Crack expansion dynamics of freeze-dried soybean curd during rehydration

Abstract

In contrast to the freeze-drying process, the rehydration process of freeze-dried foods remains unclear. This study investigated the rehydration of freeze-dried soybean curd, also known as tofu, and elucidated its breakage mechanism during rehydration, which impairs texture. The rehydration of freeze-dried tofu was observed at different water temperatures (20, 40, 70, and 100 °C), with the required rehydration time found to decrease with increasing water temperature. Furthermore, tofu was found to absorb water faster in cracks formed during production than in the porous body. Crack expansion was observed only in high-temperature water, leading to breakage of the tofu. Environmental scanning electron microscopy revealed that tofu expanded when a sufficient amount of water was absorbed. Accordingly, crack expansion in high-temperature water is attributed to the stress concentration at the tip of the crack, which is caused by differences in the rehydration rate and resulting stiffness between the porous body and cracks.

Introduction

Freeze-drying is a dehydration technique in which water is removed from frozen materials by sublimation at low pressure (George and Datta, 2002; Liapis and Litchfield, 1979), and has been applied in various fields, such as food manufacture, pharmaceuticals, and tissue engineering (Uscanga-Ramos et al., 2021; Qian and Zhang, 2011; Badylak et al., 2009; Hutmacher, 2000). In the food industry, freeze-drying is a powerful method owing to its various advantages, including achieving a long shelf-life, the retention of structural properties, and preservation of the original properties, such as taste, color, flavor, and texture (George and Datta, 2002; Liapis and Litchfield, 1979). A typical freeze-dried product is highly porous and, therefore, has a superior rehydration capacity, which is a significant characteristic of freeze-dried foods (Harnkarnsujarit et al., 2016; Ceballos et al., 2012).

The rehydration of freeze-dried foods is a key phenomenon in determining the quality of the final products (Dadmohammadi and Datta, 2022; Marabi and Saguy, 2004). This process is usually conducted by soaking the products in a large amount of water. The microscale structure, namely, the pore size and porosity, is known to affect the rehydration properties. As these porous characteristics are greatly influenced by freeze-drying processes, the effects of process conditions, such as pretreatment (Dziki, 2020; Deng and Zhao, 2008), freezing (Harnkarnsujarit et al., 2012; Harnkarnsujarit and Charoenrein, 2011; Khalloufi and Ratti, 2003), and drying (Pei et al., 2014; Oikonomopoulou et al., 2011; Khalloufi and Ratti, 2003), on the structural properties of freeze-dried foods have been investigated extensively.

The temperature of the liquid during rehydration is also important for determine the rehydration properties, especially rehydration rate. The rehydration rate is among the most important parameters of the rehydration properties, as it describes how quickly water is absorbed by a material. Several studies have been conducted on the influence of rehydration temperature on the kinetics of water uptake (Hernando et al., 2008; García-Pascual et al., 2005; Krokida and Philippopoulos, 2005; Pal and Chakraverty, 1997). Hernando et al. (2008) reconstituted freeze-dried Boletus edulis mushrooms with water at 20 and 70 °C, and evaluated the rehydration kinetics by measuring the moisture content. Krokida and Philippopoulos (2005) examined the effect of water temperature (40, 60, and 80 °C) on the rehydration rate of various freeze-dried fruits and vegetables. However, most of these studies have focused on natural and plant materials, such as fruits and vegetables, while research on protein-based and processed foods is scarce.

In recent years, there has been an increasing demand for protein-based foods, which has led to the widespread use of freeze-dried soybean curd. Soybean curd, commonly referred to as tofu, is a gel-type processed food with a high-water content, primarily composed of protein and lipid (Wang et al., 2019). Because it is a major source of protein for vegetarians and vegans, maintaining the appearance of tofu during cooking is crucial for enhancing the dietary experience. However, a report suggests that the rehydration process impacts the appearance of freeze-dried soybean curd. Specifically, Harnkarnsujarit et al. (2016) reported that the rehydration process causes cracking in the freeze-dried soybean curd. Food appearance, such as color and geometry, are crucial factors that impact food quality (Krokida and Philippopoulos, 2005). As a result, the difference in color between raw and rehydrated materials has been extensively studied (Pei et al., 2014; Henríquez et al., 2013; Babić et al., 2009). However, despite its importance, few studies have investigated changes in food geometry upon reconstitution, thus the detailed mechanism of cracking remains unknown. To develop visually appealing and appetizing soybean curd dishes, an in-depth understanding of geometrical deformation induced by water adsorption is highly desired.

This study aimed to examine the relationship between the rehydration properties and crack expansion in freeze-dried soybean curd. The rehydration process of freeze-dried soybean curd at different water temperatures was investigated and the results were theoretically analyzed using fluid dynamics equations. Furthermore, structural changes in freeze-dried soybean curd during rehydration were directly observed on a microscale by environmental scanning electron microscopy (ESEM). These results will contribute to improving the quality of rehydrated soybean curd.

