Recrystallization Behavior of Ice Crystals in Sucrose Solution Containing Sodium Caseinate

Abstract

The effect of adding sodium caseinate on ice recrystallization in 40 wt% sucrose solution at -10°C was investigated. A minor effect on the ice recrystallization rate constant was observed up to 2 wt% sodium caseinate. However, at higher concentrations, a decrease in the recrystallization rate was noted, indicating a suppressive effect on ice recrystallization. Differential scanning calorimetry revealed a linear decrease in the amount of freezable water with an increase in sodium caseinate concentration. Dielectric relaxation measurements at 25°C indicated an increase in the amount of free water with higher sodium caseinate concentrations, while the relaxation time of the water coupled with solutes tended to increase. It was suggested that the effect of adding sodium caseinate on ice crystal recrystallization was due to the combined result of the reduction in the amount of freezable water, the increase in free water content, and the decrease in water mobility.

Recrystallization Behavior of Ice Crystals in Sucrose Solution Containing Sodium Caseinate

Dan QIAO Zihao CHEN Mario SHIBATA, Tomoaki HAGIWARA

Department of Food Science and Technology, Tokyo University of Marine Science and Technology

(4-5-7 Konan, Minato-ku, Tokyo 108-0075, Japan)

Summary

The effect of adding sodium caseinate on ice recrystallization in 40 wt% sucrose solution at -10°C was investigated. A minor effect on the ice recrystallization rate constant was observed up to 2 wt% sodium caseinate. However, at higher concentrations, a decrease in the recrystallization rate was noted, indicating a suppressive effect on ice recrystallization. Differential scanning calorimetry revealed a linear decrease in the amount of freezable water with an increase in sodium caseinate concentration. Dielectric relaxation measurements at 25°C indicated an increase in the amount of free water with higher sodium caseinate concentrations, while the relaxation time of the water coupled with solutes tended to increase. It was suggested that the effect of adding sodium caseinate on ice crystal recrystallization was due to the combined result of the reduction in the amount of freezable water, the increase in free water content, and the decrease in water mobility.

Keywords: Ice recrystallization, Differential scanning calorimetry, Dielectric relaxation, Freezable water, Ice crystals, Water mobility

1. Introduction

Recrystallization of ice crystals is one of the key factors influencing changes in the quality of frozen food¹⁾. During the freezing process, the water in foods undergoes a phase transition, forming ice crystals. However, over time, these ice crystals recrystallize, resulting in an increase in their size. This phenomenon can disrupt the structure of food and impact its taste and texture. Particularly in products like ice cream²⁾, the recrystallization of ice crystals can alter the original smooth texture, resulting in a hard or uneven mouthfeel. Therefore, studying the ice recrystallization process is crucial for minimizing the degradation of quality in frozen foods.

Protein is one of the most abundant components of foods. Understanding the ice crystal recrystallization and its mechanism in the presence of proteins is important to control the ice crystal recrystallization in foods containing protein and thereby improve their quality³⁾. Although there have been many studies on the inhibitory effect of antifreeze protein on ice recrystallization^4-6), the effect of food proteins, such as those contained in foods that are primarily composed of proteins, on ice crystal recrystallization has not been extensively investigated.

Casein is commonly found in milk and its products and is one of the most important protein components in milk. It has good nutritional value and functional properties and is widely used in the dairy food industry⁷⁾. Therefore, studying the effect of casein on ice recrystallization is of great significance for our enhanced understanding of the ice recrystallization behavior in dairy food products. In particular, clarifying the ice recrystallization ability of casein would provide a foundation for understanding the recrystallization behavior of ice crystals in ice cream and similar frozen desserts. For example, it has been reported that certain polysaccharides, which are often added to these products as stabilizers, inhibit the recrystallization of ice crystals^8-10). However, the mechanism of action is not well understood. To elucidate this mechanism, experiments on the recrystallization of ice crystals in samples containing polysaccharides have been conducted^8,9); however, these studies utilized sugar solutions that do not contain milk proteins. Since it is known that protein–polysaccharide interactions can play a crucial role in the formation of structure and texture in dairy products¹¹⁾, the findings obtained from sugar solutions without milk proteins may not be directly applicable to ice cream and related dairy products. Furthermore, experiments using actual ice cream can provide data that is practically useful. However, because ice cream itself is a complex composition made up of multiple substances, it may sometimes be difficult to extract and identify the factors that contribute to the ice recrystallization behavior of ice cream and related dairy products. Given these considerations, it is significant to investigate the effect of casein, the main protein in milk that accounts for approximately 80 wt% of milk protein¹²⁾, on ice recrystallization to better understand the recrystallization behavior in ice cream and related dairy products. In fact, it has been reported that casein interacts with water and increases the viscosity of ice cream mixes and the size of ice crystals formed by freezing is reduced^13,14). These results suggest that casein also influences ice crystal recrystallization. Elucidating this effect will enhance our understanding of the recrystallization behavior of ice crystals in ice cream and similar frozen desserts. In this study, we investigated the effect of casein sodium salt (sodium caseinate) addition on the recrystallization of ice crystals in sucrose solution. Differential scanning calorimetry (DSC) and dielectric relaxation analysis were also performed to gain insight into the factors contributing to the effects of sodium caseinate addition.

