Least squares fitting-polynomials for determining

Powder Technology 342 (2019) 829–839

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Least squares fitting-polynomials for determining inflection points in adsorption isotherms of spray-dried açaí juice (Euterpe oleracea Mart.) and soy sauce powders Yunia Verónica García-Tejeda a,⁎, Víctor Barrera-Figueroa b a

Instituto Politécnico Nacional. Av. Luis Enrique Erro S/N, Gustavo A. Madero, Ciudad de Mexico C.P. 07738, Mexico Instituto Politécnico Nacional, Sección de Estudios de Posgrado e Investigación, UPIITA, Avenida Instituto Politécnico Nacional No. 2580, Col. Barrio la Laguna Ticomán, Gustavo A. Madero, Ciudad de Mexico C.P. 07340, Mexico

b

a r t i c l e

i n f o

Article history: Received 14 July 2018 Received in revised form 24 September 2018 Accepted 27 October 2018 Available online 30 October 2018 PACS: 05.70.Np64.70.Pf Keywords: Least Squares Fitting-polynomial adsorption isotherms glass transition temperature critical water activity. PACS: 05.70.Np64.70.Pf

a b s t r a c t The modelling of adsorption isotherms is an important method for analysing the stability of spray-dried powders. These properties are intimately related with the inflection points of the isotherm curve. The accurate calculation of the inflection points provides a method to determine the critical water activity (RHc) of powders. Though the inflection points can be estimated from standard models such as BET or GAB, in the present work we employ Least Squares Fitting-Polynomials for representing the adsorption isotherms, which in turn leads to a minimum error in the calculation of the inflection points. Then we investigate the physicochemical sense of the inflection points thus calculated for spray-dried açaí juice (Euterpe oleracea Mart.) and soy sauce powders and compare with the results reported in literature. Here we observe that from the LSF-polynomials, the inflection points may give an approximation of the value of the monolayer moisture content, and the critical water activity. © 2018 Elsevier B.V. All rights reserved.

1. Introduction The spray-drying of bioactive compounds gives rise to some issues owing to their high content of sugars and organic acids, which have low glass transition temperatures (Tg). The parameter Tg is defined as the temperature at which an amorphous system changes from a glassy to a rubbery state [1]. Above Tg some physical changes can be observed in biopolymers such as agglomeration, stickiness, collapsing, crystallization, etc. [2]. Some biochemical changes observed in microparticles containing bioactive compounds include enzyme inactivation, non-enzymatic browning, vitamin inactivation, protein denaturation, oxidation, and sucrose hydrolysis [3]. The usage of a carrier agent during the drying process increases the value of Tg, which in turn reduces stickiness, and produces free-flowing powders with an improved handling [4]. The main carrier agents used for the encapsulation of bioactives are: carbohydrates, gums, proteins, fibers and mixtures of them [5]. Carrier agents offer protection to bioactives, which otherwise could be influenced by adverse factors [6,7]. The choice of a carrier agent for encapsulating bioactive compounds is based on different criteria such as its ⁎ Corresponding author. E-mail address: [email protected] (Y.V. García-Tejeda).

https://doi.org/10.1016/j.powtec.2018.10.058 0032-5910/© 2018 Elsevier B.V. All rights reserved.

moisture sorption properties. Several authors agree that for spray drying the ideal carrier agent should have low viscosity at high solid levels, as well as high solubility, film forming ability, emulsifying properties [8], extended shelf-life, etc. [9]. Gum Arabic (GA) is a material commonly used as carrier agent due to its good retention of bioactives. Since GA is an expensive material, it is usually mixed with starch derivatives or proteins in order to cheapen the process. Maltodextrins of different degree of Dextrose Equivalent (DE) are cheaper than GA. DE is a measure of the hydrolysis of starch for reducing sugars, and it is defined as the percentage of anhydrous dextrose of the total dry substance. Pure dextrose has a DE equal to 100%, while in starch DE equals 0% [10]. From the adsorption isotherms it is possible to determine some thermodynamic parameters of materials [11], such as isosteric heat of sorption, differential entropy, differential enthalpy, critical water activity (RHc), among others. In spray-dried powders, RHc is determined from their equilibrium moisture content M and T g , at different values of water activity a w , at a given temperature. The value of a w in microparticles containing bioactive compounds allows an estimation of the microbial growth, lipid oxidation, enzymatic activity, and the texture of biopolymers [2]. A good fitting of an adsorption model implies, among other things, a low mathematical error associated with the physicochemical characteristics

