization [144,145] and the presence of metastable regions. Yet thermodynamic aspects are critical since equilibrium information of a fat mixture will enable the driving force of crystallization to be quantified and establish a benchmark for the kinetic behaviour. The kinetic aspects will now be addressed. 3. Kinetic aspects of the melt crystallization of fats Although a solid becomes the thermodynamically stable phase when a melt is cooled down below its melting temperature, this liquid–solid transition does not occur spontaneously. The occurrence of a solid phase in its early stages requires two distinct events: (1) the formation of nuclei in the

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mother phase followed by (2) the advancement of the faces of the nuclei resulting in crystal growth. In fat systems, it has been proposed that an ordering process of molecules into lamellae acts as a precursor to the formation of a crystalline solid phase [24,51] (see Fig. 17a). This process follows a path through transitory states that requires energy barriers to be overcome as shown in Fig. 17b for different polymorphic forms [148]. The finite diffusion rates of molecules in the liquid and solid phases and the arrangement and subsequent attachment of molecules onto the surface of growing crystals all contribute to the kinetics of the overall process [149]. Consequently, kinetic factors are as important as thermodynamic ones in determining which polymorph will form from the melt and the amount, composition and properties of the crystalline phase. Examples of these kinetic effects are described below. (a) Polymorphic occurrence Usually fats crystallize first in the least stable polymorph with the lowest energy barrier (α) and later transform or recrystallize to more stable polymorphs (β′ or β). Direct crystallization of β′- or β-forms from melts tends to occur only when no supercooling, or sometimes little, of the less stable forms is present. Fig. 18 shows the kinetic phase diagram of PPP/SSS [150] upon linear cooling at different cooling rates. Depending on the cooling rates applied, either α- or β′-forms crystallize. This illustrates the strong influence of kinetics on polymorphic occurrence in fats. (b) Composition gradients within crystals Differences in composition between the outer and inner regions of a crystal are thought to occur during a slow cooling crystallization as described in Wesdorp [23] and Los et al. [144,145]. This would be due to the higher melting components preferentially solidifying during the early stages of crystal growth which are then depleted from the liquid melt. The low

19

Fig. 18. Effect of kinetics on the polymorphic occurrence in the binary PPP/SSS system at a number of cooling rates [150]. Tm,α is the equilibrium temperature of the α-form. The β′-form crystallised at 0.5 and 1 K min− 1 in PPP-rich mixtures (shown as open symbols).

diffusion rate in the solid phase hampers the inner part of the growing crystals to reach equilibrium with the liquid phase as the composition of the liquid phase changes, whereas the surface composition is much closer to equilibrium. The crystals are ultimately inhomogeneous in composition having a concentration gradient between the centre and the surface of the crystal. However, although the concept of a composition gradient within crystals is plausible, as far as we know no experimental proof has been published. (c) Crystal perfection

Fig. 17. (a) Simplified schematic representation of ordering in the liquid state of TAGs preceding the formation of a crystalline solid phase [24,51] (reprinted with permission). (b) Energy barrier diagrams for the three main polymorphic forms of a TAG at a given conditions below their melting temperatures. Adapted from [148].

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Poorly packed crystals can result from rapid crystallization [67,151,152]. The thermal properties of such imperfect crystals deviate significantly from those of well-ordered ones. Imperfect crystals may persist for years in the absence of a liquid phase [20] but can easily recrystallize into well packed crystals via the liquid phase if a liquid phase is present [22]. Los et al. [130,144,145] extended the work of Wesdorp [23] by implementing a simple kinetic expression into the “flash” calculation of multi-component TAG mixtures. They showed, via simulation using thermodynamic parameters from Wesdorp [23], that the effect of kinetics on the prediction of the SFC of fat mixtures is substantial. However, comparisons with experimental data were not presented. It is clear that kinetic factors should be considered in order to describe properly the crystallization behaviour of fats. In the following sections aspects characterising the dynamics of fats crystallization are examined. 3.1. Nucleation and crystal growth rates — theoretical aspects 3.1.1. Thermodynamic driving force The fundamental thermodynamic driving force for the crystallization of a component i is the difference in chemical potential of i (Δμi) between the liquid (μiL ) and solid (μiS) phases. The chemical potentials are formulated as in Eq. (8), and thus: Dli ¼ lLi −lSi ¼ Dli;0 þ RT ln

gLi xLi gSi xSi

ð14aÞ

Substituting in the expression for (Δμi,0) from Eq. (11) yields: Dli DHm;i ðTm;i −T Þ DCpi ðTm;i −T Þ ¼ − RTm;i T
RT  RT
 DCpi Tm;i gL xL ln þ þ ln iS iS R T gi xi

ð14bÞ

However, in almost all cases in the literature, one of two simplified approaches is used [148,153]. (a) Liquid-solution approach The first approach represents the fat blend as a mixture of two pseudo-components that are immiscible in the solid state. The pseudo-component with the higher melting temperature is considered to be the solute, while the one with lower melting temperature is the solvent. This is normally applied when fats contain two families of distinctly different TAGs [2,31,154]. The approach is similar to most studies of industrial crystallization, where the crystallization driving force is modelled as the result of supersaturation. Thus for a liquid phase of a defined concentration of solute, the difference between the saturation concentration is evaluated (at the same temperature). The saturation composition (xiL,eq) is that which is in equilibrium with the forming solid phase (xiS), which can related by Eq. (9) thus: ln

gSi xSi gL;eq xL;eq i i

! ¼

Dli;0 RT

ð15Þ

Combining Eqs. (14a) and (15) and eliminating (Δμi,0) results in: ! ! gSi xSi gLi xLi gLi xLi Dli ¼ RT ln L;eq L;eq þ RT ln S S ¼ RT ln L;eq L;eq gi xi gi xi gi xi ð16Þ In many cases, the liquid phase of multi-component fats is nearly ideal due to the relatively similar size and structure of the component molecules [23], i.e. γiL,eq ≈ γiL ≈ 1. Eq. (16) is thus further simplified to: Dli iRT ln

xLi xL;eq i

ð17Þ

L For small supersaturations (xiL/xi,eq < 1.1) a further approximation via a Taylor expansion yields Δμi ≅ RT[(xiL − xiL,eq)/ xiL,eq]. Frequently, s ≡ (xiL − xiL,eq)/xiL,eq is used as an approximation to the crystallization driving force, particularly at low supersaturations, considering that concentrations are relatively easy to measure. Sometimes, (xiL − xiL,eq) is also used [155,156]. For high supersaturations (xiL/xiL,eq > 1.1), Eq. (16) should be used. A limitation of this method is that it is reliant on the availability of an equilibrium liquid concentration for the solid phase. This cannot be evaluated if the sample temperature is below the solidus, in which case a different approach is called for. (b) Liquid-melt approach When fats are composed of relatively similar component TAGs, it is often assumed that crystallization can be described as occurring from a pure melt. Thus the last term in Eq. (14b) is neglected. A further simplication can also be made by neglecting the second and third terms on the right-hand side of Eq. (14b), which for fats are at least two orders of magnitude smaller than the first term, where ΔT = Tm,i − T is not larger than 10 K [130]. This gives:
 Tm;i −T Dli iDHm;i ð18Þ Tm;i

