Fig. 8.1
Various water structures and slow dynamics in aqueous materials. Examples of dynamics for protein molecules are also shown
The dynamics of water molecules interacting with biopolymers in biological systems are generally restricted, and the restriction makes the relaxation frequency shift to a lower frequency. The cooperative dynamics may include not only the water molecules but also the chain dynamics of biopolymers, the overall rotation and local motion of atomic groups at various length scales, and those of counter ions. The relaxation frequencies of these dynamics are thus dependent on the time scale of the collective interactions. In the case of aqueous solutions of immunoglobulin, relaxation processes due to the overall rotation and motion of some domains, such as hinge-bending motion or elbow motion, have been recognized at low frequencies as explained in Sect. 8.4.1.1. Even in these protein dynamics, the ion and water molecules surrounding the protein surface cooperatively interact.
Figure 8.1 shows the water structures observed for various aqueous mixtures such as DNA, liposomes , cell s, tissues, and cement with processes due to the interfacial polarization and chain dynamics of solute molecules. These various water structures and physical properties are characterized in extremely wide time and space scales. BDS is one of the most effective experimental techniques for studying the dynamic behaviors of water through the correlation of dipole moment s of water and interacting molecules. A general picture of various water structures can be deduced from BDS measurements.
The observation techniques used for dynamic molecular complex systems often include serious problem s [7]. Figure 8.2 explains one of the problems from a physical viewpoint of observation. The dynamic structures of complex systems have a characteristic time, i.e., the lifetime. All observation techniques have a characteristic time scale. If the lifetime of the structure is sufficiently longer than the characteristic time scale of the observation technique, the instantaneous structure can be correctly recorded by the measuring system. On the other hand, if the lifetime of the dynamic structure is shorter than the characteristic time scale of the observation, the result obtained is simply the average during the observation period, and dynamic structures with a short lifetime apparently disappear, as shown in Fig. 8.2. For example, the technique of differential scanning calorimetry (DSC) has a characteristic time scale of 100–1,000 s, which is not the scanning rate of the temperature. If the time scale is not suitably taken into account, the dynamic structures cannot be correctly characterized. Since water has various structures with different lifetimes from ps to ms order, misinterpretation of the observation easily leads to incorrect water structures.
Fig. 8.2
Observation problem of dynamics of molecular complex systems (Adapted from Yagihara [7] with permission from the Nanofiber Society. Copyright 2013)
8.2.2 Hydrogen Bonding Network of Aqueous Systems
The liquid structure of pure water has recently been understood in terms of the behaviors of its hydrogen bonding (HB) network. The HB network obtained by molecular dynamics (MD ) simulation is shown as green dotted lines in Fig. 8.3. An MD simulation of 500 water molecules (TIP3P) in NVT ensembles was performed using Discovery Studio 3.0 (Accelrys) at the computing center of Tokai University with the particle mesh Ewald and SHAKE methods. The calculation was performed at intervals of 1 fs over 1 ns in the equilibrium state at 300 K. The length and angle were analyzed for 500 hydrogen bonds (steps, 10,000; time step, 0.001 ps; target temperature, 300, 310, 320 K; type, NVT; frequency of saved results, 100) forming a flexible HB network. The snapshot of the HB network expanding to all of molecules in Fig. 8.3 suggests the existence of intermolecular interactions with long-range correlation throughout the HB network.
Fig. 8.3
HB network obtained by MD simulation. MD simulation was performed by Discovery Studio 3.0 (Accelrys) at the computing center of Tokai University
The dielectric relaxation approach suggests the suitability of the concept of the HB network for explaining some of the experimental results for water in various aqueous systems [8]. The relaxation time for a dipole relaxation process due to the reorientation of motional units with a high molecular weight shows a discontinuous temperature dependence at the freezing temperature in the Arrhenius plot since larger motional units have a larger excluded volume effect. For example, in the case of a polymer blend of poly(vinyl pyrrolidone) (PVP) and poly(ethylene glycol) (PEG), the local chain dynamics of PVP show discontinuous behavior, but those of PEG show continuous behavior, since the local chain motion of the oxide polymer (PEG) is much more flexible because of the oxygen atoms. In the case of liquid crystal molecules with an anisotropic and rigid shape, benzene solutions of 4-cyano-4′-pentylbiphenyl (5CB) exhibit continuous behavior for molecular rotations around the long axis but discontinuous rotations around the short axis because of the large excluded volume effect. The Arrhenius plot for the relaxation process due to the chain dynamics of poly(vinyl acetate) (PVAc) in benzene solution shows both a continuous change and a discontinuous change in the slope, which corresponds to the apparent activation energy. On the other hand, the Arrhenius plot for PEG chain dynamics shows not only a discontinuous change in the activation energy but also a discontinuous change in the relaxation time. This means that the HB interaction has a considerably different molecular mechanism from the van der Waals interaction for its dynamics because of the network structure of both solute and solvent molecules, and small water molecules behave as a large motional unit.
