**5. Broadband Dielectric Spectroscopy**

Recent studies showed that molecular mobility is probably the most relevant factor for reliably predicting the crystallization behavior of amorphous pharmaceutical solids [19,70]. However, it should be noted that amorphous pharmaceutical solids exhibit complex molecular structures accompanied with various configurational topologies and a variety of molecular interactions. Consequently, molecular mobility of amorphous pharmaceutical solids is rather complex in general, and it could be reflected in various relaxation processes originated from different natures. As a result, establishing proper correlations between molecular mobility and physical stability of amorphous pharmaceutical solids is quite challenging.

In recent studies, a variety of strategies have been exploited to investigate the molecular mobility in glassy and supercooled liquid state—including light scattering, mechanical spectroscopy, temperature modulated differential scanning calorimetry (TMDSC), and nuclear magnetic resonance (NMR). Among these approaches, broadband dielectric spectroscopy is able to explore different relaxation processes and has been demonstrated to distinguish the global and local molecular motions. The measurement of broadband dielectric spectroscopy could be performed over an extremely wide frequency range from mHZ to THz, and concurrently in wide temperature and pressure ranges.

In an excellent review, Paluch and coworkers give a detailed introduction for the equipment principle and basic parameters of broadband dielectric spectroscopy [19]. In brief, dielectric measurement is based on the interactions between the electric dipole moment and the charges of the sample when an external electric field is applied. The investigated pharmaceutical material is firstly placed in a sample capacitor. Herein, a generator (e.g., sine wave generator) could apply an alternating voltage U\*(ω) to the capacitor. Consequently, the external alternating electric field *E* (ω) is generated on the sample capacitor. The complex impedance Z\*(ω) of the samples is determined by the impedance analyzer through measuring the sample capacitor complex voltage and the current. From the basic quantity of complex electrical impedance Z\*(ω) measured by BDS, other complex quantities such as complex dielectric permittivity ε\*(ω) could be derived.

In the field of pharmaceuticals, the Havriliak–Negami (HN) functions plus dcconductivity term are usually used for the analysis of measured isothermal dielectric spectrum.

$$\varepsilon^\*(\omega) = \varepsilon'(\omega) - i\varepsilon''(\omega) = \varepsilon\_{\infty} + \frac{\varepsilon\_s - \varepsilon\_{\infty}}{\left(1 + \left(i\omega \tau\_{HN}\right)^{\alpha}\right)^{\beta}} + \frac{\sigma\_U}{\mathrm{i}\omega\varepsilon\_0} \tag{3}$$

Here, ε and ε respectively represent the real and imaginary parts of the complex dielectric permittivity. *ε*<sup>∞</sup> represents the high-frequency limit permittivity and *ε*<sup>s</sup> represents the static dielectric constant. *α*, *β* are shape parameters of the dielectric peaks and they respectively represent the asymmetry and width. *σ*dc/i*ε*0ω represents the conductivity component, where *σ*dc represents the level of dc-conductivity and *ε*<sup>0</sup> represents the vacuum permittivity.

α-relaxation reflects the reorientations of entire molecules for the low-molecular weight materials. In the case of polymer, α-relaxation, also termed as segmental relaxation, is related to some segmental motions for the polymer chain, which would lead the conformational change. α-relaxation time (*τ*α), representing the relaxation time at a maximum loss (*τ*max). The value of τ<sup>α</sup> was calculated by the following equation on the basis of the parameters obtained by the HN function.

$$\tau\_{\mathfrak{a}} = \tau\_{\text{max}} = \tau\_{\text{HN}} \left[ \sin \left( \frac{\pi \mathfrak{a}}{2 + 2 \beta} \right) \right]^{-1/a} \left[ \sin \left( \frac{\pi a \beta}{2 + 2 \beta} \right) \right]^{1/a} \tag{4}$$

Empirical Vogel–Fulcher–Tamman (VFT) equation is most widely used for describing the temperature dependence of *τ*α in the supercooled liquid state.

$$\pi\_{\mathfrak{a}} = \pi\_0 \exp\left(\frac{DT\_0}{(T - T\_0)}\right) \tag{5}$$

In this equation, *τα* represents the α-relaxation time and *τ*<sup>0</sup> is the relaxation time of the unrestricted material. *D* represents the strength parameter for a measure of fragility and *T*<sup>0</sup> represents the zero mobility temperature. Moreover, the dependence of *τ*<sup>α</sup> in the glassy state could be also predicted by the extended VFT equation.

Local motions have either intra- or intermolecular origins, could be reflected in different secondary relaxations. For the secondary relaxation times, their temperature dependence in the glassy state can be commonly fitted to the Arrhenius equations

$$
\pi = \tau\_0 \exp\left(\frac{\triangle E}{RT}\right) \tag{6}
$$

Herein, Δ*E* represents the activation energy and *R* represents the universal gas constant.

