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Development of an ultrasonic shear reflection technique
to monitor the crystallization of cocoa butter
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Annelien Rigollea,c, Imogen Fouberta,c, Jan Hettlerb, Erik Verbovenb, Ruth Demuyncka,c, and Koen Van Den
Abeeleb
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a
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b
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c
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Keywords : cocoa butter, fat crystallization, ultrasound, shear reflection, in-line monitoring
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Research Unit Food & Lipids, KU Leuven Kulak, Etienne Sabbelaan 53, B-8500 Kortrijk, Belgium
Research Unit Wave Propagation and Signal Processing, KU Leuven Kulak, Etienne Sabbelaan 53, B-8500
Kortrijk, Belgium
Leuven Food Science and Nutrition Research Centre (LFoRCe), Department of Microbial and Molecular
Systems (M2S), KU Leuven, Kasteelpark Arenberg 20, B-3001 Leuven, Belgium
Corresponding author: Annelien Rigolle
Phone: +32 (0) 56 24 62 57
Fax: +32 (0) 56 24 69 99
E-mail: Annelien.Rigolle@kuleuven-kulak.be
Abstract
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The quasi-isothermal crystallization process of cocoa butter was monitored by an ultrasonic shear
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reflection technique utilizing a custom-built experimental set-up in a temperature controlled
20
environment. To facilitate the interpretation of the measurement results, the propagation of shear waves
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was first theoretically studied in different configurations of gas, liquid or solid layers with varying
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thickness for the case of normal incidence, yielding theoretical equations of the shear wave reflection
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coefficient (swRC) for different layering conditions. The typical experimentally observed pattern of the
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swRC during quasi-isothermal cocoa butter crystallization, was subsequently linked to the theoretical
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equations. The remarkable oscillatory damped response in the swRC as function of the crystallization
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time could be explained by constructive and destructive interference of a first reflection at the boundary
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between a plexiglass delay line and the crystallized cocoa butter and a second reflection occurring at the
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interface between crystallized and liquid substance. This hypothesis was supported by the excitation
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frequency dependence of the oscillations. The quality of the fit of the theoretical model to the
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experimental results was very good and also the reproducibility between different independent
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measurements was acceptable. Finally, measurements at different temperatures (18°C and 20°C)
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suggested that the technique was able to detect differences in crystallization behavior, as measurements
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at 18°C displayed faster oscillations compared to measurements at 20°C. Moreover, this was also
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confirmed by the theoretical model, as a higher value of the crystallization rate parameter K, exhibited
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more rapid oscillations.
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1. Introduction
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In the production of fat containing food products, insight in the crystallization behavior of fats is of
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utmost importance to obtain the desired product functionality and product quality. The crystallization
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process consists of triglycerides crystallizing in a particular polymorphic form and the aggregation of
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these crystals into clusters and larger microstructures until a continuous threedimensional network is
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formed (Narine & Marangoni, 1999). Several methodologies exist to monitor the crystallization process,
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with the major contemporary techniques being differential scanning calorimetry (DSC), pulsed nuclear
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magnetic resonance (pNMR), X-ray diffraction (XRD), rheology and polarized light microscopy (PLM)
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(Foubert et al., 2003). Even though the microstructure development most likely affects the macroscopic
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properties to a great extent (Narine & Marangoni, 1999), rheology and PLM are the only techniques of
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the above mentioned that can actually monitor microstructural characteristics. Furthermore, all these
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techniques are applied off-line in the assumption that the tested batch is representative of the entire
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production line, while in reality variation exists in most process streams. So, there is a need for in-line
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monitoring, which –apart from providing more accurate information- can also yield considerable
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financial benefits. Expensive rework or disposal of out-of-specification products can be avoided as the
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process variables can be adjusted by monitoring crystallization during the production process (Kress-
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Rogers et al., 2001). Moreover, primary trends in the food industry nowadays are reduced-fat products
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and trans fat replacement because of human health concerns (Acevedo & Marangoni, 2014; Ma & Boye,
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2013). However, as fat determines to a large extent the product quality, it is a challenge to produce
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healthy products while retaining many of the quality characteristics (Marangoni et al., 2012). An in-line
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method could provide important information on the structural development during processing and thus
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stimulate product innovation.
