Colloidal networks of fat crystals

Advances in Colloid and Interface Science 273 (2019) 102035

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Historical perspective

Colloidal networks of fat crystals Edmund D. Co, Alejandro G. Marangoni* University of Guelph, Guelph, ON, Canada

a r t i c l e i n f o

a b s t r a c t

Article history: Received 31 May 2019 Received in revised form 1 September 2019 Accepted 9 September 2019

The following paper traces the development of the study of colloidal networks of fat crystals. The work starts with traditional pre-fractal particle network models of fat crystal networks. Due to its central importance in the study of colloidal networks of fat crystals (and other colloidal aggregates), a short exposition of fractal geometry is provided. The development of fractal aggregation models as well as models that describe the rheology of networks of these fractal aggregates is introduced. Later sections of the paper show the application of these aggregation and mechanical models specifically to fats. Finally, recent work in elucidating the nanostructural elements of fat crystal networks and aggregates of these nanostructures is provided. © 2019 Published by Elsevier B.V.

Contents 1. 2. 3. 4. 5. 6. 7. 8. 9.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Pre-fractal network models of fat rheology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Fractals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Fractal models of aggregation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Rheology of fractal aggregate networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Fractal models of fat microstructure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 Crystalline nanoplatelets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 Aggregates of crystalline nanoplatelets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 List of symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Moduli, stress and strain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Network and particle characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Declaration of competing interest . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1. Introduction Fats are solid lipid materials consisting predominantly of triacylglycerols (TAG). Fats derive their solidity from a microscopic Fat Crystal Network (FCN), which is a network of solid fat particles where the network components assume dimensions in the micrometer length scale [1,2]. This network is assembled when supersaturated TAGs crystallize into lamellar crystallites (Fig. 1), which serve as the basic building block of the FCN. These lamellar * Corresponding author. E-mail address: [email protected] (A.G. Marangoni). https://doi.org/10.1016/j.cis.2019.102035 0001-8686/© 2019 Published by Elsevier B.V.

crystallites (may aggregate to form spherical intermediate structures commonly referred to as primary particles (or just particles) or microstructural elements or spherulites in much of the literature with a size between 1 mm and 10 mm (Fig. 2). These particles may further aggregate to form flocs or microstructures or clusters, as they are commonly referred to in the literature (Fig. 3). Typically these flocs have a size in the length scale of around 100 mm. These higher-order structures are “open” in that these aggregates do not consist of a single mass of solid particles but rather consist of linear branches of connected spherical particles with oil filling the interstitial spaces. The microstructures, however, pack in a space-filling manner (oil and all) to form the colloidal network called the Fat Crystal Network.

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Fig. 1. Crystalline Nanoplatelets isolated from a model fat consisting of tristearin and triolein.

The lamellar crystallites long known to be the basic building block of a Fat Crystal Network have never been unequivocally identified and described. Rather, their existence was always assumed in light of the X-ray data on crystalline fats. Recent advances in isolating and imaging these lamellar crystallites have opened new research avenues to examining the aggregation of the lamellar crystallites themselves (see Fig. 1) [3]. However, these lamellar crystallites are experimentally inaccessible due to their small size (in the 100 nm range). As such, much of the studies of these lamellar crystallites involve techniques such as computer simulation and ultra-small-angle X-ray scattering (USAXS). The current review builds on past work [4] published two decades ago by summarizing advances in the development of microstructural models for colloidal networks of fat crystals as well as discussing

Fig. 3. Flocs, or fractal aggregates of microstructural elements.

the relatively recent advances in the characterization of fat crystal networks at the nanostructural (< 1 mm) level. 2. Pre-fractal network models of fat rheology The observation that the microstructure of a fat plays a profound role in the material’s rheological properties [5] is not a recent observation as the reader will note throughout this review. Crystallization conditions have a significant effect on the rheological properties by way of the microstructure as noted by several authors [6e8]. Besides a model that describes the kinetics of the formation of the network of fat particles, the main challenge in this field of endeavour is to correlate some measure or feature of this microstructure to the resulting rheological characteristics of the material underpinned by this network.

Fig. 2. Aggregate of Crystalline Nanoplatelets (shown at low and high magnifications) isolated from a model fat consisting of tristearin and triolein.

E.D. Co, A.G. Marangoni / Advances in Colloid and Interface Science 273 (2019) 102035

The earliest paper to suggest a microstructural model for a fat was published by van den Tempel, who proposed the Linear Chain Model in 1961 [9]. In this particle network model, the viscoelastic character of a system consisting of flocculated solid particles was assumed to be due to the London dispersion forces component of van der Waals forces. A “flocculate” network was modeled as an assembly of linear chains in three dimensions. An expression for the shear modulus G for such a system was formulated as:

5Ah a0:5

G¼

24pl0 3:5

F

where Ah is Hamaker’s constant (which describes the polarizability of the atoms in the particle [10]), a is the diameter of the particle, l0 is the equilibrium separation distance between two particles and F is the solid particle volume fraction of the material. This model was applied by Payne to colloidal carbon black networks but the data did not agree with the model on account of the polydispersity of the carbon black suspension [11]. In 1963, Nederveen used van den Tempel’s model [12] but incorporated a repulsive force in the form of a 6e12 Lennard-Jones potential between two particles to obtain the following expression for the elastic modulus E:


E¼

Ah F 2pl0



11aε 1 2 2l0

!

where ε is the extensional strain. In 1968, Sherman proposed a modification of van den Tempel’s Linear Chain Model [13]. Instead of van den Tempel’s assumed homogeneity of the particulate network, Sherman proposed a model where the density of packing varies through the network. Regions of densely packed particles are interspersed throughout regions of less densely packed particles, the net result being that the degree of interlinking is spatially dependent. For such a system, the formulation for the shear storage modulus G0 is:

G0 ¼ ð1 þ 1:828VÞ

Ah F 36pa3 l30

where V is the total volume of the liquid continuous phase in the interstitial void spaces between particles. In 1972, Papenhuijzen developed a network model relating the network structure to the rheological properties of a dispersion of solid fat in oil using an apparatus that utilized superimposed oscillatory and steady shear to deform a material [14,15]. The relation between a “theoretical” G0 to the dispersed phase volume fraction was given as:

