*2.6. Drying Kinetics*

The kinetics of moisture loss during freeze-drying of the loaded gellan gum gels were described by five empirical models commonly employed to characterise drying kinetics in foods [15,16]: Newton, Page, Henderson and Pabis, logarithmic and Wang and Singh. Table 1 lists them all alongside their expressions.

**Table 1.** Drying kinetics models considered in this work to describe moisture loss during freeze-drying of riboflavin-loaded gellan gum gels with different pHs.


Parameter units: (*h*−*<sup>1</sup>*) for *k1*, *k3*, *k4*, *k5*; (*h*−*<sup>n</sup>*) for *k2;* (*h*−*<sup>2</sup>*) for *k6; a1, a2* and *b1* are dimensionless.

To fit the models to the experimental freeze-drying curves, the (dimensionless) moisture ratios (MRs) of the samples were calculated first from the measured water content data as follows [15]:

$$MR = \frac{X(t) - X\_{eq}}{X\_0 - X\_{eq}} \tag{2}$$

where *X(t)* is the moisture content on a dry basis for the different processing times (h), *X0* is the initial moisture content (*w*/*w* d.b.) and *Xeq* is the equilibrium moisture content (*w*/*w* d.b). The equilibrium moisture content for the dehydrated gels was calculated using the GAB model with measured water activities and parameters for gellan gums presented in [9]:

$$\frac{a\_{\rm uv}}{X\_{eq}} = 0.165 + 14.3a\_{\rm uv} - 13.2 \, a\_{\rm uv}^{-2} \tag{3}$$

For all the models in Table 2, the unknow parameters (parameters *aj* and *ki*, with *j* = 1,2 and *i* = 1, ... , (6) were estimated using regression analysis. The error (e) between the experimental (θ) and predicted (i.e., fitted) MR values (θ) [15],

$$J = \sum\_{i}^{N} c\_i^2 = \sum\_{i}^{N} \left(\theta\_i - \overline{\theta}\_i\right)^2,\tag{4}$$

was minimised for all the *i* measurements that formed the experimental data set of size *N* using a nonlinear least squares method (implemented in Matlab with tolerance <sup>10</sup>−10).


**Table 2.** Regression and goodness-of-fit results for the drying kinetics models.

Parameter units: (*h*−*<sup>1</sup>*) for *k1*, *k3*, *k4*, *k5*; (*h-n*) for *k2;* (*h*−*<sup>2</sup>*) for *k6; a1, a2* and *b1* are dimensionless.

The goodness-of-fit of each fitted model was then assessed using three statistical measures that take into account the complexity (i.e., number of parameters, *p*) of each model [20]. These were:


$$R\_{adj}^2 = 1 - \frac{N-1}{N-p}(1 - R^2) \tag{5}$$

where *R<sup>2</sup>* is the regression coefficient of determination.


$$AICC = AIC + \frac{2p(p+1)}{N-p-1} \tag{6}$$

where *AIC* is the Akaike information criterion [21,22].


$$BIC = p \ln(N) - 2 \text{ } \ln(L) \tag{7}$$

where *L* is the maximum log-likelihood of the estimated model.

The goodness of the fit (or the likelihood) can be increased by adding more parameters to the model. However, this will increase complexity and might result in overfitting (i.e., more parameters than can be estimated with the available data), all which is penalized with higher AICC and BIC values [15,20]. Therefore, the model with best performance will be the one with higher *<sup>R</sup>*<sup>2</sup>*adj* and lower *AICC* and *BIC* values [20].