One important goal is to reveal the dominant type of molecular mobility responsible for physical stability, which has been fervently debated for the past decades. Some argue that global relaxation is responsible for the physical stability while other propose that secondary relaxation is the key controlling factor of physical stability. Kothari et al. investigated the molecular mobility and crystallization kinetics of amorphous drug griseofulvin and nifedipine [71]. In their work, the crystallization kinetics of these amorphous drugs is monitored by powder X-ray diffraction technique (PXRD) above *T*<sup>g</sup> while by PXRD technique equipped with synchrotron X-ray source below *T*g [71]. They found that physical stability both in the glassy state and supercooled liquid is strongly related to the α-relaxation rather than the secondary relaxations [71]. Similar strong correlations between α-relaxation and physical stability could also been observed in several other pharmaceuticals including itraconazole [72], trehalose [73], celecoxib [74].

In addition to the pure drug system, α-relaxation has also been reported to play the controlling role for the physical stability in the polymer-based amorphous solid dispersions (ASD) [75–77]. Suryanarayanan and co-workers proposed that the formation of nifedipine-polymer hydrogen bonding interactions could translate to a high resistance to the crystallization by reducing the global mobility, as evidenced by longer system α-relaxation time [76]. Further study revealed that α-relaxation times of nifedipine ASD increase linearly with the polymer concentration increasing [75]. In addition, the established relationship between α-relaxation time and crystallization kinetics of nifedipine ASD doped with a low content of polymer could be used as a reliable predictor for the crystallization of nifedipine ASD containing a higher content of polymer [75]. The usefulness of this predictive model is well confirmed as the matching of the predictive and experimental results of physical stability [75]. Moreover, Mistry et al. reported that the stronger drug–polymer interactions could lead to a longer delay before the onset of crystallization, indicating the enhanced physical stability [78]. Interestingly, the correlation between α-relaxation and crystallization times is almost unaffected by the formation and strength of drug–polymer molecular interactions [78]. Mohapatra et al. investigated the effects of molecular weight of PVP on the molecular mobility and crystallization of PVP-indomethacin ASDs [77]. SS-NMR revealed that drug–polymer hydrogen bonding interaction is independent of the molecular weight of PVP [77]. For comparison, the dependences of viscosity and molecular mobility on temperature are reasonably similar for indomethacin ASDs containing PVP with various molecular weight [77]. It is concluded that increased viscosity would also translate to reduced global molecular mobility (e.g., α-relaxation times), and thus effectively inhibit the crystallization of ASDs [77].

Recent studies revealed that molecular mobility of amorphous drug and ASD could be enhanced by the addition of water [79,80], glycerol [81], and low-*T*<sup>g</sup> polymer, e.g.,

poly(ethylene oxide) [31,32]. Mehta et al. proposed that the increased molecular mobility of amorphous system by the sorbed water is attributed to the plasticization effect [79]. This view is strongly supported by the fact that relaxation times of systems containing different water content overlapped on a temperature scaling of *T*g/*T* [79]. Moreover, as shown in Figure 5, a single linear relationship could be observed for the temperature dependence of crystallization *t*<sup>c</sup> in both dry and water-sorbed griseofulvin systems [79]. Herein, *t*<sup>c</sup> represents the time taken for 0.5% of griseofulvin to crystallize. These results indicate that plasticization effect is also the underlying mechanism for the physical stability of amorphous solids [79]. Given that the coupling extent between α-relaxation and crystallization times of ASD remain the same in the presence of low content of water, a predictive model was built by using the sorbed water for studying the crystallization of some slow crystallizing systems [80]. Similarly, Fung et al. demonstrated that glycerol could also act as a plasticizer, which facilitates the development of an accelerated physical stability testing model of ASDs [81]. The success of this predictive model is mainly attributed to the idea that glycerol could effectively accelerate the crystallization without affecting the mobility–crystallization coupling [81].

**Figure 5.** Plot of crystallization times tc as a function of (**a**) inverse temperature (1000/T), and (**b**) *T*g/*T* for the griseofulvin dry powder (black rounds) and powder containing low content water (blue rounds). Adapted from the [79] with the permission. Copyright © 2015 American Chemical Society.