G0 ¼

Ah F 6pl3

Much of the model was previously formulated for oil-in-water emulsions. The rheological properties of the network were attributed to van der Waals forces between solid fat particles as well as hydrodynamic interactions between the solid fat particles and the liquid milieu. Hydrodynamic interactions were found to be negligible in describing the mechanical properties of solid fat dispersions. In 1979, van den Tempel revisited his earlier Linear Chain Model and proposed a new formulation: the “particles” that formed the linear chain were not particles per se but were themselves aggregates of particles [16]. We know such aggregates of “particles” as flocs or clusters or microstructures. The interaction between these flocs was due to interpenetrating chains of constituent particles “shared” by the interacting pair of particles. It was also noted in this

3

work that the storage modulus G0 scaled with the solid particle volume fraction in a non-linear manner:

G0  Fm where m is a scaling exponent that defines the scaling relationship. Of great importance was the observation that the value of m was directly proportional to the amount of particles within an aggregate (estimated to be about 106 crystallites) and that m is indicative of the state of aggregation of these particles within the aggregate. In 1984, Kamphuis et al. developed a transient-network model to describe the rheological behavior of concentrated dispersions [17]. Three-dimensional chains of particles, echoing van den Tempel’s Linear Chain Model [9] were assumed to be spontaneously built up via attractive forces between these particles. The breakage of these chains were modeled to be due to thermal motion and an applied deformation. The chains of particles were modeled as a non-linear solid spring. A probability of fracture is built into the model, with the fracture causing stress relaxation throughout the network. The authors derived an expression relating the response of the network structure to the macroscopic stress tensor, which serves as a generalized model relating the response of the network to a given type of deformation (shear, extension, etc.). Integration of this expression and appropriate modification would allow for the description of almost any macroscopic rheological property on the basis of the microscopic network structure. Kamphuis and Jongschaap later applied their model to the study of the viscoelastic behavior of suspensions of solid fat particles in oil as well as largedeformation studies of these particles [18]. All of the rheological models discussed in the section above exhibited limitations in predicting the non-linear dependence of rheological properties such as G0 on the solid volume fraction since it adopted a geometry that was ill-suited to describing colloidal aggregates. It would require the development of a new type of geometry to adequately model the microstructures of colloidal particle networks. 3. Fractals The concept of a fractal was considered to have been given wide currency by Benoit Mandelbrot’s 1982 publication of the The Fractal Geometry of Nature [19]. This monumental work is the revised and enlarged English adaptation of his earlier French publication of Les Objects Fractals: Forme, Hasard et Dimension in 1975. The concept of a fractal thus predates the commonly-accepted birth year of 1982 and indeed, some ideas peculiar to fractals such as self-similarity goes back centuries [20]. The study of fractals, however, was expedited by the advent of the computer, which made repetitive calculations feasible. Mandelbrot introduced the concept of fractal geometry as an extension to the classical Euclidean geometry used to describe the shape of nature since antiquity. Mandelbrot noted that “irregular and fragmented patterns” such as coastlines and the branch structure of a tree defy description by traditional Euclidean geometry [19]. He proposed a new class of shapes called fractals to describe these objects. Fractals exhibit a new (or heretofore underappreciated) type of symmetry referred to as self-similarity, which is symmetry across scale. A self-similar fractal exhibits the same geometry at all levels of its structure, a property referred to as scale invariance. In this sense, a fractal object is symmetrical because an operation (a change in the observer’s “magnification”) does not lead to a state that can be distinguished from the initial pre-operation state. Thus, the details that determine the higher-level geometry of a fractal are present within the lower-levels of its structure. In this sense, there is a long-range correlation between a fractal’s lower-level and higher-level structure.

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The requirement for exact self-similarity between all length scales need not be absolute. Non-fractal objects may still be modeled using fractals if they exhibit fractal behavior only within a limited range of length scales, a condition known as statistical selfsimilarity. Furthermore, statistical self-similarity implies that the fractal structures at applicable length scales do not have to be exactly self-similar since the randomness of the phenomena that gives rise to the fractal geometry may also introduce variability that breaks exact self-similarity. A fractal is a geometrical object with a dimension that assumes a non-integer (or fractional) value, hence fractal. A fractal “exists” within a larger Euclidean dimension than itself called the embedding dimension and denoted d and assumes only integer values. A putatively 2-dimensional fractal such as a Koch snowflake may have a dimension between 1 and 2 but “lives” on a 2-dimensional sheet of paper (Fig. 4). Central to the concept of fractals is dimension, a concept which is intuitive but difficult to define. Consider the simplest measure of dimension, the self-similarity dimension Ds:

Ds ¼

logNF logF

where NF is the number of self-similar objects observed when the scale is modified by a factor F. Consider a line segment with arbitrary length L0. If the scale is reduced by a factor F of 3 to a third of its length, what number NF of smaller self-similar line segments with length r ¼ 13L0 can be observed? (Fig. 5A) The answer is calculated as follows:

NF ¼

L0 L ¼10 ¼3 r 3L0

and the self-similarity dimension is calculated as:

Ds ¼

log3 ¼1 log3

The length of a line is thus its essential measure. To demonstrate the absurdity of using a measure such as an area to describe a line, consider dividing the line above with squares with side r and an area r2. The number of squares to cover the length of the line is NF and the area A of the line is:

A ¼ lim NF ∙r 2 ¼ lim r/0

r/0

L0 2 ∙r ¼ lim L0 ∙r ¼ 0 r r/0

The following treatment, however, makes sense for an object such as a square (Fig. 5C). Consider a square with side length L0 and an area L20. If the scale is again reduced by a factor F of 3 to create squares with side r ¼ 13L0 , what number of squares NF can be observed?

NF ¼

L0 2 L 2 L 2 ¼
0 2 ¼ 0 2 ¼ 9 2 1L r 1L 9 0 3 0

The self-similarity dimension is calculated as:

For any given length L0, the length L of the line segment as r approaches 0 is:

L ¼ lim NF ∙r ¼ lim r/0

Fig. 5. Reduction of the scale by 1/3 for a (A) line, (B) Koch curve and (C) square.

r/0

L0 ∙r ¼ L0 r

Ds ¼

log9 ¼2 log3

Likewise, consider the area of this square as it is subdivided into smaller and smaller squares:

L0 2 2 ∙r ¼ L0 2 r/0 r 2

A ¼ lim NF ∙r 2 ¼ lim r/0

Now, consider a segment of the fractal Koch snowflake (Fig. 5B). Unlike the Euclidean line segment, reducing the scale by a factor of 3 produces not 3, but 4 self-similar Koch curves with r ¼ 13L0 . The self-similar dimension is thus:

Ds ¼

log4 ¼ 1:2619 log3

A Koch snowflake is thus “too much” to be a line but “not enough” to be a surface. Interestingly, a fully enclosed Koch snowflake has an infinitely long length but a finite area. Another measure of dimension is the capacity dimension Dcap, which is a generalization of the self-similarity dimension Ds. Measurement of this dimension involves the use of a “measuring stick” (or ball). The capacity dimension involves finding the exponent that defines the scaling relationship between the minimum number N(r) of a “measuring stick” with a characteristic size r required to measure out an object in a given space:

Fig. 4. A Koch Snowflake of Order ¼ 5.