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In the last decade, several feasible techniques for in-line monitoring of fat crystallization have been
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suggested: NMR-MOUSE (Nuclear Magnetic Resonance - mobile universal surface explorer) (Martini
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et al. 2005b), Laser backscattering (Hishamuddin et al., 2011), Near-Infrared (NIR) Spectroscopy
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(Bolliger et al., 1999) and ultrasonic techniques (Saggin & Coupland, 2004; Coupland, 2004; Martini et
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al., 2005b; Häupler et al., 2014). Because the topic of this paper is the development of an ultrasonic
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shear reflectance technique to monitor fat crystallization, the focus in the following brief literature
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review will be on the application of ultrasonic techniques in fat crystallization monitoring.
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Ultrasonic waves are mechanical or elastic waves with a frequency higher than the upper limit of human
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hearing (16 kHz) for which two important types of waves can be distinguished (Povey & McClements,
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1988). In compressional waves (or longitudinal waves) the particles move in the same direction as the
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propagating wave (McClements, 1997). This type of propagation is supported by solid as well as fluid
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media (Létang et al., 2001). On the contrary, in shear waves (or transversal waves) the movement of the
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particles is perpendicular to the direction of the propagating wave (McClements, 1997). Therefore, shear
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waves only propagate in materials having shear elasticity (Létang et al., 2001).
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Ultrasonic inspection is capable of performing rapid and precise measurements which are non-
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destructive, relatively inexpensive, non-hazardous and can be fully automated, which make it highly
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suitable for in-line monitoring (McClements & Povey, 1992; Martini et al., 2005a). However, this
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technique also has some drawbacks and limitations for particular applications, which explain why
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ultrasonic monitoring, although very promising, is not widely used in the food industry to date.
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Especially the application of ultrasonics to monitor fat crystallization is hampered by the high
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attenuation of the ultrasonic signal in lipids with high solid fat content (SFC) (Coupland, 2004).
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Furthermore, pure solid triglycerides tend to form voids during cooling, and at the same time, systems
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with high SFC tend to detach from the container wall, causing an air gap (McClements & Povey, 1992;
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Saggin & Coupland, 2002). Air bubbles and layers of air scatter and reflect ultrasound very strongly
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because of the large difference in acoustic impedance between gas bubbles and fat, which leads to
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substantial amplitude reduction of the transmitted signals (Coupland, 2004). So, the main problem of
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performing ultrasonic measurements on fat samples is to get the ultrasonic pulse through the material
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(Martini et al., 2005a).
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The research reported in literature about this topic, varies in the type of ultrasonic waves used, the
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applied waveforms for excitation, and the chosen ultrasonic measurement set-up. As compressional
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waves are the easiest to generate and detect, a lot of research has been published on their use for fat
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monitoring applications (Povey & McClements, 1988). The pulse-echo technique is the simplest and
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therefore most widely used ultrasonic measurement technique (McClements, 1997). In this technique, a
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transducer which is directly connected to a sample sends out a short pulse which travels across the
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sample until it reaches the opposite cell wall, where it is reflected back to the transducer. The same
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transducer acts as receiver and records the reflected signal. By analyzing the resulting echoes, several
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ultrasonic parameters can be determined: the ultrasonic velocity (v), the attenuation coefficient () and
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the acoustic impedance (Z), the latter being the product of density () and v (McClements, 1995;
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McClements, 1997). The ultrasonic velocity measurements can be used to calculate the SFC, because
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of the substantial difference in the speed of sound in the solid phase (approximately 2000 m/s) as
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compared to the liquid phase (approximately 1400 m/s) (Bijnen et al., 2002). However, Singh et al.