Nomenclature

k recrystallization rate constant μm³･min^-1

r number-averaged size of ice crystals μm

r₀ value of r at t=0 μm

∆H_s melting enthalpy of sample solution J/g

∆H_ice melting enthalpy of pure

water J/g

j imaginary unit -

f frequency Hz

t elapsed time after sample temperature reached -10°C min

Greek symbols

β Cole–Cole parameter related

to distribution of relaxation time -

${\epsilon }^{*}$ Complex dielectric permittivity -

${\epsilon }^{\prime }$ real part of ${\epsilon }^{*}$ -

\varepsilon" imaginary part of ${\epsilon }^{*}$ -

${\epsilon }_{\infty }$ high frequency permittivity limits -

${\epsilon }_0$ dielectric permittivity of vacuum F･m^-1

$\Delta \epsilon$ dielectric relaxation intensity -

τ dielectric relaxation time ps (10^-12s)

σ direct current conductivity S･m^-1

Subscript

1 fast process

2 slow process

2. Materials and Methods

2.1 Materials

The casein sodium salt (sodium caseinate), manufactured by Sigma-Aldrich (Darmstadt, Germany) from bovine milk was used. Sucrose of reagent grade was purchased from Fujifilm Wako Pure Chemical Corp.(Osaka, Japan). Milli-Q water (Merck KGaA, Darmstadt, Germany) was used to prepare sample solution.

2.2 Preparation of sodium caseinate solution

Sodium caseinate solution was prepared using a 40wt% sucrose solution as the solvent at room temperateure (about 25 ℃). The concentration were set at 1, 2, 3, 4 and 5 wt%, respectively. Additionally, 40 wt% sucrose solution containing no sodium caseinate was prepared for comparison and used as a control.

2.3 Ice recrystallization observation

The similar method used in previous studies^2,5,6,15-17) was employed. A 2 µL of the sodium caseinate solution was sandwiched between two circular glass cover slips (diameter: 16 mm, Linkam Scientific Instruments, Tadworth, UK) and placed on a heating and freezing microscope stage (THMS600, Linkam Scientific Instruments, Tadworth, UK) equipped with an optical microscope (BX-53, Olympus Corp., Tokyo, Japan). Initially, the sample temperature was maintained at 30℃ for 10 min. Subsequently, the sample was frozen by reducing the temperature to -30℃ at a rate of 90℃/min and kept at that temperature for 10 min. The sample temperature was then raised from -30℃ to -10℃ at a rate of 10℃/min and kept constant at -10℃. Upon reaching -10℃, ice crystal image obtained by the microscope were taken with a microscopic camera (DS-Fi2, Nikon Corporation, Tokyo) at specific time interval for up to 120 min.

The images were then analyzed by two image analysis software programs (PopImaging; Digital being Kids Corp., Yokohama, Japan and Image J¹⁸⁾) as in previous studies^5,6,15). The size of each ice crystal was evaluated as the radius of a circle with the same area as the ice crystal on the image (equivalent size) and the number-averaged value was obtained. The recrystallization rate constant k was determined based on the Ostwald ripening equation¹⁹⁾ as

𝑟³ = 𝑟₀³ + 𝑘𝑡 (1)

where r is the number-average ice crystal size (mean equivalent size), t corresponds to the elapsed time after sample temperature reached -10℃, and r₀ represents the value of r at t=0. The value of k was evaluated from the slope of r³ versus t. In this study, the recrystallization rate constants of ice crystals in 40wt% sucrose solution containing 1, 2, 3, 4, and 5 wt% sodium caseinate were experimentally determined. The value for 40 wt% sucrose solution without sodium caseinate was obtained from our previous research¹⁵⁾.

2.4 Differential scanning calorimetry (DSC)

The ice melting enthalpy of the sample solutions was determined by using a differential scanning calorimeter (Diamond DSC, Perkin Elmer U.S. LLC, Connecticut, USA). The DSC equipment was calibrated using indium and deionized water. The DSC system was purged with dry nitrogen. A mechanical cooler (Intracooler 2P; Perkin Elmer U.S. LLC, Connecticut, USA) was used to cool the system. Approximately 9-16 mg of the sample solution was hermetically enclosed in an aluminum sample cell (Perkin Elmer U.S. LLC, Shelton, Connecticut, USA) and set in the DSC equipment. Sample was cooled from 25℃ to -35℃ at 10℃/min and annealed at -35℃ for 30 min, following a similar procedure to previous studies^20,21). It was then cooled to -50℃ and subsequently heated to 25℃ at 10℃/min. Heat flow changes during this heating process were recorded. Figure 1 shows the typical DSC thermogram data of the sample solutions (sodium caseinate concentration: 1%). A stepwise change around -32℃ (the onset of ice melting) in the baseline was observed, followed by the endothermic peak corresponding to ice melting as in previous studies^22-24). Similar stepwise and endothermic peak were observed at other sodium caseinate concentrations. In this study, the midpoint of the stepwise baseline shift was taken as a start of ice melting and the enthalpy of ice melting ∆𝐻_s (J/g) was evaluated as preceding study²⁴⁾. The percentage of freezable water content in the total sample solution weight was calculated as follows:

Percentage of freezable water =(∆𝐻_s)/(∆𝐻_ice)×100%　 (2)

where ∆H_ice is the enthalpy of ice melting for pure water (=333.7 J/g).