NðrÞ 

1 rD

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The space is a metric space, which, to speak plainly, is a set of points for which the concept of a distance can be defined. The most familiar such metric space is the 3-dimensional Euclidean space. The capacity dimension is calculated as:

Dcap ¼ lim

r/0

logNðrÞ log 1r

5

more information on the measurement of fractal dimensions by image analysis, please refer to the works by Awad and Marangoni [22], Tang and Marangoni [23,24], Campos [25] and Russ [26,27]. For a more general description of the mathematics of fractals as it applies to practical analysis, please refer to Bassingthwaighte et al. [28].

4. Fractal models of aggregation

A simple example to demonstrate this scaling relationship involves finding the scaling relationship between the minimum number N(r) of “yardstick line segments” with length r required to construct a longer line segment with arbitrary length. It should be obvious that the number of such line segments scales linearly (that is, D ¼ 1) with the length of the line segment, that is, the shorter the line segment, the more of them are required to construct the longer line segment. Thus, it can be said that the longer line segment has Dcap ¼ 1. Likewise, the number of circles required to completely cover an arbitrary rectangular area increases by a factor of 25 if the radius is reduced to a fifth. Yet another example is the number of spheres required to fill a cubic volume: the number of such spheres increases by a factor of 8 if the radius is halved. In the latter two examples, the rectangle and the cube can be said to have capacity dimensions of 2 and 3, respectively, since N scales with r2 and r3. If the capacity dimension utilizes contiguous boxes (instead of balls) to measure the dimension, the resulting dimension is called the box-counting dimension Dbox. To use the “measuring stick” analogy, a Koch snowflake is a fractal since the number of circles required to completely cover the area of this putatively 2-dimensional object does not scale with the radius of the “measuring circle” via an exponent of 2. Since the Koch snowflake occupies less area than a rectangular area, a proportionally smaller amount of smaller circles is required to completely cover the Koch snowflake than if a larger circle were used (Fig. 6). Another commonly used example is the West coast of Britain, where the number of line segments required to “measure out” the coast increases by the exponent D ¼ 1.25 as the length of the measuring “ruler” decreases. The number of required line segments increases as shorter measuring sticks can fit every nook and cranny of the coastline. Strangely, this would suggest that the length of the coastline progressively becomes longer and longer (towards infinity) the shorter the measuring stick gets [21]. A description of the mathematical features of fractals is important since the determination of the fractal dimension from images of fractals invariably rely on measuring one of these dimensions. For

It did not take long for researchers to start applying concepts from fractal geometry to describe the microstructure of colloidal aggregates [29]. In the context of the microstructure of materials, the fractal dimension is a non-integer scaling exponent that describes how the mass of a floc aggregate (often assumed to be spherical) built up from primary particles scales with its size, denoted as the diameter of the aggregate x: D

Mass  x

With regards to aggregation, the fractal dimension D in a 2dimensional embedding space typically takes on two values: (1) D ¼ 1.75 and (2) D ¼ 1.53. A fractal dimension D ¼ 1.75 typically corresponds to diffusion-limited aggregation (DLA) [30]. This is the case when the rate of aggregation of the particles is much greater than the rate at which the particles can diffuse to the reaction site (called the cluster). This often occurs when the repulsive forces between particles is low or non-existent. In a 3-D embedding space, D ¼ 2.5 for DLA. For the more complicated diffusion-limited clustercluster aggregation (DLCA), the fractal dimension typically takes on values of D ¼ 1.7e1.8 in a 3-dimensional embedding space. The second fractal dimension D ¼ 1.53 corresponds to reactionlimited cluster-cluster aggregation (RLCA) in a 2-dimensional embedding space, where the rate of aggregation is much lower than that of diffusion due to the presence of repulsive forces between particles (typically electrostatic surface charges). A low fractal dimension suggests that the aggregate is more open and less compact than an aggregate with a high fractal dimension. Thus, aggregates formed via RLCA are less “dense” than aggregates formed via DLCA. In a 3-dimensional embedding space, RLCA typically exhibits a D ¼ 2.0e2.1. Even before the publication of The Fractal Geometry of Nature, Forrest et al. noted a class of dendritic aggregates (Fig. 7) with longrange density correlations [31]. These aggregates consisted of ultrafine iron metallic smoke particles. The density correlations were described by a power-law with an ‘anomalous’ fractional exponent.

Fig. 6. Koch Snowflakes Covered by N ¼ 51 balls of Radius ¼ 50 and N ¼ 182 balls of radius ¼ 25.

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particles and with other clusters. Both papers independently showed that the geometry of the resulting aggregates were scaleinvariant. Colloidal aggregates can therefore be modeled as stochastic mass-fractals with properties of a length scale beyond that of the primary aggregating particle. The fractality of aggregates formed by DLA has been demonstrated in colloidal gold [36,37] and silica particles [38]. Brown and Ball later modeled reaction-limited (a.k.a. chemically-limited) aggregation using an extension of the Witten-Sander model and found that the resulting aggregates also behaved as mass fractals [39]. Bremer et al. introduced a fractal aggregation model to describe the elasticity of casein particulate gels [40]. Their model of a colloidal network of aggregates was independently developed earlier by Brown [41] and also de Gennes [42] but is favored here for its simplicity and elegance. Consider a 2-dimensional (d ¼ 2) “square” floc with “diameter” x (Fig. 8). The number of spherical particles N with size a that would fit within this floc is given by the following scaling relationship:

 x 2 N¼s a

Fig. 7. Dendritic Fractal Iron Aggregates. Reproduced from: Forrest and Witten [31].