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(2004) showed that the correlation between the ultrasonic velocity measurements and the SFC was only
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applicable for samples with an SFC lower than 20% at a thickness of 1.6 cm. By upgrading the excitation
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waveform to a chirp, Martini et al. (2005a) reported measurements through an 8.11 cm thick sample
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with an SFC of ~20% in a through transmission mode. The attenuation coefficient also depends on the
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SFC, but is found to be more sensitive to the microstructure than velocity measurements (McClements
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& Povey, 1992). Moreover, Häupler et al. (2014), using a chirp wave excitation, demonstrated that the
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attenuation of tempered cocoa butter is higher than the attenuation of untempered cocoa butter at the
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same SFC level at frequencies of 1.7 and 3 MHz , suggesting that the attenuation is also polymorph
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dependent. Saggin and Coupland (2002) used a modified pulse echo technique to measure the
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compressional wave reflection coefficient (cwRC) of a series of confectionery coating fats and cocoa
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butter dispersions in corn oil. The main advantage of this reflection approach was that it could be used
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with highly attenuating materials, since the waves do not have to travel across the material. However, a
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disadvantage is that reflectance only depends on the surface properties of the sample and can be
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misleading if this region is not representative of the bulk (Coupland, 2004).
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Contrary to compressional waves, shear waves have much less frequently been investigated in a fat
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crystallization context. To our knowledge, only Saggin and Coupland (2004) have proposed a shear
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reflectance technique to calculate the shear modulus and dynamic viscosity of different solid fat
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dispersions with varying SFC, but these properties were not measured during the crystallization process
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itself. Interestingly, they observed that the shear ultrasonic properties revealed a sensitivity to the sample
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microstructure. In other reports, a similar shear reflectance technique has also been suggested to study
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the crystallization of forming biopolymer films (Peura et al., 2008) and polychloroprene films (Alig &
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Tadjbakhsch, 1988).
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Most of the recent studies concerning ultrasonic measurements of crystallizing fats deal with emulsions
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or dispersions of fat crystals in a liquid oil, rather than with bulk fats, presumably because of the very
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high attenuation of the latter. However, because of the difference in physical state between bulk fats and
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dispersions, and therefore also in the interaction with the propagating wave, the results of dispersions
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cannot be readily transferred to bulk fats.
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Based on this literature review, it was decided to study the potential of an ultrasonic shear reflectivity
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technique to continuously monitor the crystallization process of pure cocoa butter. Because the wave-
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matter interaction in semi-crystalline fats is less well known than in emulsions (Saggin & Coupland,
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2002), this work started with the construction of a theoretical model of the propagation of shear waves
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in a layer of crystallizing cocoa butter. Subsequently, ultrasonic shear reflectivity experiments were
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performed at different crystallization temperatures (18°C and 20°C) and the evolution of the observed
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shear wave reflection coefficient (swRC) was linked to the theoretical model.
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2. Materials and Methods
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The cocoa butter used in this study was a standard factory product of West-African origin which was
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kindly provided to us by Barry Callebaut (Wieze, Belgium).
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Sample preparation
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The cocoa butter was melted in a furnace at 85°C for 30 min to erase all crystal memory. Subsequently,
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20g of the sample was air-cooled statically until 30°C was reached. This temperature was chosen as a
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compromise between avoiding the presence of air bubbles and preventing a high temperature increase
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of the plexiglass delay line (see experimental set-up). Both features would have a negative effect on the
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ultrasonic measurements as air bubbles attenuate the signal whereas temperature gradients influence the
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ultrasonic properties. Furthermore, completely melted cocoa butter does not crystallize statically above
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30°C.