2.5 Dielectric relaxation measurement

The similar method used in previous studies¹⁵⁾ was employed. A schematic description of the experimental apparatus is shown in Fig.2¹⁵⁾. It included a network analyzer (8720B; Hewlett-Packard Company, Palo Alto, USA), a dielectric probe (85070B; Hewlett-Packard Company, Palo Alto, USA), a coaxial cable, a PC for controlling the network analyzer, a temperature sensor device for monitoring the sample temperature (Model 7563; Yokogawa Test & Measurement Corp., Tokyo, Japan), and a thermostatic chamber for controlling the sample temperature (MC-710, Espec Corp., Osaka, Tokyo). The thermostatic chamber temperature was set to 25℃ and left to stabilize for 1h. Subsequently, calibration of network analyzer was performed using air, short, and water as standards. A 15mL sample solution was placed in the measurement vessel and positioned in the thermostatic chamber to equilibrate the sample temperature to 25°C. The sample complex dielectric permittivity at frequencies ranging from 130 MHz to 19.5 GHz were then collected 20 min after the sample temperature reached 25°C. As same as the ice recrystallization rate constant, the sample complex dielectric permittivities of 40wt% sucrose solution containing 1, 2, 3, 4, and 5 wt% sodium caseinate were experimentally measured in this study. That for the 40wt% sucrose solution without sodium caseinate was obtained from our previous research¹⁵⁾.

Similar to the recrystallization rate constant, the complex dielectric permittivity of 40wt% sucrose solution containing 1, 2, 3, 4, and 5 wt% sodium caseinate solution were experimentally measured in the present study. The data for the 40% sucrose solution without casein was obtained from our previous research¹⁵⁾.

In the sample solutions containing sodium caseinate, a trend of increase in \varepsilon" was observed with decreasing frequency in the lower frequency range, despite no corresponding relaxation in ${\epsilon }^{\prime }$ . This phenomenon was attributed to the presence of ions originating from sodium caseinate in the solutions, leading to direct current (DC) conductivity²⁵⁾. Therefore, to analyze the experimental data of sodium caseinate solution, the complex dielectric permittivity ${\epsilon }^{*}$ was fitted by the double Cole–Cole equation which includes the term of the DC conductivity^15,25), as shown in Eqs.(3) and (4)

\varepsilon^{*} = \varepsilon^{'} + \varepsilon" (3)

${\epsilon }^{*}\left(f\right)={\epsilon }_{\infty }+\frac{\Delta {\epsilon }_1}{1+{\left(2\pi j{\tau }_1\right)}^{{\beta }_1}}+\frac{\Delta {\epsilon }_2}{1+{\left(2\pi j{\tau }_2\right)}^{{\beta }_2}}+\frac{\sigma }{2\pi j{\epsilon }_0}$ (4)

where j is the imaginary unit, Δε and τ represent the relaxation intensity and relaxation time, β (0 <β ≤ 1) is the Cole–Cole parameter related to the distribution of relaxation time, ${\epsilon }_{\infty }$ is the high frequency permittivity limits, σis the direct current conductivity, and ${\epsilon }_{\infty }$ indicates the dielectric permittivity of vacuum. The subscripts 1 and 2 indicate the fast and slow processes (τ₁<τ₂), respectively; the relaxation process 1 is faster than that of process 2.

40 wt% sucrose solution containing no sodium caseinate was analyzed using the double Cole–Cole equation, excluding the DC conductivity term as below.

${\epsilon }^{*}\left(f\right)={\epsilon }_{\infty }+\frac{\Delta {\epsilon }_1}{1+{\left(2\pi j{\tau }_1\right)}^{{\beta }_1}}+\frac{\Delta {\epsilon }_2}{1+{\left(2\pi j{\tau }_2\right)}^{{\beta }_2}}$ (5)

In order to discuss the effect of sodium caseinate addition on the recrystallization behavior of ice crystals based on the change in water mobility, it was preferable to set the sample temperature to -10°C as in the recrystallization experiment. However, the preliminary experimental data of the sample in a frozen state at -10°C was not reproducible and there was a lot of noise in the data, so we only utilized the data obtained at 25°C in this study. It should be noted that the effect of sodium caseinate addition at -10°C may differ from that at 25°C, as the concentration of sucrose and sodium caseinate in the freeze-concentrated phase was higher.

2.6 Statistical analysis

In each experiment, three or more measurements were taken under identical conditions, and the mean value and standard deviation were computed.