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calculated from the model, such as RHc and M. For example, the GAB model presented a better fitting than Oswin, Smith, Caurie, and Iglesias and Chirife models in adsorption isotherms of spray dried tamarind pulp powder stored in three different packaging materials, and describes more accurately the water adsorption characteristics of the microparticles [12]. Least Squares Fitting-Polynomials (LSF-polynomials for short) are a good option for a better fitting in adsorption isotherms of spray-dried powders, and they have been used in several branches of applied sciences. For instance, in the work [13] it has been studied the use of LSF-polynomials for the assessment of direct image processing methods to measure the apparent contact angle of liquid drops. On the other hand, in the fields of computer vision and edge detection, LSF-polynomials were used in [14] for measuring the contact angle on micro-patterned surface using sessile drop shape fit profile detection. For decomposition of powder diffraction patterns and profile analysis of pair correlation functions a least-squares algorithm were developed in [15]. This algorithm leads to determining the individual constituents of overlapping peaks of X-ray diffraction patterns of powder materials. In the field of chemical thermodynamics, in [16] a LSF analysis was used for evaluating solubility data of proline–leucine dipeptide in water and in aqueous NaCl solutions. LSF algorithms have also been employed in other fields in which a reduced set of data points is available, see, e.g. [17–19]. To our best knowledge, there exist no publications about the use of LSF-polynomials for fitting adsorption isotherms of powders. In the present work it is highlighted the importance of having a good fitting between a model that represents the adsorption isotherm and the experimental data. The main objective of this paper consists in the accurate calculation of RHc, and the monocape of the adsorbent (denoted by M0) from adsorption isotherms of spray-dried açaí juice (Euterpe oleracea Mart.) and soy sauce powders modelled by LSFpolynomials, and to provide some physicochemical sense to the inflection points calculated from the LSF-isotherms. The outline of this paper is as follows. In Section2 the materials and experimental are described. In Section3 the theoretical background related to the models used for the adsorption isotherms is presented. In this section the LSF-polynomials are applied for the determination of the inflection points in adsorption isotherms of spray-dried açaí juice [20], and soy sauce powders [21]. In Section 6 a discussion of the obtained results is made, and a physicochemical sense to the inflection points calculated from LSFpolynomials is provided. Finally, in Section 7 some concluding remarks are shown.

2. Materials and experimental 2.1. Materials Previously reported values of equilibrium moisture content (M) for soy sauce [21] and açaí juice [20] microparticles prepared by the spray drying process with different encapsulating materials were considered for the analysis. Açaí juice was obtained from fruits harvested in the city of Igarapé-Miri (Pará, Brazil); its composition included 97.14% of moisture, 0.42% of proteins, 1.6% of total sugars, and 0.3% of lipids, with a solid content of approximately 14%. Naturally brewed soy sauce was obtained from Kikkoman Pte. Ltd. (Singapore); its composition included 10.3% of proteins, 8.1% of carbohydrates, 16.5% of NaCl, and the rest water. Three types of maltodextrins (MD) with different DE content were used as carrier agents for soy sauce, namely, maltodextrin with 5% DE (MD5DE), maltodextrin with 10% DE (MD10DE), and maltodextrin with 15% DE (MD15DE). For encapsulating açaí juice the following carrier agents were used: MD10DE, MD20DE (i.e., maltodextrin with 20% DE), tapioca starch (TS), and GA.

2.2. Spray drying The drying of açaí juice [20] was carried out in a pilot-scale spray dryer Mobile Minor™ (GEA, China). The dimensions of the drying chamber included 62 cm of cylindrical height, 80 cm of diameter and a 60° conical base. The carrier agents were then added to the açaí juice, in a concentration of 6% (w/w), the solutions were fed to a twofluid nozzle (diameter, 1 mm) using a peristaltic pump (Masterflex, USA). The spray dryer was operated in a co-current airflow mode. The inlet temperature of drying air was at 185°C, and its outlet temperature was maintained at 85°C by adjusting the feed flow rate via the peristaltic pump. The compression air pressure for atomization was controlled at 0.2 MPa, with an airflow rate of 4 m3/h. The drying of soy sauce [21] was carried out in a laboratory scale spray dryer LabPlant SD-05 (Huddersfield, England) with a 0.15 cm diameter nozzle, and main spray chamber of 50 × 21.5 cm. Maltodextrins (i.e. MD5DE, MD10DE and MD15DE) were dissolved into soy sauce to form a viscous fluid. Two solutions that contained 20% and 40% maltodextrin were prepared for each of the three types of maltodextrins. Solutions were fed into the main chamber through a peristaltic pump, one at time. The drying air flow rate was 74 m3/h and compressor air pressure was 0.06 MPa. The feed flow rate used was 15 g/min, inlet and outlet air temperature were 140 ± 2°C and 78 ± 2°C. 2.3. Adsorption isotherms Experimental data of equilibrium moisture content were determined by the static gravimetric technique [20,21], which consists in placing the powders into sealed flasks containing saturated salt slurries at different aw, at 25°C. Approximately 1-2 g of powders were put into aluminium dishes, which were placed into jars with clamp lids of 10 cm diameter, each containing one of the following saturated solutions: LiCl, CH3CO2K, MgCl2, K2CO3, Mg(NO3)2, NaCl, KCl, BaCl. The corresponding values of aw for the different solutions were: 0.113, 0.226, 0.328, 0.432, 0.529, 0.689, 0.753, and 0.843, respectively. The powders into the different jars were equilibrated up to a constant weight. This process took about three weeks for each jar. The values of Tg of powders equilibrated as described above were determined by Differential Scanning Calorimetry (DSC) as the midpoint in proximity of change of their specific heats (ΔCp) [22]. 3. Fitting models for adsorption isotherms It is usual to analyse the water adsorption characteristics of materials by means of mathematical models, which are derived from empirical and/or theoretical criteria. Some models are commonly used to describe the water adsorption in microparticles. Among the models found in the literature, Brunauer-Emmett-Teller (BET) [23], and GuggenheimAnderson de Boer (GAB) [24] are the most used for studying biopolymer microparticles. In this section we provide a brief review of the most common models for representing the adsorption isotherms of microparticles. 3.1. Mean absolute percentage error model The fitting of isotherm models to the experimental data can be evaluated by means of the mean absolute percentage error that is defined as follows [25].