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Experimental set-up
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Figure 1 presents a schematic overview of the custom-built experimental set-up. A shear wave
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transducer is attached with a shear wave couplant to the bottom side of a plexiglass delay line, above
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which the aluminum sample holder (60 mm diameter) is placed. Liquid phase cocoa butter (20g) at 30°C
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is poured into the sample holder to form a layer of 7.7 to 8 mm, directly in contact with the plexiglass
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delay line. The same transducer is used for sending and receiving the signals. The delay line with a
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thickness of 15 mm creates a sufficient time delay between the signal sent by the transducer and the
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signal reflected from the interface between the plexiglass and the sample. It is important to emphasize
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that the delay line is composed of one piece in order to avoid additional (irrelevant) reflections that have
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nothing to do with the crystallization process. To allow adequate temperature control, which is essential
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for fat crystallization applications and to minimize temperature influences on the ultrasonic parameters,
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the sample holder was surrounded by a container filled with water and the experiments were carried out
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in a temperature controlled room. The water was kept at constant temperature by refreshing it, using a
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pumping system (Masterflex L/S, Metrohm, Belgium), with water from a cryostat (RC6 LAUDA,
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Lauda-Königshofen, Germany) at a fixed temperature. The temperature of the water bath and of the fat
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sample was logged by two 80TK thermocouple modules (N.V. Fluke Belgium S.A, Gent, Belgium).
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Ultrasonic measurements
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A PXI-5412 arbitrary waveform generator card (National Instruments, Austin, Texas, USA) was used
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to produce an electrical signal which was transferred to a shear wave transducer (V154-RB, 12.7 mm
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active diameter, 2.25 MHz central frequency, Olympus Corporation, Tokyo, Japan) and subsequently
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converted into ultrasonic shear waves. Excitations were performed at 1 and 2MHz whereby the signal
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consisted of a sinusoidal burst containing 10 and 20 cycles at the respective frequencies. The reflected
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signal from the plexiglass-sample system above the transducer was detected using the same transducer
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and registered by a PXI-5122 data acquisition card (National Instruments, Austin, Texas, USA) within
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the same PXI-chassis. To improve the signal-to-noise ratio (SNR), the received signals were averaged
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over 64 bursts. The ultrasonic measurements were fully controlled by a LabVIEW® script, which sets
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all excitation and acquisition parameters, and monitored and analyzed the acoustic responses.
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The determination of the swRC was based on a relative measurement. First, a reference measurement
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was performed with an empty sample holder. Here, the signal reflected from the interface between the
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plexiglass delay line and air was detected, corresponding to the case of total reflection because air does
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not support shear waves. Subsequently, the cocoa butter at 30°C was poured into the sample holder and
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statically and quickly cooled to the pre-set crystallization temperature (18°C or 20°C). Next, the signal
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reflected from the plexiglass - cocoa butter system was monitored every 10s during the quasi-isothermal
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crystallization process. At each instance, the current signal was processed to obtain the swRC of the
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composite sample by calculating the ratio of the maximum amplitude of the frequency spectrum of the
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current signal to the reference signal, as expressed in Equation 1.
swRC 
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max(FFT(sample signal))
max(FFT(reference signal))
(1)
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All measurements were carried out at least in quadruplicate.
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Theoretical model
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The analytical equations for the theoretical swRC in different configurations of layered systems (single
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or multiple interfaces) were determined using the symbolic Maple® software. Since normal incidence is
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considered, the expressions remain quite simple (see next section for more details). The analytical
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formulations were then transferred to Matlab®, and a Matlab® script was developed to simulate the
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evolution of the swRC during crystallization for any set of experimental parameters defining the density,
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velocity, attenuation, crystallization rate, etc.
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3. Results and Discussion
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3.1 Theoretical considerations
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Before discussing the characteristic results of monitoring fat crystallization with the ultrasonic shear
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reflection technique, the propagation and reflection of shear waves through and from parallel interfaces
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under normal incidence will be briefly reviewed from a theoretical point of view. This will provide more
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insight into the typical observations that can be expected from the measurement technique, and ensures
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that the experimental results will be correctly interpreted.
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To explain the different situations, we distinguish between three cases, as illustrated in Figure 2:
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
Case A: In case of a liquid (e.g. non-crystallized cocoa butter) or gas (e.g. air) layer above the
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plexiglass plate, the determination of the swRC is very simple: as the weak molecular bonds of
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liquids and gasses does not allow propagation of shear waves, this results in total reflection and
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a swRC magnitude of 1.