Materials and Methods

Structural observation of freeze-dried soybean curd　Cubes of freeze-dried soybean curd (approx. 15 × 15 × 15 mm³) were purchased from ASUZAC FOODS Co., Ltd, Japan. The ingredients of the silken curd were soy milk, starch, stabilizers, and coagulants. The reason for the addition of starch was to improve the texture of the freeze-dried products. The freeze-dried samples were produced from silken curd by rapid freezing below −22 °C, followed by dehydration. Their surfaces contained cracks of approx. 0.1 mm in width, created during the manufacturing process (Fig. 1). The distribution of cracks differed among individual cubes.

Fig. 1

Freeze-dried soybean curd with a crack approx. 0.1-mm wide. No surface treatment has been performed.

Microstructures inside the samples were characterized by scanning electron microscopy (SEM). Samples were cut into thin slices (approx. 1 mm thick) and observed by SEM with an acceleration voltage of 10 kV. Generally, nonconductive samples are coated with a conductive thin film to prevent distortion of the SEM images induced by charge buildup (Joy and Pawley, 1992). However, our sample was deformed by the temporal temperature increase during chemical vapor deposition (see supplemental materials Fig. S1). As this charge buildup problem was solved by fixing the sample on an aluminum plate with conductive carbon tape, the samples were observed without any coating. Fig. 2 shows the porous structures of samples consisting of numerous microscale pores. No difference in appearance was observed between the inner and outer surface structures. The average pore diameter was 11.5 ± 2.5 µm (error bars represent the 95 % confidence interval), measured from 60 pores using the same method as in the previous study (Harnkarnsujarit et al., 2016). This pore diameter was smaller than those previously reported for freeze-dried soybean curd frozen at −90 °C (48 ± 6 µm) and that frozen with liquid nitrogen (15 ± 3 µm) (Harnkarnsujarit et al., 2016).

Fig. 2

SEM images of interior structure of freeze-dried soybean curd taken at (a) 1 000 × and (b) 2 000 × magnification. Area surrounded by red dashed line in (a) corresponds to the area of (b).

Rehydration observation at four different temperatures　To investigate the effect of water temperature on the rehydration process, the samples were reconstituted in water at four different temperatures, namely, 20 (room temperature), 40, 70, and 100 °C. We selected these four water temperatures based on two factors: firstly, to examine the effects of temperature within the commonly used range of water temperatures, specifically between 20 and 100 °C, and secondly, to investigate the impact of vapor. The experiment was conducted according to the following procedure: first, pure water (200 mL) obtained using a water purifier (RFP742HA, Advantec, Japan) was placed in a beaker, and heated to the specified temperature using a hot plate (SLR, SCHOTT, Germany). Additionally, degassed water was prepared by boiling at 100 °C for 10 min to investigate the influence of dissolved gases. Immediately after removing the beaker from the heat source, a curd sample was gently placed in the water. This procedure was repeated for 15 samples each at 20, 40, 70, and 100 °C. The water temperature was monitored using a thermometer to ensure that it remained almost constant during rehydration. This rehydration was observed and recorded by a CMOS camera. Wetted and dried areas of the sample were identified by the difference in image contrast.

In conventional methods, the sample weight during rehydration is measured as a function of time by gravimetric analysis. However, because the wetted soybean curd was soft and fragile, it was not feasible to measure its mass without affecting its shape. Furthermore, the wetted area extends during removing the sample from the water and measuring its mass, making a direct comparison with its appearance difficult. Therefore, instead of the conventional mass recording method, we directly measured the rehydration time, defined as the time required to completely wet the entire surface of the sample, from its appearance (refer to supplemental materials Fig. S2).

We note that the glass transition phenomenon does not need to be taken into consideration for our study. The glass transition temperature, T_g, which is defined as the temperature at which an amorphous structure changes from a glassy state to a rubbery state (Khalloufi and Ratti, 2003), affects rehydration capacity (Marques et al., 2009). The T_g of freeze-dried soybean curd in a dry state was determined to be −33 °C, as investigated through differential scanning calorimetry measurement (refer to supplemental materials Fig. S3 for detailed information). Because the T_g of foods decreases with increasing water content (Hashimoto et al., 2003; Khalloufi et al., 2000), our samples can be assumed to remain in a rubbery state throughout the rehydration experiment, indicating that glass transition has no effect on the water rehydration process.

Microscale observation of the rehydration process using ESEM　To observe the rehydration of curd samples on a microscale, in situ observation was conducted using ESEM (Versa 3D LoVac, FEI, USA). First, the samples were crushed into fine pieces of less than 0.5 mm. Next, a small piece was fixed to a copper substrate with carbon tape and placed on a sample stand. The chamber pressure was set to 250 Pa and gradually increased to 570 Pa while cooling the sample stand to 0 °C using the Peltier element. Finally, water droplets with diameters of several hundred micrometers were dropped directly onto the samples by microinjection (Kleindiek, Germany). Owing to the relatively high pressure and low substrate temperature, stable water could be observed under electron beam irradiation conditions.

Statistical analysis　The free software R v4.2.2 (Institute for Statistics and Mathematics of WU, n.d.) was used for statistical analysis of the experimental data. Significant differences were determined using the Tukey-Kramer test at p < 0.05.