According to the latter equation, the driving force is thus proportional to the difference between the actual temperature and the melting temperature. Note, however, that for very complex systems such as natural fats which have many different TAG components, the definition of the above crystallization driving force becomes ambiguous as melting typically occurs over a broad range [51] and a single representative melting point is difficult to establish in a way that can be consistently reliable under different conditions. Different polymorphic forms can also crystallize concomitantly to hamper accurate melting temperature identification. A reasonable strategy in some circumstances is to apply a global supercooling approximation [5]. The global melting temperature of the complex melt mixture is defined to be the highest temperature at which solid phases can exist and are about to disappear. The difference between the crystallization temperature and this global melting temperature is regarded as the

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driving force of crystallization [28]. If this does not appear to work satisfactorily then recourse should be made to Eq. (14b). 3.1.2. Nucleation thermodynamics, kinetics and mechanisms The formation of nuclei is an early stage of solid phase formation. Theoretical models are well known for nucleation from a solution [157,158], and from a melt [159,160]. Classical nucleation theory visualises the event as bimolecular reactions of growth units. The Gibbs free energy of the system, ΔGhom, changes due to the decrease of free energy per unit volume arising from the enthalpy of fusion, − ΔGV, and the increase of the surface energy due to the surface tension, ΔGS. For spherical nuclei of isotropic pure substances undergoing homogeneous nucleation this yields the familiar equation: 4 DGhom ¼ −DGV V þ DGS S ¼ − pr3 DGV þ 4pr2 r 3

ð19Þ

where V, S and r are the volume, surface and radius of the cluster respectively; σ is the surface energy. ΔGhom increases with r until a critical (maximum) value ΔGhom⁎ is reached at a critical size r⁎, i.e. when dΔGhom/dr = 0. Any clusters larger than r⁎ = − 2σ/ΔGV decrease the free energy when they grow and hence become more stable. Eq. (17) gives for ΔGV ≅ − ΔH (ΔT/TmVm), where Vm is the molar volume of the clusters, and ΔT = Tm − T is the supercooling. The critical free energy, the activation energy barrier, of nucleation can thus be written as: DGhom ⁎ ¼

16 pr3 Vm2 Tm2 3 ðDHm DT Þ2

ð20Þ

Thermodynamic considerations yield the energy barrier for nucleation and the critical nucleus size, but not the nucleation rate (the number of nuclei formed per unit volume per unit time). It is normally postulated that for a particular value of Δμ (= ΔGhom) a cluster size distribution arises which follows the Boltzmann distribution and thus the density of the critical size clusters (Chom⁎) can be expressed as Chom⁎ = Noexp (− ΔGhom⁎/kT), where No is the number of molecules per unit volume, and k is the Boltzmann constant [6,153]. As only clusters greater than the critical size are able to grow into a stable crystal, the frequency of nuclei formation (Jhom) turns out to be proportional to Chom⁎, as well as the maximum molecular frequency of collision, given by kT/h where h is Planck's constant: Jhom ¼


 NA kT −DGhom ⁎ exp h kT

ð21Þ

and where NA is the Avogadro number. Note, however, that there are other barriers to nucleation as molecules must diffuse to the nucleus site and adopt the appropriate configuration to the surface of the growing nuclei. These barriers lead to additional diffusive and entropy terms [159]. The diffusive term reflects the fact that as the temperature is lowered the diffusion rate falls caused by an increase in the viscosity of the melt or solution. The entropy term can be significant for long and flexible TAG molecules.

21

The loss of entropy due to the incorporation of molecules into a nucleus is given by ΔSm = ΔHm/Tm. The probability of the fraction, αS, of molecules in the melt with suitable conformation to incorporate to the surface of nuclei is exp(− αSΔS/R). However, one often assumes this conformation barrier is ⁎ ), included in the expression for the diffusion barrier (− ΔGdiff hence Eq. (21) becomes:

 NkT −DGdiff ⁎ −DGhom ⁎ exp exp ð22Þ Jhom ¼ h kT kT In real solutions, nucleation is substantially accelerated due to the presence of impurities which act as catalytic nucleation sites [6,148,153]. In fat processes these can be the vessel wall, impellers, mono- or diglycerides and other minor lipids, as well as dust particles. TAGs thus almost always undergo heterogeneous nucleation since they are normally impure [5]. The activation energy is lower than that of homogeneous type (a result of the catalytic action of foreign substances). Consequently, the supercooling required is also reduced. The activation energy for heterogeneous nucleation can be related to that for homogeneous nucleation as ΔGhet⁎ = ΔGhom⁎f(θ), with θ represents the wetting characteristics of foreign solid impurities by the supercooled melts [6]; thus a similar expression to Eq. (22) applies. Another nucleation mechanism is secondary nucleation which is caused from (1) fragments of growing crystals that are mechanically chipped off and which act as new nuclei, (2) the generation of small crystals due to collisions of crystals with other crystals as well as with parts of the crystallizer, and (3) the disturbance of the (pseudo) static condition of the liquid lamellae by the presence of crystal lattices leading to a lamellae alignment which enhances the nucleation event [7,151,161]. These mechanisms are more likely to be important in industrial scale crystallizers. A potential tool to control the polymorphic crystallization of fats is to provide “ready made” crystal nuclei via the addition of crystal seeding. The polymorphic forms of the seeds must be in the same forms of the targeted forms to nucleate from the mother phase which are mostly the β′- or βforms [24]. Additionally, at molecular level, the chemical structure of the seed TAGs is also of paramount importance. Takiguchi et al. [162] suggested, postulating from the template effects of fatty acid thin films on the crystallization of long chain n-alcohols, that the chain length of the fatty acid moieties should not differ by more than 4 carbon numbers. Sato [24] found that in order to crystallize TAGs which contain unsaturated fatty acids, the seed material should also have an unsaturated fatty acid moiety to be effective. This reflects the immiscibility of those TAGs which are fully saturated with those which contain unsaturated fatty acid moieties, as discussed in Section 2.4.1 (Fig. 14). Lastly, and perhaps this is obvious, the melting temperature of the seed material must be such that it does not itself melt when added to the liquid phase of the crystallizing material. BOB and SOS have been shown to be quite effective for controlling the polymorphic nucleation of cocoa butter [163–166].