#### *2.7. Kinetics and Mechanisms of Vitamin Release*

The Korsmeyer–Peppas model [17] has been used to describe release kinetics and identify delivery mechanisms. It is a semi-empirical power law that relates the fractional release of vitamin/drug to the release time [17,18]:

$$\frac{M(t)}{M\_{\odot}} = k\_{rel} t^{\eta\_{rel}} \tag{8}$$

where the *M*(*t*) and *M* ∞ are the cumulative amounts of drug released at time *t* (measured in hours *h*) and infinite time, respectively; the constant *krel* (in *h*−*nrel* units) relates to the structure and geometry of the delivery matrix (in this case, the freeze-dried gels); and the dimensionless exponent *nrel* is the release mechanism indicator. For cylindrical substrates *nrel* ≤ 0.45 defines Fickian mechanisms, while anomalous/non-Fickian delivery is described by nrel > 0.45 [17,18]. Experimental release curves were fitted to Equation (8) using a nonlinear least squares method [15], and parameters *krel* and *nrel* were estimated within 95% confidence intervals.

#### **3. Results and Discussion**

#### *3.1. E*ff*ect of pH on Moisture Losses during Freeze-Drying*

Figure 1 shows a comparison of the drying curves, in terms of the moisture ratio (MR) and freeze-drying processing times (h), for gellan gum gels at pH 2.5, 4 and 5.2 (natural) loaded with riboflavin. Gels at pH 2.5 exhibit the fastest drying rates, with most of the moisture content (MR∼ 0.25) removed during the first 6 h of the freeze-drying process. On the other hand, samples at pH 5.2 and 4 followed a very similar drying trend up to the first 4 h of processing. From this time onwards, the samples at pH 4 present a significantly slower drying rate; i.e., MR∼ 0.3 at t = 8 h compared to MR∼ 0.15 for pH 5.2 at the same time. The three samples were completely dried (i.e., free water totally removed) by the end of the freeze-drying experiments at t= 18 h, independent of their pHs.

**Figure 1.** Moisture ratio evolution along time for 2% ( *w*/*w*) gellam gums with pH 2.5 (black dots), pH 4 (blue squares) and pH 5.2 (magenta triangles) loaded with riboflavin during the conducted freeze-drying experiments.

The fastest drying rates observed for the gels at lower pHs can be explained by the e ffects of acidifying the gel solution. As reported by Cassanelli et al., [14], the acidification step both enhances ice crystal nucleation and weakens the gel structure at pH values as low as pH 2.5. The combination of these two e ffects might favour a more interconnected pore structure—i.e., more nuclei will lead to more crystals that will find lower resistance in the weak gel structure to form a network. This could lead to faster drying rates and also a ffect the strength of the rehydrated structure.

#### *3.2. Freeze-Drying Kinetics: Parameter Estimation and Model Discrimination*

Estimated parameters for all the drying models considered in this work (i.e., Newton, Page, Henderson and Pabis, logarithmic and Wang and Singh) are listed in Table 2, together with the RMSE (root mean square error) for each fitting and the results corresponding to the goodness-of-fit of each model. According to these results, the models that provide more accurate descriptions for the drying kinetics are the Page (Table Equation 1) and the Wang and Singh (Table Equation 5) models, both presenting *<sup>R</sup>*<sup>2</sup>*adj*∼ 0.99 (in average) for all pH values.

For samples at modified pHs (i.e., pH 2.5 and 4), the Wang and Singh model presents the lowest RMSE (3.52 × 10−<sup>4</sup> for pH 2.5) and the highest *<sup>R</sup>*<sup>2</sup>*adj*, while for the freeze-dried gels at pH 5.2, the Page model is the model that presents the best fitting (RMSE = 0.040 and *<sup>R</sup>*<sup>2</sup>*adj* = 0.986). This is in agreemen<sup>t</sup> with fittings reported in Cassanelli et al. [19], which showed the Page model as the best option to describe the freeze-drying kinetics of non-loaded gellan gels at natural pH.

The goodness-of-fit for all models is illustrated in Figure 2, where experimental values are plotted against predicted moisture ratios for each drying model at all pH studied. This graph also shows the accuracy of the Page and Wang models, for which most of the predicted points lie on the correlation line (see Figure 2b,d).