Results and Discussion

Rehydration and breakage of soybean curd　Cracks were found in most of the rehydrated samples, regardless of water temperature, because they had originally formed during production. Meanwhile, the breakage phenomenon, where the curd split into two or more pieces by developing cracks, as shown in Fig. 3(a), occurred only in high-temperature water. In particular, 1 out of 15 samples and 6 out of 15 samples broke up at 70 and 100 °C, respectively, while no breakage was observed in 15 samples each at 20 and 40 °C (shown in Fig. 4). The rehydration rate in the cracks also varied with temperature. Specifically, water was absorbed through cracks in about 10 s at 100 °C (Fig. 3(b)), but took more than 70 s at 20 °C (Fig. 3(c)). In addition, as shown in Figs. 3(b) and 3(c), water uptake by cracks was much faster than that by the porous body at 70 and 100 °C, while the difference was small at 20 and 40 °C.

Fig. 3

Rehydration of freeze-dried soybean curd (a, b) at 100 °C and (c) at 20 °C: (a) Crack developed and finally split the sample into two pieces after 40 s (broken face is vertically downward); (b) sample after 13 s, where the blue circle indicates a bubble nucleated from the crack; (c) sample after 75 s. In (b) and (c), the green dashed line indicates the region wetted by water absorbed through the cracks.

Fig. 4

Relationship between rehydration time and water temperature. The rehydration time is defined as the time taken to completely wet the whole sample, where 0 s is the time at which the sample is placed on the water surface. The average rehydration time was 146 ± 23^A s at 20 °C, 95 ± 13^B s at 40 °C, 62 ± 5^C s at 70 °C, 59 ± 10^C s at 100 °C, respectively. Different superscripts indicate significant differences (p < 0.05). Error bars represent the 95 % confidence interval. The rehydration time gradually approaches around 30 seconds with increasing temperature due to the temperature dependence of the viscosity coefficient. Detailed results are shown in supplemental materials Table S1.

Furthermore, small bubbles were observed to be generated from inside the cracks (Fig. 3(b)). This was observed even in water at 70 °C or degassed water at 100 °C. Therefore, bubble formation was not due to vapor or gases dissolved in water, but air trapped inside the sample, which escaped into the water due to differences in the rehydration speed between the porous body and cracks. The bubbles were also observed regardless of whether the samples were broken or not. Therefore, we concluded that the bubbles were not the cause of the breakage. Furthermore, breakage even occurred in water at 70 °C, which suggested that boiling and the phase-change-induced volume increase did not cause the breakage. The reason for the breakage is explained later in relation to the rehydration rate. Moreover, it was observed that the wetted soybean curd became soft. As mentioned above, glass transition temperature is not related to this phenomenon. Thus, this may be attributed to the hydrogel properties of soybean curd, making it softer as the water content increases (Li et al., 2020).

Next, the rehydration time at each water temperature was investigated. The rehydration time decreased significantly (p < 0.05) with increasing water temperature from 20 to 70 °C, while no significant difference (p > 0.05) was observed between the values at 70 and 100 °C, as shown in Fig. 4. This difference in the decreasing amount is attributed to the temperature dependence of the viscosity coefficient, discussed later. Compared with the mean rehydration time at 20 °C, those at 40, 70, and 100 °C were about 1.5, 2.3, and 2.5 times faster, respectively. This temperature dependence on the rehydration time is in agreement with previous studies on the rehydration of dehydrated foods (Krokida and Philippopoulos, 2005). Furthermore, there was a significant difference (p < 0.05) in the rehydration time between the broken and unbroken samples: all broken samples, as indicated by a horizontal black line in Fig. 4, were wet in less than 58 s. This implied that rapid rehydration might be a cause of the breakage.

Notably, the crack distribution, as well as rapid rehydration, might also be a factor affecting breakage. These cracks are formed during manufacturing, and their distribution and depth varies among individual curd samples. This would explain the different results for the breakage and bubble generation at the same water temperature.

Estimation of rehydration ratio by simplified modeling　Although dispersion originating from individual differences, such as the distribution of cracks, was observed, the rehydration time was dependent on the water temperature, as shown in Fig. 4. To investigate the relationship between the rehydration kinetics and sample properties, water flow in the freeze-dried soybean curd during rehydration was modeled. Specifically, water flow inside the porous body was described by Darcy’s law, which is applied to fluid transport in porous media and has previously successfully modeled fluid flow inside freeze-dried foods (Uscanga-Ramos et al., 2021; Feng et al., 2004). In contrast, water flow through cracks was modeled by two-dimensional flow in parallel plates. Water was defined to only flow in the x-direction parallel to the pore wall. Notably, capillary pressure has a dominant effect, compared to atmospheric pressure, on the flow in highly unsaturated products such as freeze-dried foods (Datta, 2007).