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
Case B: In case of an infinitely thick solid layer (e.g. crystallized cocoa butter) above the
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plexiglass plate, continuity of the shear displacement ux and of the shear stress xz at the
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plexiglass-solid interface should be required. As a consequence, the swRC becomes a function
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of the (complex) acoustic impedances of the solid and the plexiglass, as expressed in Equation
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2 (McClements, 1997):
swRC 
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Ar Z s2  Z s1 2vs2 (1  ias2 )  1vs1


Ai Z s2  Z s1 2vs2 (1  ias2 )  1vs1
(2)
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where Ai (m) and Ar (m) are the amplitudes of the incident and reflected waves respectively, Zs
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(kg/(m2s)) is the shear wave acoustic impedance, kg/m3) is the density, vs (m/s) is the shear
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ultrasonic velocity, and as (-) a parameter controlling the shear ultrasonic attenuation.
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Introducing  as the angular frequency (rad/s), which is equal to 2*frequency (Hz), the shear
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ultrasonic attenuation is defined as  asvsNepers/m). The subscripts 1 and 2 correspond to
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medium 1 (plexiglass) and medium 2 (infinitely thick solid layer) respectively. The plexiglass
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delay line is considered to be non-attenuative.
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It is important to note that the very same expression holds for a sufficiently thick layer of a
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strongly damping solid for which the back-reflection from its free end is not strong enough to
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reach the solid-plexiglass interface with a non-zero amplitude.
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
Case C: When the material above the plexiglass is a solidified layer (e.g. crystallized cocoa
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butter) with a finite thickness and a liquid (e.g. non-crystallized cocoa butter) on top of it, the
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shear waves that are transmitted through the plexiglass-solid interface and reflected from the
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solid-liquid interface (total reflection), back-propagate towards the solid-plexiglass interface
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and interfere with the directly reflected shear wave. Obviously, the determination of the swRC
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becomes more complicated. Requiring continuity of the shear displacement ux and of the shear
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stress xz at the plexiglass-solid interface, and absence of shear stress at the interface between
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solid and liquid material (stress-free state), the swRC becomes a thickness (and therefore also a
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frequency) dependent function, which can be expressed as follows:
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



  2vs 1  ias  1vs     2vs 1  ias  1vs  exp  2i  d  exp  2  as d 
2
2
1 
2
2
1 


 vs 
 vs 2 

2


2

swRC 




  2vs 1  ias  1vs     2 vs 1  ias  1vs  exp  2i  d  exp  2  as d 
2
2
1 
2
2
1 
2


 vs 
 vs


2


2

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(3)
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where , vs and as are the same material parameters as considered in Eq.(2). The parameter d
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(m) is the thickness of the solidified layer of medium 2 and  is the angular frequency (rad/s)
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which is equal to 2*frequency (Hz).
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3.2 Experimental results








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3.2.1 Typical evolution of the swRC during crystallization
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During the fat crystallization process, the material evolves progressively from a fully liquid substance
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to mostly solidified fat. The solidification commonly starts at the outer boundaries of the sample within
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the sample holder, and progresses inwards. As a consequence, a solidified layer with gradually
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increasing thickness is formed on top of the plexiglass delay line, meaning that the propagation and
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reflectivity of shear waves will gradually change as well. Figure 3 shows a typical experimental result
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of the swRC during quasi-isothermal crystallization of cocoa butter. As outlined above, three different
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phases can be distinguished with regard to the propagation of shear waves. At the start of the monitoring,
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the magnitude of the swRC approximates 1, because the sample is fully liquid and total reflection occurs
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as shear waves cannot be transmitted in liquids (Phase 1, corresponding to Case A discussed in section
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3.1). When the layer of cocoa butter (medium 2) on top of the plexiglass (medium 1) gradually solidifies
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during the crystallization process, the transmitted pulse can travel across the plexiglass-crystallized
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cocoa butter interface. Then, as soon as the transmitted pulse inside the crystallized cocoa butter sample
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reaches the boundary between solidified cocoa butter and liquid substance, it is totally reflected and
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propagates back towards the plexiglass interface. Consequently, the reflected pulse from the boundary
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between the crystallized cocoa butter layer and the liquid cocoa butter and the reflection at the plexiglass
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- cocoa butter interface travel together towards the transducer and may interfere with each other.