**Figure 2.** Correlation between predicted and experimental moisture contents for freeze-dried 2% (w/w) gellan gum samples for: (**a**) Newton model (Table Equation 1), (**b**) Page model (Table Equation 2), (**c**) Henderson and Pabis model (Table Equation 3) and (**d**) Wang model (Table Equation 5).

When comparing models with similar accuracies, the *AICC* criterion constitutes the best measure to discriminate models, with more negative *AICC* values indicating better model performances. According to this, if the Newton and Page models were compared at pH 4—the pH at which both models show very similar RMSE, *<sup>R</sup>*<sup>2</sup>*adj* and *BIC*—the lower AICC (−21.39) of the Newton model would make it the preferred one. This criterion is also an indicator of the complexity (e.g., number of parameters) of the

assessed models—the Newton model involves a single parameter (*k1*), compared to the two needed in the Page model (*k2, n*). On the other hand, the logarithmic model (Table Equation 4), with the highest number of parameters considered (*p* = 3), presents the less negative *AICC* values at each pH.

#### Effect of pH on the Drying Kinetic Parameters

The effect of pH on drying kinetic parameters has been determined by analysing the values of the constants in Newton (Table Equation 1) and Henderson and Pabis (Table Equation 3) models. These two models are derived from Newton's cooling law and Fick's Second law [16], respectively, so their constants enclose physical meaning—as opposite to the Page and Wang and Singh models that are purely empirical [16].

Parameters *k1* (Newton) and *k3* (Henderson) in Table 2, both time constants (*h*−*<sup>1</sup>*), characterise the drying rates of the system, while *a1* (Henderson) is a dimensionless parameter related to the shape and structural properties of the samples [16].

For gellan gum gels at different pHs, both Newton and Henderson rate parameters (i.e., *k1* and *k3*, respectively) show very similar trends. The higher values (*k1* = 0.225 h−<sup>1</sup> and *k3* = 0.232 <sup>h</sup>−1) correspond to samples at pH 2.5, indicating a faster dehydration process. On the other hand, rate constants for samples at pH 4 are the lowest (*k1* = 0.157 h−<sup>1</sup> and *k3* = 0.158 <sup>h</sup>−1), which relates to the slower drying rate of these samples. This is in agreemen<sup>t</sup> with differences on moisture ratios (MR) at different pHs discussed in Section 3.1.

The values of constant *a1* are again similar for samples at pH 2.5 and 5.2 (*a1* = 1.036 and *a1* = 1.043), which suggests no significant structural differences at those pH values. However, the value of *a1* for freeze-dried gels at pH 4 (*a1* = 1.006) suggests changes in microstructure that might be behind the different drying rates observed at this particular pH. These findings are in agreemen<sup>t</sup> with the mechanical properties (i.e., higher gel strength and Young's modulus) reported in Cassanelli et al. [14] for gellan gels at pH 4 before and after freeze-drying—"stronger" gels might make ice nucleation and growth difficult, and therefore affect the freeze-dried microstructures of the gels.

#### *3.3. Riboflavin Release from Freeze-Dried Gellan Gels at Di*ff*erent pHs*

Figure 3 presents experimental riboflavin release curves from freeze-dried gels prepared at different pHs, plotted as normalised vitamin released (NVR) across time. Data in this graph show significant differences in release times: freeze-dried gels at pH 4 completed the vitamin release in approximately 9.5 h; gels at natural pH (5.2) were fully unloaded after 6h, and total vitamin delivery took 3h for gels at pH 2.5. Samples at pH 2.5 presented a weak structure—in accordance with strength at fresh and freezing stages—that lead to breakage during the release experiments. This increased the surface area of the gels in contact with the release medium, which explains the shorter delivery times.