The mean flow rate inside porous media (u_por) is described by Darcy’s law (Datta, 2007; Feng et al., 2004), as follows:

  

where k is the intrinsic permeability, k_lr is the relative permeability of the liquid phase, µ is the viscosity coefficient of the water, and p_c is the capillary pressure of the water. The capillary pressure gradient is expressed by Eq. 2 (Camp and Figliola, 2011):

  

where γ_Λv is the liquid–vapor interfacial tension, θ is the contact angle, R is the pore radius, and L is the pore length. From Eqs. 1 and 2, the mean rehydration rate in the porous media is given by Eq. 3:

  

Next, the mean flow rate through the crack was derived. Assuming incompressible, steady, two-dimensional, and constant viscosity flow, the mean flow rate through the crack (u_cra) is expressed by Eq. 4 (Camp and Figliola, 2011):

  

where b is the crack width. The capillary pressure gradient is expressed by Eq. 5 (Camp and Figliola, 2011):

  

where l is the crack length. From Eqs. 4 and 5, the mean rehydration rate through the crack is given by Eq. 6:

  

The permeability is known to take a wide range of values depending on the porous properties, such as pore size and porosity (Corrochano et al., 2015). The intrinsic permeability is usually determined by measuring the flow rate while varying the pressure loss (Feng et al., 2004), although no data have been measured for freeze-dried soybean curd. Furthermore, treating the permeability as a constant value for this material is difficult because the porous structure deforms during rehydration. Therefore, in this study, we utilized the nondimensionalized mean rehydration rates, represented by u_por^* and u_cra^*. These values were obtained by dividing the mean rehydration rates at each temperature by the values at 20 °C for the porous media and the crack, respectively, as described by Eq. 7:

  

Note that the contact angle (θ) of the sample was assumed to be 0° because the measurement was difficult owing to the water absorbency of the sample (see supplemental materials Fig. S4).

Table 1 shows the dimensionless mean rehydration rates (u_por^*, u_cra^*) at each water temperature obtained by substituting the values of γ_LV and µ (Kestin et al., 1978; Vargaftik et al., 1983) into Eq. 7. For either the porous portion or cracks, the mean rehydration rates at 40, 70, and 100 °C were 1.5, 2.2, and 2.9 times faster, respectively, than that at 20 °C. This was qualitatively consistent with the experimental values of about 1.5, 2.3, and 2.5 times at 40, 70, and 100 °C, respectively, shown in Fig. 4.

Table 1. Surface tension of water, γ_LV (Vargaftik et al., 1983), viscosity coefficient of water, µ (Kestin et al., 1978), and the dimensionless mean rehydration rate (u_por^*, u_cra^*) at each water temperature.

Temperature	γLV (mN/m)	µ(µPa ·S)	upor*, ucra*
20 °C	72.8	1002	1
40 °C	69.6	652.7	1.5
70 °C	64.5	404.6	2.2
100 °C	58.9	282.1	2.9

In Eq. 7, the numerator represents the dimensionless quantity of surface tension, while the denominator represents the dimensionless quantity of the viscosity coefficient. As shown in Fig. S5 in the supplemental materials, the viscosity coefficient decreases more than the surface tension with temperature when both are nondimensionalized by the values at 20 °C. In addition, the surface tension changed almost linearly with temperature, while the slope of the viscosity coefficient with respect to temperature decreases as temperature increases. Therefore, the change in the viscosity coefficient has a more dominant effect on the rehydration rate. Consequently, the mean rehydration rate increased with increasing water temperature, particularly between 20 to 70 °C, as discussed earlier. In short, the increase in the mean rehydration rate with increasing water temperature was attributed to a decrease in the viscosity coefficient of water.

The difference in rehydration rate between the cracks and porous body is expressed by Eq. 8 from subtraction of Eqs. 3 and 6:

  

Eq. 7 was derived based on the assumption that the crack width (b) is constant and independent of the water temperature. However, when the samples were rehydrated with hot water, the crack widths increased more than 10-fold immediately after water absorption (see supplemental materials Fig. S6). Furthermore, the rehydration rate of the porous body (u_por) increased with increasing water temperature. Therefore, from Eq. 8, the difference in rehydration rate between the crack and porous body should increase with increasing temperature of the rehydration water, which is consistent with the results shown in Figs. 3(b) and 3(c).

Microscale observation of the rehydration process using ESEM　Fig. 5 shows the ESEM images of freeze-dried soybean curd. When a droplet was put on the relatively large sample (around 3 mm), both expansion and shrinkage of the pores were observed. The pore marked by the red dashed circle in Fig. 5(a) shrank by about half in 10 s, and then more gradually decreased in size. This was because the water–vapor interface formed in the pore pulled the pore wall inward during transportation due to interfacial tension. A similar phenomenon was previously reported in the food drying process, where internal stress induced by water transfer leads to pore shrinkage (Niamnuy et al., 2007). In contrast, some pores became larger, such as that marked by the yellow circle in Fig. 5(a). This might be attributed to the shrinkage of adjacent pores.