P¼

N  0  100 X Y i −Y i ; N i¼1  Y i 

ð1Þ

where N is the number of data points, Yi denotes the experimental data, and Yi′ is the forecast value obtained from the model. The mean absolute percentage error has been widely used as a measure of the forecast

Y.V. García-Tejeda, V. Barrera-Figueroa / Powder Technology 342 (2019) 829–839

accuracy for models, and has been recommended in most textbooks for its reliability [26,27]. 3.2. BET model According to the BET theory of adsorption, the water present in the absorbent is absorbed layer-wise in some active sites. The monomolecular film of water M0 predicted by this theory at the active sites is firmly bound to the macromolecular constituents of the substances [23]. The BET model is described by the formula MBET ðaw Þ ¼

M0 C BET aw ; ð1−aw Þð1−aw þ C BET aw Þ

ð2Þ

where M = MBET(aw) (g water per 100 g of dry solids) is the equilibrium moisture content, M0 (g water per 100 g of dry solids) is the monolayer moisture content, and CBET is a parameter related to the sorption energy. The BET model is only applicable over the range 0.04 ≤ aw ≤ 0.5 [28], since the isotherm fits well with an approximate linear behaviour having a low slope. However, if aw N 0.5 the isotherm suddenly grows and diverges from the BET model, losing the fitting. 3.3. GAB model The GAB model is the most used expression for modelling the adsorption isotherms of spray-dried microparticles, and it is considered a fundamental model by a number of authors for the characterization of water adsorption in biopolymers. Moreover, it is accepted as the most versatile model available [11]. The GAB model is defined by the formula MGAB ðaw Þ ¼

M0 C GAB Kaw ; ð1−Kaw Þð1−Kaw þ C GAB Kaw Þ

ð3Þ

where M = MGAB(aw) is the moisture content in the sample (g water per 100 g of dry solids), M0 is the monolayer moisture content (g water per 100 g of dry solids), CGAB is a Guggenheim parameter, and K is a dimensionless parameter. The GAB model is applicable in a wider range 0.1 ≤ aw ≤ 0.9. The GAB model introduces a second well differentiated adsorption stage resulting of the addition of an extra degree of freedom (the K parameter) in its formula. This may give a better fitting to the experimental data in some cases. 3.3.1. Analysis of GAB model applied to adsorption isotherms The GAB model can be used to predict the monolayer moisture content M0. This value is closely linked to the physicochemical stability of spray-dried microparticles. In edible materials M0 has a direct influence on the lipid oxidation, enzyme activity, non-enzymatic browning, flavour preservation, and product structure [29]. Some researches have shown a correlation between M0 and the number of polar groups. For instance, in wheat starch it is reported that M0 = 0.55 mol, and the corresponding polar group number (that is, one water molecule per anhydroglucose monomer) is 0.62 [24]. As another example, consider the β-cyclodextrin polymer that has only 7 glucose units bounded by α-1,4-bounds; β-cyclodextrin is linked to 12 water molecules that form a dodecahydrate. If it is used as a wall material for encapsulating limonene, then M0 is 12.36 mol of water absorbed [30]. The parameter CGAB leads to a classification of adsorption isotherms [31]. If CGAB N 2 the adsorption isotherm has an inflection point in the curve. This corresponds to an isotherm of the type II. When CGAB b 2, the inflection point moves to the fictitious space of negative values of aw [32]. This corresponds to an isotherm of the type III, according to the Brunauers classification [33]; adsorption isotherms of the types IV and V are characteristic of vapour adsorption by capillary condensation into small adsorbent pores, but are not considered in the present paper. In the range 5.67 ≤ CGAB b ∞ the value of M0 calculated from the isotherm has an error not greater than 15.5% [34]. In general, the adsorption

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isotherms of dried powders fall into the classification of the type II, since these possess the typical sigmoidal shape. The parameter K of the GAB model determines the rate of growth of the isotherm curve for the higher values of aw. In order for the error P to be lower than 15.5%, the value of K should lie in the allowed interval 0.24 ≤ K ≤ 1.00. The values of K N 1 are not allowed [34]. If K lies in the allowed interval, it is observed that the lower values of K give the larger values of aw at which the inflection points occur. 4. Stability of powders by the use of Tg and moisture adsorption isotherms Determining the moisture sorption at different water activities can be used for analysing the stability of encapsulated materials. The structural stability of microparticles is highly influenced by their moisture adsorption characteristics and the glass transition temperature. 4.1. Glass transition temperature The glass transition temperature Tg is defined as a second-order phase change temperature at which a solid is transformed to a liquidlike “rubber” [1]. Once the temperature of amorphous powders exceeds the glass transition, viscous flow appears and cake strength increases [35]. The conditions that cause stickiness or cohesion in an amorphous material are related to Tg. According to [36], the sticky-point temperature coincides with Tg. Glass transition temperature Tg is accompanied by changes in enthalpy, dielectric properties, and viscosity, among others. In addition, Tg is related to the adsorption of water in both the surface and the bulk of the amorphous sample [37], so that the lower the value of Tg the greater the absorbed water. The value of Tg for microparticles depends on the corresponding values of Tg for both the wall material and the encapsulated bioactive. Bioactive compounds such as fruit juices, possess a high content of sugars and organic acids. For example, the main components in orange juice are the fructose, glucose and citric acid. In a pure, dry state the values of Tg for these components are 5, 31 and 16°C, respectively. [38] produced spray dried orange juice with Tg = 5.7°C, but when orange juice is mixed with MD DE 6 the value increased to Tg = 167.9°CC. 4.2. Deliquescence of powders Water adsorption of powders often results in the loss of powder flowability due to the agglomeration of particles. This phenomenon is called caking, and begins by the formation of crystals when powders are exposed to water vapour wit a pressure higher than the water vapour pressure of saturated solutions of solids, at a given temperature [39]. The powders from rosemary essential oil [40] encapsulated by spray drying with GA or a mixture of GA/inulin were subjected to different humidity levels at 25°C. The physical characteristics of powders prepared with GA are shown in Fig. 7a. Here we observe the agglomeration of powders at aw = 0.64. Apparently, no phase change is observed in the particles produced with inulin. A similar observation has been reported for freeze dried beetroot extracts [41] encapsulated in MD, GA, succinylated starch (n-OSA), and chitosan. According to Fig. 7b, beetroot microencapsulated with MD at aw = 0.66 was in a liquid form with high viscosity, while microparticles with GA presented a similar effect at aw = 0.82 [42]. In general, powders stored at low values of aw remain as free-flowing powders [2]. Under this condition, a small amount of water is adsorbed by the surface of the material, owing to the available discrete hydrogen binding sites, and also to the accessible interstitial spaces of the dense, rigid glassy material [37]. Approximately, at aw ≈ 0.5 the material continues adsorbing water (bulk water), which gives rise to a sharp inflection point in the isotherm curve [43]. At aw ≈ 0.6 the morphology of powders is altered, producing the swelling of the biopolymer matrix. At aw = 0.7 the initial structure of the microparticles