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3.1.3. Polymorphic-dependent nucleation In monotropic polymorphism, as is the case with TAGs, there is only one truly stable solid polymorph below the melting temperature, the others being only meta-stable. However, if more than one polymorph possesses a positive driving force for crystallization then nucleation and growth rates are decisive in determining polymorphic occurrence (see Fig. 18). Empirically, this is described by Ostwald's rule of stages [6], which states that the thermodynamically less stable phase is always formed first and a step-by-step phase change may then occur towards the most stable one. Three main factors influence the polymorphic-dependent nucleation of TAGs: • Supercooling (ΔT), which if increased leads to higher nucleation rates (Eq. (20)) • The interfacial free energy of the crystals (σ) which is normally smaller for the less stable forms (α < β′ < β) • The ordering dynamics from a random conformation of liquid TAG molecules to a densely packed conformation of the crystalline state [6,123]. ΔT must be positive for a polymorph to form but of those polymorphs which possess a positive ΔT it is the interfacial free energy term that is usually decisive. Lower melting polymorphs have lower values of σ and so they will form if they can. Ostwald's rule of stages is thus explained by the competition to nucleate among the polymorphic forms; the phase with the highest nucleation rate will form preferentially [6]. The Ostwald rule of stages in TAGs, however, can be overridden if external influences are present such as local pressure and temperature fluctuations (e.g. by ultrasonic stimulation), template and seeding [24]. These are used in industry to manipulate the crystallization of fats. Tempering, which applies certain temperature-time protocols and frequently involves shear, is often employed in order to induce the nucleation of the more stable polymorphs. As the more stable phases are too slow to crystallize from the melt directly, a tempering protocol (e.g. thermal annealing) allows the formation of the less stable form followed by its transformation to the desired form either by direct transformation or via melting of the less stable phase. Additionally, if a mixture of unstable and stable polymorphs occurs, then raising the temperature will melt out unstable polymorphs and leave seed crystals of the stable polymorph. This procedure is the most common technique for controlling polymorphic fat crystallization in chocolate confectionery. The introduction of shearing has been found to significantly accelerate the formation of certain polymorphic forms [8,9,10,11,12,45]. Mazzanti et al. [11] argued that macroscopic shearing can provoke the alignment of less stable polymorph crystals that in turn will induce a higher transformation rate into a more stable phase. Mazzanti et al. [12] then reported that increasing the shearing rate does not influence the nucleation rate of the less stable phase but it does affect the rate of the formation of the more stable phase which is transformed from the less stable phase. So far, only limited explanations have

been made of the effect of shearing on fat crystallization kinetics [24]. Most studies on the effect of shearing have been conducted on natural fats. A better understanding might be obtained by conducting a systematic study on pure or well-defined TAG mixtures. Recently, ultrasound and magnetic fields have been examined as other potential candidates to control the polymorphic nucleation of fats [48,167,168]. The creation of local pressure and thus higher local supersaturation may induce faster nucleation rates. 3.1.4. Induction time According to nucleation theory (see Section 3.1.2), a critical cluster size distribution must be established before nucleation starts. It can be imagined that when the melt is cooled down rapidly from well above its melting temperature, the number of clusters will barely have exceeded zero by the time the crystallization temperature is reached [169]. An induction or incubation time, tind, is required to develop such a cluster size distribution [170]. This induction time can be long (hours) for relatively large TAG molecules [148]. In many nucleation studies, including those involving TAGs, researchers have tended to correlate tind as inversely proportional to the theoretical homogeneous/heterogeneous nucleation rate [7,171], although induction time and nucleation rate represent distinctly different physical phenomena. From Eq. (22), this yields:

 h −DGdiff ⁎ −DGhom ⁎f ðhÞ tind i1=J ¼ exp exp : NA kT kT kT ð23Þ

3.1.5. Growth rate and mechanisms The growth rate from melts can be controlled by either the attachment rate of growing units at the crystal surface (surface kinetics) or by the transport of mass to or heat from the growing surface [161]. The factors affecting the growth of TAGs can be grouped into two main categories, namely factors that are governed by the bulk crystal structure and those which are dictated by the nature of the mother phase [6]. Fig. 19 shows these relationships schematically. In fat systems it is regularly presumed that surface kinetics are rate controlling [2]. The primary evidence for this is that dissolution or melting rates (which are heat and mass transfer limited, but not surface kinetics limited) can often be orders of magnitude faster than the growth rate for the same driving force. There are also mechanistic arguments that surface kinetics are important as a great number of conformational changes are needed before a molecule can properly fit into the crystal lattice, and this also allows a chance for detachment to occur in the meantime. This conformational hindrance of TAG molecules causes relatively slow crystal growth rates of the order of 0.01–0.1 μm/s at Δμ/RT values between 0.5 and 1.5 [2]. It has been observed using AFM that the advancement of a smooth crystal front in TAGs progresses

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Fig. 19. Flow chart showing the interrelationships of factors affecting the morphology and growth rate of TAG crystals.

via the creation of two-dimensional nuclei on the surface [172]. The effect of chain length on the growth rate of TAGs is significant. Fig. 20 shows that the linear growth rates of monosaturated TAGs (CCC, LLL, PPP, and SSS) reduce as the chain length increases. This might be attributed to the time needed for the ordering of the methyl chains — the longer the chain, the longer the ordering time. The crystallization of fats does also release a considerable quantity of latent heat (as much as 200 kJ kg− 1) [2]. At a microscopic level, the transport of this heat away from the crystal surface into the bulk liquid can be of importance. Los and Mitovic [173] recently showed from computer simulations that heat transport in molecular systems, such as in fat mixtures, may be more important than mass transport, as opposed to atomic systems such as the solidification of alloys. Kloek et al. [31] roughly estimated that heat transfer may increase the interfacial temperature by about 1 K at the half crystallization time (i.e. when the solid content has reached 0.5), which is a significant amount. Mass transport can reduce the overall growth rate if, for instance, a significant increase in viscosity occurs leading to a decrease in diffusivity [5,174]. Walstra et al. [2] argued that this is unlikely in fat system as only moderate supercoolings are normally applied in fat crystallization. Hollander et al. [149], however, showed that at high supercooling, the solution and

melt crystallization of pure and binary mixtures of TAGs is transport limited. An interesting example is shown in Fig. 21. The growth of pure CLC at low Δμ shows a non-linear

Fig. 20. Growth rates of monosaturated TAGs with different chain lengths. The growth rates of tricaprin (CCC) and trilaurin (LLL) were collected from [212] and those of tripalmitin (PPP) and tristearin (SSS) (β′-form) from [182]. The growth rate of 30% SSS in PPP/SSS system is also shown [182].

