Complex heat capacity of non-Debye process in glassy glucose and fructose

Fluid Phase Equilibria 256 (2007) 123–126

Complex heat capacity of non-Debye process in glassy glucose and fructose Yuji Ike ∗ , Yuichi Seshimo, Seiji Kojima Graduate School of Pure and Applied Sciences, University of Tsukuba, Tsukuba, Ibaraki 305-8573, Japan Received 2 October 2006; received in revised form 15 April 2007; accepted 15 April 2007 Available online 20 April 2007

Abstract Complex heat capacity, Cp∗ = Cp − iCp , of pure glucose and fructose glasses have been investigated by modulated-temperature differential scanning calorimetry (MDSC). According to the linear response theory, real and imaginary parts of complex heat capacity have obtained as a function of temperature through a glass transition. The non-Debye property of thermal relaxation process has examined by means of the Cole–Cole plot. The Cole–Cole plots of two monosaccharide glasses exhibit the non-Debye nature. The results indicate the distribution of relaxation times in fructose is broader than that of glucose. It reflects the broader degree of freedom in motions and conformations in fructose with comparing the glucose. © 2007 Elsevier B.V. All rights reserved. Keywords: Complex heat capacity; Liquid glass transition; Saccharide; Thermal relaxation; Non-Debye

1. Introduction Glassy state of saccharide is found in nature as biopreservation agents of biological tissues and life itself under the extreme circumstances of cold and drought [1,2]. The biopreserving mechanism is mainly on the packing of reactive biomaterials in a rigid matrix of glasses. These properties are imitated in the food processing and pharmaceutical applications. Saccharides are often used as the packing agents for biomaterials [3]. Although various applications are available, the fundamental understandings of physical properties and molecular dynamics of glass transitions are not fully understood yet [4–6]. Glass has the random structures without any translational symmetry, and undergoes a liquid–glass transition from supercooled liquid to glassy states at a glass transition temperature (Tg ). Glass is formed by cooling a viscous liquid in a rapid rate enough to avoid crystallization. In the vicinity of Tg , the motions of atoms or molecules in a liquid phase become quite slow and finally the liquid is “frozen” in practical time scale. The numerous investigations clarified that various relaxation processes have the dominant role in glass transitions, such as ␣-relaxation,

∗

Corresponding author. E-mail address: [email protected] (Y. Ike).

0378-3812/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.fluid.2007.04.017

␤-relaxation, boson dynamics. These relaxation processes are well explored by several investigation techniques of dielectric relaxation, calorimetry, nuclear magnetic resonance, light scattering, neutron scattering and other techniques [5]. Since the glass transitions are considered as relaxation phenomena, the physical quantities exhibit time- or frequency-dependent, and the behaviors are well expressed by dynamical or complex susceptibility χ(ω). The complex susceptibility of χ(ω) is related to the relaxation function, ψ(t), of the system by the relation of linear response theory [7]. Therefore, ψ(t) represents the characteristic feature of relaxations and non-Debye behavior is prevalent in the distribution function through glass transitions. One of the major methods, dielectric relaxation spectroscopy provides detail in both relaxation time and distribution in wide temperature and frequency ranges. The dielectric relaxation properties have been widely studied [8]. The electric perturbation affects the response of polarization, thus the χ(ω) is deduced to the complex dielectric susceptibility in the case of dielectric spectroscopy. Since dielectric techniques investigate the behavior of polar motions, the dielectric relaxation relates only to the orientational motions of polar groups of molecules. Moreover, when the applied perturbation is temperature, the complex specific heat capacity Cp∗ (ω, τ(T )) can be defined. To compare with the dielectric relaxations, the thermal relaxations are originated from “total” degree of freedom in motions, translational,