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is completely lost because of the dissolution of the wall material, which leads to an extensive fusing of the microparticles that avoids their rehydration, see Fig. 1. 4.3. The Gordon-Taylor model A state diagram of Tg characterizes the state changes of a material as a function of the temperature, moisture or content of solids. State diagrams of materials can be used for selecting suitable conditions of temperature and moisture content for processing powders [44], and for predicting the stability of spray-dried powders during storage. The Gordon-Taylor model [45] analyses the plasticization effect of water on microparticles. It is described by the formula Tg ¼

w1 T g1 þ kw2 T g2 ; w1 þ kw2

where Tg is the glass transition temperature of a mixture of solids and water, w1 is the anhydrous fraction having a glass transition temperature Tg 1, w2 is the water fraction having a glass transition temperature Tg 2, which is often taken as −135∘C, corresponding to pure water; and k is a parameter [20]. If k = 1, the relationship between Tg and the anhydrous fraction w1 is linear, resulting in a straight line plotted in the w1-Tg plane. If k N 1 the resulting plot is concave, while if k b 1 the plot is convex [46].

4.4. A classical method for determining the critical water activity (RHc) The adsorption isotherms of powders and the Gordon-Taylor model representing their corresponding glass transition temperatures Tg can be used to determine the critical conditions for storing the powders. Let Tg = Tg(aw) and M = M(aw) be two functions depending on the water activity aw. The first one represents the glass transition temperature, and the second is the moisture equilibrium content at aw. Let the plots of these functions share the same horizontal axis aw. Let RHc be the value of aw at which Tg(RHc) = 25°C. Here RHc represents the critical aw at which the glass transition occurs at room temperature. Graphically, RHc is the abscissa of the intersection point of a horizontal line at 25°C and the curve Tg(aw), see Fig. 2. Now, consider a vertical line at aw = RHc and its intersection with the curve M(aw). The ordinate of such intersection point is the absorbed moisture M.

5. Polynomial interpolation and least squares fitting polynomials Interpolation is a well known mathematical method that provides an approximate continuous representation of certain phenomenon from a discrete set of samples. The interpolating function that approximates the phenomenon must pass through the data points, so that in between a pair of adjacent points the interpolating function gives some value that does not necessarily have the lower error.

Fig. 1. Physical characteristics of spray dried rosemary (A) and freeze dried beetroot (B) microparticles produced with different carrier agents at different relative humidities during the determination of the moisture adsorption isotherms. Adapted from [40,42]

Y.V. García-Tejeda, V. Barrera-Figueroa / Powder Technology 342 (2019) 829–839

833

which were experimentally obtained. Let R2 ¼

N h  i2 X yi − a0 þ a1 xi þ ⋯ þ ak xki i¼1

be the residual to be minimized, that is, the sum of the squares of the vertical offsets between the data points and the polynomial y. In the minimization process we take the partial derivatives of R2 with respect to the coefficients ai, that is   ∂ R2

Fig. 2. Variation of the glass transition temperature and water activity with equilibrium moisture content. Arrows indicate the RHc(aw) value and moisture M.

When using polynomials as interpolating functions, the degree of the interpolating polynomial depends on the number of data points. More precisely, if N data points are to be interpolated (where N N 1), then the degree of the interpolating polynomials is exactly N − 1. One may expect that increasing the number of data points could reduce the error between the interpolating polynomial and the function representing the phenomenon. However, if the data points are equally spaced then larger the number of data points, the larger the error of interpolation. This error is presented as an oscillation in the interpolating polynomial whose amplitude rapidly increases near the ends of the interval in which the independent variable lives. This is the well-known Runge phenomenon [47] that accounts for the divergence of the interpolating polynomial near the ends of the interval. One way to deal with this drawback when treating with a large set of data points is to take samples of the phenomenon at non-equidistant points. More specifically, if the samples are taken according to the roots of Tchebishev or Legendre polynomials the error associated with the interpolating polynomial near the ends of the interval will reduce [48]. Another way consists in using piecewise polynomials. In this method, low-degree polynomials are used for each pair of adjacent data points, so that the ends of the polynomial coincide with the pair of data points. This give a continuous representation for the approximate interpolating function. In order for this representation to be smooth, the coefficients of the polynomials are chosen so that the derivatives of the interpolating polynomials coincide with the derivative of the interpolated (original) function at the data points. Smoother interpolating polynomials can be obtained by securing the continuity of the derivatives of higher order. This method is often called spline interpolation [49]. Given a set of data points, it is possible to construct a best-fitting curve by minimizing the sum of the squares of the offsets of the points to the curve. The offsets can be vertical of perpendicular to the curve, though in practice vertical offsets are frequently minimized [50]. If the fitting curves are polynomials these are called least squares fittingpolynomials (LSF-polynomials for short). Note that LSF-polynomials do not necessarily pass through the data points as interpolating polynomials do. Moreover, the degree of a LSF-polynomial can be much lower than the number of data points. Let