This clearer model of HB network behavior is supported by the results of dielectric spectroscopy as shown in Fig. 8.4 [7, 9]. In this figure, the relaxation time of the relaxation process due to liquid solvents, such as water and various n-alcohol s in solutions of PVP at various concentrations, is plotted against the HB acceptor density, which was simply estimated from the chemical structures. The relative behaviors of the trajectories in the figure are not affected by the approximate estimation. Figure 8.4 suggests that a high HB density reduces the relaxation time since the rate of exchange of HB in the network is higher. Small arrows indicate the direction of the trajectory with increasing polymer concentration for each solvent. The HB densities of the n-alcohols and the repeat unit VP appear to be the same for 1-butanol and 1-pentanol . The plots obtained for all solutions lie on the same line. This result means that the mechanism of solvent molecule dynamics in n-alcohol solutions of PVP can be explained from the HB network. However, the plots for different alcohols such as dihydric and trihydric alcohols are not located on this line. In that sense, it is also possible to remove plots for aqueous solutions from the line in Fig. 8.4, as shown by the dotted line. This ambiguity does not contradict the molecular mechanism of the HB network mentioned above.
Fig. 8.4
The relaxation time for solvent dynamics vs. HB acceptor density. Open circles indicate values for the pure solvents and closed circles were obtained for 10, 20, 30, and 40 wt% polymer solutions (Adapted from Yagihara [7] with permission from the Nanofiber Society. Copyright 2013)
8.2.3 Fractal Concept of Water Structures
It is not easy to devise an effective approach to describing water structures from the concept of an HB network . Although the relaxation time characterizes the average behavior of the dynamics, the dynamic behaviors of an HB network cannot be explained by only the average behavior. Thus, the new concept of the fractal behavior of water structures is useful. The expression of the relaxation process for fractal water structures considers not only the average behavior but also the fluctuations evaluated using the relaxation time distribution [10–12]. To evaluate this distribution in aqueous mixtures, if an exponential hyperbolic-type dipolar correlation function corresponding to the Cole –Cole equation is compared with the memory function, the exponent and the spatial–temporal fractal dimension are found to be in good agreement. The Cole–Cole relaxation function is here expressed as
where ε* is the complex dielectric permittivity, ε ∞ is the high-frequency limit of the dielectric constant, Δε is the relaxation strength , j is the imaginary unit, ω is the angular frequency, τ is the relaxation time , and β is the relaxation time distribution parameter . Debye ’s single relaxation process is expressed by setting β = 1. The Cole –Cole relaxation function has been applied to the relaxation process of water in polymer aqueous solutions and also to the process induced by confining molecular liquids in porous systems. The common feature of the molecules contributing to the symmetric loss spectrum is that they are spatially confined and exist under a geometrical constraint [13]. Monte Carlo simulations of the random walk of a particle in a space with a geometrical constraint of the fractal structure also lead to a symmetric Cole–Cole type relaxation function [14].
(8.1)
(8.2)
Both the relaxation time , τ, and the distribution parameter, β, had been treated as independent quantities until our recently developed fractal approach, although the parameters τ and β are closely related to each other in the case of water structures. The conventional investigation carried out without considering the relationship between τ and β might be a result of the fact that the dynamics of water molecules have not been directly observed so far because of the lack of a suitable observation technique. The space fractal dimension used to scale the volume division size of the relaxation unit, reflected in Eq. (8.1), can be used to obtain the relationship between the relaxation time and the exponential in the Cole –Cole relaxation function. This new concept suggests that the fluctuation of the characteristic time of the dynamics, β, can be explained by the geometrical self-similarity of the polymer network [12] as follows:
where d G is the spatial fractal dimension for relaxed units interacting with the surrounding environment, τ o is the cutoff of the scaling time, and ω s is given by
with Euclidean dimension d E geometrical coefficient G (usually taking a value of about unity), self-diffusion coefficient D s , and cutoff size of the scaling in the space R o.