The fractional exponent was ‘universal’ in that it did not vary considerably when the conditions of the aggregating system (including the type of metal) were altered. Subsequent computer simulation work by Witten and Sander showed that diffusion-limited aggregation (DLA), where particles diffuse to a single point of aggregation, resulted in density correlations described by power-laws such as that observed in the earlier work on metallic smoke particles [32]. Later work showed that the geometry of the objects formed by DLA are scale-invariant and possessed longrange order. Despite this, the entire aggregate is assembled, not via long-range forces, but via short-range interactions [33]. Meakin [34] and Kolb et al. [35] extended the Witten-Sander model using computer simulations to study diffusion-limited cluster-cluster aggregation (DLCA). Cluster-cluster aggregation is an extension of previous DLA models since in DLCA, the clusters themselves can diffuse and aggregate upon first contact with

where s is a constant of proportionality that takes into account the fact that spherical particles will not totally occupy the entire volume circumscribed by a square element of the grid. More generally, the total number of sites that would fit particles with size a scales with the floc size according to:

 x d N¼s a Consider a fractal aggregate (with the mass fractal dimension D) of particles with size a within the floc with diameter x. This is analogous to overlaying a grid of size x over the fractal aggregate. More generally, this is analogous to embedding a fractal object within a space of a higher integer dimension. The number of particles within the fractal aggregate scales with the size of the floc x according to:

Np ¼ s

 x D a

The particle volume fraction F of the floc can be expressed as the ratio between the number of particles to the total number of particles that can be accommodated:

Fig. 8. Embedding a Putative 2D Aggregate within a 2D Embedding Space. Adapted from: [1].

E.D. Co, A.G. Marangoni / Advances in Colloid and Interface Science 273 (2019) 102035

 D x  x Dd Np s a ¼  d ¼ F¼ N a s ax The relationship holds for length scales beyond x all the way to macroscopic dimensions if the material itself is considered a spacefilling collection of fractal aggregates. Thus, the macroscopic particle volume fraction of the material Ft can be substituted for the particle volume fraction of a floc in the relationship above, a result that is familiar in polymer physics [42] but also shown to be applicable to colloidal gels [43]. Re-arrangement of this equation, performing the appropriate substitutions and assigning d ¼ 3 (to describe a real-world macroscopic material) gives the following relationship between the floc size and the particle volume fraction of the material: 1

x  ðFt ÞD3

7

transition regime between reversible elastic behavior and irreversible flow in low-concentration colloidal gels of silica [51]. Bremer et al. developed a model of the scaling behavior between the shear storage modulus G0 and the particle volume fraction of acid casein gels [52,53] by considering two modes of stress transmission within a particulate colloidal network: (1) straight (or fully stretched) strands and (2) curved strands. Stress transmission through straight strands is straightforward and assumes that the stress on a string of particles causes the elongation of those particles. Stress transmission through curved strands is slightly more complicated since the application of stress first straightens out the curved strand (analogous to straightening out a wire with some slack before the wire stretches) before the particles are elongated as in straight strands. For straight strands, the scaling relationship was formulated as: 2

G0  F3D whereas, for curved strands, the scaling relationship is formulated as: 3

G0  F3D 5. Rheology of fractal aggregate networks With the fractal geometry of colloidal aggregates being firmly established, researchers now turned their attention to how a network of such fractal aggregates would behave under mechanical stress. The idea that fractal scaling concepts could be applied to model the elasticity of colloidal gels was first suggested by Sonntag and Russel’s study of polystyrene latex colloids, where the storage modulus G0 was noted to exhibit a power-law dependence with the particle volume fraction [44]. While not performing the analyses themselves, these authors suggested that the models for the elasticity of percolated networks developed by Kantor and Webman [45] and by Feng et al. [46,47] can be used to explain the scaling behavior they observed. Brown developed a model for the elasticity of a fractal colloidal network by calculating the elastic constant of a fractal aggregate [41]. The elasticity of this fractal aggregate was assumed to survive at a scale related to the particle volume fraction via the mass fractal dimension D: 1

x  FD3 Stress transmission through this aggregate was assumed to pass through a single “backbone” chain of particles. However, Edwards and Oakeshott pointed out that the stress transmission has to be equally distributed across branches in order to maintain the stability of the aggregate [48]. This result was later independently arrived at by Bremer et al. [40]. Brown also noted that the elastic storage modulus of a fractal colloidal network should scale in a power-law manner with the particle volume fraction with an exponent m equal to:

m¼

The next major advance in modeling the elasticity of fractal colloidal networks came in 1990 with the development of Shih et al.’s scaling theory [54], which built on previous developments and essentially presents a general case for previous models. Indeed, the reader will note that the models by Brown and Bremer et al. are essentially specific cases of Shih’s strong-link model. In Shih et al.’s work, the elasticity of an aggregated particle network is attributed to two elastic regimes: (1) the intra-floc elasticity within the flocs and (2) the inter-floc elasticity between the flocs. Fig. 9 shows such an arrangement under a shear stress. The interaction between two flocculated particles within a floc can be modeled as a “particle spring” and described by the floc elastic constant Kx, which is shown to vary with the particle volume fraction. Likewise, the “link” between two flocs can also be modeled as a “floc spring” and described by the link elastic constant Kl. However, unlike Kx, Kl is largely constant regardless of the particle volume fraction. The elasticity of a floc/cluster can be attributed to a main “backbone” chain of particles. This backbone chain spanned the length of the floc and is therefore described by the same size x as the floc. The backbone itself can be modeled as a chain of springs. The floc elastic constant (due to the backbone) can be formulated as:

Kx ¼

K0 Nbb x

2

3 þ dchem 3D

where dchem is the chemical length exponent, which describes the fractal structure of the main stress transmission pathway in an aggregate. Buscall et al. utilized Brown’s model to model the elasticity of a colloidal network of silica particles [49]. Later work by Ball [50] explained the existence of higher fractal dimensions in silica aerogels by considering that past a certain critical cluster size of particles, the assumption that the particles are rigid no longer holds true. Rather, the particles deform and “partially coalesce” to form a new kind of strong bond, which would dramatically raise the fractal dimension. Uriev and Ladyzhinsky developed a fractal model to describe the

Fig. 9. The Floc (Kx) and Link (Kl) Elastic Constants of a Colloidal Aggregate Network Under Shear Stress. Adapted from: [1].