y ¼ a0 þ a1 x þ ⋯ þ ak xk

be a polynomial of degree k depending on the independent variable x. The polynomial coefficients ai (i = 0, 1, ⋯, k) are determined such that the polynomial y best fits a set of N data points of the form (xi, yi),

N h  i X 1 ¼ −2 yi − a0 þ a1 xi þ ⋯ þ ak xki ; ∂a i¼1  0 N h  i ∂ R2 X ¼ −2 yi − a0 þ a1 xi þ ⋯ þ ak xki xi ; ∂a1 i¼1 ⋮   2 N h  i ∂ R X ¼ −2 yi − a0 þ a1 xi þ ⋯ þ ak xki xki : ∂ak i¼1

By equating these equations to zero we obtain a system of equations for the polynomial coefficients ai (i = 0, 1, ⋯, k), that is a0 N þ a1 a0

a0

N X

xi þ a1

N X i¼1 N X

i¼1

i¼1

N X

N X

xki þ a1

i¼1

xi þ ⋯ þ ak x2i þ ⋯ þ ak ⋮

N X i¼1 N X

N X

yi ;

i¼1

xkþ1 ¼ i

i¼1

xkþ1 þ ⋯ þ ak i

i¼1

xki ¼

N X

N X

xi yi ;

i¼1

x2k i ¼

i¼1

N X

xki yi :

i¼1

Let us introduce the matrices 2

3 a0 6 a1 7 7 a¼6 4 ⋮ 5; ak

2

3 y1 6 y2 7 7 y¼6 4 ⋮ 5; yN

2

1 6 x1 6 X¼4 ⋮ xk1

1 x2 ⋮ xk2

3 ⋯ 1 ⋯ xN 7 7; ⋱ ⋮ 5 k ⋯ xN

where a and y are column vectors representing the polynomial coefficients, and the ordinates of the data points, respectively, and X is the (k + 1) × N Vandermonde matrix associated with the data points. Then previous system of equations reads XΤ Xa ¼ XΤ y where XΤ denotes the transpose of matrix X. Hence, the polynomial coefficients are given by the equation  −1 XΤ y: a ¼ XΤ X By using standard mathematical software, e.g., Wolfram Mathematica ® (Illinois, U. S. A.) or MatLab ® (Massachusetts, U. S. A.), it is possible to calculate the polynomial coefficients with instructions already available. LSF-polynomials best fit to a given set of data points according to the procedure above indicated, however this technique have some drawbacks when data points are ill-estimated. For instance, when data points are not normally distributed, LSF-polynomials may lead to unreliable results. In this sense, LSF-polynomials are sensitive to outliers, so more robust technique should be used in such cases (see, e.g. [51–53]). In addition, LSF techniques have a tendency to overfit data, for which LASSO or Ridge regression might be advantageous (see, e.g., [54,55])

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5.1. A new approach for determining RHc and M0 by means of LSF-polynomials When modelling moisture adsorption isotherms by means of LSFpolynomials it is possible to minimize the fitting error by increasing the degree k of the polynomial instead of increasing the number of data points. These allows to obtain the best-fitting approximation when the number of moisture adsorption samples is scarce. We observe that the fitting error associated with LSF-polynomials is smaller than the corresponding errors obtained with classical models such as GAB o BET. It is worth mentioning that polynomial models have been barely applied in biopolymers. To the best of our knowledge, the works of [37,43,56–58] seem to be the only that use polynomials for modelling adsorption isotherms, though the employed polynomials are not of the LSF kind. These works consider large sets of experimental data generated by the Dewpoint Isotherm method (DDI) of Decagon Devices (Pullman, WA, USA), and show a existing relation between the inflection point of an isotherm and the value of Tg. In the present work we found a simple way to determine the values of RHc and M0 in the selected powders from the calculation of the inflection points of the corresponding adsorption isotherms. The essence of our approach consists in constructing a LSF-polynomial from a reduced set of data points and then calculating the inflection points from its second derivative. The data points can be obtained from an experimental setting using desiccators in case a DDI device is not available in laboratory. The usage of LSF-polynomials is justified mainly in two ways, namely, they provide a polynomial expression with a controlled error so that the calculations based on it carry small errors as well; and the resulting expression is simple enough so the calculation of its second derivative is almost trivial. Moreover, no DSC analysis is needed, that is, no experimental data of Tg(aw) are necessary. Let Mk ðaw Þ ¼ b0 þ b1 aw þ ⋯ þ bk akw be a LSF-polynomial of degree k depending on the real variable aw. This polynomial is used to represent moisture M as a continuous function of the water activity aw. The polynomial coefficients bi (i = 0, 1, ⋯, k) can be calculated according the previous matrix procedure or by the instruction Fit[] of Wolfram Mathematica. This polynomial has at most k roots, which satisfy the equation Mk(aw) = 0. In addition, the polynomial Mk has at most k − 1 critical points, which satisfy the equation Mk′(aw) = 0. Moreover, the polynomial has at most k − 2 inflection points, which satisfy the equation Mk″(aw) = 0. Note that Mk″ is also another polynomial of degree k − 2 that has the form M″k ðaw Þ ¼ 2b2 þ 2  3b3 aw þ ⋯ þ kðk−1Þbk ak−2 w ; where k ≥ 2, otherwise Mk″(aw) ≡ 0. Hence, calculating the inflection points of an adsorption isotherm reduces to calculate polynomial roots, which can be determined numerically, for instance, with the instruction Roots[] of Wolfram Mathematica®. We stress that the inflection points can also be calculated from the BET or GAB model, by calculating the second derivative of Formula (2) or (3), respectively, and calculating the zeros of the resulting expressions equated to zero. Indeed, the second derivatives of BET (2) and GAB (3) models are given by the formulas