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Fig. 21. Growth rate versus crystallization driving force curves of pure CLC and with 10%-w addition of LLL [149].

dependency on the driving force, which suggests that surface kinetics are dominant [6]. The growth rate is thus proportional to the surface nucleation rate on the crystal surface [5,175]. This explains the non-linear growth rates at low Δμ (in Fig. 21) as nucleation rates, as a rule, vary exponentially with driving force. This trend changes to a linear relationship at high Δμ. Furthermore the addition of a small amount (10%) of trilaurin has a significant effect on reducing the growth rate at high Δμ, but not at low Δμ. These are both evidence for mass transfer control. It is argued that because a trilaurin molecule is too large to fit into CLC crystals it does not affect surface integration. However, trilaurin does build up a layer surrounding the growing crystals and this is important when mass transport becomes controlling at higher driving forces. It is not clear yet whether this behaviour applies generally for all TAGs as Hollander et al. [149], on the other hand, also found that the growth rate of PSP showed a linear dependency on Δμ even at reasonably low driving forces. The reason for these apparently contradictory findings is unclear. The solid–liquid interface, being dependent on both the crystal structure (internal) and the nature of the mother phase (external), plays a critical role in crystal growth. Two distinct interfaces are recognised: (i) smooth or faceted surfaces and (ii) rough interfaces. The former are characterised by an atomically immediate change in the degree of crystalline order across the solid–liquid boundary. On the other hand, the rough interface can be described as being structurally diffuse, with a degree of crystalline order that varies continuously over the scale of a few atomic planes across the solid–liquid boundary [176]. An entropy factor, defined as ΔHm/kTm, can be used as an internal parameter to indicate the likely smoothness of a growing surface [177]. Highly anisotropic materials with high entropy factors such as TAGs (with ΔHm/kTm ∼ 60) exhibit smooth surfaces if the planes are densely packed, but planes which are much less densely packed can easily experience roughening [5]. Hollander et al. [172] showed using AFM that different surfaces of a TAG under the same crystallization

conditions can grow by different mechanisms. The highly asymmetric shape of TAG molecules causes different packing densities in different crystal planes [178]. Accordingly, different surfaces of a TAG crystal can exhibit significantly different surface roughnesses so that the growth rates of different planes at the same conditions can differ greatly, resulting in needle shape crystals [179]. This is in contrast to other materials which have a relatively low entropy factor, and generally possess rough surfaces. The attachment of growth units is relatively easy in such cases and results in fast and unstable growth, for example dendritic growth. Two distinctly different mechanisms for roughening have been observed. “Thermal roughening” occurs when a surface is exposed above a particular temperature called the “roughening temperature” [161]. Above this temperature, growth units attach to the surface very quickly as they are more easily able to overcome the attachment energy. Alternatively, rough surfaces can be created at high crystallization driving forces, where mass transport is controlling, and growth units will easily attach onto the crystal surface wherever they come into contact. This is well known as “kinetic roughening” [180]. As rough surfaces grow much faster than smooth ones, in the frame of morphology development, the rough surfaces ultimately disappear and are not observed in the final habit. In multi-component systems there is competition between similar TAG molecules for vacant sites, and thus the growth rates are generally slower than that of pure TAGs. Kellens et al. [181] observed that PPP/SSS mixtures with an equal composition of PPP and SSS had a half-time of crystallization of the β′ solid solution in between those of pure PPP and pure SSS. Himawan et al. [182] confirmed this behaviour after measuring growth rates using light microscopy (see Fig. 20). The exception to this rule is when molecular compounds are formed [2]. Takeuchi et al. [123], and Sato and Ueno [59] showed that the crystallization rate of molecular compounds in binary TAG mixtures is significantly faster than the rates for pure components, presumably because steric hindrances are much reduced. As mentioned previously, solid solutions of TAG molecules are frequently formed especially in metastable polymorphs. The composition can vary between the inner and outer regions of a crystal as higher melting TAGs preferentially deposit first and diffusion rates in the solid phase are low, so that such composition gradients persist. There is a further issue of segregation kinetics of TAG components during the growth of TAG solid solutions. Kirwan and Pigford [183] presented an approach to estimate the composition of a growing solid solution from multi-component melts. They assumed spiral growth to be the growth mechanism and derived the segregation kinetics accordingly. Los and Floter [130] proposed a method to obtain kinetic phase diagrams for multi-component TAG systems. The model was derived for growth at rough surfaces, thus using linear supersaturation dependency on crystal growth (linear kinetic segregation). Los et al. [144,145] showed significance differences between simulated equilibrium solid solution compositions and those estimated when also including kinetic factors. Subsequently, Los and Matovic [173] also