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Constructive as well as destructive interference may occur, depending on the thickness of the
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crystallized fat layer and thus the phase difference between both pulses. This explains the oscillatory
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behavior of the swRC during the crystallization process (Phase 2, corresponding to Case C, section 3.1),
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and links the observed periodicity to the accumulation in time of thin layers of solidified fat with
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thicknesses in the order of magnitude of the shear wavelength, hence confirming the hypothesis of a
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gradually increasing thickness. With reference to Equation 3, this corresponds to the contribution of the
256
factor exp 2i d  / vs2 in numerator and denominator. Another interesting point of observation is that
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the amplitude of the oscillation is decreasing in time. This can be explained by the attenuation of the
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ultrasonic pulse travelling across the crystallized sample. Indeed, as the crystallized layer increases, the
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accumulated damping effect on the pulse will also increase. This means that the reflected pulse,
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generated at the boundary between the crystallized cocoa butter layer and the liquid cocoa butter,
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gradually becomes lower in amplitude, and therefore, the interference effect between the two back-
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propagating pulses becomes less and less pronounced. Eventually, when the trajectory and/or the
263
attenuation through the crystallized cocoa butter become critically large, there is no energy left in the
264
second reflected pulse and therefore no interference occurs at all: the swRC thus stagnates (Phase 3,
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corresponding to Case B, section 3.1). At this instance, the propagation of the shear waves is similar to
266
the case of an infinitely thick layer, as illustrated in Figure 2B. Indeed, with reference to Equation 3, the
267
analytical expression of the swRC simplifies to Equation 2 at a certain critical value of the product


8


268
between thickness and attenuation coefficient due to the factor exp 2 d  as2 / vs2
269
and denominator.
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To confirm the hypothesis that the oscillatory pattern is caused by constructive and destructive
271
interferences, the swRC was measured simultaneously at two different frequencies of 1 and 2 MHz
272
(Figure 4). A higher frequency means a smaller wavelength, which should lead to more frequent
273
alternations of constructive and destructive interferences. Figure 4 shows that the oscillations are indeed
274
frequency dependent with measurements at 2 MHz showing a larger number of oscillations with a
275
smaller period compared to measurements at 1 MHz, which fully supports the hypothesis of constructive
276
and destructive interferences.
277
In addition, these consecutive interferences ensure that not only information about the surface properties
278
is provided by means of the presently proposed shear reflectivity technique. Indeed, based on 4 to 5
279
visible local maxima in the measurement results at 1 MHz, and a shear wave velocity for solidified
280
cocoa butter of approximately 500 m/s, we can infer that the swRC is affected by a range of the order
281
of 1 mm inside the sample within which secondary reflections are generated at the boundary between
282
crystallized and liquid fat causing interference effects with the first reflection at the plexiglass – sample
283
boundary.
284
in numerator
3.2.2. Modelling the evolution of the swRC
285
Now that we understand the different phases of the measurement results using the shear wave reflectivity
286
technique during the crystallization process, we focus on the modelling of the time evolution. Therefore,
287
we will link the Foubert model (Foubert et al., 2002) to the previously determined theoretical swRC
288
(Equation 3), in order to introduce a natural time dependence in the analytical results. More precisely,
289
since we simplify the crystallization process as an accumulation of thin layers of solidified fat, the
290
characteristic solution of the Foubert model will be applied to quantify the evolution of the thickness of
291
the solidified layer during the crystallization process.
292
Assuming that the time dependence of the quasi-isothermal crystallization process is governed by the
293
differential equation shown in Equation 4, its algebraic solution is a sigmoidal function which can be
294
expressed as in Equation 5 (Foubert et al., 2002).
295
dy
 K  yn  y 
dt
296
y (t )  1   y01 n  1 e 1 n  Kt  1 n
(4)
1
(5)
9
297
Here, y(t) denotes the fraction crystallizable fat at time t and y0 (-) the initial fraction crystallizable fat.