3. Results and Discussion

3.1 Ice recrystallization process

Figure 3 shows the example of ice crystal images of 40 wt% sucrose solutions containing 1 to 5 wt% sodium caseinate at -10°C. In all sample solutions, the ice crystal sizes at 120 min were larger than those at 20 min, demonstrating that ice recrystallization proceeded within 120 min. The example of plots between the cube of the number-averaged ice crystal size r³ and elapsed time t are shown in Fig. 4. The plot exhibited a linear trend for all sodium caseinate concentrations, and the ice recrystallization rate constant k was calculated using Eq. (1), with correlation coefficients (R²) all over 0.95, most over 0.99. Figure 5 illustrates the plot of the ice recrystallization rate constant as a function of sodium caseinate concentration. Up to a sodium caseinate concentration of 2 wt%, there was little effect on the ice recrystallization rate constant. At the concentrations above 4 wt%, it was notably smaller than that of the control sample, and the progress of recrystallization was suppressed.

3.2 DSC analysis

Figure 6 shows the effect of sodium caseinate concentration on the percentage of freezable water in the total sample weight. The freezable water percentage exhibited a linear decrease as the sodium caseinate concentration increased. This indicates that the formation of ice in the sample decreased as the sodium caseinate concentration increased. The reduction in ice formation can decrease the frequency of contact between ice crystals, thereby slowing down the progression of recrystallization. In fact, this slowing down by the ice formation reduction has been supported by both theoretical^26-28) and experimental^27,29) researches.

3.3 Dielectric Relaxation Measurement

Figure 7 shows the typical complex dielectric permittivity (up: ${\epsilon }^{\prime }$ , below: \varepsilon" , sodium caseinate concentration: 5 wt%). Both ${\epsilon }^{\prime }$ and \varepsilon" were well-fitted well by Eq. (4). This result indicated that there were two relaxation processes (fast process 1 and slow process 2). Table 1 summarizes the dielectric relaxation parameters obtained from the fitting.

The dielectric relaxation time is correlated with the rotational mobility of electric dipoles and equivalent to the time needed for the electric dipoles to align in the direction of the electric field^15,25). It has been generally recognized that in aqueous solutions, fast relaxation is due to water molecules less coupled with solutes, while slow relaxation is attributed to the water molecules that are strongly coupled with solutes^15,30-33). Therefore, the origins of the two relaxation processes identified in this study were also interpreted to be the same as in previous studies¹⁵⁾, namely, free water (process 1: fast) and the relaxation of water molecules coupled with solutes (process 2: slow).

Many dielectric relaxation measurements of solutions containing proteins have been conducted under conditions similar to those of this study, including frequency range and temperature^34-41). It is commonly reported that three relaxation processes—β, δ, and γ—are observed in the presence of protein³⁴⁾, beginning from the low-frequency side. The γ-relaxation is associated with the reorientation of free water molecules, exhibiting a peak at approximately 18 GHz in the imaginary part of the dielectric permittivity ( \varepsilon" ) at room temperature. The β-relaxation process pertains to the rotation of protein molecules, corresponding to a peak in the range of 1 MHz to 100 MHz in \varepsilon" . The δ process, which occurs at a frequency between γ and β, remains a topic of debate³⁴⁾; however, it is strongly suggested that it is related to bound water. We will demonstrate that there was no contradiction between these existing interpretations and the data interpretation employed in this study. To demonstrate this, we will compare our research results with the dielectric relaxation measurements of lysozyme aqueous solutions conducted by Wolf et al³⁴⁾. They performed dielectric relaxation measurements on lysozyme aqueous solutions at similar concentrations (approximately 4%) and temperatures (22°C and 32°C) across a frequency range from 1 MHz to 40 GHz. According to their findings, the γ-relaxation was observed at around 20 GHz, the β process at approximately 10 MHz, and the δ process at around 200 MHz. In this study, the fast process was interpreted as being associated with the reorientation of free water, which woud correspond to γ-relaxation. However, the peak in the dielectric loss attributed to the fast process appeared at several GHz (see Fig. 2), which was much lower than the peak frequency of the γ-relaxation of the lysozyme solution (approximately 20 GHz) reported by Wolf et al³⁴⁾. This discrepancy can be explained by the effects of slow dynamics due to the presence of 40 wt% sucrose, as noted in numerous previous studies^15,25,30). Regarding the slow process in this study, it was interpreted as originating from the bound water of the solute (sucrose or casein), similar to existing studies of dielectric relaxation in sugar solutions¹⁵⁾, which would correspond to δ-relaxation. We did not consider the existence of β-relaxation, even in samples with relatively high concentrations of sodium caseinate, for two reasons. First, the measured values were successfully fitted using the double Cole-Cole equation (Eq. (3)). Second, since casein (molecular weight: 20 kDa-25 kDa) has a larger molecular weight than lysozyme (14 kDa) and 40 wt% sucrose was present, it was expected that the β-relaxation, which reflects the rotation of the casein molecule, would appear at a lower frequency than that of lysozyme (10 MHz)³⁴⁾, and that this frequency would be well below the lower limit of the frequency range in this study (130 MHz). Therefore, we judged that the effect of the β-relaxation of casein on the dielectric relaxation data was small. It should be noted that the slow process in this study may be affected by the relaxation based on the rotational motion of the sucrose molecules, i.e., β-relaxation of the sucrose molecule, but following previous studies^15,30-33), the slow process was interpreted as the the relaxation of water molecules coupled with solutes only.