Fig. 5

ESEM images of microstructures of freeze-dried soybean curd showing (a) deformation of the pores at 1 000 × magnification and (b) expansion of the entire sample at 500× magnification. Note that the time at which water was dropped by microinjection was set as 0 s. In (a), the pore marked by the red dashed circle gradually shrank, while that marked by the yellow circle became larger. In (b), a sufficient amount of water was retained at 54 s and the entire sample expanded.

Next, a smaller sample (approx. 400 µm in size) was observed. As shown in Fig. 5(b), when a droplet larger than the sample was applied, the entire sample expanded due to absorbing the water (13 % by area). Raw soybean curd is known to retain water well owing to the three-dimensional network structure (composed of protein) being able to trap a large amount of water (Shen and Kuo, 2017). These results indicated that, although pores can shrink or expand owing to surface tension, as shown in Fig. 5(a), the entire wetted region expands by adsorbing water. Moreover, because the thermal expansion coefficient is commonly in the range of 10⁻⁴ – 10⁻⁶ K⁻¹, the thermal expansion of the medium induced by a temperature increase from 20 to 100 °C is typically limited to 1 % or less. Therefore, the wetted area of the samples was concluded to expand as a result of rehydration.

Mechanism of crack expansion　Rehydration experiments at different water temperatures showed that sample breakage occurred only when rehydration was completed rapidly in high-temperature water, as shown in Fig. 4. Furthermore, cracks were observed to absorb water significantly faster compared with the porous body in hot water. Additionally, ESEM observation confirmed that the wetted area of the sample expanded due to retaining the water. Using these results, we propose a crack expansion mechanism. When a sample is reconstituted in hot water, rehydration proceeds rapidly, especially through the crack. As mentioned in Eq. 8, the rehydration rate becomes faster in cracks compared with the porous body as the viscosity coefficient decreases with increasing water temperature. Furthermore, the wetted region becomes soft, while the dry region remains hard, and these regions coexist in the same sample as depicted in Fig. 6. Owing to the difference in stiffness between these regions and expansion in the wet portion, stresses are concentrated at the tip of the crack, resulting in crack development and sample breakage. Notably, the difference in rehydration rate between the porous body and cracks was small in low-temperature water, which explained why breakage did not occur upon rehydration at 20 and 40 °C.

Based on the above experimental results, we propose that the optimal rehydration temperature for freeze-dried soybean curd is around 70 °C. At that temperature, the soybean curd experienced minimal breakage while maintaining water rehydration rates almost identical to that at 100 °C. For practical purposes, it is suggested to use water that has been boiled and then cooled for a brief period.

Conclusions

Freeze-dried soybean curd was observed during rehydration using two experiments. First, samples were rehydrated by immersion in water at four different temperatures (20, 40, 70, and 100 °C). The samples rehydrated faster in hot water compared with cold water, while rapid rehydration induced sample breakage. Furthermore, cracks were found to absorb water faster than the porous body as the water temperature increased. Our simplified models of the rehydration rate indicated that the change in rehydration rate with water temperature was due to the viscosity coefficient of water. Next, deformations of pores and the entire sample during rehydration were observed on a microscale using ESEM. This confirmed that the sample expanded, regardless of shrinkage and enlargement of the pores when wetted. According to these experimental results, crack expansion of the freeze-dried soybean curd during rehydration was attributed to stress concentration at the crack tip, caused by the difference in rehydration rate between the porous body and cracks. These findings will provide valuable insight into the optimization of the rehydration process, which is important for utilizing freeze-dried tofu in cooking without compromising its appearance.

Acknowledgements　The authors would like to sincerely thank C. Inoue and Z. Wang for their invaluable technical support and generous provision of the experimental equipment. This work was supported by JSPS KAKENHI Grant Nos. JP20H02089 and JP22K14193. The sponsors had no role in study design; in the collection, analysis, and interpretation of data; in the writing of the report; and in the decision to submit the article for publication.

Conflict of interest　There are no conflicts of interest to declare.

Supplemental materials　Supplemental materials associated with this article can be found in the online version at doi:

Author contributions　Mai Hirakawa: Conceptualization, Formal analysis, Investigation, Methodology, Writing – Original Draft, Visualization. Hideaki Teshima: Conceptualization, Methodology, Resources, Validation, Writing – Review and Editing, Supervision, Funding acquisition. Tatsuya Ikuta: Investigation, Methodology, Visualization. Koji Takahashi: Conceptualization, Writing – Review and Editing, Supervision, Funding acquisition.