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included the transport mechanisms into the kinetic expressions and estimated the importance of mass and heat transfer on different systems. However, experimental validation of this procedure has not been carried out. The segregation issue, however, has been tackled in metallurgy regarding the solidification of alloys. The definition of thermodynamic driving forces and the segregation kinetics have been well developed in this field [184–186]. The challenge to apply such approaches to fat systems is still open. 3.1.6. Morphology of TAG crystals The overall morphology (shape) of a crystal is fundamentally determined by the relative growth rates of different crystal surfaces. The slower the advancement rate of a crystal surface, the higher the probability that the surface will have a large surface area in the final crystal habit. As mentioned in the previous section, the anisotropic nature of TAG crystals results in large differences in growth rate between surfaces. TAG crystals of different polymorphs, as the term suggests, exhibit various morphologies. Under the microscope the α-form produces an amorphous mass of very tiny crystals, the β′-form is generally a bulky shape or spherulitic, while the β-form is usually a needle shape [59]. Although it is often found that fat crystals grow as spherulites, enormous variations can occur depending on the crystallization conditions, even for the same TAG system [95]. Fig. 22 shows morphology-maps depicting the habit variations during the isothermal crystallization of binary TAG mixtures (POS/SOS) as functions of TAG composition and temperature [146]. Traditionally, the morphology of a crystal is estimated by employing surface energy theories such as the Bravais-FriedelDonnay-Harker (BFDH) theory [161,180], which assume that the crystal faces advance at rates proportional to their surface energies. The equilibrium shape of a crystal is ultimately achieved when the total surface free energy of existing crystal surfaces is minimised. The free energy of a surface can thus be related to the distance between the face and the central point of the crystal body [6,187]. Alternatively, morphology can be predicted using the attachment energy as the growth-controlling parameter, as in the Hartman-Perdok (HP) theory [180,188]. The attachment energy here represents the amount of energy that is lost when a crystal is cut along the plane of a crystallographic orientation. The growth rate of the plane of that orientation is assumed to be proportional to its attachment energy. However, the surface energy and the attachment energy methods, which are solely based on thermodynamics (knowledge of crystal structure), have not been able to completely explain experimentally observed TAG crystal morphologies [178,189]. Hollander et al. [149,178] showed that the aspect ratio of the needle β- and β′-forms of different saturated TAGs model systems was underestimated by the morphology estimation procedure using the attachment energy. Therefore in order to reproduce the experimentally observed morphology of TAG crystals, the growth mechanism (kinetics) and roughness of each crystal surface must be taken into

Fig. 22. A map of volume fractions of the various crystal morphologies upon isothermal crystallization of the POS/SOS system with different compositions (a) 75/25, (b) 50/50, (c) 25/75 [146]. Reprinted with permission from the American Oil Chemists' Society.

account [149,178–180,190]. Fig. 23 shows a comparison between experimental TAG morphologies and those estimated from the surface energy theory (BFDH), the attachment energy algorithm (HP), and the attachment energy algorithm combined with the Burton-Cabrera-Frank screw dislocation theory (HPBCF) [161] for the β′-form of PSP and for the β-form of PPP. Clearly, kinetic factors are predominant since the experimental morphologies are much better predicted by the latest methods. The application of this approach for multi-component fats, however, is a challenge. 3.1.7. Spherulitic growth In most cases, pure TAGs and mixtures of TAGs grow as spherulites. A spherulite is an aggregate made up of many crystalline ribbons that grow radially from the same central nucleus (Fig. 24) [148,149]. The ribbons that build up the

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surfaces where the completion of a layer is not necessary, thus secondary nucleation can occur simultaneously in different layers resulting in more diffuse structures (Fig. 24b). The spherulitic growth of TAGs can be described by the Hoffman-Lauritsen or surface nucleation theory that has been widely applied to polymer crystallization [191]. The model assumes that the progression of the spherulite front is controlled by secondary nucleation at the frontal surfaces of the spherulites. The relation between the spherulitic growth rate and the driving force is given by:
Rspherulite ¼ R0 exp −

Fig. 23. Morphology of the β-form of PPP and the β′-form of PSP as (i) calculated using the surface energy theory (BFDH), (ii) calculated using the attachment energy theory (HP), (iii) calculated by means of the attachment energy combined with screw dislocation growth mechanism (HP-BCF), and (iv) observed from experiments [172].

spherulite are in many cases needle-like (Fig. 24a). The reasons for this morphology to exist in TAGs have been discussed earlier. Often irregular structures are observed where some distortions are possible and the interface with the liquid may be diffuse (see Fig. 24b). The difference between the morphologies of the TAG spherulites shown in Fig. 24 is due to differences in the magnitude of the driving force, which causes changes to the mechanism of the secondary nucleation of crystal layers. At low to moderate driving force the progression of the front is layer by layer. This means secondary nucleation is slow enough to ensure that one layer is completed before the next layer is created (Fig. 24a). Higher driving forces can induce rougher


 U⁎ KR exp − RðT −Tl Þ T ðTm −T Þ

ð24Þ

where Rspherulite is the linear growth rate of the spherulite, R0 is the pre-exponential factor, U⁎ is the activation energy related to entropy and diffusion barriers (equivalent to ΔGdiff in Fisher-Turnbull equation, see Eq. (22)), T∞ is a hypothetical temperature where the motion associated with viscous flow ceases, and KR is the kinetic parameter. A linear correlation is obtained when plotting lnRspherulite versus 1/T(Tm − T), which assumes a relatively constant value of [lnRo − U⁎/R(T − T∞)]. Both Rousset et al. [146,148] and Himawan et al. [182] have used this equation to model TAG spherulitic growth. Fig. 25 shows the equation applied to the PPP/SSS system showing two different correlating fits arising from the different nucleation mechanisms (and morphologies) at different driving forces. 3.1.8. Polymorphic transformation Transformations between phases (liquid, α, β′, and β) are an important factor in the processing and storage of fats (see Introduction). As fats exhibit monotropic polymorphism, the transformation among polymorphs is irreversible in the direction of less stable to more stable phases. This can proceed via a solid–solid or a melt-mediated transformation. The solid–solid transformation mechanism in fat systems is still poorly understood. Hagemman [20] suggested that the change from the vertical α-form to the tilted β′-form is suspected to occur via a collapse of hydrocarbon chains or from

Fig. 24. Spherulites of β′-polymorph crystallized from 30%-w SSS of PPP/SSS binary mixture at (a) 52.5 °C, (b) 49 °C. The field of view of the pictures is 200 μm across [182].

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in SOS significantly increased the complexity of the transformation. Of particular interest was that: • During the transformation from the α-form into the γ- or β′forms, the SR-XRD long spacing spectra occurred earlier than the short spacing spectra. This indicated that the formation of lamellar ordering occurred more rapidly than that of subcell packing. The time lags between the spectra became shorter for the less stable polymorphs (for instance for the γ-form compared to the β′-form). • Liquid crystal phases were observed if a very rapid meltmediated transformation occurred from the α-form to the β′-form.