The slow relaxation time had a tendency of increase with an increase with sodium caseinate concentration, but there is no such trend in the fast relaxation time. These results suggested that the effect of increasing sodium caseinate concentration on water mobility differed between the free water and those coupled with solutes.

The relaxation intensity of the fast relaxation (Δε₁) demonstrated an increase with an increase in sodium caseinate concentration, while that of the slow relaxation (Δε₂) exhibited a decrease. The dielectric relaxation intensity is generally a measure of the intensity of the response of a dielectric material to an external alternating electric field^15,42,43). An increased relaxation intensity indicates that a greater number of molecules or ions within the material are engaged in the reorientation or rearrangement process. However, it has been demonstrated that different coexisting substances can have different dielectric intensities despite the presence of an equivalent quantity of water molecules⁴⁴⁾. It is therefore essential to proceed with caution when discussing the potential correlation between the observed change in relaxation intensity and sodium caseinate concentration differences. In spite of this uncertainty, this study assumed that the relaxation intensity was positively correlated with the amount of free water or solute- coupled water as previous study¹⁵⁾. The trend of Δε₁ and Δε₂ suggested that as the sodium caseinate concentration increased, the amount of free water increased. At the same time, that of water coupled with solutes decreased. Similar behavior has been observed in sucrose solution containing sodium alginate¹⁵⁾; increasing sodium alginate concentration resulted in increase of Δε₁ and decrease of Δε₂. Previous research⁴⁵⁾ using nuclear magnetic resonance and molecular dynamics techniques indicated that in concentrated aqueous sucrose solutions, sucrose molecules interact with one another to form a cage-like structure. Within this structure, certain water molecules become confined^15,45). Due to the constrained mobility of these water molecules, some of them may exhibit characteristics of solute-coupled water¹⁵⁾. The addition of sodium caseinate may potentially disrupt the integrity of the cage-like structure in a similar way as sodium alginate¹⁵⁾, which could result in a reduction in the amount of water molecules confined within the cage-like structure and an increase in the amount of free water molecules¹⁵⁾.

In this study, the measurement data at -10°C was not used because it was not reproducible and there was a lot of noise. According to previous studies, there are examples of successfull measurements of frozen aqueous solutions containing proteins (gelatin⁴⁶⁾ and bovine serum albumin⁴⁷⁾). According to these^46-48) and other previous studies about dielectiric properties of ice^48-50), the relaxation time of the ice contained in the frozen aqueous solution at -10°C was generally larger than 10^-4 sec. Such the relaxation would occur at a frequency sufficiently lower than the frequency range accessible by this study (130 MHz-20 GHz). Therefore, by accurately removing noise and factors that interfere with reproducibility, it should be possible to obtain information about the relaxation behavior of the substances contained in the froze-concentrated phase. In particular, since previous research has shown that the mobility of water in the froze-concentrated phase was one of the factors controlling the ice recrystallization rate constant¹⁷⁾, it is desirable to elucidate its relaxation behavior. In general, measurements of dielectric properties are strongly affected by the measurement temperature, so strict temperature control is required. In addition, condensation may occur on the surface of the cable and probe in the subzero temperature range. This can also cause noise. The researches menthioned above, which succeeded in the measurement of frozen aqueous protein solution, may carefully suppress the factors that could cause such noise and low reproducibility. In order to truly clarify the effect of adding sodium caseinate on the ice recrystallization behavior, it is still necessary to obtain the acceptable data of the frozen sample solution, and for this reason, trials for such as strictly controlling the temperature and suppressing noise and other factors that could lead to a decrease in reproducibility should be conducted in the future.

3.4 Effect of sodium caseinate addition and its mechanism

Up to a sodium caseinate concentration of around 2 wt%, there was little effect on the rate constant of ice recrystallization. Above 4 wt%, a clear decrease in the recrystallization rate constant was observed. It was not easy to explain this behavior simply using the results of DSC measurements or dielectric relaxation measurements. The DSC results showed that the addition of sodium caseinate decreased the amount of freezable water and ice, which would work towards suppressing ice recrystallization. The dielectric relaxation measurements indicated that the addition of sodium caseinate increased the amount of free water. Increasing the amount of free water would promote the recrystallization process. However, no clear decrease in the fast processes was observed as the sodium caseinate concentration increased.

As described above, the effect of sodium caseinate concentration on ice recrystallization as predicted from the DSC or dielectric relaxation measurement, does not match the experimental results. Thus, it was suggested that the factors contributing to the effect of sodium caseinate concentration on ice recrystallization were not straightforward. The recrystallization of ice crystals may be influenced by a multitude of factors, including a decrease in freezable water, an increase in the amount of free water, and a decrease in water mobility. When the sodium caseinate concentration increased sufficiently, the effects of reduced water mobility and decrease in freezable water content may be strong enough to overcome the effect of an increase in free water content, resulting in clear recrystallization suppression.