References

•  Babić,  J.,  Cantalejo,  M. J., and  Arroqui,  C. (2009). The effects of freeze-drying process parameters on Broiler chicken breast meat. LWT-Food Sci. Technol., 42, 1325–1334. doi: 10.1016/j.lwt.2009.03.020
•  Badylak,  S. F.,  Freytes,  D. O., and  Gilbert,  T. W. (2009). Extracellular matrix as a biological scaffold material: Structure and function. Acta Biomater., 5, 1–13. doi: 10.1016/j.actbio.2008.09.013
•  Ceballos,  A. M.,  Giraldo,  G. I., and  Orrego,  C. E. (2012). Effect of freezing rate on quality parameters of freeze dried soursop fruit pulp. J. Food Eng., 111, 360–365. doi: 10.1016/j.jfoodeng.2012.02.010
•  Corrochano,  B. R.,  Melrose,  J. R.,  Bentley,  A. C.,  Fryer,  P. J., and  Bakalis,  S. (2015). A new methodology to estimate the steady-state permeability of roast and ground coffee in packed beds. J. Food Eng., 150, 106–116. doi: 10.1016/j.jfoodeng.2014.11.006
•  Dadmohammadi,  Y. and  Datta,  A. K. (2022). Food as porous media: a review of the dynamics of porous properties during processing. Food Rev. Int., 38, 953–985. doi: 10.1080/87559129.2020.1761376
•  Datta,  A. K. (2007). Porous media approaches to studying simultaneous heat and mass transfer in food processes. I: Problem formulations. J. Food Eng., 80, 80–95. doi: 10.1016/j.jfoodeng.2006.05.013
•  Deng,  Y. and  Zhao,  Y. (2008). Effect of pulsed vacuum and ultrasound osmopretreatments on glass transition temperature, texture, microstructure and calcium penetration of dried apples (Fuji). LWT-Food Sci. Technol., 41, 1575–1585. doi: 10.1016/j.lwt.2007.10.018
•  Dziki,  D. (2020). Recent trends in pretreatment of food before freeze-drying. Processes, 8, 1–18. MDPI AG. doi: 10.3390/pr8121661
•  Feng,  H.,  Tang,  J.,  Plumb,  O. A., and  Cavalieri,  R. P. (2004). Intrinsic and relative permeability for flow of humid air in unsaturated apple tissues. J. Food Eng., 62, 185–192. doi: 10.1016/S0260-8774(03)00231-0
•  García-Pascual,  P.,  Sanjuán,  N.,  Bon,  J.,  Carreres,  J. E., and  Mulet,  A. (2005). Rehydration process of Boletus edulis mushroom: Characteristics and modelling. J. Sci. Food Agric., 85, 1397–1404. doi: 10.1002/jsfa.2126
•  George,  J. P. and  Datta,  A. K. (2002). Development and validation of heat and mass transfer models for freeze-drying of vegetable slices. J. Food Eng., 52, 89–93. doi: 10.1016/S0260-8774(01)00091-7
•  Harnkarnsujarit,  N. and  Charoenrein,  S. (2011). Influence of collapsed structure on stability of β-carotene in freeze-dried mangoes. Food Res. Int., 44, 3188–3194. doi: 10.1016/j.foodres.2011.08.008
•  Harnkarnsujarit,  N.,  Charoenrein,  S., and  Roos,  Y. H. (2012). Microstructure formation of maltodextrin and sugar matrices in freeze-dried systems. Carbohydr. Polym., 88, 734–742. doi: 10.1016/j.carbpol.2012.01.028
•  Harnkarnsujarit,  N.,  Kawai,  K.,  Watanabe,  M., and  Suzuki,  T. (2016). Effects of freezing on microstructure and rehydration properties of freeze-dried soybean curd. J. Food Eng., 184, 10–20. doi: 10.1016/j.jfoodeng.2016.03.014
•  Hashimoto,  T.,  Hagiwara,  T.,  Suzuki,  T., and  Takai,  R. (2003). Study on the glass transition of Katsuobushi (boiled and dried bonito fish stick) by differential scanning calorimetry and dynamic mechanical analysis. Fish. Sci. (Tokyo, Jpn.), 69, 1290–1297. doi: 10.1111/j.0919-9268.2003.00757.x
•  Henríquez,  M.,  Almonacid,  S.,  Lutz,  M.,  Simpson,  R., and  Valdenegro,  M. (2013). Comparison of three drying processes to obtain an apple peel food ingredient. CyTA-J. Food, 11, 127–135. doi: 10.1080/19476337.2012.703693
•  Hernando,  I.,  Sanjuán,  N.,  Pérez-Munuera,  I., and  Mulet,  A. (2008). Rehydration of freeze-dried and convective dried Boletus edulis mushrooms: Effect on some quality parameters. J. Food Sci., 73, 356–362. doi: 10.1111/j.1750-3841.2008.00913.x
•  Hutmacher,  D. W. (2000). Scaffolds in tissue engineering bone and cartilage. Biomaterials, 21, 2529–2543. doi: 10.1016/B978-008045154-1.50021-6
• Institute for Statistics and Mathematics of WU (Vienna University of Economics and Business). (n.d.). The Comprehensive R Archive Network. https://cran.r-project.org/