(8.3)
(8.4)
Fractal analysis with Eq. (8.3) is performed by considering the τ–β diagram. The values of τ and β obtained from experimental results for the water content dependence of the relaxation process due to water are plotted and the hyperbolic behaviors of the trajectories of the plots are evaluated. Fractal water structures are found to be characterized by the trajectory reflecting the characteristic slow dynamics of water for each material. By examining various aqueous systems, some pattern of the dominant area was shown for each kind of molecular behavior. Dispersion and gel systems appear to have water structures with a lower fractal dimension. The solution and dispersion systems clearly appear in different areas of the diagram. In other words, this fractal analysis is essentially different from conventional analysis, in which the relaxation time and its fluctuations are compared at the same water content. It is natural to consider that the same concentration will not have the same effect on the slow dynamics in different aqueous systems. Thus, conventional analytical methods are not suitable for the comparison of solution and dispersion aqueous systems. Some examples of the application of fractal analysis are given in the following sections.
8.3 Experimental Techniques of Dielectric Spectroscopy for Biomaterials
8.3.1 Broadband Dielectric Spectroscopy for Various Water Structures
BDS involves the use of different subsystems to carry out measurements in different frequency regions. For example, in the BDS equipment used at Tokai University, the frequency range of all subsystems is from 1 μHz to 50 GHz. All subsystems are classified into two kinds of measuring systems for the frequency domain and time domain measurements of response functions. The time and frequency response functions are related to each other in accordance with linear response theory [15]. Cole expressed a general relationship with the dielectric relation function as the frequency response function and the dipole correlation function as the time response function using linear response theory [16]. Generally, for biomaterials, frequency domain measurements of the time response function result in less damage to electrodes, but simple and quick procedures can be used in time domain measurements. The more suitable of the two experimental techniques is selected to obtain the time and frequency response functions.
8.3.2 Time Domain Reflectometry Method in Microwave Frequency Region
The time domain reflectometry (TDR) method is a useful and effective technique for observing water structures in the GHz frequency region. It has also been developed as a nondestructive measuring tool for biological systems including the living body through easy and simple treatments. TDR is a time domain spectroscopy (TDS) method, which was first developed in electrical engineering, for example, as a technique to detect damage to undersea cables by comparing the transmitted and reflected signals from damaged cables with the incident pulse [17, 18]. This technique was subsequently applied to dielectric measurements in the physicochemical analysis of materials [19–22]. Cole and coworkers rearranged the equations used in the TDS method to obtain more general mathematical expressions without approximations restricting measurements by numerical analysis using computers [23, 24]. A different method of TDR measuring techniques was also developed [25–27], in which the difference between reflected pulses from unknown and reference samples was analyzed with a precision two orders of magnitude higher than that in the conventional direct method. The TDR method has also been refined through the use of newer devices, greater computer power, high-quality microwave cables, better-designed electrodes, new analytical methods, and so forth, since the basic establishment of its architecture [28–31].
A typical example of a TDR setup for relaxation measurements in the frequency range from 0.1 to 30 GHz is shown in Fig. 8.5a. The main frame of a digitizing oscilloscope (HP54120B, Hewlett-Packard) and a four channel test set (HP54124A, Hewlett-Packard) are employed here. A step pulse with a rapidly increasing voltage step having a 35 ps rise time and a 200 mV pulse height is applied via a flexible coaxial cable to a sample probe, in which the flat-end transmission line is terminated by the sample, and the pulse reflected from the sample is observed and analyzed. The highest frequency of such measurements has recently been extended to 50 GHz.
