Water Sorption Isotherm and Critical Water Activity of Amorphous Water-Soluble Carbohydrates Characterized by the Glass Transition Temperature

Abstract

Water-soluble carbohydrates commonly exist in an amorphous state in foods and undergo glass-rubber transition (glass transition) at the glass transition temperature (T_g). The critical water content (W_c) and critical water activity (a_wc) are the water content and water activity (a_w) at which the glass transition occurs at 298 K (typical ambient temperature), respectively. For amorphous water-soluble carbohydrates, W_c can be predicted from the T_g of anhydrous solid (T_gs) using previously reported equations. However, an approach for predicting a_wc is still lacking. This study aimed to establish an a_wc-predictive approach for amorphous water-soluble carbohydrates based on T_gs. First, the water sorption isotherms of four hydrogenated starch hydrolysates were investigated, and the results were analyzed using the Guggenheim-Anderson-de Boer (GAB) model. Second, the effect of T_gs on the GAB parameters (C, K, and W_m) was evaluated using the T_gs values reported in previous literatures. C and W_m decreased and increased logarithmically, respectively, with increasing 1/T_gs. K was fixed to 1 (constant), as it showed little variation. These results enabled the prediction of the GAB parameters from T_gs. The GAB model could then predict a_wc from W_c, which was determined using the previously established equations. The predicted a_wc values were in good agreement with the experimentally determined a_wc. Additionally, we demonstrated that this a_wc-prediction approach is also applicable to amorphous water-soluble electrolytes and partially water-insoluble carbohydrates. Thus, this approach can be used for the quality control of amorphous water-soluble carbohydrates and carbohydrate-based foods.

Abbreviations

a_wc, critical water activity; DM, dry matter; T_g, glass transition temperature; T_gs, T_g of anhydrous solute; T_gw, T_g of water; W_c, critical water content.

INTRODUCTION

Water-soluble carbohydrates are commonly present in an amorphous form in food products and undergo glass-rubber transition (glass transition) at their glass transition temperature (T_g). At temperatures below T_g, amorphous carbohydrates in food matrices exist in the glassy state and have a hard, brittle, and/or crisp texture [1,2]. Meanwhile, above T_g, amorphous carbohydrates become rubbery and have a texture that is soft, ductile, and/or sticky. Caking (aggregation) of amorphous carbohydrate-based powders readily occurs in the rubbery state [3,4,5,6,7,8]. Because the T_g of amorphous hydrophilic carbohydrates decreases with increasing water content and/or water activity (a_w) owing to water plasticization, water sorption can result in glass transition even at a constant temperature. The water content and a_w at which glass transition can occur at ambient temperature (commonly 298 K) are termed as the critical water content (W_c) and critical water activity (a_wc), respectively [9,10,11,12], which are useful parameters for the quality control of amorphous water-soluble carbohydrates and carbohydrate-based foods.

The effect of water content on the T_g of amorphous materials is described by the Gordon-Taylor (GT) equation (Eq. 1),

  

$$T_g=\frac{M_s{T}_{\mathit{gs}}+k{(1-M_s)T}_{\mathit{gw}}}{M_s+k(1-M_s)}$$(Eq. 1)

where M_s is the mass fraction of the solid, T_gs is the T_g of the anhydrous solid (K), T_gw is the T_g of water (K), and k is a constant (dimensionless). T_gw can be set to 136 K [13,14]. With two constants (T_gsw and k), T_g-curve (water content-dependence of T_g) and W_c can be determined using the GT equation. In addition, a_wc can be determined using the Guggenheim-Anderson-de Boer (GAB) water sorption equation (Eq. 2),

  

$$W=\frac{W_m{\mathit{CKa}}_w}{(1-{\mathit{Ka}}_w)(1+(C-1){\mathit{Ka}}_w)}$$(Eq. 2)

where W and W_m are the equilibrium water content (g/g-DM, dry matter) and monolayer water content (g/g-DM), respectively; C is a factor correcting the sorption properties of the monolayer with respect to the bulk liquid; and K is a factor correcting the sorption properties of the multilayer with respect to the bulk liquid. From the GAB equation with three parameters (W_m, C, and K), W_c can be converted to a_wc.