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orientational and vibrational movements [9,10]. Therefore, the relaxation provides more general insights into the relaxation phenomena in glass transitions. In the early 1990, the technical improvement of Calorimetry was performed by Reading which is called modulated-DSC (MDSC) [11]. It has been successfully applied to the study of the glass transition dynamics [12–14]. While Birge et al. observed the Cp∗ (ω) of some organic glass forming liquids such as glycerol, the method has not been widely used due to their technical difficulties [9,10]. Since the invention of MDSC equipments, MDSC is commercially available and the investigation of Cp∗ (ω) has been familiar in the field of thermodynamics. From the modulated contribution of heat flow signals, one can easily obtain the frequency-dependent complex specific heat. Therefore, the new technique of MDSC may have an important role in the study of dynamics in glass transitions. It is relatively new technique, thus the fundamental aspects of thermal relaxations in several glass forming materials are scarce. Therefore, we focused on the simple biopreserving agent of lower saccharide glasses to clarify the basic nature of the relaxation behavior in the vicinity of Tg . In the present study, we have investigated Cp∗ (ω) of glucose and fructose glasses to attempt a new approach for the characterization of non-Debye nature in relaxation approach. 2. Experimental We used melt-quenching method to make a sample of glucose glass. To eliminate the water contents in the sample, chemical reagent of anhydrous glucose (Sigma) and fructose were used. They must be carefully heated around melting temperature since it is easily caramelized at higher temperature. Then the melted glucose was quenched to vitrify and immediately investigated. MDSC measurements were performed by TA Instruments DSC2920 with Liquid Nitrogen Cooling Accessory. The instrumental calibrations were preliminary done for accurate measurements in temperature, enthalpy and heat capacity. The same mass of aluminum pans for reference and sample were prepared within the 0.1 mg error. Apiece of prepared thin glass sample, around 7 mg mass, was placed in the sample pan. To contact between the sample and the aluminum pan, once melted the sample at 100 ◦ C and cooled. Then put the reference and sample pans into the MDSC sample chamber together. The temperature procedure is started from 90 to 0 ◦ C with cooling. The underlying cooling rate was 0.1 ◦ C/min, the amplitude of modulation is ±0.5 ◦ C and the period was 60 s. Although MDSC can apply the period of modulation, T (ω = 2π/T), from 10 to 100 s, we chose 60 s for separating the influence of calorimetric glass transition. Dry nitrogen gas was flowed in the sample chamber as purge gas with the flow rate of 10 ml/min during whole experiments. 3. Results MDSC measurement is the extension of the heat flux type of conventional DSC, thus the measured quantity is the temperature difference between sample and reference. The temperature difference converts into the differential heat flow. Resultant heat flow is analyzed by the deconvolution of the sample’s response

Fig. 1. The curves of absolute value of complex heat capacity |Cp∗ | and phase lag of glucose obtained by MDSC.

into underlying heating rate. The raw resultant heat flow (HF) is averaged over the period of more than one modulation. Then the averaged heat flow is subtracted from the resultant HF, the modulated contribution to the resultant HF is obtained. This modulated contribution is analyzed by using discrete Fourier transformation (DFT) to obtain the amplitude of modulated contribution (AMHF ), and the phase lag(ϕ) between the modulated contribution and modulation in the heating rate as shown in Fig. 1. The correction of the phase angle is performed by using the method described in ref. [15]. The Cp∗ (ω) is defined as the ratio of the amplitude of modulated contribution (AMHF ) and the amplitude of modulation in the heating rate (AMq ). We can separate Cp∗ into the real Cp and imaginary Cp parts using phase lag(ϕ) by Cp = |Cp∗ | cosϕ,

(1)

Cp = |Cp∗ | sinϕ,

(2)

where Cp is related to an in-phase component of the modulation and Cp relates to an out-of-phase component. Fig. 2 shows the Cp and Cp of the glucose as a function of temperature in the vicinity of Tg . The frequency of the applied temperature perturbation to the system is fixed and the relaxation times of the studied sample drastically change as a function of temperature. As shown in Fig. 2, the curve of Cp indicates a step-like behavior and Cp has a peak around Tg . Both of them are the characteristic features of the complex (dynamic) susceptibility of a relaxation process. 4. Discussion The temperature procedure of MDSC induces modulated temperature profile where a small sinusoidal perturbation is superimposed onto the underlying linear temperature ramp used in a conventional DSC. Birge et al. successfully observed the frequency-dependent specific heat capacity of glass forming liquid by 3ω method [9,10]. As to the work of Birge et al., the dynamical specific heat Cp∗ (ω) is related to the ψ(t) of enthalpy by  ∞ ∗ ∞ 0 ∞ ˙ {−ψ(t)} eiωt dt, (3) Cp (ω) = Cp + (Cp − Cp ) 0

Cp∞

is the contribution of total degrees of freedom that where equilibrates instantaneously and Cp0 is the static specific heat in

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Fig. 2. The real and imaginary parts of complex heat capacity in glassy (a) glucose and (b) fructose.