4. Conclusions

Up to a concentration of 2 wt% sodium caseinate, there was little effect on the ice recrystallization rate constant. However, at higher concentrations, a decrease in the recrystallization rate constant was observed, confirming the effect of inhibiting ice recrystallization by sodium caseinate. The effect of sodium caseinate on ice recrystallization may be attributed to a combination of multiple factors, such as reduced freezable water, increased free water content, and decreased water mobility. The extent to which these multiple factors contribute to the actual ice recrystallization process would be a topic for future research.

Finally, it should be noted that the effect of fat, another main ingredient, should also be examined in order to fully understand the recrystallization behavior in ice cream and related ice desserts. The amount and type of fat have been reported to affect the sensory properties, melting, color, and hardness of ice cream^51-53). In order to investigate the effect of milk fat on recrystallization, it would be necessary to conduct experiments using samples in which milk fat is added to sucrose solutions, but to the best of my knowledge, we have not seen any such experiments. As mentioned above, since milk fat is one of the main ingredients of ice cream, the effect of milk fat should also be investigated in future research.

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27. 27)   Budke, C., Heggemann, C., Koch, M., Sewald, N. and Koop, T., Ice Recrystallization Kinetics in the Presence of Synthetic Antifreeze Glycoprotein Analogues Using the Framework of LSW Theory, Journal of Physical Chemistry B, 2009, 113, pp.2865-2873
28. 28)   Voorhees, P.W., Glicksman, M.E. Ostwald Ripening during Liquid Phase Sintering—Effect of Volume Fraction on Coarsening Kinetics, Metallurgical and Materials Transactions A, 1984, 15, pp.1081–1088.
29. 29)   Sutton, R.L., Lips, A., Piccirillo, G. and Sztehlo, A., Kinetics of Ice Recrystallization in Aqueous Fructose Solutions. Journal of Food Science, 1996, 61, pp.741-745
30. 30)   Fuchs, K. and Kaatze, U., Dielectric Spectra of Mono- and Disaccharide Aqueous Solutions, Journal of Chemical Physics, 2002, 116, pp.7137–7144.
31. 31)   Yamamoto, W., Sasaki, K., Kita, R., Yagihara, S. and Shinyashiki, N. Dielectric Study on Temperature Concentration Superposition of Liquid to Glass in Fructose-Water Mixtures, Journal of Molecular Liquids, 2015, 206, pp. 39–46.
32. 32)   Mashimo, S., Kuwabara, S., Yagihara, S. and Higasi, K., Dielectric Relaxation Time and Structure of Bound Water in Biological Materials, Journal of Physical Chemistry, 1987, 91, pp. 6337–6338.
33. 33)   Mashimo, S., Umehara, T., Ota, T., Kuwabara, S., Shinyashiki, N. and Yagihara, S., Evaluation of Complex Permittivity of Aqueous Solution by Time Domain Reflectometry, Journal of Molecular Liquids, 1987, 36, pp. 135–151.
34. 34)   Wolf, M., Gulich, R., Lunkenheimer, P. and A. Loid A., Relaxation Dynamics of a Protein Solution Investigated by Dielectric Spectroscopy, Biochimica et Biophysica Acta (BBA) - Proteins and Proteomics, 2012, 1824, pp.723-730.
35. 35)   Cametti, C., Marchetti, S., Gambi, C.M.C. and G. Onori, G., Dielectric Relaxation Spectroscopy of Lysozyme Aqueous Aolutions, Journal of Physical Chemistry B, 2011, 115, pp.7144-7153