Fig. 25. The application of the Hoffman-Lauritzen theory to model spherulite growth in 30/70 PPP/SSS mixtures. The units of Rspherulite are 10− 3 μm s− 1 [182]. Two different fits are possible depending on morphology, resulting from different nucleation mechanisms at different driving forces.

a bending of each molecule in the glycerol region. Dafler [152] interpreted his DSC data to suggest a two step mechanism during the β′- to β-form transformation. The first part, which is fast, may be regarded as a thermally activated process; the second, slower, process is speculated to be dependent on the concentration of the β-form already present. Due to the more restricted molecular arrangement in the solid state the rate of this transformation mode is much slower than that of meltmediated transformations. Melt-mediated crystallization can be viewed as the melting of the less stable form followed by the subsequent nucleation and growth of the more stable forms. Mass transfer occurs in the liquid formed by the melting of the less stable form [24]. These events most probably occur concurrently within a sample which makes it difficult to measure them separately. Traditionally, DSC is used for characterising polymorphic transformations, but recently synchrotron radiation (SR) XRD has been applied to pure TAG systems [66,93,94,110], binary TAG mixtures [120–122,192], and natural fats [12,44–46]. This has been able to provide molecular-level insights of the polymorphic transformation of fats. A review on this subject can be found in Sato and Ueno [59]. Kellens et al. [93,192] presented SR-XRD spectra of the melt mediated polymorphic transformation from the α-form into the β-form in a pure PPP system. During the α→β transformation, the β′-form could be seen for a very short time. This is barely detectable by DSC, although this is more obvious in mixtures of PPP and SSS [181,193]. SR-XRD was also able to show the sequential ordering in the β-form subcell arrangement from time lags between the appearance of the three XRD spacings associated with the β-form. Ueno et al. [66,110] studied the melt-mediated transformation of SOS, and found out that the presence of the double bond

It is implied from observations in the PPP and SOS systems that there are potentially enormous variations of mechanism during the melt-mediated polymorphic transformation of fats. A number of researchers have reported that the crystallization rate, for instance, of the β′-form, from the melt of the α-form (the α-melt) is much faster than that from the supercooled melt. The explanations, however, vary. Some workers argued that this is due to a molecular arrangement in the liquid phase [20,194]. Hagemman [20] reported that the tuning-fork conformation of TAGs associated with the lamellar arrangement already exists above the melting temperature. Relaxation data on primary glycerol carbons of a rapidly heated α-form were different from those from an undercooled liquid at the same temperature. Another possibility is that nonmelted high melting TAGs may act as seed materials, accelerating the crystallization rate of the more stable phase [195]. For some fats involving unsaturated moieties, it has also been suggested that liquid crystalline phases might be involved [66–68,110,196]. 3.2. Measurement of fat crystallization kinetics In order to characterise crystallization kinetics, nucleation and crystal growth rates should ideally be measured separately. Neither nucleation nor growth rate measurements are straightforward in TAGs. Only a few attempts to directly measure nucleation and growth rates have been reported [31,146,154,197]. Inadequate resolution of existing instruments to identify solid particles at the length scale of nuclei limits nucleation rate measurements, whilst the irregularity and aggregative form of TAG crystal morphologies hampers accurate crystal growth rate characterization. This remains a major challenge in the characterization of lipid crystallization [198]. 3.2.1. Induction time and nucleation rate The induction time, tind, has been regularly chosen as a mean of characterising the crystallization kinetics of TAGs and fats in general [55,92,198–200]. A diverse range of measurement techniques have been employed to measure tind ranging from DSC [28], light transmittance [201], turbidimetry [198], pulsed nuclear magnetic resonance [198], laser-polarised light turbidimetry [33,202], polarised light microscopy (PLM)

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[171,200,203], viscometry [27], PLM supplemented by a CdS photo sensor [106,199], and diffusive light scattering (DLS) [30]. PLM and DLS are considered to be the most sensitive. This does not guarantee, however, that the measured tind reflects the real induction time since, for instance, the smallest identifiable crystal by PLM is approximately 0.2 μm [200], while Timms [151] has estimated that the nuclei critical size is about 0.01 and 0.001 μm at 3 and 7 °C supercooling, respectively. Measured tind values thus depend on the instrument employed. They have also been interpreted in one of two ways. Some investigators regard tind to contain both nucleation and growth information: tind ≅ tnucl + tgrowth. In this sense, tind is the total time needed to form the nuclei and then to grow to a size, which is observable by the instruments used, and as such is often used as a representative measure of the overall crystallization kinetics. Example data are shown in Fig. 26 from Sato and Kuroda [92] who used PLM with a CdS sensor to measure induction times for PPP at various isothermal temperatures. This shows fast, intermediate, and slow kinetics for α, β′, and β polymorphs, respectively, and reflects Ostwald's rule whereby nucleation favours the least stable polymorphs. Similar findings were obtained for POP and SOS [106], and in POS [199]. The induction time has also been utilized for comparing the rates of melt and melt-mediated crystallizations. Other researchers assume that tind predominantly represents the nucleation rate. Takeuchi et al. [123] for instance interpreted tind as nucleation dominated and used it to qualitatively show the faster nucleation of the α-form of the SOS/SSO molecular compound compared to pure αSOS and αSSO. Frequently, tind is correlated inversely with the expression for nucleation rate, J, from the Fisher-Turnbull relation (see Eq. (22)). Combining Eqs. (20) and (23) results in: !
 h DGdiff ⁎ 16 pr3 Vm2 Tm2 tind ¼ exp f ð hÞ exp NA kT kT 3 kT ðDHm DT Þ2 ð25Þ A further assumption for TAG systems is that the main barrier to diffusion is entropic and arises from molecular conformation issues (i.e. ΔT is moderate or even low, but the flexibility of the three fatty acid chains is enormous) leading to ΔGdiff⁎/kT = αSΔS/R. Rearranging Eq. (25) results in: Ttind

!
 h aS DS 16 pr3 Vm2 Tm2 exp ¼ f ð hÞ exp NA k R 3 kT ðDHm DT Þ2

ð26Þ

Thus from a plot of ln(Ttind) against 1/T(ΔT)2, a slope, sind, can be obtained which allows the estimation of the activation energy of nucleation from ΔGnucl = ΔGhom⁎ f (θ) = simdk/(ΔT)2. However, this approach has weak theoretical and practical foundations when considering that (i) the Fisher-Turnbull equation was derived for pure systems and (ii) the induction time is a physically distinct phenomenon from nucleation rate. Nevertheless, it is commonly applied to characterize the

Fig. 26. Inverse induction times for the crystallization of PPP polymorphs. The melting temperatures of the polymorphs are indicated (αm, βm′, βm) [92]. Reprinted with permission from the American Oil Chemists' Society.