•  Joy,  C, D. and  Pawley,  B, J. (1992). High-resolution scanning electron microscopy. Ultramicroscopy, 47, 80–100. doi: 10.1016/0304-3991(93)90044-X
•  Kestin,  J.,  Sokolov,  M., and  Wakeham,  W. A. (1978). Viscosity of liquid water in the range −8 °C to 150 °C. J. Phys. Chem. Ref. Data, 7, 941–948. doi: 10.1063/1.555581
•  Khalloufi,  S.,  El-Maslouhi,  Y., and  Ratti,  C. (2000). Mathematical model for prediction of glass transition temperature of fruit powders. J. Food Sci., 65, 842–848. doi: 10.1111/j.1365-2621.2000.tb13598.x
•  Khalloufi,  S. and  Ratti,  C. (2003). Quality deterioration of freeze-dried foods as explained by their glass transition temperature and internal structure. J. Food Sci., 68, 892–903. doi: 10.1111/j.1365-2621.2003.tb08262.x
•  Krokida,  M. K. and  Philippopoulos,  C. (2005). Rehydration of Dehydrated Food. Drying Technol., 23, 799–830. doi: 10.1081/DRT-200054201
•  Li,  Z.,  Liu,  Z.,  Ng,  T. Y., and  Sharma,  P. (2020). The effect of water content on the elastic modulus and fracture energy of hydrogel. Extreme Mech. Lett., 35, 100617. doi: 10.1016/j.eml.2019.100617
•  Liapis,  A. I. and  Litchfield,  R. J. (1979). Optimal control of a freeze dryer-I Theoretical development and quasi steady state analysis. Chem. Eng. Sci., 34, 975–981. doi: 10.1016/0009-2509(79)85009-5
•  Marabi,  A. and  Saguy,  I. S. (2004). Effect of porosity on rehydration of dry food particulates. J. Sci. Food Agric., 84, 1105–1110. doi: 10.1002/jsfa.1793
•  Marques,  L. G.,  Prado,  M. M., and  Freire,  J. T. (2009). Rehydration characteristics of freeze-dried tropical fruits. LWT-Food Sci. Technol., 42, 1232–1237. doi: 10.1016/j.lwt.2009.02.012
•  Munson,  B. R.,  Okiishi,  T. H.,  Huebsch,  W. W.,  Rothmayer,  A. P. (2013). Fluid mechanics (7th ed. SI Ver.). Wiley.
•  Niamnuy,  C.,  Devahastin,  S., and  Soponronnarit,  S. (2007). Effects of process parameters on quality changes of shrimp during drying in a jet-spouted bed dryer. J. Food Sci., 72, 553–563. doi: 10.1111/j.1750-3841.2007.00516.x
•  Oikonomopoulou,  V. P.,  Krokida,  M. K., and  Karathanos,  V. T. (2011). The influence of freeze drying conditions on microstructural changes of food products. Procedia Food Sci., 1, 647–654. doi: 10.1016/j.profoo.2011.09.097
•  Pal,  U. S. and  Chakraverty,  A. (1997). Thin layer convection-drying of mushrooms. Energy Convers. Manage., 38, 107–113. doi: 10.1016/0196-8904(96)00020-9
•  Pei,  F.,  Yang,  W. J.,  Shi,  Y.,  Sun,  Y.,  Mariga,  A. M.,  Zhao,  L. Y.,  Fang,  Y.,  Ma,  N.,  An,  X. X., and  Hu,  Q. H. (2014). Comparison of Freeze-Drying with Three Different Combinations of Drying Methods and Their Influence on Colour, Texture, Microstructure and Nutrient Retention of Button Mushroom (Agaricus bisporus) Slices. Food Bioprocess Technol., 7, 702–710. doi: 10.1007/s11947-013-1058-z
•  Qian,  L. and  Zhang,  H. (2011). Controlled freezing and freeze drying: A versatile route for porous and micro-/nano-structured materials. J. Chem. Technol. Biotechnol., 86, 172–184. doi: 10.1002/jctb.2495
•  Shen,  Y. R. and  Kuo,  M. I. (2017). Effects of different carrageenan types on the rheological and water-holding properties of tofu. LWT-Food Sci. Technol., 78, 122–128. doi: 10.1016/j.lwt.2016.12.038
•  Uscanga-Ramos,  M. A.,  Lopez-Sanchez,  E.,  Martínez-Navarrete,  N.,  García-Alvarado,  M. A., and  Salgado-Cervantes,  M. A. (2021). Analytical solution of freeze-drying mathematical model based in Darcy’s law: application to an orange juice-based cake. CyTA-J. Food, 19, 265–272. doi: 10.1080/19476337.2021.1892195
•  Vargaftik,  N. B.,  Volkov,  B. N., and  Voljak,  L. D. (1983). International Tables of the Surface Tension of Water. J. Phys. Chem. Ref. Data, 12, 817–820. doi: 10.1063/1.555688
•  Wang,  C.,  Du,  Q.,  Yao,  T.,  Dong,  H.,  Wu,  D.,  Qin,  W.,  Raheem,  D., and  Zhang,  Q. (2019). Spoilage Bacteria Identification and Food Safety Risk Assessment of Whole Soybean Curd. Indian J. Microbiol., 59, 250–253. doi: 10.1007/s12088-019-00778-1