M″BET ðaw Þ ¼ −

M″GAB ðaw Þ

¼−

  2C BET M0 2 þ 3aw ðC BET −1Þ þ a3w ðC BET −1Þ2 −C BET ðaw −1Þ3 ð1 þ aw ðC BET −1ÞÞ3

;

ð4Þ

  2C GAB K 2 M0 a3w K 3 C GAB ðC GAB −2Þ þ ðaw K−1Þ2 ðaw K þ 2Þ þ C GAB ð3aw K−1Þ 3

ðaw K−1Þ ð1 þ aw ðC GAB −1ÞK Þ

3

respectively, which were obtained by using the symbolic calculus of Wolfram Mathematica®. However, the low fitting of these models to the data points will render in larger errors in the calculation of the inflection points, not to mention that calculating the zeros of the equation ″ ″ MGAB (aw) = 0 or MBET (aw) = 0 will be much more difficult than calculating polynomial roots of the polynomial Mk″(aw) = 0.

6. Results and discussion In this section we consider two examples of water adsorption isotherms of microparticles. In the first one we apply a 5-th order LSF-polynomial, while in the second we apply a 6-th order LSFpolynomial. Finally, we discuss the physical properties of the microparticles derived from these LSF-polynomials. It is worth mentioning that we are not proposing a new isotherm model, since LSF-polynomials are used only for improving the fitting and hence reducing the error of the inflection points calculated from the resulting equations. However, LSF-polynomials may lead a general model that can be applied in more general situations. This, of course, shall follow in forthcoming works. This opens an interesting path for interpreting the physicochemical parameters in terms of the inflection points of isotherms.

6.1. A 5-th order LSF-polynomial In the work of [20] açaí juice powders were encapsulated using four carrier agents, namely, maltodextrins of 10 (MD 10) and 20 (MD 20) degree of dextrose equivalent, GA, and tapioca starch (TS). For each carrier agent up to eight data points of the form (aw, M) derived from the corresponding adsorption isotherm were reported. In addition, the parameters of the GAB model (i.e., CGAB and KGAB), the values of RHc at Tg = 25°C, and the associated error PGAB were also reported, see Table 1. Here we observe that the GAB model renders large errors, ranging from 6.08% (GA) to 14.73% (TS). We can see in Figure 7 that for each carrier agent the GAB model does not fit well in the lower points (aw b 0.3), but for the upper points (aw N 0.4) the GAB model adequately fits to the experimental points.

Table 1 Parameters of the GAB model and 5-th order LSF-polynomial for açai juice powder [20] encapsulated with different carrier agents. Carrier agents Parameters

MD10DE

MD20DE

GA

TS

GAB model CGAB K M0 RHcGAB PGAB(%) awM 0a I nota1

2.83 0.962 0.05 0.574 6.29 0.39 0.15

1.51 0.981 0.063 0.535 9.05 0.45 0.32

3.07 0.996 0.053 0.571 6.08 0.37 0.17

5.75 0.98 0.032 0.554 14.73 0.30 0.24

0.080867 1.60794 −8.72373 22.2052 −25.7942 11.676 0.243 0.541 0.539 1.77

−0.118635 2.15532 −11.0873 26.8585 −30.0714 13.1348 0.262 0.556 0.577 3.00

−0.102175 1.67476 −8.14682 19.2052 −21.3561 9.26644 0.283 0.550 0.546 3.46

5-th order LSF-polynomial b0 −0.133055 b1 2.35018 b2 −12.0941 b3 28.6545 b4 −31.0343 b5 12.9133 I1b 0.264 I2 nota2 0.589 RHcPolc 0.573 PPol(%)d 1.68 a

;

b c

ð5Þ

d

Values calculated from the GAB model; these values are not reported in [12]. Inflection points calculated from the second derivative of a 5-th order LSF-polynomial. RHc determined from the inflection point I2. Error calculated as described in Formula (1).