Fig. 8.5
(a) Schematic diagram of TDR measuring subsystem. (b) Flat-end coaxial design of electrodes (upper) and semirigid coaxial electrodes (middle), triaxial electrodes (below)
The reflected signal is observed by the electrodes in the setup and averaged. One of the difficulties in the use of flat-end electrodes is obtaining good contact between the surfaces of the electrodes and the materials. The good contact should always be confirmed during measurements by a careful procedure. The complex permittivity of the sample can be expressed by the basic equation in the TDR method [23, 24] as
where c’ is the speed of propagation in the coaxial line; v 0 and r are the Laplace transforms of the incident and reflected pulse waveforms, respectively, X cot X expresses the multiple reflections from the interfaces between the sample and air and the coaxial line geometry; d is the length of the inner electrode; and γd is the effective electric length of electrodes. Using a dielectric material with a known complex permittivity ε* S as a standard sample, the complex permittivity of an unknown sample ε* X can be expressed as
![$$ {\varepsilon}_X^{*}\left(\omega \right)={\varepsilon}_S^{*}\left(\omega \right)\frac{1+\left\{\left(c{f}_S\right)/\left[j\omega \left(\gamma d\right){\varepsilon}_S^{*}\left(\omega \right)\right]\right\}\rho {f}_X}{1+\left\{\left[j\omega \left(\gamma d\right){\varepsilon}_S^{*}\left(\omega \right)\right]/\left(c{f}_S\right)\right\}\rho f{}_S}, $$](https://i0.wp.com/basicmedicalkey.com/wp-content/uploads/2018/03/A314557_1_En_8_Chapter_Equ7.gif?w=960)
where
Here, r S and r X are the Fourier transforms of pulses reflected from the standard sample R S (t) and the unknown sample R X (t), respectively. This equation was originally derived for TDR studies [22–25] with the use of a cylindrical electrode [32, 33].
(8.5)
(8.6)
(8.7)
(8.8)
(8.9)
8.3.3 Flat-End Electrodes for Dielectric Relaxation Measurements
Flat-end electrode s have been widely used, particularly in biological systems [32, 34], since they allow convenient nondestructive testing for biomaterials and the living body [32]. Even if flat-end electrodes cannot fix the electric field for samples, any ambiguity can be corrected by a suitable procedure using a reference sample. A typical schematic drawing of a flat-end coaxial electrode used at higher frequencies is shown in Fig. 8.5b (upper). Examples of electrodes modified from semirigid coaxial cables that are directly connected to the TDR-sampling head via an SMA connector are shown in Fig. 8.5b (middle). An example of triaxial electrodes used at low frequencies is also shown in Fig. 8.5b (lower). The actual size of each electrode is listed in Table 8.1.
Table 8.1
Examples of characteristic diameters of flat-end coaxial and triaxial electrodes with the electric length of penetration depth of the electric field determined for a two-layer model with acetone–Teflon system
Type | Code | Diameter of outer conductor/mm | Diameter of dielectric/mm | Diameter of inner conductor/mm | Electric length L 0/mm | Inner Diameter of G electrode/mm | Outer Diameter of G electrode/mm | Large scale electric length L 0L/mm |
|---|---|---|---|---|---|---|---|---|
Semi-rigid electrodes | S09 | 0.86 | 0.66 | 0.20 | 0.0806 | – | – | – |
S12 | 1.19 | 0.94 | 0.29 | 0.110 | – | – | – | |
S22 | 2.20 | 1.68 | 0.51 | 0.215 | – | – | – | |
S36 | 3.58 | 2.98 | 0.92 | 0.319 | – | – | – | |
S64 | 6.35 | 5.31 | 1.63 | 0.577 | – | – | – | |
Triaxial electrodes | K3 | 30.00 | 20.00 | 18.00 | 1.351 | 34.00 | 40.00 | – |
K2 | 30.00 | 19.00 | 10.00 | 1.586 | 32.00 | 40.00 | 4.642 | |
K4 | 30.00 | 25.00 | 5.00 | 1.851 | 31.00 | 40.00 | 6.101 |
The effective measuring depth depends on the size of the flat-end probe since the field contour lines depend on this size [35]. Naito et al. suggested that the effective measuring depth in the case of human skin is approximately 1/2γd [36–38]. The characteristic depth of the electric field inside a material has been examined using various definitions. The dielectric constant, ε X(l), obtained from measurements with a flat-end electrode perpendicularly applied to the surface of a Teflon block sunk in a liquid at a frequency of approximately 1 GHz, which is the lower side of the relaxation frequency of the liquid was described as
where ε L and ε T are the static dielectric constants for the liquid and for Teflon, respectively, l is the gap length between the surface of the Teflon block and the electrode, and L 0 is the characteristic measuring depth of the electric field in the material. The L 0 values thus obtained with acetone as the liquid are shown in Table 8.1. The L 0 value obtained from Eq. (8.10) is dependent on the combination of materials used for the liquid and block. The definition of the characteristic measuring depth is applicable to triaxial electrodes used at lower frequencies as listed in Table 8.1. The examination of Eq. (8.10) suggests that the two exponential terms reflecting short-scale dipolar and long-scale ionic behaviors are characterized by two different L 0 values.