nucleation kinetics of natural fats such as palm and other vegetable oils [26–30,171,203], cocoa butter [46,49], and milk fat [37,39,40,43,53]. Using this approach, ΔGnucl values were estimated and used to illuminate the significant differences in nucleation energy barriers between polymorphic forms. Surprisingly, there have been few attempts to apply such a method to pure TAGs or well-defined TAG mixtures. We recently performed such an attempt for the PPP/SSS mixtures as shown in Fig. 27, which resulted in a much larger value of energy barrier for the β form compared to the β′ form [127]. It should be noted that the use of t ind for kinetic characterization is also limited to a certain range of isothermal crystallization temperatures, as below a certain temperature the tind value becomes so short that isothermal conditions cannot be established in the sample before nucleation begins [200]. This is especially true for the α-form crystallization [127]. Nonisothermal crystallization methods have been introduced to try to characterize the crystallization kinetics of this polymorph [150,193,204]. 3.2.2. Overall crystallization rates A plot of total solid content versus time during an isothermal crystallization of fats exhibits a sigmoidal shape [205]. This is a result of simultaneous nucleation, growth and the so-called impingement effect, i.e. the colliding of growing crystal faces of neighbouring crystals that slows down growth and thus the overall crystallization process. Nucleation is the predominant event at the initial stage, growth in the following stage, but impingement is the predominant factor towards the end of the crystallization process [155]. In order to describe properly the whole crystallization process, each event must be well-characterised. However, it is difficult to measure these overlapping events separately. The overall sigmoidal curve is reasonably quantifiable using various measurement techniques such as DSC, pulsed NMR, turbidity, time resolved XRD, viscosity change, and ultrasound velocity measurements (see reviews in Foubert et al. [205] and Marangoni [15]). Hence, the characterisation of crystallization

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crystal volume can be calculated. A correction is finally made to allow for impingement effects assuming that volume increases are proportional to the liquid volume fraction still remaining. After some mathematical arrangements, a simple equation can be derived as follows: Xsolid ¼ 1−expð−kt n Þ

Fig. 27. Linear correlation between ln(Ttind) and 1/T(ΔT)2 for β′ and β polymorphs in PPP/SSS mixtures [127]. Reprinted with permission from the American Oil Chemists' Society.

kinetics of fats is routinely carried out via the analysis of the experimental sigmoid curve. The interpretation of the sigmoid curve has been performed using various models. Rousset [148] and Foubert et al. [205] have compiled a review of the existing models used in fat crystallization. The existing models can be divided into deterministic, numerical, and stochastic approaches, Rousset [148]. The deterministic approach incorporates nucleation, growth, and impingement events with some assumptions leading to a simple equation to fit the time evolution of the solid fraction. The Avrami model (also known as the JohnsonMehl-Avrami-Kolmogorov equation), the Gompertz model, and the aggregation-flocculation model are examples of this approach [205]. An analytical model that incorporates the evolution of size distribution along the melt crystallization is also available [155,156]. The numerical approach is theoretically similar to the deterministic one but uses more complicated expressions for nucleation and growth rates. The resulting equations cannot be solved analytically and thus numerical integration must be applied [148]. Stochastic models visualise nucleation and growth as statistical/probabilistic events. A 2- or 3-dimensional space is defined in which the evolution of the solid/liquid interface, and thus the evolution of solid content, can be obtained as time progresses [148]. The Avrami model [206,207] is frequently used to describe the isothermal crystallization kinetics of fats [205]. This model is based upon a geometrical framework which mimics the actual crystallization process of nucleation, crystal growth and impingement. Nucleation is modelled by the number of nuclei per unit volume (NJ), and this is assumed to be either “sporadic” where NJ linearly increases with time or “instantaneous” where NJ is fixed number created at time zero (equivalent to a “seeded” crystallization). Once formed, nuclei grow at a constant rate in one, two, or three dimensions from which an unimpinged

ð27Þ

where Xsolid is the solid fraction, k is a constant depending on the nucleation rate, nucleation mode (instantaneous/ sporadic) and on the growth rate, while n depends on the nucleation mode and the growth morphology (see Foubert et al. [205]). Violations of the assumptions made in the Avrami model are likely to occur in fat crystallization. For instance, the growth velocities are assumed to be constant throughout the transformation which is not the case in multi-component systems where the crystallization driving force can vary substantially. Fitting the sigmoidal crystallization curve with the Avrami model often gives a non-integer n parameter which is difficult to explain physically. A mathematical model mimicking reversible reaction kinetics has also been proposed by Foubert et al. [47]. This generally provides acceptable fits to data but the fitted parameters do not have as much physical meaning as those of the Avrami model. The challenge thus remains to develop more realistic models which will undoubtedly require numerical rather than algebraic integration. This must be obviously in accordance with the development of reliable measurement techniques. In many multi-component fat systems, it is frequently observed that the sigmoid curve appears to exhibit a two-step transformation [12,15,44]. It has been proposed that the isothermal crystallization of multi-component fats is initiated by the formation of the α-form followed by a subsequent transformation to a more stable phase β′-form. Modifications of the Avrami model to fit this crystallization curve have been proposed [12,15]. 4. Concluding remarks TAGs are not only commercially useful materials but their crystallization behaviour is also academically interesting. TAG systems can be made to be very simple (such as a pure TAG system where all the fatty acid moieties are identical and saturated, such as PPP) to very complex, naturally derived fats containing hundreds of TAGs with seemingly limitless combinations of fatty acid moieties. The behaviour of TAGs is ultimately dictated by the molecular structures of the TAGs in the system (the fatty acid mix) and how they are able to pack together in crystals in the various polymorphic forms available. Heterogeneity of fatty acid moieties in a TAG results in more complicated behaviour, in particular different lengths of fatty acids in a mixed-acid TAGs, and TAGs containing both saturated and unsaturated fatty acid moieties. Incompatibility of TAGs due to differences in chain length and degree of saturation leads to immiscibility in the solid state. However, in special cases where the molecular