Y.V. García-Tejeda, V. Barrera-Figueroa / Powder Technology 342 (2019) 829–839

From the experimental points reported by Tonon et al. (2009), we construct a 5-th order LSF-polynomial of the form M5 ðaw Þ ¼ b0 þ b1 aw þ b2 a2w þ b3 a3w þ b4 a4w þ b5 a5w

ð6Þ

for each carrier agent. The polynomial coefficients were calculated with the instruction Fit[] of Wolfram Mathematica®. The fitting of the 5-th order LSF-polynomials is superior to the GAB model for each carrier agent, with errors PPol lower or equal than 3.46%. For a 5th-order polynomial, at most three inflection points arise from the roots of the equation M5″(aw) = 0, which in general may be complex numbers. In our case, one real root and two complex conjugate roots were obtained for each carrier agent. We observed that the complex roots have negligible imaginary parts, so that they were conveniently treated as one real root, whose value is given approximately by the real part of these complex roots. Hence, we can say that two effective inflection points I1 and I2 result from the 5-th order LSFpolynomials representing each carrier agent. These points are depicted by arrows in Figure 3, and their values for each carrier agent are reported in Table 1.

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Let awM 0 denote the water activity at M0. This value can be easily obtained from the GAB isotherm by finding the water activity at which the equilibrium moisture content equals M0. From the LSFpolynomials, the inflection point I1 may give an approximation of the value of awM 0. This is particularly evident for the TS carrier in which I1 = 0.283 and awM 0 = 0.30, but it is not evident for the other materials. In order to support this hypothesis observe that the value of CGAB for MD 10DE, MD 20DE, and GA lies outside the required interval 5.67 ≤ CGAB b ∞, hence the error P associated to awM 0 is higher than 15.5%, as was established by Lewicki [34]. On the other hand, since the value of CGAB for TS does lie in the required interval, the error PGAB associated to awM 0 is not higher than 15.5%. In other words, the value of awM 0 from the GAB model is more accurate in TS, thereby the difference with inflection point I1 from the corresponding LSF-polynomial is smaller. This hypothesis, however, should be proved in other cases. With respect to the inflection point I2 we observe in Table 1 that their values are in agreement with the values of RHcGAB reported by [20], for the four materials. Note that the values of RHcGAB were experimentally obtained by means of DSC and adsorption isotherms, but the values of I2 come from a 5th order LSF-polynomial constructed from eight experimental data points. Therefore, there exists an intimately

Fig. 3. Adsorption isotherms of açaí juice powder produced with different carrier a gents: MD10DE, MD20DE, GA, and TS.

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A 6-th order polynomial has at most four inflection points, which can be calculated from the roots of M6″(aw) = 0. In our case, four real inflection points arose from the equation M6″(aw) = 0, which are denoted by I1 b I2 b I3 b I4, see Table 2. From the plots of Figure 4, only two inflection points are evident with a naked eye, namely, I1 and I3. We also consider the inflection point I4 that seems to be related with the solubilization of the powders, but it is not evident in the plots. The remaining inflection point (I2) is not considered at all. The possible physical sense of the selected inflection points is described below. Let MI 1 denote the moisture content calculated at the inflection point I1, that is, MI 1 = M6″(I1). We observe that the values of MI 1 are closer to the corresponding values of M0, which were calculated with the BET model. Note that results reported [21] may not accurately represent the real M0 of the soy sauce powders because moisture adsorption data were restricted to the smaller range 0.11 ≤ aw ≤ 0.529. Owing to this similarity, up to the prescribed accuracy, we can use MI 1 as an approximate value for the monolayer moisture content M0. In turn, the inflection point I3 takes values in the interval 0.35 ≤ aw ≤ 0.50 (see Table 2). This interval coincides with the absorbed bulk water (Carter and Schmidt, 2012). Hence, an estimate of RHc is given by the inflection point I3. The inflection point I4 seems to be related to the dramatic moisture adsorption at aw = 0.689, which is reported by [21]. In order to support this assertion we can see in Fig. 7b that powders prepared with MD become liquid at aw = 0.66. It is highlighted that LSF-polynomials are meant to work with few data points, as in the cases considered in this paper. On the other hand, it is always possible to calculate LSF-polynomials of orders higher than six, however the resulting inflection points may lack of the proper physical sense. Moreover, arbitrarily increasing the order of a LSF-polynomial may not reduce the associated fitting error in the same fashion. In addition, high-order polynomials have a larger number of concavities (undulations) than low-order polynomials, which would not fit the set of data points in the best way.

link between the inflection point I2 of a 5th order LSF-polynomial and the critical water activity RHc. A similar approach based on the use of the second derivative can be found in the work [56] where the Savitsky-Golay 2nd derivative curve is used for calculating the value of RHc but using DDI with extremely high data resolution (of the order of thousands of data points). Nonetheless, using few data points can give accurate values of RHc provided that the employed model would fit the experimental points with a smaller error, as is the case of using LSF-polynomial. Obtaining such few data points can be accomplished in laboratories with basic facilities. 6.2. A 6-th order LSF-polynomial It may happen that neither BET nor GAB model adequately fits the adsorption isotherms of powders. This is the case of highly hygroscopic materials, whose experimental points are not well fitted by these models, particularly at the lower values of aw, since the isotherms have relatively high slopes. Hence, the values of M0 and RHc calculated from these models may be inexact and the associated errors will be higher. For instance, in the work of [21], the adsorption isotherms of soy sauce powders showed a dramatic increase of moisture adsorption at aw ≈ 0.689. In order to have a well-fitting model [21] approached the problem piece-wisely. Namely, in the interval 0.11 ≤ aw ≤ 0.529 they used the BET model, which barely behaves linearly, while in the interval 0.53 ≤ aw ≤ 0.84 they used the GAB model. By representing the isotherm in this way it was possible to calculate M0 as usual with the BET model. These values are shown in Table 2, at column 2. By restricting the BET model to the range 0.11 ≤ aw ≤ 0.529 they were able to calculate RHc by means of DSC with a low error. In their work, [21] determined that Tg for soy sauce powders is lower than in other powders stored at similar aw conditions, which leads to an isotherm with a higher slope. The approach used in this paper for this case consists in using a 6-th order LSF-polynomial for modelling the experimental data of soy sauce powders [21], that is M6 ðaw Þ ¼ b0 þ b1 aw þ b2 a2w þ b3 a3w þ b4 a4w þ b5 a5w þ b6 a6w ;