(8.10)
8.4 Water Structures and Various Dynamics
8.4.1 Dynamic Structures Observed in Suspensions of Biological Materials
8.4.1.1 Proteins: Hierarchical Structures
Figure 8.6 shows the molecular weight, M n, dependence of the relaxation time for a process due to the overall rotation of protein s and other molecules in 5 wt% aqueous solutions. The Debye –Stokes–Einstein equation for the rotational motion of spheres with radius a in a liquid with viscosity η is
where k is Boltzmann’s constant and T is the absolute temperature. Thus, the slope of the straight line in Fig. 8.6 should be unity, in good agreement with the value of 0.97 obtained from the experimental results for protein s and nucleotides. On the other hand, the slope of 3.10 obtained for oligosaccharides is considered to be due to the linear shape of the molecules. Plots obtained for proteins denatured by urea lie on the larger-relaxation-time side of the straight line, indicating their larger radii than the native values. When the density of proteins is assumed to be 1.3 g/cm3 and the viscosity of water is 1 cP or larger, the relaxation times calculated for native proteins indicate that their radii are 10 % smaller than the experimental values. Since the same result has also been obtained by X-ray analysis [39], the existence of bound water surrounding the protein molecules is suggested. The relaxation strength associated with the electric dipole moment for the overall rotation does not only depend on the molecular weight. Takashima calculated the dipole moment from structures obtained by X-ray and nuclear magnetic resonance (NMR) studies and confirmed good agreement with the values obtained from dielectric measurements [40].
Fig. 8.6
The relaxation time for the process due to the overall rotation of globular protein and other molecules vs. molecular weight. Aqueous solutions of (a) trehalose (342), (b) maltose (360), (c) maltotriose (504), (d) maltoteraose (667), (e) maltopentaose (829), (f) maltohexaose (991), (g) maltoheptaose (1153); (A) adenosine monophosphate (347), (B) adenosine diphosphate (427), (C) adenosine triphosphate (507); (1) cytochrome C (12384), (2) ribonuclease A (13,700), (3) lysozyme (14,300), (4) myoglobin (17,800), (5) trypsin inhibitor (20,100), (6) trypsin (24,000), (7) pepsin (34,700), (8) ovalbumin (45,000), (9) hemoglobin (65,750), (10) albumin (66,000), (11) γ-globulin (150,000), (12) α-chymotrypsin (25,000) with 8 M-urea (U), (13) ovalbumin with 8 M-U, (14) ovalbumin with 1 wt% mercaptoethanol and 8 M-U, (15) albumin with 8 M-U, (16) albumin 1 wt% mercaptoethanol and 8 M-U. Numbers in parentheses mean the molecular weight
(8.11)
Nakasako reported that water molecules located at suitable positions on the surface of protein s can be clearly observed by low-temperature X-ray analysis at 100 K [39]. Corresponding results were also obtained by Higo from MD simulations [41], in which water molecules surrounding proteins were oriented for a longer time than those in bulk water. The density of water molecules is particularly high in the clefts where active sites and structural changes in proteins are often found [42]. Water molecules on the surface of protein molecules cannot withstand the high density of HB network s and exhibit cooperatively slow dynamics with peptide chains and ions. These restricted and bound water molecules may be closely related to the stability and functions of proteins.