36. 36)   Pethig, P., Protein–Water Interactions Determined by Dielectric Methods, Annual Review of Physical Chemostry, 1992, 43, pp.177-205.
37. 37)   Nandi, N., Bagchi, B., Anomalous Dielectric Relaxation of Aqueous Protein Solutions, Journal of Physical Chemistry A, 1998, 102, pp.8217-8221.
38. 38)   Bonincontro, A., De Francesco, A., Onori, G., Temperature-Induced Conformational Changes of Native Lysozyme in Aqueous Solution Studied by Dielectric Spectroscopy, Chemical Physics Letters, 1999, 301, pp.189-192.
39. 39)   Knocks, A. and Weingärtner, H., The Dielectric Spectrum of Ubiquitin in Aqueous Solution, Journal of Physical Chemistry B, 2001, 105, pp.3635-3638.
40. 40)   Feldman, Y., Ermolina, I. and Hayashi, Y., Time Domain Dielectric Spectroscopy Study of Biological Systems, IEEE Transactions on Dielectrics and Electrical Insulation, 2003, 10, pp.728-753.
41. 41)   Oleinikova, A., Sasisanker, P. and Weingartner, H., What Can Really be Learned from Dielectric Spectroscopy of Protein Solutions?, Journal of Physical Chemistry B, 2004, 108, pp. 8467-8474.
42. 42)   Kishikawa, Y., Seki, Y., Shingai, K., Kita, R., Shinyashiki, N. and Yagihara, S., Dielectric Relaxation for Studying Molecular Dynamics of Pullulan in Water, Journal of Physical Chemistry, 2013, 117, pp. 9034–9041.
43. 43)   Yagihara, S., Oyama, M., Inoue, A., Asano, M., Sudo, S. and Shinyashiki, N. Dielectric Relaxation Measurement and Analysis of Restricted Water
44. 44)   Structure in Rice Kernels, Measurement Science and Technology, 2007, 18, pp. 983–990.
45. 45)   Naito, S., Hoshi, M. and Mashimo, S. In Vivo Dielectric Analysis of Free Water Content of Biomaterials by Time Domain Reflectometry, Analytical Biochemistry, 1997, 251, pp. 163–172.
46. 46)   Ekdawi-Sever, N., De Pablo, J. J., Feick, E. and Von Meerwall, E., Diffusion of Sucrose and α, α-Trehalose in Aqueous Solutions. Journal of Physical Chemistry A, 2003, 107, pp. 936-943.
47. 47)   Yasuda, T., Sasaki, K., Kita, R., Shinyashiki, N. and Yagihara, S., Dielectric Relaxation of Ice in Gelatin–Water Mixtures, Journal of Physical Chemistry B, 2017, 121, 2896-2901.
48. 48)   Tsukahara, T., Sasaki, S., Kita, R. and Shinyashiki, N., Dielectric Relaxations of Ice and Uncrystallized Water in Partially Crystallized Bovine Serum Albumin–Water Mixtures, Physical Chemistry Chemical Physics, 2022,24, 5803-5812.
49. 49)   Sasaki, K., Kita, R., Shinyashiki, N., Yagihara, S., Dielectric Relaxation Time of Ice-Ih with Different Preparation, Journl of Physical Chemistry B, 2016, 120, pp.3950– 3953.
50. 50)   Johari, G. P., Whalley, E., The Dielectric Properties of Ice Ih in the Range 272−133 K, Journal of Chemical Physics, 1981,75, pp.1333-1340.
51. 51)   Auty, R. P., Cole, R. H., Dielectric Properties of Ice and Solid D₂O. Journal of Chemical Physics, 1952, 20, pp.1309−1314.
52. 52)   Roland, A. M., Phillips, L. G. and Boor, K. J., Effects of Fat Content on the Sensory Properties, Melting, Color, and Hardness of Ice Cream, Journal of Dairy Science, 1999, 82, pp.32-38.
53. 53)   Li, Z., Marshall, R., Heymann, H. and Lakdas Fernando, L., Effect of Milk Fat Content on Flavor Perception of Vanilla Ice Cream,　 Journal of Dairy Science, 1997, 80, pp.3133-3141
54. 54)   Hyvönen, L., Linna, M., Tuorila, H. and Dijksterhuis, G., Perception of Melting and Flavor Release of Ice Cream Containing Different Types and Contents of Fat, Journal of Dairy Science, 2003, 86, pp.1130-1138.