For amorphous water-soluble carbohydrates, k in the GT equation (Eq. 1) can be predicted from T_gs (K) using the following empirical equation [12]:

  

$$k=0.0293T_{\mathit{gs}}-4.39$$(Eq. 3)

After determining the T_gs and k, W_c can be predicted by rearranging the GT equation (Eq. 1) as follows:

  

$$M_{\mathit{sc}}=\frac{162k}{162k{+T}_{\mathit{gs}}-298}$$(Eq. 4)

  

$$W_c=\frac{1-M_{\mathit{sc}}}{M_{\mathit{sc}}}$$(Eq. 5)

where M_sc is the mass fraction of the solute at T_g = 298 K. The T_gs values of various amorphous water-soluble carbohydrates have been widely reported by previous studies. If no data are available in the literature, the T_gs of such carbohydrates can be easily evaluated using differential scanning calorimetry (DSC). Additionally, the T_gs of an amorphous water-soluble carbohydrate mixture can be predicted from the T_gs and ΔC_p (heat capacity change induced by glass transition) of the individual components [11]. Thus, the fact that W_c can be predicted from T_gs has an important basis in food quality control. However, no study has reported an approach for predicting a_wc. Because the quality of food and food ingredients is commonly controlled by a_w, determining a_wc is more important than determining W_c. Like W_c, it is useful if a_wc can be also predicted by T_gs. The water sorption behavior is affected by T_gs [15,16], as shown in Fig. 1; further explanation is provided in Appendix a. Thus, there is a possibility that GAB parameters can be characterized by T_gs.

Fig. 1. Water sorption isotherm of amorphous water-soluble carbohydrates with varying T_gs (A), and three regions in the type II isotherm (B).

The present study aimed to establish a method to predict the a_wc of amorphous water-soluble carbohydrates from their T_gs. First, we evaluated the water sorption isotherms of four types of hydrogenated starch hydrolysate (HSH) and analyzed the results using the GAB model. Second, the effect of T_gs on the GAB parameters of amorphous water-soluble carbohydrates was investigated using T_gs values reported in the literature, and an a_wc-predictive approach was proposed. Finally, we assessed the applicability of this a_wc-prediction method to amorphous water-soluble electrolytes and partially water-insoluble carbohydrates.

MATERIALS AND METHODS

Materials. Four types of HSH syrup (water content 30 %) were obtained from B Food Science Co., Ltd. (Aichi, Japan). The sugar alcohol composition of these HSHs [11] is shown in Supplemental Data (Table S1; see J. Appl. Glycosci. Web site). Based on which, they were categorized as low molecular weight (Mw)- (LoHSH), lower-middle Mw- (LMiHSH), middle Mw- (MiHSH), and high Mw-enriched HSH (HiHSH).

Preparation of freeze-dried amorphous powder samples. Amorphous LoHSH, LMiHSH, MiHSH, and HiHSH powder samples were prepared by freeze-drying their aqueous solutions. Briefly, the HSHs were diluted to 20 % (w/w) by adding distilled water and frozen in a freezer at 253 K. The frozen samples were then transferred to a pre-cooled chamber and freeze-dried at approximately 11 Pa by increasing the temperature from 238 to 278 K gradually over 48 h. Subsequently, the freeze-dried solids were ground manually to obtain the powders. The LoHSH solid was ground in liquid nitrogen in an aluminum container to obtain fine powder. The freeze-dried samples were then stored in a refrigerator until use.

Water sorption isotherms. Freeze-dried amorphous powder samples (0.5 g) were placed in an aluminum dish (diameter 40 mm), and the residual moisture in the samples was removed by vacuum-drying at 80 ºC (stage temperature) for 6 h. Next, the dry samples were maintained at 298 K for longer than 7 days in a desiccator with saturated salts: Mg(NO₃)₂ (a_w = 0.529), NaBr (a_w = 0.576), KI (a_w = 0.688), NaCl (a_w = 0.753), and KCl (a_w = 0.843). We have previously reported the T_g and equilibrium water content of the HSHs in the low a_w-region (0-0.432) [11]. The water content in the samples was determined gravimetrically by oven-drying at 378 K for 16 h. The measurements were performed in triplicate, and the results were averaged.

Statistical analysis. Regression analyses were conducted using the root mean squared error (RMSE), mean absolute percentage error (MAPE), and coefficient of determination (R²).

RESULTS AND DISCUSSION

Water sorption isotherms of amorphous HSHs.