Dielectric absorption curves observed at −9 °C for 5 wt% immunoglobulin G (IgG) aqueous solutions and an aqueous solution of two fragments, F ab and F c, obtained by the hydrolysis of IgG with papain are shown in Fig. 8.7a, b, respectively. The contributions of the DC conductivity and electrode polarization have already been removed from these curves. Large relaxation processes such as those due to bulk water and overall rotations emphasized by ion behaviors are prohibited below the freezing temperature. By employing conventional fitting procedures, four relaxation processes for IgG and three processes for F ab and F c solutions were indicated. The thus obtained permittivity is given by
![$$ \begin{array}{ll}{\varepsilon}^{*}-{\varepsilon}_{\infty }=\hfill & \frac{\Delta {\varepsilon}_{\mathrm{l}1}}{1+j\omega {\tau}_{\mathrm{l}1}}+\frac{\Delta {\varepsilon}_{12}}{1+j\omega {\tau}_{\mathrm{l}2}}+\frac{\Delta {\varepsilon}_{\mathrm{m}}}{{\left[1+{\left(j\omega {\tau}_{\mathrm{m}}\right)}^{\beta_{\mathrm{m}}}\right]}^{\alpha_{\mathrm{m}}}}\hfill \\ {}\hfill & \begin{array}{ll}+\frac{\Delta {\varepsilon}_{\mathrm{h}}}{1+{\left(j\omega {\tau}_{\mathrm{h}}\right)}^{\beta_{\mathrm{h}}}},\hfill & 0<{\alpha}_{\mathrm{m}},{\beta}_{\mathrm{m}},{\beta}_{\mathrm{h}}\le 1\hfill \end{array}\hfill \end{array} $$](https://i0.wp.com/basicmedicalkey.com/wp-content/uploads/2018/03/A314557_1_En_8_Chapter_Equ12.gif?w=960)
where the subscripts l1, l2, m, and h indicate the relaxation processes l1, l2, m, and h, respectively, as shown in Fig. 8.7 [43, 44]. These relaxation processes were identified as the dynamic behaviors of the unfreezable water around 5 GHz (h process), the peptide chain with bound water and ions at approximately 20 MHz (m process), the elbow motion of two F ab fragments at approximately 3 MHz (l2 process), and the hinge-bending motion between F ab fragments at approximately 800 kHz (l1 process). Domain motion such as elbow and hinge-bending motion [45, 46] is also correlated with the ion behavior. When the hinge-bending motion vanished due to hydrolysis by papain, the low-frequency relaxation process l2 for the mixture of F ab and F c fragments was only caused by the elbow motion.
Fig. 8.7
Dielectric absorption curves for (a) 5 wt % aqueous solution of IgG at −9 °C and (b) 5 wt % aqueous solution of Fab and Fc fragment mixture at −9 °C
(8.12)
Peptide chain motion is usually restricted in native protein s and is invisible in dielectric relaxation curves owing to the large contributions of interacting water molecules and ion behavior to the permittivity. To confirm the existence of peptide chain motion, dielectric relaxation measurements were performed for a glass egg [47]. The decreasing HB network density with decreasing amount of water reduces the hydrophobic and hydrophilic interactions of globular protein and induces a globule–coil transition. Simultaneously, the gel–glass transition is promoted. Actually the glass transition phenomenon has also been observed by performing DSC for the glass egg [47]. The change in the water structure in the gel of the heat-denatured globular protein is significant, and the diffusion of water molecules is gradually restricted in the shrinking networks of the gel [48, 49]. Similar results have been reported for keratin–water mixtures [37].
As explained in Sect. 8.2.3, the τ–β diagram obtained for a process due to the dynamic behaviors of the rotational diffusion of water molecules is a possible tool for discussing a universal treatment for water structures. Figure 8.8 shows the τ–β diagram for protein solutions and various kinds of cheese s [50]. The plots for the present samples are shown in the diagram in an extended lengthwise form, and they lie in the area for dispersion systems. The values of the fractal dimension obtained were 0.03 ± 2.45 for BSA, 0.03 ± 1.70 for CEWL, and 0.22 ± 0.89 for OVA. These values are much smaller than those obtained for ordinary aqueous solution systems. The fractal dimensions obtained for water structures in aqueous polymer solutions are usually about 1.3–1.5 with an error of 0.02–0.71. Water structures with a small fractal dimension exhibit only a small increase in the relaxation time with increasing concentration. It turns out that the physical meaning of a fractal dimension of less than 1 is the HB network structure fractionized in dispersed system from the knowledge of our estimation of the fractal dimension for geometric models [51, 52]. Such a small fractal dimension should be explained from the spatiotemporal scale used in the technique for the observation and analysis of the slow dynamics. The process of the slow dynamics reflects the conformation of the polymer chains and the agglomeration structures and also the hydrogen bonding ability. Therefore, the characterization of the slow dynamics is determined not only by the absolute value of the fractal dimension but also by the geometric features of the domain in which the locus of a plot appears in the τ–β diagram .