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shape of TAG components is compatible to each other, chainchain interactions can provoke the formation of molecular compounds at a composition of 50/50. Both thermodynamic and kinetic aspects need to be considered to gain a full understanding of fat crystallization behaviour, and it is useful to be able to classify behaviour according to whether they are caused by thermodynamics, kinetics or both. Thermodynamics is able to establish the stability (or metastability) of different polymorphs, and the miscibility or immiscibility of different TAGs in mixed crystals. It also enables the driving force for crystallization to be quantified, which provides a benchmark for the modelling of nucleation and growth rates and the surface roughness of crystals. To date either supersaturation or supercooling has generally been used for this purpose, but for fat systems which display an almost continuous spectrum of melting points a definition based upon the rigorous expression for chemical potential differences (using pure component melting data and activity coefficients) is required to model individual driving forces for each component. Only a very limited number of studies can be found regarding the description of solid–liquid equilibria of polymorphic systems via activity coefficient expressions. This is largely also due to the lack of reliable experimental data for the estimation of thermodynamic parameters. Kinetics is however required to explain many other phenomena, apart from the obvious practical consideration of the timescale over which crystallization proceeds. Polymorphic occurrence, for example, cannot be explained by thermodynamics which would always predict the most stable polymorph to be formed. Instead it is governed by the kinetics of the polymorph which is able to nucleate first. Kinetics factors also dictate the relative advancement rates of different crystal surfaces and thus the overall crystal morphology. Although our understanding of fat crystallization is extensive there remain many gaps in our knowledge. Much of this is due to our inability of make precise measurements of the nucleation and crystal growth rates of TAGs. This is partially solved by the determination of overall crystallization kinetics and the use of models such as the Avrami equation, but the assumption that the driving force, nucleation and crystal growth rates do not vary is often too simplistic. It is additionally worth stating that in a number of industrial applications fat crystallization occurs in dispersions (emulsions) which introduces its own complications [208–211]. In nucleation studies, induction time measurements are commonly made, whereas nucleation rates (the number of nuclei appearing per unit time and per unit volume) are seldom measured. It is often assumed that induction times represent the reciprocal of the nucleation rate despite the fact that they are distinctly different physical phenomena. Spherulitic growth is common in TAGs but crystal growth mechanisms are not well understood. Most of the work in the past has assumed that surface integration kinetics are dominant, yet more recent data have shown the importance of transport processes. A general theory is not yet currently available.

The segregation of components during the growth of TAGs solid solutions needs to be investigated in order to describe properly the growth and overall crystallization kinetics of TAGs. This requires some method of following the crystallization of individual components in fats. This is difficult as TAGs are chemically very similar to one another. Other than retrospective analyses of fractionation experiments few studies have investigated this aspect of fat crystallization. A great deal has been learned about the crystallization behaviour of fats in the past half century, but it is clear that much remains to be achieved. For example, food engineers do not have access to the sort of simulation software which designers of distillation columns are now routinely able to turn to, whereby merely entering the composition of an input stream and a choice of processing parameters allows a complete simulation of a process to be made via vapour–liquid equilibrium calculations. A similar tool to predict solid–liquid equilibria in fat crystallization processes, in which TAG composition of a fat blend is entered in a similar fashion along with process conditions, is still a long way off. Although, the theoretical framework for the equilibrium phase behaviour of binary systems is now well established, this still needs to be extended and properly validated for multi-component systems. These solid–liquid equilibria can (in theory) be handled in an analogous manner to vapour–liquid equilibria in distillation column simulators, and it is quite feasible (again, in theory) to extend the database to cover the various different fat interaction parameters required. However, as already mentioned, a proper model of fat crystallization will need to extend beyond that of solely predicting phase equilibria if it is to be a truly useful tool, as the kinetic aspects of nucleation and growth also need to be taken into account. An additional consideration is how many of the hundreds of different TAG and non-TAG components present in natural fats need to be included in a working model, or can one make do with grouping many of the less abundant components into “pseudo-components” without causing problems? Trace components can cause problems in nucleation, for example, which is liable to be heavily influenced by small quantities of high melting fats which act as seed crystals and it may be difficult to establish their presence by analytical techniques. The surface integration of molecules into growing crystals can similarly be upset by small quantities of impurities. Add to that the need to predict polymorphic occurrence, inter-polymorphic transformations (if any), crystal morphology and size distribution and one can readily conclude that such a model would need to be highly sophisticated. Nomenclature Abbreviations AFM Atomic Force Microscopy BCF Burton-Cabrera-Frank BFDH Bravais-Friedel-Donnay-Harker DLS Diffusive Light Scattering

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DSC Differential Scanning Calorimetry FTIR Fourier Transform Infra Red HP Hartman-Perdok NMR Nuclear Magnetic Resonance PLM Polarised Light Microscopy SFC Solid Fraction Content SR-XRD Synchrotron Radiation X-ray Diffraction TAGs Triacylglycerols XRD X-ray diffraction

Acknowledgements The authors wish to acknowledge and thank the BBSRC (UK) for funding this work (grant reference D20450). References

Symbols Aij G H h Jhom k KR NA NJ P R r r⁎ Ro Rspherulite S sind T tind Tm U⁎ V Vm xpi Xsolid

Binary parameters in the 3-suffix Margules equation Gibbs free energy Enthalpy Planck constant Homogeneous nucleation rate Boltzmann constant Kinetic parameter in the Lauritsen-Hoffman equation of spherulite growth rate Avogadro number Number of nuclei per unit volume Pressure Universal gas constant, (8.314 J mol− 1K− 1) Cluster radius Critical radius Pre-exponential factor in Lauritsen-Hoffman equation of spherulite growth rate Linear growth rate of spherulite Entropy Slope in the determination of ΔGnucl Temperature Induction time Melting temperature Activation energy related to entropy and diffusion barriers in the Lauritsen-Hoffman equation Volume Molar volume of clusters in the nucleation theory Mole fraction of component i in phase p Solid fraction

– J mol− 1 J mol− 1 Js # m− 3 s− 1 J K− 1 J mol− 1 # mol− 1 # m− 3 kPa J mol− 1 K− 1 m m – m s− 1 J mol− 1 K− 1 K s K J mol− 1 m3 m3 – –

Greeks αS γpi μpi σ θ ΔCpi ΔG ΔGdiff ΔGE ΔGhet⁎ ΔGhom⁎ ΔGnucl ΔGS ΔGV ΔH ΔHm ΔS ΔSm ΔT

Fraction of molecules in the melt with suitable conformation to incorporate to the surface of a growing nuclei Activity coefficient of component i in phase p Chemical potential of component i in phase p Surface energy Wetting characteristics of foreign solid impurities by the supercooled melts Molar heat capacity difference between liquid and solid for component i Free energy difference Activation energy related to diffusion barriers Excess free energy difference The critical free energy (the activation free energy) of heterogeneous nucleation The critical free energy (the activation free energy) of homogeneous nucleation The activation energy of nucleation The surface free energy The volume free energy Enthalpy difference Melting enthalpy Entropy difference Melting entropy Supercooling, Tm − T

31

– – J mol− 1 J mol− 1 m− 2 – J mol− 1 K− 1 J mol− 1 J J mol− 1 J J J mol− 1 J m− 2 J m− 3 J mol− 1 J mol− 1 J mol− 1 K− 1 J mol− 1 K− 1 K

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