Water sorption isotherms of the HSH samples maintained at 298 K are shown in Fig. 2. LMiHSH, MiHSH, and HiHSH exhibited “type II” water sorption isotherms (Appendix a). Their T_gs values were considerably higher than 298 K (Table 1); thus, glass transition occurred at a certain a_w in the water sorption isotherms. However, LoHSH exhibited a “type III” water sorption isotherm (Appendix a). The T_gs of LoHSH (283.5 K) was lower than 298 K (Table 1); thus, a rubbery or liquid state was observed in all the a_w-region. Type III is commonly observed for amorphous water-soluble carbohydrates that have a T_gs lower than or close to 298 K; thus, a_wc cannot be defined intrinsically for these materials (Appendix a). LoHSH exhibited a type III curve like we expected. Thus, a_wc-prediction based on T_gs was conducted for only the samples showing type II isotherms (LMiHSH, MiHSH, and HiHSH).

Fig. 2. Water sorption isotherm of LoHSH, LMiHSH, MiHSH, and HiHSH.

　The solid curves were obtained by fitting the GAB model (Eq. 2) to the data. The values are expressed as the mean ± SD (n = 3).

Table 1.　 Glass transition and water sorption properties of amorphous water-soluble carbohydrates.

Sample	C	K	W m (g/100g)	T gs (K)	k	W c (g/100g)	a wc	Ref.
LoHSH	0.572	0.853	25.583	283.5	2.23a	NA	NA	11
LMiHSH	1.424	0.965	9.156	324.9	4.31a	2.43b	0.173c	11
MiHSH	2.725	0.946	7.257	344.8	12.13a	3.95b	0.227c	11
HiHSH	9.639	0.949	5.325	402.0	24.25a	8.74b	0.482c	11
Maltodextrin (MD)	9.192	0.995	4.3	435.8	9.00	9.4	0.575	4
Mango solute (MS)d	1.686	0.920	16.1	310.4	3.86	1.9	0.074	3
MS-MD (0.2 : 0.8)	2.303	0.922	6.4	366.1	7.18	5.8	0.345	3
MS-MD (0.4 : 0.6)	2.268	0.958	8.1	335.3	5.57	4.1	0.215	3
MS-MD (0.6 : 0.4)	2.640	0.950	9.8	325.5	4.95	3.4	0.160	3
MS-MD (0.8 : 0.2)	3.414	0.915	13.0	320.3	4.07	3.3	0.120	3
Maltose	4.05	1.085	5.33	368.6	7.4	5.89b	0.337c	17
Glucose-MD (0.2 : 0.8)	6.40	0.933	6.47	370.1	8.11	7.20	0.343	7
Inulin (low MW)	3.66	0.944	6.87	384.4	6.43	8.29b	0.431c	18
Inulin (native)	6.95	0.852	6.77	398.0	5.93	10.41b	0.525c	18

^a The value was calculated by the approximated Couchman-Karasz model [11]. ^b The value was calculated by the Gordon-Taylor model with T_gs and k (Eq. 1). ^c The value was calculated by the Guggenheim, Anderson, and de Boer model with W_c (Eq. 2). ^d Water-soluble materials (manly sucrose, glucose, and fructose) were extracted from mango puree [4].

Effect of T_gs on the GAB parameters of amorphous water-soluble carbohydrates.

To establish an a_wc-predictive approach based on T_gs, the GAB and GT parameters for amorphous water-soluble carbohydrates were taken from the literature [3,4,7,11,17,18] and are listed in Table 1. GAB parameters are very sensitive to experimental values; a small deviation in the experimental values can cause a large difference in the GAB parameters [4,19,20]. In addition, the water contents at each a_w are different between water sorption and water desorption (i.e., hysteresis effect). Therefore, the water content in samples needs to be removed fully prior to water sorption. However, these problems are not always elucidated in the literature. To avoid employing uncertain data, we only included the data reported by our research group at this stage and the data reported by other studies were applied later for comparison.

The relationship between C and T_gs is shown in Fig. 3A. The C values increased with increasing T_gs. The effect of the temperature on C can be analyzed using the Arrhenius-type (or van’t Hoff-type) equation [21,22,23,24]:

  

$$C=C_0\exp \left(\frac{{\mathit{\Delta H}}_C}{\mathit{RT}}\right)$$(Eq. 6)

where T and R are absolute temperature and gas constant, respectively, and C₀ and ΔH_C are pre-exponential factor and activation energy, respectively. Temperature is directly proportional to molecular mobility; the higher the temperature, the higher the molecular mobility. Conversely, T_gs is inversely proportional to molecular mobility; the higher the T_gs, the lower the molecular mobility at a constant temperature (298 K in this study). In other words, T and T_gs are interchangeable parameters that reflect molecular mobility. Accordingly, the Arrhenius-type plot for C (Eq. 6) can be rearranged as follows:

  

$$C={C_0}^{\text{*}}\exp \left(\frac{C^{\text{*}}}{T_{\mathit{gs}}}\right)$$(Eq. 7)

where C₀^* and C^* are constants. This equation is referred to as “T_g-based Arrhenius-like equation” in this paper.