298
The parameter K (1/min) represents the crystallization rate and the parameter n (-) indicates the
299
asymmetry of the sigmoidal curve. Using this solution, we can define a function S(t) which is the
300
normalized fraction crystallized fat (with values between 0 and 1) at time t:
301
S (t ) 
1
 y0  y(t ) 
y0
(6)
302
In view of a gradually increasing thickness of the solidified layer, the normalized function S(t) is used
303
to estimate the time dependence of the thickness of the crystallization layer (Equation 7).
304
305
d  t; n, y0 , K   d *. S t; n, y0 , K 
(7)
306
The parameter d* (m) in Equation 7 is linked to the thickness of the crystallization layer whereby
307
complete attenuation of the secondary pulse arises.
308
To run a forward prediction model of the evolution of the swRC for a given crystallization process,
309
Equation 7 is inserted in Equation 3, in which 1 and v1 are set to respectively 1180 kg/m³ and 1200 m/s
310
for plexiglass and in which 2 is assumed to be 920 kg/m³ for crystallized cocoa butter according to
311
Maleky and Marangoni (2011). The parameters vs2 (m/s) and as2 are the shear ultrasonic velocity and
312
shear ultrasonic attenuation parameter of crystallized cocoa butter respectively, which are assumed to
313
have a constant value at each instance and over the entire solidified layer extending between 0 and d(t).
314
Consequently, the theoretical model consists of 6 variable parameters: n, y0, K, d*, as2 and vs2.
315
Figure 5 clearly shows that the newly developed theoretical model fits the experimental results very
316
well. This suggests that the assumptions made in the model (the time evolution of the solidified layer
317
thickness according to the Foubert model and the material properties vs2 and as2 of crystallized cocoa
318
butter being constant) are allowed to simplify the full-blown crystallization process. Only in the
319
beginning one can notice a somewhat larger deviation between model and experiment, which is due to
320
a small divergence of the experimental result from the theoretical value of 1, most probably caused by
321
a temperature fluctuation in the initial equilibration phase.
322
3.2.3 Validation of the measurement technique
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In order to verify the reproducibility of the experiment, four independent measurements were performed
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at 20°C. Figure 6 shows that the first drop in the swRC (corresponding to the first occurrence of a
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destructive interference) coincides for all measurements. The small variation in the beginning of the
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measurement is due to the equilibration to the pre-set crystallization temperature. Moreover, it should
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be noted that the samples consisted of 20g cocoa butter and were independently sampled, which could
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induce a small natural variation, creating small deviations during the crystallization process. Overall,
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we can conclude that the reproducibility is acceptable.
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In order to verify the sensitivity of the technique to differences in the crystallization process, Figure 7
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presents an overlay plot of experimental measurements performed quasi-isothermally at 18°C and at
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20°C. The results clearly show that the oscillation pattern starts earlier and is damped more rapidly at
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18°C compared to 20°C. Furthermore, the end swRC is lower for the curve at 18°C compared to the
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curve at 20°C, which means that the sample at 18°C is more acoustically similar to the plexiglass delay
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line and therefore suggests that this sample is more elastic (Higher shear modulus results in a higher
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shear velocity, which gives rise to a lower value of the swRC at the interface between two semi-infinite
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solids, cfr Case 2, Eq.(2) section 3.1). These trends were observed in all measurements, but for the sake
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of clarity, only one exemplary measurement is shown. In our hypothesis of accumulating solidified
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layers with time, the faster oscillations at 18°C correspond to a more rapid formation of thin layers of
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solidified fat with thicknesses in the order of magnitude of the shear wavelength observed by faster
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constructive and destructive interferences in the swRC pattern. Therefore, it also suggest that the
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crystallization process will be faster at 18°C than at 20°C which coincides with reality. To link this
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experimental observation to the theoretical model, the influence of parameter K, representing the
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crystallization rate, was examined. Figure 8 shows the theoretical swRC for different values of K (while
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all other parameters remained constant) and the corresponding crystallization curve (displayed as the
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fraction crystallized fat, 1  y (t ) ) which was used to calculate the swRC during crystallization. It can be
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concluded that a higher crystallization rate (higher K) indeed demonstrates faster oscillations and hence
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a faster occurrence of destructive and constructive interferences. This validates the measurement
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technique and also implies that it will be possible in the future to deduce quantitative parameters of the
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crystallization process from the evolution of the swRC during crystallization by means of an inverse
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model.