7. Concluding remarks In this work, it is explored the usage of LSF-polynomials for calculating the inflection points of adsorption isotherms of açaí juice and soy sauce powders with a high accuracy. For the selected cases, the usage of LSF-polynomials leads to a better fitting of the experimental data than the common models, resulting in lower mean absolute errors. The possible physical sense of the inflection points calculated from the LSF-polynomials was explored. In the selected cases, the values of the monolayer moisture content M0, the critical water activity RHc, and the solubilization of the material can be derived from the calculated inflection points. For highly hygroscopic materials it is better to use 6-th order LSF-polynomials, otherwise 5-th order LSFpolynomials maybe suitable for a correct fitting of the experimental points.

ð7Þ

where aw takes values in the full interval 0.1 ≤ aw ≤ 0.84. In this case, a 6th order LSF polynomial was chosen since it yielded a better fitting for the experimental data than a 5-th order LSF polynomial. We consider seven carrier agents, namely, three types of maltodextrins with different dextrose equivalent (DE = 5, 10, and 15), each of which has a concentration of 20% and 40%; and also the soy sauce powder without MD (the without case of Table 2). The polynomial coefficients for each carrier are shown in Table 2. The error PPol associated with these polynomials ranges from 0.93% to 4.09%, as is shown in the table. In Figure 4 we observe the plots of both the LSF-polynomial and the GAB model for each carrier agent.

Table 2 Comparative analysis of the parameters RHc and M0 obtained from BET model and 6-th order LSF-polynomial. BET model

6-th order LSF-polynomial

Carrier agent

Polynomial coefficients

Without MD5DE, 20% MD10DE, 20% MD15DE, 20% MD5DE, 40% MD10DE, 40% MD15DE, 40%

Inflection points

M0

RHc

b0

b1

b2

b3

b4

b5

b6

I1

I3

I4

MI 1

RHcPol

PPol(%)

0.130 0.116 0.124 0.127 0.114 0.100 0.113

0.032 0.132 0.123 0.114 0.241 0.219 0.212

−2.02167 −1.10864 −0.958744 −1.10804 0.336624 0.343864 0.167202

43.8436 24.3457 20.9247 23.9117 −6.0044 −5.99069 −2.99643

−331.185 −189.235 −161.222 −182.2 43.9517 42.0532 24.3596

1209.25 717.587 610.965 682.276 −147.307 −130.26 −79.553

−2279.69 −1405.65 −1199.19 −1326 250.392 197.323 120.138

2130.43 1363.66 1166.88 1280.65 −205.874 −137.338 −76.3466

−774.579 −513.798 −440.681 −481.316 66.6701 35.3264 15.4883

0.203 0.196 0.198 0.201 0.216 0.220 0.192

0.500 0.484 0.474 0.476 0.348 0.373 0.354

0.762 0.740 0.743 0.741 0.747 0.821 0.787

0.156 0.087 0.092 0.102 0.061 0.070 0.071

0.098 0.110 0.111 0.111 0.193 0.220 0.195

4.094 2.877 2.761 2.886 1.275 1.241 0.931

Y.V. García-Tejeda, V. Barrera-Figueroa / Powder Technology 342 (2019) 829–839

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Fig. 4. Adsorption isotherms of soy sauce powders produced with different kinds of maltodextrins: without MD; DE5, 20%; DE10, 20%; DE15, 20%; DE5, 40%; DE10, 40%; DE15, 40%.

838

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In this work we used experimental data obtained from a gravimetric method, which provides a fewer number of points than other techniques such as the DDI method. Indeed, with only eight points it was possible to calculate the LSF-polynomials, and accurately calculate the parameters above discussed. It was shown that the usage of LSF-polynomials leads to interesting results, so it becomes a natural framework for treating a reduced number of points when modelling adsorption isotherms.

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Yunia V. García-Tejeda is Biochemical Engineer, she obtained a MSc degree in the Development of Biotic Products from the National Institute Politechnic of México (IPN).She is PhD in Material Science from the CINVESTAV-IPN (2015), where shedeveloped new materials for microencapsulation of anthocyanins.Her research focuses on the synthesis and characterization ofwall materials for encapsulation of different bioactives. She hasskills in the use of different materials characterization techniques,which give valuable information about the physicochemicalproperties of microparticles.She is Professor in the Food Academyof the ENCB-IPN.

Víctor Barrera-Figueroa was born in Mexico City in 1980. He received a Bachelor Engineering (B.Eng.) degree, and a Master of Science (M.Sc.) degree from the National Polytechnic Institute (IPN) of Mexico. Later he earned a Doctor of Philosophy (pH.D.) degree in Mathematics from the Department of Mathematics of CINVESTAV-IPN, at Quere-taro City, where he specialized in applied mathematics, among other fields. He currently works as a lecturer and researcher at UPIITA-IPN in Mexico, where he lectures sev-eral graduate courses.