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• 26) Sutton, R.L., Evans, I.D. and Crilly, J.F. (1994), Modeling Ice Crystal Coarsening in Concentrated Disperse Food Systems. Journal of Food Science, 1994, 59, pp.1227-1233.
• 27) Budke, C., Heggemann, C., Koch, M., Sewald, N. and Koop, T., Ice Recrystallization Kinetics in the Presence of Synthetic Antifreeze Glycoprotein Analogues Using the Framework of LSW Theory, Journal of Physical Chemistry B, 2009, 113, pp.2865-2873
• 28) Voorhees, P.W., Glicksman, M.E. Ostwald Ripening during Liquid Phase Sintering—Effect of Volume Fraction on Coarsening Kinetics, Metallurgical and Materials Transactions A, 1984, 15, pp.1081–1088.
• 29) Sutton, R.L., Lips, A., Piccirillo, G. and Sztehlo, A., Kinetics of Ice Recrystallization in Aqueous Fructose Solutions. Journal of Food Science, 1996, 61, pp.741-745
• 30) Fuchs, K. and Kaatze, U., Dielectric Spectra of Mono- and Disaccharide Aqueous Solutions, Journal of Chemical Physics, 2002, 116, pp.7137–7144.
• 31) Yamamoto, W., Sasaki, K., Kita, R., Yagihara, S. and Shinyashiki, N. Dielectric Study on Temperature Concentration Superposition of Liquid to Glass in Fructose-Water Mixtures, Journal of Molecular Liquids, 2015, 206, pp. 39–46.
• 32) Mashimo, S., Kuwabara, S., Yagihara, S. and Higasi, K., Dielectric Relaxation Time and Structure of Bound Water in Biological Materials, Journal of Physical Chemistry, 1987, 91, pp. 6337–6338.
• 33) Mashimo, S., Umehara, T., Ota, T., Kuwabara, S., Shinyashiki, N. and Yagihara, S., Evaluation of Complex Permittivity of Aqueous Solution by Time Domain Reflectometry, Journal of Molecular Liquids, 1987, 36, pp. 135–151.
• 34) Wolf, M., Gulich, R., Lunkenheimer, P. and A. Loid A., Relaxation Dynamics of a Protein Solution Investigated by Dielectric Spectroscopy, Biochimica et Biophysica Acta (BBA) - Proteins and Proteomics, 2012, 1824, pp.723-730.
• 35) Cametti, C., Marchetti, S., Gambi, C.M.C. and G. Onori, G., Dielectric Relaxation Spectroscopy of Lysozyme Aqueous Aolutions, Journal of Physical Chemistry B, 2011, 115, pp.7144-7153
• 36) Pethig, P., Protein–Water Interactions Determined by Dielectric Methods, Annual Review of Physical Chemostry, 1992, 43, pp.177-205.
• 37) Nandi, N., Bagchi, B., Anomalous Dielectric Relaxation of Aqueous Protein Solutions, Journal of Physical Chemistry A, 1998, 102, pp.8217-8221.
• 38) Bonincontro, A., De Francesco, A., Onori, G., Temperature-Induced Conformational Changes of Native Lysozyme in Aqueous Solution Studied by Dielectric Spectroscopy, Chemical Physics Letters, 1999, 301, pp.189-192.
• 39) Knocks, A. and Weingärtner, H., The Dielectric Spectrum of Ubiquitin in Aqueous Solution, Journal of Physical Chemistry B, 2001, 105, pp.3635-3638.
• 40) Feldman, Y., Ermolina, I. and Hayashi, Y., Time Domain Dielectric Spectroscopy Study of Biological Systems, IEEE Transactions on Dielectrics and Electrical Insulation, 2003, 10, pp.728-753.
• 41) Oleinikova, A., Sasisanker, P. and Weingartner, H., What Can Really be Learned from Dielectric Spectroscopy of Protein Solutions?, Journal of Physical Chemistry B, 2004, 108, pp. 8467-8474.
• 42) Kishikawa, Y., Seki, Y., Shingai, K., Kita, R., Shinyashiki, N. and Yagihara, S., Dielectric Relaxation for Studying Molecular Dynamics of Pullulan in Water, Journal of Physical Chemistry, 2013, 117, pp. 9034–9041.
• 43) Yagihara, S., Oyama, M., Inoue, A., Asano, M., Sudo, S. and Shinyashiki, N. Dielectric Relaxation Measurement and Analysis of Restricted Water
• 44) Structure in Rice Kernels, Measurement Science and Technology, 2007, 18, pp. 983–990.
• 45) Naito, S., Hoshi, M. and Mashimo, S. In Vivo Dielectric Analysis of Free Water Content of Biomaterials by Time Domain Reflectometry, Analytical Biochemistry, 1997, 251, pp. 163–172.
• 46) Ekdawi-Sever, N., De Pablo, J. J., Feick, E. and Von Meerwall, E., Diffusion of Sucrose and α, α-Trehalose in Aqueous Solutions. Journal of Physical Chemistry A, 2003, 107, pp. 936-943.
• 47) Yasuda, T., Sasaki, K., Kita, R., Shinyashiki, N. and Yagihara, S., Dielectric Relaxation of Ice in Gelatin–Water Mixtures, Journal of Physical Chemistry B, 2017, 121, 2896-2901.
• 48) Tsukahara, T., Sasaki, S., Kita, R. and Shinyashiki, N., Dielectric Relaxations of Ice and Uncrystallized Water in Partially Crystallized Bovine Serum Albumin–Water Mixtures, Physical Chemistry Chemical Physics, 2022,24, 5803-5812.
• 49) Sasaki, K., Kita, R., Shinyashiki, N., Yagihara, S., Dielectric Relaxation Time of Ice-Ih with Different Preparation, Journl of Physical Chemistry B, 2016, 120, pp.3950– 3953.
• 50) Johari, G. P., Whalley, E., The Dielectric Properties of Ice Ih in the Range 272−133 K, Journal of Chemical Physics, 1981,75, pp.1333-1340.
• 51) Auty, R. P., Cole, R. H., Dielectric Properties of Ice and Solid D2O. Journal of Chemical Physics, 1952, 20, pp.1309−1314.
• 52) Roland, A. M., Phillips, L. G. and Boor, K. J., Effects of Fat Content on the Sensory Properties, Melting, Color, and Hardness of Ice Cream, Journal of Dairy Science, 1999, 82, pp.32-38.
• 53) Li, Z., Marshall, R., Heymann, H. and Lakdas Fernando, L., Effect of Milk Fat Content on Flavor Perception of Vanilla Ice Cream,　Journal of Dairy Science, 1997, 80, pp.3133-3141 38.
• 54) Hyvönen, L., Linna, M., Tuorila, H. and Dijksterhuis, G., Perception of Melting and Flavor Release of Ice Cream Containing Different Types and Contents of Fat, Journal of Dairy Science, 2003, 86, pp.1130-11