The observed differences in the riboflavin release times to the medium can be related to the different microstructures and mechanical properties of the gels. Both Norton et al. [23] and Cassanelli et al. [14] reported that freeze-dried gellan gum gels at pH 4 exhibited an aggregated and rigid structure. This can impede mass transfer within the gel, increasing the time needed to release the vitamin completely from the substrate and leading to longer delivery processes. A much lower level of aggregation and very weak structures were reported for freeze-dried gels at pH 2.5 [14,23], which is also in agreemen<sup>t</sup> with our experimental observations. According to the same authors, unloaded freeze-dried gels at natural pH (pH 5.2) exhibit intermediate levels of aggregation [14,24], explaining the also intermediate release times shown in Figure 3.

**Figure 3.** Release curves for the riboflavin encapsulated in freeze-dried 2% (*w*/*w*) gellan gums with different pHs. The vitamin content in the release medium is expressed as normalised vitamin released (NVR). Error bars correspond to triplicate tests.

#### *3.4. Delivery Mechanisms at Di*ff*erent pHs*

To estimate the value of the dimensionless parameter *nrel* in Equation (8), and therefore determine the release mechanism governing riboflavin delivery, the portion of the release curves (Figure 3) corresponding to the first 60% of the total released vitamin—i.e., release curve portions such that *M*(*t*) *M*∞ = *NRV* ≤ 0.60—were fitted to the Korsmeyer–Peppas model [17,18]. Samples at pH 2.5 were not considered in this analysis, as they broke into several pieces during the release tests, leading to delivery conditions out of the scope of this work. Table 3 lists the parameters *krel* and *nrel* (95% CI) estimated at pH 5.2 and pH 4 (see Table 3 for parameter units). These results are discussed next.

**Table 3.** Fitted parameters (95% CI) for the Korsmeyer–Peppas release model and release mechanisms found.


Parameter units: (*h*−*nrel*)*; nrel* is dimensionless.

#### 3.4.1. Release from Freeze-Dried Gellan Gels at pH 5.2

Riboflavin delivery from gels at natural pH (pH 5.2) is characterised by a shape constant *krel* = 0.509 diffusional coefficient *nrel* = 0.131 (see Table 3 for the corresponding confidence intervals). According to the classification given in [21,22], this indicates that the governing release mechanism is purely Fickian, as *nrel* = 0.131 < 0.45, which is the limiting value of the diffusional coefficient for Fickian transport mechanisms. Therefore, we can define an apparent diffusion coefficient *Dapp* (m<sup>2</sup>/s) for samples at pH 5.2 using a short-time approximation of Fick's Second law [18]:

$$\frac{M(t)}{M\_{\infty}} = 4 \left[ \frac{D\_{app}t}{\pi a^2} \right]^{\frac{1}{2}} \tag{9}$$

As the aspect ratio of the cylindrical gels is approximately in the order of 1, the predictive capabilities of the short-time solution include up to the 85% of the total vitamin release [22]. Thus, values such that *M*(*t*) *M*∞ = *NRV* ≤ 0.85 in the release curve at pH 5.2 were fitted to Equation (9). This

gave an estimated *Dapp* = 1.325 × 10−<sup>9</sup> m<sup>2</sup>/s with 95% CI defined by [1.086 × <sup>10</sup>−9, 1.564 × <sup>10</sup>−9] m<sup>2</sup>/s. Figure 4 presents the comparison between the experimental and predicted release curve using *Dapp*, showing a good agreemen<sup>t</sup> between the modelled Fickian delivery mechanism and the experimental data—and thus confirming Fickian transport for riboflavin released from freeze-dried gellan gels at pH 5.2

**Figure 4.** Predicted release curve for encapsulated riboflavin at pH 5.2 using estimated *Dapp* (dash –) compared to experimental curve (blue dots).

#### 3.4.2. Release from Freeze-Dried Gellan Gels at pH 4

As shown in Table 3, the estimated shape constant and diffusional exponent at pH 4 were *krel* = 0.287 and *nrel* = 0.472, respectively. These estimates (i) confirm the structural difference of samples at pH 4 discussed in previous subsections, as the value of *krel* at pH 4 is almost half of the corresponding to pH 5.2, and (ii) indicate an anomalous delivery mechanism from freeze-dried gellan matrices at pH 4, since we estimated *nrel* > 0.45.