8

E.D. Co, A.G. Marangoni / Advances in Colloid and Interface Science 273 (2019) 102035

where Nbb is the number of springs in the backbone and K0 is the bending constant between two adjacent springs. Since the floc is a fractal object, by virtue of scale-invariant self-similarity, the backbone itself is a fractal object with a fractal dimension (the tortuosity) x and thus, the following scaling relationship exists between the number of springs in the backbone and the size of the backbone:

E

x

Nbb  x

If the particle density within the fractal floc is independent of concentration, the bending elastic energy K0 can be treated as a constant. This is not an unreasonable assumption since the fractal structure within a floc should remain unchanged with concentration. Alternatively, this means that as the particle density within a floc increases, the effective size of the floc must increase to maintain the same scaling relationship as given by the fractal dimension. Considering this, the scaling relationship between Kx and x can thus be written as:

Kx ¼

K0

xxþ2

As the size of the floc x increases, the floc elastic constant decreases. This indicates that larger flocs behave as weaker springs. Bremer et al.’s formulation of the scaling relationship between the floc size and the particle volume fraction shows that the floc size increases with particle volume fraction with scaling exponent ¼ 1/ (D-3). Shih’s description of the relationship between the floc elastic constant Kx and the floc size shows that Kx decreases with increasing floc size with scaling exponent (x þ 2). Combining these two formulations gives an expression for the floc elastic constant in terms of the material’s particle volume fraction:

Kx ¼ 

xþ2 K0 ¼ K0 F3D 1 xþ2

FD3

Note that (D e 3) has been reformulated to (3 e D). The relationship shows that as the particle volume fraction increases, the floc elastic constant Kx increases accordingly. Since the elastic constant between flocs Kl does not vary with the particle volume fraction, above some critical particle volume fraction Fc, the floc elastic constant becomes greater than the link elastic constant (Kx > Kl). Likewise, below Fc, the converse is true, where the link elastic constant is greater than the floc elastic constant (Kl > Kx). The former case, where (Kx > Kl) is known as the weak-link regime and is applicable to materials with a high particle volume fraction. The latter case, where (Kl > Kx) is known as the strong-link regime and is applicable to relatively dilute materials. In the strong-link regime, the “floc spring” is stiffer than the “particle spring” within the flocs (Kl > Kx). Thus, the site of the material’s response to an applied stress under the strong-link regime occurs within a floc and is described by the floc elastic constant Kx and it is assumed that there is no deformation in the links between flocs. The macroscopic elastic constant of the material E is therefore given as:

E

 L Kx

x

where L describes some macroscopic dimension of the material along which the elastic deformation occurs. L/x is effectively the number of flocs. The following expression for E under the stronglink regime can be arrived at from the previous formulations: xþ3

E  F3D

In the weak-link regime, the “particle springs” within a floc are stiffer than the “floc springs” between two flocs (Kx > Kl). Thus, the deformation in response to an applied stress occurs between flocs and is dependent on the link elastic constant Kl. The material’s elastic constant E can therefore be written as:

 L

x

Kl

Using the appropriate substitutions and combinations, the following expression for E under the weak-link regime can be arrived at: 1

E  F3D Both the strong- and weak-link regimes have been extensively demonstrated by experiments. However, a quick examination of the scaling relationships under the strong- and weak-link regimes show that the formulation for the strong-link regime contains an additional tortuosity term x that is considered difficult to measure. Indeed, this value is generally estimated to be between 1 and 1.3, which was suggested by Shih et al.’s conductivity measurements [55]. After its introduction, Shih et al.’s scaling relationships was experimentally verified by numerous workers in various systems. Chen and Russel’s demonstrated Shih’s model on work on gels of silica microspheres (coated with octadecyl chains) in hexadecane [56]. de Rooij et al. demonstrated Shih’s model in a polystyrene latex colloid [57]. Vreeker et al. utilized Shih’s strong-link model in the analysis of whey protein gels and found that the fractal dimension obtained using this model agreed very well with the fractal dimension obtained from dynamic light scattering measurements [58]. Hagiwara et al. utilized Shih’s weak-link model to analyze various food protein gels and found that the fractal dimension values obtained from Shih’s weak-link regime agreed very well with values obtained from the image analyses of micrographs of the gels [59,60]. Stading et al. [61] utilized Bremer et al.’s [40] formulation for straight stress-carrying strands to obtain fractal dimensions of particulate gels of b-lactoglobulin. More recently, Wu and Morbidelli [62] introduced a case where both the “particle springs” and “floc springs” yield under an applied stress in what they called the transition regime. Under this regime, the expression for the shear storage modulus G0 is:

G0TR  F

ðd2Þþð2þxÞð1aÞ dD

where a is a dimensionless constant that describes the share of the stress borne by the particle and floc springs. A high value of a (> 0.5) means that a greater share of the stress is borne by the links between flocs, with a ¼ 1 corresponding to the weak-link regime. A lower value (a < 0.5) implies that most of the stress is borne by the interactions between the particles within a floc with a ¼ 0 corresponding to the strong-link regime. Mellema et al. developed a more generalized scheme to classify rheological scaling models for particulate colloidal casein gels into five categories by synthesizing much of the previous work in the literature [63]. The strong- and weak-link regimes form the extreme ends of these categories with three intermediate categories. These five categories range in the level of disorder of the aggregates within a floc. At one end (the weak-link regime), the strands (or chains) within the floc are described as “random”. The second category contains “curved” strands, followed by “hinged”, “straight” and “rigid” strands, with the last case corresponding to the strong-link regime.

E.D. Co, A.G. Marangoni / Advances in Colloid and Interface Science 273 (2019) 102035

6. Fractal models of fat microstructure In 1992, Vreeker et al. introduced scaling theories developed for colloidal gels to model the structure of “dilute” fat crystal networks [64]. Vreeker et al. determined the fractal dimension of tristearin aggregates in olive oil using static light scattering. The fractal dimension is the exponent for the scaling relationship between the scattering intensity I(q) and the scattering vector q:

IðqÞ  qD At the initial stages of the experiment, Vreeker et al. determined the fractal dimension to be D ¼ 1.7, indicative of DLCA. Later measurements found the fractal dimension to be D ¼ 2.0, indicative of structures formed via RLCA. It was reasoned that the initial aggregation step is diffusion-limited as fat particles do not have significant repulsive interactions, thus the aggregation step proceeds at a much faster rate than diffusion. The later measurements of D ¼ 2.0 indicated that the aggregates reorganized into a more compact structure over time, perhaps via a post-crystallization process. Vreeker et al. also demonstrated that the storage modulus G0 of the tristearin dispersion in olive oil scaled with the solid fat content FSFC according to a power law formulation:

G0  FSFC m

where m is the scaling exponent and found to be m ¼ 4.1. Using Shih’s strong-link regime formulation with x ¼ 1.3, the fractal dimension D was determined to be 2.0, which agrees very well with the results from the light scattering experiments. Johansson utilized simple sedimentation techniques [65] to study the relationship between the relative floc size (x/a) and the solid particle volume fraction (F) using an expression for the sedimentation velocity of fractal aggregates previously derived by Bergenståhl [66] for food emulsions:

x a



F

max

1  3D

F

where Fmax is the maximum particle volume fraction of the sediment. Using this formula, the size of the aggregates could be studied using sedimentation techniques. Large aggregates were found to form weak gels while smaller aggregates tended to sediment out. The influence of temperature on gelation was assumed to be through the size of the aggregates, with higher temperatures reducing the floc size. Marangoni and Rousseau utilized the developed concepts to analyze blends of butterfat and canola oil as well as chemicallyinteresterified (CIE) blends of these oils [67]. It was noted that the non-interesterified (NIE) and CIE blends did not show appreciable differences in the solid fat content and the polymorphic form and yet the hardness of the CIE blend was markedly lower than the hardness of the NIE blend. A quick look at the microstructure of these fats under a polarized light microscope showed that the fat crystal morphology and the microstructural arrangement was different. But, how then to quantify such microstructural differences? The authors proposed using the fractal dimension as a measure of the microstructure. Assuming a weak-link regime, the scaling relationship between the storage modulus G0 and the solid fat content was determined. The fractal dimension for the NIE blend was found to be D ¼ 2.46 while D ¼ 2.15 for the CIE blend. This, however, did not explain the decrease in hardness observed upon interesterification. A decrease in the fractal dimension D is typically accompanied by an increase in the storage modulus and hardness of a fat. A variation of this study by the same authors examining enzymatically-interesterified (EIE) blends of butterfat

9

and canola oil yielded D ¼ 2.50 for the EIE blend and D ¼ 2.59 for the NIE blend [68,69]. Narine and Marangoni applied the weak-link model to the study of other fats as well as devising techniques (rheology, confocal and light microscopy) to measure the fractal dimension after noting the presence of statistically self-similar structures (Fig. 10) in fats [70e72]. Marangoni and Rousseau also used the weak-link model to examine the microstructure and rheology of CIE blends of lard/ canola oil and of palm oil/soybean oil [73e75].These authors also reformulated the scaling relationship between the storage modulus G0 and the particle volume fraction by introducing a constant of proportionality l, which describes the interparticle forces responsible for the interactions between the flocs: 1

E  lF3D The fractal dimensions for lard/canola oil was found to be D ¼ 2.82 and for palm oil/soybean oil, D ¼ 2.88. These did not change appreciatively upon interesterification. However, the constant of proportionality l registered a four-fold increase upon interesterification of the lard/canola oil blend, which would explain the increase in hardness upon interesterification of this blend. Indeed, the authors performed a re-analysis of their initial work from 1996 [67] and concluded that the decrease in the hardness of butterfat/canola oil blends upon interesterification was due to a decrease in l and not due to changes in the distribution of mass as given by the fractal dimension. Narine and Marangoni utilized Nederveen’s integration of the 6e12 Lennard-Jones potential to obtain an expression for l [76].For the weak-link regime, the following expression for the elastic modulus E can be obtained:

E

1 mAh F3D 2cpaxl30

where m is the number of neighboring particles at the junction between two flocs/microstructures, Ah is Hamaker’s constant (about 5  1020 J for alkanes), c is a proportionality constant in the scaling relationship between the number of particles in a floc with diameter x (N ¼ c(xD)), a is the diameter of a microstructural element or particle, and l0 is the average equilibrium distance between two microstructural elements. The authors posit that Hamaker’s constant is highly dependent on the triglyceride composition and the polymorphic form of the lamellar crystallites that form the microstructural element. This model ignored the negligible inertial and hydrodynamic contributions to elasticity. An equivalent expression for the shear modulus G can be obtained by formulating E ¼ 2G(1 þ n), where n is Poisson’s ratio, which is equal to 0.5 for an incompressible material. This yields an expression for G as G ¼ 3E:

G

1 mAh F3D 6cpaxl30

Difficulties in estimating parameters in the pre-exponential term serves to limit the practical application of such structural models. To improve on this, Marangoni approached the problem from a thermodynamic perspective [77] and obtained an expression for E for the weak-link regime with the following expression for the pre-exponential factor:




12 E

DUx Dl

paxε



1

F3D

where DUx is the change in the total internal energy of the inter-floc link (assumed to be equal to the change in free energy DGx when

10

E.D. Co, A.G. Marangoni / Advances in Colloid and Interface Science 273 (2019) 102035

Fig. 10. Statistical Self-Similarity of Fat Crystal Networks Imaged with Polarized Light Microscopy within the mm length scale. Adapted from: [1].

the entropic contribution is negligible [78]), Dl is the change in the inter-particle separation upon the application of a stress and is equivalent to (l0l) where l0 is the interparticle separation at equilibrium and l is the inter-particle separation under stress, ε is the extensional strain and is equal to Dxl. DU The term Dlx is essentially an expression for a force describing the inter-floc links. The force can be assumed to take the form of a colloidal van der Waals force. For a spherical floc, the applicable force is:

DUx Ah x ¼ Dl 12l20 which would yield the following expression for the elastic modulus E:

E

Ah

1

F3D

paεl20

Marangoni then used the general expression for the elastic modulus to derive an expression for the yield stress s*, which is the stress beyond which the material flows [79]. The yield stress can be taken as E times ε*, which is the strain at the limit of linearity or the yield strain:




s* 

12

DUx Dl

pax



s* 

1 6d 3D F a

Awad et al. later applied Shih’s weak-link regime, as well as the developments by Marangoni et al. to describe the scaling behavior between the elastic modulus and the solid fat content [81]. Awad et al. found the weak-link regime adequately describes fats diluted with oil. The changes in the microstructure were attributed to changes in crystallization kinetics brought about by dilution. Tang and Marangoni [82] later developed a modified fractal model (under a weak-link regime) using probabilistic arguments to describe the heterogeneous distribution of stress through a fat material, where the load-carrying capacity of a material is not evenly distributed throughout the entirety of the solid mass but is rather concentrated across a small part of the fat network. This led to the formulation of an “effective” solid volume fraction Fe:

C

1 3D

F

The force term can be approximated via the Derjaguin approximation (which requires x > l0), which formulates the force between two particles as an adhesion force Fad arising from surface interactions [80]:

Fad ¼ pdx

where d is the solid-liquid interfacial free energy per unit area and, in the case of fat crystal networks, is equivalent to the crystal-melt DU interfacial tension. Substituting Dlx ¼ 12Fad , the following expression for the yield stress can be obtained:



Fe ¼ Ff 1  ekF r

b



where r is the density of the fat material, C is the constant of proportionality in the scaling relationship between the mass of a floc and the solid volume fraction: Mx ¼ CFf with f being the scaling exponent, where k is the probability that an inter-floc bond becomes the only stress-bearing bond between two clusters multiplied by Cr and b ¼ 1 e f. Substituting Fe for F in Shih’s weak-link formulation yields the following scaling relationship for the shear storage modulus G0 :

E.D. Co, A.G. Marangoni / Advances in Colloid and Interface Science 273 (2019) 102035

 1   b 3D G0  Ff 1  ekF The developed model showed a good fit to the G0 and SFC data, allowing for estimates of the relevant parameters. 7. Crystalline nanoplatelets Much of the previous has focused on structures at the micrometer length scale. The advent of techniques such as transmission electron microscopy and ultra-small angle X-ray scattering has elucidated much lower levels of structure. The single-crystal lamellar crystallites long thought to be the microstructural element of a Fat Crystal Network were first imaged in 2010 by gently de-oiling a fat made of fully‑hydrogenated canola oil (FHCO) and high-oleic sunflower oil (HOSO) in cold isobutanol [83,84]. The lamellar crystallites were found to have platelet-like morphology and thus christened Crystalline Nanoplatelets (CNP) (Fig. 1). Such CNPs formed from a melt of FHCO were found to have average lateral dimensions of 150 nm  60 nm (length  width) with a height of 30 nm. Larger CNPs were formed by crystallizing FHCO out of a 30/70 FHCO/HOSO solution: lateral dimensions of 370 nm  160 nm with a height of 40 nm. These dimensions were highly sensitive to added emulsifiers [85,86], crystallization conditions as well as external forces such as applied shear [87]. The application of shear caused an increase in the aspect ratio of the CNPs as well as an increase in the size of the CNPs. The model developed by Marangoni and Rogers [79] as well as Shih’s weak-link regime was applied to study the scaling behavior of the mechanical properties of fats with the parameters corresponding to the primary particle diameter being the CNP dimensions. Sheared fats were found to have a lower shear storage modulus G0 and yield stress s*. This was attributed to the inverse proportionality in the model between these properties and the size of the microstructural element since the sheared fats exhibited CNPs with larger sizes [88]. The fractal dimension of fats crystallized in the absence of shear was determined to be D ¼ 2.5e2.7, which corresponds to diffusion-limited aggregation of the primary particles. Fats crystallized under an applied shear field exhibited fractal dimensions of D ¼ 2.9e3.0, which suggest that the primary particles packed into Euclidean, non-fractal aggregates. 8. Aggregates of crystalline nanoplatelets Once the basic structural features of CNPs were described, attention turned to how these CNPs aggregated. Pink et al. modeled CNPs as rectangular arrays of spherical subunits in a Metropolis Monte Carlo simulation of CNP aggregation via “weak”, “intermediate” and “strong” van der Waals interactions [89]. The interaction force between two CNPs was modeled as the total of all the pairwise van der Waals interactions between any two pairs of spherical subunits between the CNPs. In the Metropolis Monte Carlo simulation, each CNP was allowed to translate and rotate randomly in each simulation step. If the resulting translation and rotation minimized the interaction free energy, the new position of the CNP was accepted while translations and rotations that increased the interaction free energy were rejected. At the end of the simulation, the fractal dimension of the resulting structures were obtained by generating a hypothetical Xray spectra. The relation between the scattering intensity and the scattering vector q was assumed to be through two functions: a form function P(q) and a structure function S(q), where P(q) describes the scattering caused by the CNPs themselves (and is treated as a constant) while S(q) describes the scattering due to

11

spatial correlations between CNPs. The fractal dimension is thus the scaling exponent between the S(q) and q:

SðqÞ  qD Structures were only observed when “intermediate” or “strong” van der Waals forces were used. Under intermediate van der Waals forces, cylindrical stacks of CNPs were observed. These structures were christened TAGwoods (in homage to their resemblance to a dagwood sandwich). A higher particle volume fraction (by including more CNPs in the simulation) resulted in shorter TAGwoods. The fractal dimension was found to be D ¼ 2.0e2.1, suggesting the formation of the TAGwoods was reaction-limited. For the simulation utilizing “strong” van der Waals forces, D ¼ 1.75, suggesting diffusion-limited aggregation. Both fractal dimensions were independent of the particle volume fraction. The aggregation of the TAGwoods were themselves simulated. It was noted that at initial stages of aggregation, structures were formed with D ¼ 1.7, indicative of DLCA. At later stages of the aggregation process, D assumed values between 2.0 and 2.1, which is indicative of RLCA. This observation is very similar to that noted by Vreeker nearly two decades previously, where the aggregation of a fat crystal network first proceeds along lines of DLCA, followed by a slower relaxation phase to a structure indicative of RLCA [64]. Smaller TAGwoods were also found to proceed to the RLCA phase much faster. CNPs with modified surfaces [90] were also modeled using the same methodology to observe the effects of molecules (liquid-state and solid-state TAGs) adsorbed to CNP surfaces. CNP surfaces orthogonal to the TAG axis were assumed to be the coated surfaces due to the denser aliphatic groups along these surfaces. The simulations showed that a thick coating on these surfaces inhibits the formation of TAGwoods. Aggregation of CNPs into TAGwoods (albeit at a slower rate) may proceed if the coating on the CNP surface is thin enough. The aggregation of CNPs a model fat of tristearin in triolein was studied in real-time using ultra-small-angle X-ray scattering [91]. The data was fitted to the Unified Fit Model (UFM) and the Guinier Porod (GP) model. The results showed that scatterers in the length scale of the CNPs (< 100 nm) were found to have a radius of gyration (effectively, the size of the CNPs) between 50 and 80 nm. As the concentration of tristearin was increased, the size of the CNPs were found to decrease. At length scales corresponding to the size of TAGwods (< 1000 nm), the scatterers were found to have a radius of gyration between 300 nm and 700 nm. By way of a Porod exponent (P ¼ 4) in the UFM model, the TAGwoods were found to distributed in a non-fractal, space-filling manner. Fitting of the data to the Guinier-Porod (GP) allows for the treatment of multiple scatterers within the same structural level as well as the use of non-spherical geometries. Fitting of the USAXS data to the GP model showed the existence of cylindrical aggregates with a length between 650 nm and 950 nm, which is consistent with the length of TAGwoods determined in earlier simulations as well as by treatment of the data with the UFM model. The radius of the cylindrical aggregate was found to be between 60 and 100 nm, which is consistent with the size of a CNP. USAXS was also employed to study more complex mixtures of fats [92]. The more diverse chemical composition of these fats was assumed to lead to the modification of CNP surfaces with different types of TAGs. Most interestingly, a mixture of partially‑hydrogenated canola oil (PHCO) and triolein exhibited CNPs with a rough surface, presumably caused by the crystallization of high-melting TAGs on the surfaces of the CNPs. Interestingly, this rough surface led to a higher extent of aggregation into TAGwoods such that longer TAGwoods were observed in the PHCO/triolein mixture than in a model fat of tristearin/triolein. For this system, the USAXS evidence also showed that the TAGwoods aggregated side-by-side to form structures with a lower aspect ratio than the cylindrical