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4. Conclusions
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This work presents a shear ultrasonic reflection technique as a new non-destructive technique to monitor
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the crystallization behavior of cocoa butter, offering at the same time opportunities for in-line
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application. It was found that the shear wave reflection coefficient (swRC) exhibits an oscillating and
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damped response during the quasi-isothermal crystallization of cocoa butter, which could be explained
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by the occurrence of constructive and destructive interferences caused by a secondary reflection at the
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interface between crystallized and liquid fat. The observed oscillating and damped pattern could be well
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fitted by the developed theoretical model, coupling the analytical expression for the swRC of a bilayered
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system to the Foubert crystallization model. The measurement technique showed an acceptable
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reproducibility and was sensitive to differences in the crystallization process, induced by different
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crystallization temperatures. In future work, relevant ultrasonic parameters will be derived from the
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measurements by an inverse model.
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Acknowledgments
This work was funded by the Fund for Scientific Research-Flanders, Belgium (FWO) by a scholarship
for Annelien Rigolle and the KU Leuven BOF STRT1/10/015 project. We are grateful to Barry
Callebaut for the supply of cocoa butter.
Conflict of Interest
The authors declare that they have no conflict of interest.
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Fig. 1 Schematic representation of the experimental set-up with the sample holder placed into a water
bath connected to a cryostat for temperature control by means of a pumping system. The ultrasonic
transducer transmits and receives shear waves on the bottom side of a plexiglass plate which is in direct
contact with the sample.
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Fig. 2 The propagation of shear waves in different configurations of layered systems under normal
incidence. A) liquid or gas layer above a plexiglass plate B) infinitely thick solid layer above a plexiglass
plate C) solidified layer with a finite thickness and a liquid layer on top of it above a plexiglass plate.
Continuity conditions at the interfaces are expressed with respect to the displacement in the x-direction
ux and the resulting stress in the xy-plane xz (perpendicular to the z-direction). Ai denotes the incident
shear wave and Ar the reflected shear wave in medium 1. S+ and S- indicate respectively the upgoing and
downgoing shear waves in medium 2.
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Fig. 3 Typical pattern of the evolution of the ultrasonic swRC in function of time (min) during the quasiisothermal crystallization of cocoa butter at 20°C, measured at 1 MHz.
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Fig. 4 Overlay of the evolution of the swRC in time (min) during the quasi-isothermal crystallization of
cocoa butter at 20°C measured at two different frequencies (1 MHz and 2 MHz). The two measurements
were simultaneously performed on the same sample.
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Fig. 5 Overlay of an experimental and model result of the evolution of the swRC in function of time
(min) during the quasi-isothermal crystallization of cocoa butter.
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Fig. 6 Overlay of the evolution of the swRC in time (min) during the quasi-isothermal crystallization of
cocoa butter at 20°C for 4 independent samples measured at 1 MHz.
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Fig. 7 Overlay of the evolution of the swRC in time (min) during the quasi-isothermal crystallization of
cocoa butter measured at 18°C and 20°C. Both measurements were carried out on independent samples
and a frequency of 1 MHz was used.
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Fig. 8 Representation of the influence of the crystallization rate parameter K on the crystallization curve
(displayed as the fraction crystallized fat, 1  y (t ) ) and on the swRC as function of the crystallization
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time (min). The parameter K is varied between 0.004; 0.005 and 0.006. The other model parameters are
fixed: n=8; y0=0.98; d*=0.002; vs2=600; as2=0.15. The shear wave frequency equals 1MHz.
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