an equilibrium. Moreover, in terms of the non-equilibrium statistical mechanics, the complex specific heat Cp∗ (ω) is based on the fluctuation-dissipation theorem and defined by the correlation function of the fluctuation of enthalpy by  ∞ iωV ∗ 0 ˆ ˆ eiωt dt, (4) Cp (ω) = Cp + {h(t) − h}{h(0) − h} kB T 2 0 where kB is the Boltzmann constant, V the volume, h(t) the enthalpy as a function of time at a constant pressure,  means an expected value and hˆ is the average value of h(t). From the modulated contribution of heat flow signals, the frequencydependant complex specific heat, Cp∗ (ω) = Cp (ω) − iCp (ω), can be obtained. The analytical method to obtain Cp∗ (ω) has been proposed by Schawe et al. on the basis of the linear response theory [16,17]. The analytical expressions of the relaxation function ψ(t) are important to understand the dynamics of glass. The Cole–Cole plot, plot of imaginary part in observed χ(ω) against real part, has been widely used in the field of dielectric relaxation study in order to evaluate the non-Debye nature schematically. It is very useful to determine the non-Debye parameters with the deviation from semi-circle. We apply the Cole–Cole plot in order to analyze the line shape and determine the non-Debye parameters of the relaxation functions in the glassy glucose and fructose. Fig. 3 shows the Cole–Cole plot of glucose and fructose. The simplest model of relaxation process is the Debye’s single relaxation. The single Debye response is not general characteristics in glass transitions, and they exhibit non-Debye behavior. The shape of the curve is certainly a distorted and asymmetric semicircle. The shape indicates thermal relaxations in glucose are not single Debye relaxation. The distributed relaxation process is in time domain well characterized by Kohlrausch–Williams–Watts

(KWW) function [18,19] expressed as   t β , ψ(t) = ψ(0) exp − τ

(5)

where τ is the relaxation time and β is a stretched exponential parameter with 0 < β ≤ 1. The diviation from the value of β = 1 means the deviation from single Debye relaxation. When analyse the relaxational behavior in frequency domain, Fourier transformation (FT) is necessary. However, the transformation of KWW function to frequency domain is analytically impossible. Therefore, there are three empirical equations which describe the non-exponentiality; Cole–Cole (CC) [20], Davidson–Cole (DC) [21] and Havriliak–Negami (HN) [22] equations. HN equation is the combination of CC and CD expressed as Cp∗ (ω) = Cp∞ +

Cp0 − Cp∞

[1 + (iωτ)α ]γ

,

(6)

where α and γ are the empirical parameters with 0 ≤ α, γ ≤ 1. In the case of α = 1 is equivalent to the CD equation and the case of γ = 1 is CC equation. Regarding to Eq. (6), scan of ωτ gives the Cole–Cole plot curve and we can determine Cp∞ , Cp0 , α and γ independently. In dielectric spectroscopic measurements, the frequency dependence of χ(ω) is measured under the isothermal conditions. Since the relaxation time of the sample is in constant during the scanning of frequency ω, the parameters determined from the Cole–Cole plot are valid at fixed temperature. As to the frequency range of MDSC is strictly limited, it is difficult to scan ω in wide frequency range. However, the glass transition region is sufficiently narrow and the distribution parameters of relaxation are slightly temperature independent near glass transition temperatures [23]. Therefore, the scans of

Fig. 3. Cole–Cole plots of (a) glucose and (b) fructose.

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Table 1 The obtained non-Debye parameters of HN and KWW equations

Glucose Fructose

5. Conclusion

α

γ

β

Cp0

Cp∞

0.86 0.76

0.49 0.48

0.50 0.44

1.78 1.58

2.45 2.38

ωτ with isochronal, scanning of τ, are quite valid to apply to the present substances. Fig. 3a and b shows the Cole–Cole plot of glucose and fructose by scanning τ at the fixed frequency ω under the assumption of constant parameters Cp∞ , Cp0 , α and γ in the vicinity of Tg . The distorted semi-circle shape clearly indicates that the thermal relaxation in monosaccharide glasses is not single Debye relaxation. The non-Debye parameters, α and γ, of the relaxation functions in the monosaccharide glasses could be determined from the fitting of HN equation within the experimental error. In the study of dynamics in a glass transition, the time-domain KWW function has been widely used for the relaxation processes, especially in ␣-relaxation process. Alvarez et al. have reported the numerical relationship among the parameters of the HN equation, α, γ, and that of KWW equation, β, below [24]. αγ ∼ = β1.23 .

(7)

The relation gives us the direct transformation from the parameters of the HN equation into that of the KWW equation. Value of β = 0.50 ± 0.05 for glucose and 0.44 ± 0.05 for fructose are obtained from the parameters of HN. These values of KWW parameter are reasonable with comparing the other investigation techniques, such as dielectric measurement of β = 0.47 for pure glucose [25]. The smaller value of β means broader distribution of τ. The obtained non-Debye parameters of HN equation and KWW equation are listed in Table 1. As shown in Fig. 3 and Table 1, distorted shapes of the Cole–Cole plot in both monosaccharides are observed. The skewed shape of Cole–Cole plot and the deviation from the Debye behavior can be originated from the larger degrees of freedom in motions and the conformations of molecules. The origin of broad distribution is the presence of molecules in ring and chain conformations, also the existence of boat and chair types of ring conformations. Therefore, both monosccharides of glucose and fructose have several modes of relaxation and the broad distribution of relaxation times. In addition, the distribution of relaxation time in fructose is larger than that of glucose. There are various molecular conformers, fivemembered and six-membered rings, while the conformers of glucose has only six-membered ring. Two of these structures are five-membered rings of ␣- and ␤-d-fructofuranose and the other two are six-membered rings of ␣- and ␤-dfructopyranose, while the crystalline ␤-d-fructose consists of only ␤-d-fructopyranose [26]. The different structures of fivemembered and six-membered rings in fructose may cause distribution of relaxation times. Therefore, the broader distribution of relaxation time is derived from the wide varieties of tautomeric forms of fructose molecule.