Anomalous mass transport can be defined as a mix between Fickian and non-Fickian mechanisms, for which the general form of the Korsmeyer–Peppas model presented in Equation (8) can be split into two contributions [24]:

$$\frac{M(t)}{M\_{\infty}} = k\_{\rm rel} t^{n\_{\rm rel}} = k\_1^{\rm rel} t^{m\_{\rm rel}} + k\_2^{\rm rel} t^{2m\_{\rm rel}}.\tag{10}$$

The first one (*kre<sup>l</sup>* 1 *t<sup>m</sup>*) represents the Fickian part, while the second term (*kre<sup>l</sup>* 2 *t*2*<sup>m</sup>*) accounts for the relaxational contribution [24], which is related to stresses and state transitions of polymeric matrices. The dimensionless diffusional exponent *mrel* in Equation (10) can be determined from aspect ratio (i.e., height/diameter of the sample) correlations. For the ratio characterising the cylindrical samples used here (∼1.5), *mrel* = 0.43 [24].

Estimated values for both *kre<sup>l</sup>* 1 = 0.264 (0.240, 0.289) and *kre<sup>l</sup>* 2 = 0.022 (0.007, 0.038) were then obtained by fitting Equation (10) to the experimental release curves at pH 4 for NMC <0.60 with 95% confidence intervals (values in the parenthesis). Units for *kre<sup>l</sup>* 1 and *kre<sup>l</sup>* 2 are (*h*−*mrel*). These estimates give an idea of the relevance of each contribution. For samples at pH 4, *kre<sup>l</sup>* 1 *kre<sup>l</sup>* 2 , showing that the release of riboflavin from the freeze-dried gels at pH 4 is mostly Fickian. This is confirmed by the vitamin release percentages due to each contribution calculated as Peppas and Sahlin [24]:

$$P\_{\text{Fickian}} = \frac{1}{1 + \frac{k\_2^{rel}}{k\_1^{rel}} t^{m\_{rel}}} ; P\_{mm-\text{Fick}} = 100 - P\_{\text{Fickian}} = P\_{\text{Fickian}} \frac{k\_2^{rel}}{k\_1^{rel}} t^{m\_{rel}} \tag{11}$$

They are presented in Figure 5a. Overall, the Fickian contribution is predominant along the release process, i.e., overall *PFickian* > 80%, with values closer to 90% at the initial times of the delivery test, while relaxation effects are more important towards the end of the experiment, as the delivery of riboflavin is closer to completion.

**Figure 5.** (**a**) Fickian and non-Fickian release percentages for riboflavin corresponding to sample with pH 4 when anomalous transport mechanism was considered. (**b**) Predicted release curve for encapsulated riboflavin at pH 4 considering pure Fickian mechanism and estimated *Dapp* (dash –) compared to experimental curve (blue dots).

Given the relevance of Fickian transport during the release process at pH 4, and with the estimated diffusional exponent *nrel* so close to the Fickian limiting value of 0.45—confidence intervals for this parameter are actually cross-boarding this limit, i.e., (0.441, 0.504) as shown in Table 3—a hypothetical pure Fickian riboflavin delivery at pH 4 has been also assessed. Following the procedure explained in Section 3.4.1, an apparent diffusion coefficient Dapp = 5.626 × 10−<sup>10</sup> m<sup>2</sup>/s was estimated within a 95% confidence interval (5.409 × <sup>10</sup>−10, 5.842 × <sup>10</sup>−10) m<sup>2</sup>/s. This estimate together with the short time approximately described in Equation (9) was used to obtain a predicted release curve, which is presented in Figure 5b alongside the experimentally obtained curve. As the comparison reveals, the hypothetical pure Fickian mechanism describes the behaviour observed during the release tests well, and it could be used to predict riboflavin delivery—neglecting relaxation effects—from freeze-dried gellan gels at pH 4 with high accuracy.