Supplemental materials

Crack expansion dynamics of freeze-dried soybean curd during rehydration

Mai Hirakawa¹, Hideaki Teshima^1,2*, Tatsuya Ikuta^1,2, Koji Takahashi^1,2

¹ Department of Aeronautics and Astronautics, Kyushu University, 744 Motooka, Nishi-Ku, Fukuoka 819-0395, Japan

² International Institute for Carbon-Neutral Energy Research (WPI-I2CNER), Kyushu University, 744 Motooka, Nishi-Ku, Fukuoka 819-0395, Japan

^* Corresponding author: Email: [e-mail]

Figure S1

An SEM image of freeze-dried soybean curd deposited with Pt–Pd metal (composition in 8:2 ratio) at 1200× magnification. The sample structure was deformed by thermal energy.

Figure S2

Rehydration process of freeze-dried soybean curd with water at 20 °C. The sample was gradually rehydrated from the bottom and eventually became moistened completely on the upper surface. We define the rehydration time as the time between the moment when the sample is placed on water surface (0 s) and when the entire surface of sample is completely wet (135 s).

Figure S3

Differential Scanning Calorimetry (DSC) thermograms for freeze-dried soybean curd using a differential scanning calorimeter (X-DSC7000, Hitachi High-Tech, Japan). The sample was crushed and subjected to overnight vacuuming for water removal. Subsequently, 4.6 mg of the sample was hermetically sealed in an aluminum pan using a sealer. The sample was scanned twice ranging from −80 °C to 170 °C with a heating and cooling rate of 10 °C/min under dry nitrogen atmosphere. The experimental data was analyzed using software for NESTA (Hitachi High-Tech, Japan). In the first scan (represented by the blue line), an endothermic shift was observed around 50 °C, accompanied by an endothermic peak at approximately 140 °C, reflecting the sample’s thermal history during processing and storage (Farahnaky et al., 2005; Truong et al., 2004). In the second scan (illustrated by the red line), the DSC curve maintained a constant slope above 0 °C. The glass transition temperature (T_g) was determined to be −33 °C by identifying the onset temperature of the endothermic shift observed in both scans, indicated by arrows.

Figure S4

Measurement of the contact angle between freeze-dried soybean curd and silicon substrate. Generally, a sessile droplet is used to measure the contact angle as shown that on a silicon substrate in the left side. However, due to its porous structure, freeze-dried soybean curd instantly absorbs water, which makes it difficult to measure the contact angle.

Figure S5

Temperature dependence of (a) liquid–vapor surface tension (Vargaftik et al., 1983) and (b) viscosity coefficient of water (Kestin et al., 1978). Both are nondimensionalized with the values at 20 °C.

Figure S6

Crack expansion of freeze-dried soybean curd during rehydration with water at 100 °C. The crack width gradually expanded from approximately 0.1 mm to about 1.4 mm in 10 s.

Table S1Summary of rehydration results corresponding to Figure 4 in the main manuscript. BR, breakage; BU, bubble; t, rehydration time (s)

20 °C			40 °C			70 °C			100 °C			100 °C Degassed
BR	BU	t	BR	BU	t	BR	BU	t	BR	BU	t	BR	BU	t
no	no	74	no	no	52	no	no	42	no	no	38	no	no	56
no	no	86	no	no	53	no	yes	50	yes	yes	40	no	no	62
no	no	109	no	no	66	yes	yes	53	yes	yes	43	no	no	63
no	no	123	no	no	87	no	no	53	yes	yes	45	no	no	65
no	no	125	no	no	90	no	no	60	no	yes	45	no	no	69
no	no	130	no	no	93	no	no	62	yes	yes	50	no	no	70
no	no	131	no	no	95	no	no	62	no	no	52	no	yes	70
no	no	134	no	no	95	no	no	64	yes	yes	53	no	no	70
no	no	141	no	no	96	no	no	65	yes	yes	58	no	yes	79
no	no	171	no	no	105	no	yes	65	no	yes	62	no	no	93
no	no	180	no	yes	108	no	yes	65	no	yes	72	no	no	95
no	no	185	no	no	115	no	no	67	no	no	73	no	no	96
no	no	190	no	no	118	no	no	71	no	yes	73	no	no	97
no	no	196	no	no	120	no	no	77	no	no	92	no	no	110
no	no	216	no	no	139	no	yes	79	no	no	93	no	no	122

References

• Farahnaky, A., Badii, F., Farhat, I. A., Mitchell, J. R., and Hill, S. E. (2005). Enthalpy relaxation of bovine serum albumin and implications for its storage in the glassy state. Biopolymers, 78, 69–77. doi: 10.1002/bip.20265
• Kestin, J., Sokolov, M., and Wakeham, W. A. (1978). Viscosity of liquid water in the range - 8 °C to 150 °C. J. Phys. Chem. Ref. Data, 7, 941–948. doi: 10.1063/1.555581
• Truong, V., Bhandari, B. R., Howes, T., and Adhikari, B. (2004). Glass transition behaviour of fructose. Int. J. Food Sci. Technol., 39, 569–578. doi: 10.1111/j.1365-2621.2004.00817.x
• Vargaftik, N. B., Volkov, B. N., and Voljak, L. D. (1983). International Tables of the Surface Tension of Water. J. Phys. Chem. Ref. Data, 12, 817–820. doi: 10.1063/1.555688