Fig. 3. Effect of T_gs on the C of amorphous water-soluble carbohydrates (A) and T_g-based Arrhenius-like plot of the C (B).

　The values are listed in Table 1. The relationship between lnC and 1/T_gs (Fig. 3B) was analyzed by the linear fitting, and C* (slope) and lnC₀* (intercept) values were determined. The linear proportion was transformed to C vs. T_gs as solid line in Fig. 3A.

The T_g-based Arrhenius-like plot (lnC vs. 1/T_gs) for amorphous water-soluble carbohydrates is shown in Fig. 3B. There was a clear linearity in the plot (R² = 0.857), and C^* (slope) and lnC₀^* (intercept) values were determined to be −2022.7 and 6.9514, respectively, from the linearity (Eq. 7). Linearity also describes the solid curve shown in Fig. 3A.

The effect of temperature on K has also been analyzed using the Arrhenius-type equation previously [21,22,23,24]:

  

$$K=K_0\exp \left(\frac{{\mathit{\Delta H}}_K}{\mathit{RT}}\right)$$(Eq. 8)

where K₀ and ΔH_K are the pre-exponential factor and activation energy, respectively. Thus, it is expected that the T_g-based Arrhenius-like equation can be also applied to K in addition to C (Eq. 7). However, there was little variation in K values of the amorphous water-soluble carbohydrates (Table 1), and mean ± SD of the K values was confirmed to be 0.942 ± 0.057 (n = 13). To simplify the a_wc-predictive approach, K was fixed to 1 (constant) for amorphous water-soluble carbohydrates.

When K is fixed to 1, the GAB model is equivalent to the Brunauer-Emmett-Teller (BET) model. The BET model is commonly employed to analyze water sorption behavior up to the intermediate a_w-region [17]. Because the a_wc values of amorphous water-soluble carbohydrates are almost in the a_w-range applied by the BET model (Table 1), the setting of K as 1 simplified the a_wc-predictive approach. Electrolyte polymers (e.g., proteins) and samples containing water-insoluble materials and/or parts have been reported to exhibit a much lower value of K than 1 [7,25]. However, this study focused only on water-soluble carbohydrates, which simplified the approach.

The temperature dependence of W_m has often been undetermined by previous studies, although there are some examples in which the Arrhenius-type equation has been applied [19,21]. This will be because the monolayer water content is independent of temperature, and temperature has a negligible effect on the surface area of materials. In contrast, W_m intrinsically depends on T_gs (Appendix a), because higher the T_gs, larger is the molar mass [11,18] and thus lower the surface area per g-DM. In other words, it is expected that a higher T_gs corresponds to a lower W_m.

The effect of T_gs on the W_m of amorphous water-soluble carbohydrates is shown in Fig. 4A. As expected, W_m decreased with increasing T_gs. As discussed above, the effect of T_gs on W_m can be analyzed using the T_g-based Arrhenius-like equation.

  

$$W_m={W_{m0}}^{\text{*}}\exp \left(\frac{{W_m}^{\text{*}}}{T_{\mathit{gs}}}\right)$$(Eq. 9)

where W_m0^* and W_m^* are constants. The T_g-based Arrhenius-like plot (lnW_m vs. 1/T_gs) is shown in Fig. 4B. Good linearity was observed in the plot (R² = 0.812), and W_m^* (slope) and lnW_m0^* (intercept) values were determined to be 1159 and −5.830, respectively. Linearity also describes the solid curve shown in Fig. 4A.

Fig. 4. Effect of T_gs on the W_m of amorphous water-soluble carbohydrates (A) and T_g-based Arrhenius-like plot of the W_m (B).

　The values are listed in Table 1. The relationship between lnW_m and 1/T_gs (Fig. 4B) was analyzed by the linear fitting, and W_m^* (slope) and lnW_m0^* (intercept) values were determined. The linear proportion was transformed to W_m vs. T_gs as solid line in Fig. 4A.