MDSC is a powerful tool to investigate the relatively slow dynamics of a glass transition through the thermal relaxation. In the present study, the complex specific heat capacity Cp∗ (ω) of glucose and fructose glasses are studied. The real and imaginary parts of the complex specific heat clearly show the remarkable temperature dependences. We apply the Cole–Cole plot of Cp∗ (ω) to investigate the non-Debye behavior of molecular dynamics. The Cole–Cole plot of Cp∗ (ω) by the fixed frequency method is reasonable to the present study because of the limited temperature range and temperature independence of relaxation parameters. The non-Debye parameters are determined from the Cole–Cole plot. Since these parameters are determined from the thermal relaxation, “total” degrees of freedom in motions are included. The obtained parameters indicate that the distribution of fructose is broader than that of glucose. The several forms of conformers in fructose show the variation of relaxation times, thus the broader distribution of relaxation time in fructose is appeared. Acknowledgement This research was partially supported by the Ministry of Education, Science, Sports and Culture, Grant-in-Aid for Scientific Research (A), Japan, 2005, No. 15654057. References [1] L.M. Crowe, J.F. Carpenter, L.M. Crowe, Annu. Rev. Physiol. 60 (1998) 73–103. [2] J.H. Crowe, J.S. Clegg, Anhydrobiosis, Dowden, Hutchinson & Ross, Stroudsburg, PA, 1973. [3] F. Franks, Biophysics and Biochemistry of Low Temperatures, Cambridge University Press, Cambridge, 1985. [4] C.A. Angell, Science 267 (1995) 1924–1935. [5] C.A. Angell, K.L. Ngai, G.B. McKenna, P.F. McMillan, S.W. Martin, J. Appl. Phys. 88 (2000) 3113–3157. [6] A. Hunt, J. Non-Cryst. Solids 160 (1993) 183–227. [7] R. Kubo, Rep. Prog. Phys. 29 (1966) 255–284. [8] S. Park, K. Saruta, S. Kojima, J. Phys. Soc. Jpn. 67 (1998) 4131–4138. [9] N.O. Birge, Phys. Rev. B 34 (1986) 1631–1642. [10] N.O. Birge, S.R. Nagel, Phys. Rev. Lett. 54 (1985) 2674–2677. [11] M. Reading, Trends Polym. Sci. 1 (1993) 248–253. [12] I. Park, S. Kojima, Thermochim. Acta 352/353 (2000) 147–152. [13] L. Carpentier, O. Bustin, M. Descamps, J. Phys. D 35 (2002) 402–408. [14] O. Bustin, M. Descamps, J. Chem. Phys. 110 (1999) 10982–10992. [15] S. Weyer, A. Hensel, C. Schick, Thermochim. Acta 304/305 (1997) 267–275. [16] J.E.K. Schawe, Thermochim. Acta 261 (1995) 183–194. [17] J.E.K. Schawe, G.W.H. Hohne, Thermochim. Acta 287 (1996) 213– 223. [18] F. Kohlrausch, Prog. Ann. Phys. 119 (1863) 337–368. [19] G. Williams, D.C. Watts, Trans. Faraday Soc. 66 (1970) 80–85. [20] K.S. Cole, R.H. Cole, J. Chem. Phys. 9 (1941) 341–351. [21] D.W. Davidson, R.H. Cole, J. Chem. Phys. 19 (1951) 1484–1490. [22] S. Haviliak, S. Negami, Polymer 8 (1967) 161–210. [23] Gangasharan, S.S.N. Murthy, J. Chem. Phys. 99 (1995) 12349–12354. [24] F. Alvarez, A. Algeria, J. Colmenero, Phys. Rev. B 44 (1991) 7306–7312. [25] R.K. Chan, K. Pathmanathan, G.P. Johari, J. Phys. Chem. 90 (1986) 6358–6362. [26] J. Fan, C.A. Angell, Thermochim. Acta 266 (1995) 9–30.