However, some studies found that change in the global molecular mobility characterized by BDS might not be sufficient for explaining the accelerating or inhibitory effects in the crystallization of amorphous drugs [31,32,82]. For instance, PEO plasticizes amorphous drug systems and increases their global mobility from the liquid dynamic perspective, which is evidenced by the overlapping of α-relaxation time at the scale of *T*g/*T* [31,32]. However, from the perspective of crystallization kinetics, the accelerating growth rates of griseofulvin crystals cannot be simply explained by the increased molecular mobility, as evidenced by the fact that growth rates of pure griseofulvin could not overlap with that of griseofulvin containing low content PEO at the scale of *T*g/*T* [31,32]. Moreover, the increase in global mobility (i.e., the decrease in α-relaxation time) also has difficulty in explaining the selective accelerating effects of PEO on the crystallization of different polymorphs of a drug [32]. In addition, the coupling between crystallization kinetics and α-relaxation times could also be affected by the addition of some excipients, indicating that factors other than global mobility for governing the physical stability [82]. In addition, an increasing number of studies proposed that local mobility rather than global mobility could be the major factor for influencing the physical stability in the glassy state [83,84]. For instance, in the case of glassy celecoxib and indomethacin, strong correlations could be observed between the physical instability and Johari–Goldstein (*β*) relaxation time rather than the α-relaxation time [84]. This intermolecular secondary β-relaxation is proposed

to be a precursor of the global molecular mobility, which indicates that these small-angle reorientations would lead the cooperative α-relaxation process.

Analogous to the polymer-based ASDs, BDS has also been performed to study the molecular dynamics in coamorphous formulations [85–89]. Knapik et al. investigated the molecular mobility of ezetimib-indapamide coamorphous systems and its correlations with physical stability [85]. With the increase of indapamid content, physical stability of this binary coamorphous system was progressively enhanced, as evidenced by the longer α-relaxation time and smaller fragility [85]. In addition, *T*gs of ezetimib-indapamid coamorphous systems rose with the increasing content of indapamid in accordance with the prediction by Gordon-Taylor equations [85]. They proposed that antiplasticizing effect exerted by indapamid is the main mechanism for improving system physical stability [85]. Fung et al. explored the effects of organic acid for stabilizing amorphous ketoconazole, a weakly basic active pharmaceutical ingredient (API) [88]. With an increase in the strength of drug–acid molecular interactions, molecular mobility of these coamorphous systems decreases, as evidenced by the longer α-relaxation time [88]. However, in the case of ketoconazole–tartaric acid and ketoconazole–citric acid, the decreased global molecular mobility was not sufficient to explain the enhanced physical stability [88]. They proposed that structural factors would also enhance the physical stabilization of these drug–acid coamorphous systems [88]. Unlike oxalic and succinic acids, each critic acid molecule has three carboxylic acid groups, which are more beneficial to the formation of drug–acid hydrogen bonding interaction [88]. The hydroxyl group in tartaric and citric acid would also act as the donors of hydrogen bonds and further facilitate the formation of stronger drug–acid hydrogen bonding interactions [88].

BDS can also be applied for studying the molecular mobility of amorphous drug under the nanoconfinement effects [90,91]. Knapik et al. investigated the effects of nanoconfinement on the molecular mobility and crystallization of amorphous drug ezetimibe [90]. Amorphous ezetimib would still exhibit the tendency to crystallize in the pores of Aeroperl 300 (~30 nm pore size), while no crystallization occurs once the drug was incorporated in the pores of Neusilin US2 (<5 nm pore size) [90]. As shown in Figure 6, compared to the ezetimib-Aeroperl 300 system, α-relaxation time of ezetimib increases once incorporated into the pores of Neusilin US2 [90]. Moreover, BDS experiments also revealed the distinguishable phases of the loaded ezetimib in these two commercially used porous materials [90]. One is associated with the molecules at the pore surface–liquid interface while the other one is connected to the molecules in the inner of pores [90]. Herein, the dramatic stabilization of amorphous ezetimib in the pores of Neusilin US2 could be attributed to an interplay of three factors [90]. One factor is the decreased global molecular mobility of amorphous ezetimib under the nanoconfinement [90]. The other two factors are mainly attributed to immobilization effects of the pore wall and the smaller pore size in comparison to the critical nucleation size of amorphous ezetimib, respectively [90]. In a recent study, Zhang et al. explored the molecular mobility of amorphous drug griseofulvin and indomethacin in anodic aluminum oxide (AAO) templates as a function of pore size [91]. A typical two-layer model was also observed in the indomethacin/AAO system, as evidenced by two separated *T*gs and interfacial polarization relaxation in BDS experiments [91]. For the core–shell two-layer model, shell molecules interacting with the walls of nanopores show the higher *T*g, while the core molecules exhibit the fast dynamics with the lower *T*g. In the case of griseofulvin/AAO system, fast cooling would lead to a metastable three-layer model, featuring the existence of thermodynamic nonequilibrium interlayer in addition to the core and interfacial layer [91]. For comparison, stable core–shell two-layer model instead of unstable three-layer model was observed for griseofulvin in AAO templates using the slow cooling process (0.5 ◦C/min) [91].

**Figure 6.** Temperature dependence of α and α relaxation times determined by using BDS technique for bulk ezetimibe, ezetimibe-Aeroperl 300 and ezetimibe-Neusilin US2 systems. Adapted from [90] with the permission. Copyright © 2016 American Chemical Society.