12

E.D. Co, A.G. Marangoni / Advances in Colloid and Interface Science 273 (2019) 102035

TAGwoods. USAXS was also used to study pure tristearin and tripalmitin melts without any liquid oil phases [93]. USAXS patterns revealed that the absence of liquid oil resulted in the formation of nano-voids between the boundaries of two crystal grains.

V

m D d

x 9. Conclusions The development of microstructural models to describe colloidal networks of fat crystals was shown in this paper. These microstructural models started out with basic descriptions of the microstructure as linear and branching chains of microstructural elements held together by non-covalent van der Waals interactions. The advent of fractal geometry allowed for a new geometrical description of the particle network that comprises the microstructure of a fat. Without fractal geometry, the development of network models that accurately describe the rheology of a fat material would not be possible. Recent advances in analytical techniques have allowed for the study of the nanostructural level of fats. Models that describe the aggregation and structures of crystalline nanoplatelets are only in their infancy. In particularly, the rheological properties (and implications thereof) of these particles and their aggregates at the nanostructural level presents a rich, untapped field of study.

N Mx s Nbb

a I(q) q

l m c v C f k b P

Total volume of interstitial spaces between particles Scaling exponent in G ~ Fm Fractal dimension Embedding dimension Floc/Aggregate Diameter Number of primary particles in a floc Mass of a floc Constant of proportionality in N ~ (x/a)d Number of particles in the “backbone” of a floc Share of the stress borne by links between flocs Intensity of scattering Scattering vector Constant of proportionality in E ~ F(1/(3-D)) Number of neighboring particles at the junction between two flocs Constant of proportionality in N ~ xD Poisson’s ratio Constant of proportionality in Mx ~ Ff Scaling exponent in Mx ~ Ff Probability that a link between flocs becomes the stressbearing link 1ef Porod Exponent

List of symbols

Declaration of competing interest

Moduli, stress and strain

The authors declare no competing financial or non-financial interests.

E

s ε

*

s

ε* G G0

t g Kx Kl K0 Ah DUx

DGx Fad

d r

Elastic/Young’s modulus Extensional stress Extensional strain Extensional yield stress Extensional yield strain Shear modulus Shear storage modulus Shear stress Shear strain Elastic Constant of a floc Elastic Constant between flocs Bending Constant between particles in a floc Hamaker’s constant Change in the total internal energy of the link between flocs upon deformation Change in the total free energy of the link between flocs upon deformation Adhesion force arising from surface interactions between two particles Solid-liquid/Crystal-melt interfacial free energy per unit area Density of a material

Network and particle characteristics a l0 l

F Ft Fc FSFC Fmax Fe L

Primary particle diameter Equilibrium inter-particle separation Inter-particle separation Floc solid volume fraction Material solid volume fraction Critical solid volume fraction for cross-over between weak- and strong- link regimes Solid fat content Maximum floc solid volume fraction Effective solid volume fraction Characteristic dimension of material along which deformation occurs

Acknowledgements The authors acknowledge the financial support of the Natural Sciences and Engineering Research Council of Canada, grant number 05715-2015. References [1] Marangoni AG. Fat crystal networks: New York. New York: Marcel Dekker AG; 2005. [2] Marangoni AG, Wesdorp LH. Structure and properties of fat crystal networks. 2nd ed. Boca Raton: CRC Press; 2012. [3] Acevedo NC, Peyronel F, Marangoni AG. Nanoscale structure intercrystalline interactions in fat crystal networks. Curr Opin Colloid Interface Sci 2011;16:374e83. [4] Narine SS, Marangoni AG. Relating structure of fat crystal networks to mechanical properties: a review. Food Res Int 1999;32:227e48. [5] Deman JM, Beers AM. Fat crystal networks: structure and rheological properties. J Texture Stud 1987;18:303e18. [6] Heertje I, Leunis M, WJMv Zeyl, Berends E. Confocal scanning laser microscopy in food research: some observations. Food Microstruct 1987;6:1e8. [7] Heertje I, Eendenburg Iv, Cornelissen JM, Juriaanse AC. The effect of processing on some microstructural characteristics of fat spreads. Food Microstruct 1988;7:189e93. [8] Shukla A, Rizvi SSH. Relationship among chemical composition, microstructure and rheological properties of butter. Milchwissenschaft 1996;51:144e8. [9] Van den Tempel M. Mechanical properties of plastic-disperse systems at very small deformations. J Colloid Sci 1961;16:284e96. [10] Hamaker HC. The Londondvan der Waals attraction between spherical particles. Physica 1937;4:1058e72. [11] Payne AR. The elasticity of carbon black networks. J Colloid Sci 1964;19:744e54. [12] Nederveen CJ. Dynamic mechanical behavior of suspensions of fat particles in oil. J Colloid Sci 1963;18:276e91. [13] Sherman P. The influence of particle size on the viscoelastic properties of flocculated emulsions. 5th International Conference on Rheology, Kyoto, Japan. 1968. p. 327e38. [14] Papenhuijzen JMP. Superimposed steady and oscillatory shear in dispersed systems. Rheologica Acta 1971;10:493e502. [15] Papenhuijzen JMP. The role of particle interactions in the rheology of dispersed systems. Rheologica Acta 1972;11:73e88. [16] Van Den Tempel M. Rheology of concentrated suspensions. J Colloid Interface Sci 1979;71:18e20. [17] Kamphuis H, Jongschaap RJJ, Mijnlieff PF. A transient-network model describing the rheological behaviour of concentrated dispersions. Rheologica Acta 1984;23:329e44.

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