Although the physical significance of the constants (C^*, C₀^*, W_m^*, and W_m0^*) in the T_g-based Arrhenius-like equation is unclear at present, the T_g-based Arrhenius-like plot can be effectively used to express the effect of T_gs on the C and W_m of amorphous water-soluble carbohydrates. This is a valuable result for establishing an a_wc-predictive approach based on T_gs.

Applicability of the a_wc-predictive approach to amorphous water-soluble carbohydrates.

As C and W_m can be predicted using T_gs, a_wc can also be predicted using T_gs. Briefly, k was obtained using Eq. 3 from T_gs, and then, W_c was obtained using Eq. 5 using T_gs and k. Meanwhile, C and W_m were obtained using Eqs. 7 and 9, respectively, using T_gs. Thereafter, W_c was converted to a_wc using Eq. 2 with obtained C and W_m values. Subsequently, using the a_wc-predictive approach, a_wc was calculated from the experimentally determined T_gs listed in Table 1 [3,4,7,11,17,18]. The relationship between the experimentally determined a_wc and the predicted a_wc is shown in Fig. 5. The calculated a_wc values were in good agreement with the predicted a_wc values (RMSE = 0.035, MAPE = 9.05, and R² = 0.948; n = 13).

Fig. 5. Relationship between experimentally determined a_wc and predicted a_wc of amorphous water-soluble carbohydrates.

　The experimentally determined a_wc values are listed in Table 1.

To further validate our proposed method, the applicability of the a_wc-prediction approach to other water-soluble carbohydrates reported in the literature was tested. The experimentally determined T_gs and a_wc and predicted a_wc are listed in Supplemental Data (Table S2; see J. Appl. Glycosci. Web site) [9,10,26,27,28]. The relationship between the experimentally determined and predicted a_wc is shown in Fig. 6. There was relatively good agreement between the values (RMSE = 0.058, MAPE = 16.7, R² = 0.906, n = 12). Based on these results, we concluded that the a_wc-predictive approach based on T_gs can successfully determine the a_wc of amorphous water-soluble carbohydrates.

Fig. 6. Relationship between a_wc values obtained from previous literature and predicted a_wc values of amorphous water-soluble carbohydrates.

　The values are listed in Supplemental Data (Table S2; see J. Appl. Glycosci. Web site).

The a_wc-predictive approach can predict the GAB parameters, which implies that water sorption isotherm can also be predicted from T_gs. For confirmation, we evaluated the relationship between the experimentally determined and predicted water content at each a_w (Fig. 7) and found the values were in moderate agreement (RMSE = 7.66, MAPE = 21.2, and R² = 0.777; n = 90). A large difference between the calculated and predicted values was observed in the high-water content (high a_w) region. This is because the BET model (applicable in the low and intermediate a_w-regions) was applied as the predictive approach. The prediction was greatly improved when the water content range was limited to < 20 g/100 g-DM (RMSE = 1.68, MAPE = 17.5, and R² = 0.886; n = 73).

Fig. 7. Relationship between experimentally determined water content and predicted water content of amorphous water-soluble carbohydrates at each a_w.

　The GAB parameters reflecting experimentally determined water content are listed in Table 1.

Effect of amorphous water-soluble electrolytes and water-insoluble carbohydrates on the a_wc-predictive approach.

The a_wc-predictive approach was established with the assumption that the materials were amorphous water-soluble carbohydrates. Thus, we evaluated whether this a_wc-predictive approach is also applicable to other carbohydrates and carbohydrate-based materials.

The experimentally determined T_gs and a_wc and predicted a_wc of the amorphous electrolytes and carbohydrate-electrolyte mixtures are listed in Supplemental Data (Table S3; see J. Appl. Glycosci. Web site) [17,29]. The relationship between the experimentally determined and predicted a_wc is shown in Fig. 8. The experimental and predicted a_wc values of calcium maltobionate (CaMb) and CaMb-maltose mixtures were almost similar. CaMb is composed of a divalent cation (Ca²⁺) and two maltobionate ions. Maltobionate consists of glucose and gluconate. Because the CaMb and CaMb-maltose systems comprise carbohydrate and carbohydrate-related (hydrogen-bonding system) parts, the contribution of electrostatic interactions in them is weak. Thus, the a_wc-predictive approach established by amorphous water-soluble carbohydrates can be applied to them as well. The predicted a_wc of HMb (maltobionic acid), however, showed a large deviation from its experimentally determined value owing to large electrostatic interactions.

Fig. 8. Relationship between experimentally determined a_wc and predicted a_wc of amorphous electrolytes and carbohydrate-electrolyte mixtures.

　The values are listed in Supplemental Data (Table S3; see J. Appl. Glycosci. Web site).

The experimentally determined T_gs and a_wc and the predicted a_wc for partially water-insoluble carbohydrates are listed in Supplemental Data (Table S4; see J. Appl. Glycosci. Web site)[4,7,18]. The relationship between the experimentally determined a_wc and the predicted a_wc is shown in Fig. 9. The higher the water-insoluble content, the larger the deviation in the prediction. For example, the values of semi-crystalline inulin (high-MW), which contains water-insoluble (crystal) parts, showed a larger deviation than those of pre-melted inulin (high-MW). In the case of (maltodextrin (MD)-glucose)-cellulose samples (MD-glucose mixture at 0.2 dry weight fraction of glucose was mixed with crystalline cellulose at 0.8, 0.6, 0.4, and 0.2 dry weight fractions of cellulose), the a_wc of the sample containing 0.8 dry weight fraction of cellulose showed the largest deviation from the predicted value. These large deviations originate from the different physical significance of water content and a_w. Water content (g/g-DM) is expressed as the amount of water per amount of total dry matter, including water-insoluble materials. However, a_w is mainly affected by hydrophilic materials and/or parts; there is little or no contribution from water-insoluble materials and/or parts. Thus, the water content at each a_w value decreased intrinsically with an increase in the ratio of water-insoluble materials and/or parts. As a result, the effect of T_gs on the GAB parameters differed from that of the a_wc-predictive approach established using water-soluble carbohydrates. If the unit of equilibrium water content is changed from “g/g-DM” to “g/g-hydrophilic matters,” a better a_wc prediction will be achieved.

Fig. 9. Relationship between experimentally determined a_wc and predicted a_wc of partially water-insoluble carbohydrates.

　The values are listed in Supplemental Data (Table S4; see J. Appl. Glycosci. Web site).

CONCLUSION

In this study, we established an a_wc-predictive approach for amorphous water-soluble carbohydrates. This approach enabled us to predict a_wc from T_gs. In addition, the water sorption isotherm (equilibrium water content at each a_w) at 298 K could be predicted under limited conditions (below a water content of 20 g/100 g-DM). Furthermore, we demonstrated that the a_wc-predictive approach is also applicable to amorphous electrolytes, amorphous carbohydrate-electrolyte mixtures, and partially water-insoluble carbohydrates, depending on the ion and water-insoluble material contents. As the T_gs values of amorphous carbohydrates have been widely reported previously or can be readily evaluated, this predictive approach will be useful for characterizing the physical properties of amorphous carbohydrates and carbohydrate-based foods.

CONFLICTS OF INTERESTS

This study was financially supported by B Food Science Co., Ltd. for collaborative research. Hydrogenated starch hydrolysates used in this study were manufactured by B Food Science Co., Ltd. Authors, Yuichi Kashiwakura and Tomochika Sogabe are employee of B Food Science Co., Ltd.

Appendix a

As shown in Fig. 1, the water sorption isotherm of amorphous water-soluble carbohydrates commonly exhibits type II or III curve. Type II curve can be divided into three regions. In the low a_w-region, monolayer water adsorption occurs. The monolayer water content (W_m) depends on the surface area of the carbohydrates; the higher the T_gs, the larger the molar mass [11,18] and thus lower the surface area per g-DM (i.e., lower W_m). In the intermediate a_w-region, multilayer water adsorption occurs, and the a_w-dependence of water content becomes too sensitive; a slight change in water content results in a large change in a_w. In the high a_w-region, the a_w-dependence of water content becomes much weaker because of the appearance of bulk water; the system behaves as an aqueous solution. Around the turning point between the low and intermediate a_w-region, glass to rubber transition is expected to occur [15]. In other words, the water content and a_w become W_c and a_wc, respectively, at this point. T_gs will be directly proportional to W_c and a_wc. Based on these interpretations, we expected that type II water sorption isotherm can be characterized by T_gs. Type III, however, is commonly observed for the amorphous water-soluble carbohydrates having a low T_gs (lower than or close to 298 K). Thus, we assumed that a_wc cannot be intrinsically defined for these materials.

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