Determination of Fragility in Organic Small Molecular Glass Forming

Jan 31, 2018 - ABSTRACT: The fragility index (m) and conversely the strength ... supercooled liquid state and D. However, the crystallization propensi...
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Article Cite This: Mol. Pharmaceutics XXXX, XXX, XXX−XXX

Determination of Fragility in Organic Small Molecular Glass Forming Liquids: Comparison of Calorimetric and Spectroscopic Data and Commentary on Pharmaceutical Importance Paroma Chakravarty, Keyur Pandya, and Karthik Nagapudi* Small Molecule Pharmaceutical Sciences, Genentech, Inc., 1 DNA Way, South San Francisco, California 94080, United States S Supporting Information *

ABSTRACT: The fragility index (m) and conversely the strength parameter (D) are widely used to categorize glass forming liquids and are used to characterize temperature dependency of viscosity and relaxation time as the supercooled liquid approaches glass transition. The currently used calorimetric methods in pharmaceutical literature lead to wide variability in measured values of m. In this work, a modulated differential scanning calorimetry (DSC) method is introduced that can directly determine m with minimal variability. Although calorimetric fragility is easy to measure due to availability and ease of use of DSC, there is no correlation between calorimetric and dielectric fragility (calculated spectroscopically from relaxation times). In addition, there is also no correlation between calorimetric fragility and the so-called “thermodynamic fragility” that can be calculated using only thermodynamic parameters. No relationship can be found between the crystallization propensity in the supercooled liquid state and D. However, the crystallization propensity shows a reasonable correlation with the Kohlrausch distribution parameter βk, which defines the breadth of the relaxation time distribution. KEYWORDS: amorphous, mobility, fragility, relaxation time, calorimetry, dielectric spectroscopy, crystallization, glass transition

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INTRODUCTION Glass forming liquids are often classified as “strong” or “fragile” based on the temperature dependence of intrinsic properties such as viscosity (η) or relaxation times (τ) as they approach the glass transition temperature (Tg). This classification of strong/fragile liquids was originally proposed by Angell for nonpolymeric glass formers.1 In this classification, liquids that are considered “strong” are those with minimal structural changes around Tg with their viscosity or relaxation time following an Arrhenius-like behavior with temperature. Strong liquids are often characterized by self-reinforcing tetrahedral networks that are resistant to temperature induced structural changes. On the other hand, “fragile” liquids deviate considerably from Arrhenius behavior in the viscosity or relaxation time versus temperature profiles. Fragile liquids usually lack directional bonds and are often ionic or aromatic in nature.1 Although factors contributing to fragility of liquids are not yet well understood, estimation of fragility values is nevertheless considered important since fragility has been postulated to be closely correlated with the aging behavior and nonexponential relaxation of viscous liquids.2 A numerical estimation of fragility or magnitude of departure from Arrhenius behavior above Tg is provided by the fragility index, m, which is the “steepness” of the viscosity/relaxation time versus temperature curve as T→ Tg. Value of m ≈ 200 denotes considerable fragile behavior, while m ≈ 16 and less indicates a strong glass.2,3 Most compounds of pharmaceutical © XXXX American Chemical Society

interest lie between these two extremes, that is, 16 < m < 200, and are categorized as “moderately fragile”. Fragility index calculated using changes in viscosity or relaxation time (kinetic parameters) as a function of temperature has been referred to as kinetic fragility. Kinetic fragility can be measured using dynamic mechanical analysis (DMA), thermally stimulated depolarization current spectroscopy (TSDC), or dielectric spectroscopy (DES).4−6 The values of viscosity (η) or relaxation time (τ) obtained in the supercooled region are fitted to the Vogel−Tammann−Fulcher (VTF) equation (eq 1) to obtain the constants T0 and the strength parameter D:7−9

⎛ DT0 ⎞ τ = τ0 exp⎜ ⎟ ⎝ T − T0 ⎠

(1)

Here, τ0 is the time scale of atomic vibrations (value ranges from 10−13−10−18 s), and D is the strength parameter of glass forming liquids, with D > 30 indicating strong glass forming behavior and D < 10 denoting fragile glass formers. T0 is temperature where all molecular mobility ceases to exist. “m” is then calculated from D using eq 2 or 3:10 Received: Revised: Accepted: Published: A

November 29, 2017 January 12, 2018 January 31, 2018 January 31, 2018 DOI: 10.1021/acs.molpharmaceut.7b01068 Mol. Pharmaceutics XXXX, XXX, XXX−XXX

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Molecular Pharmaceutics m=

D=

D( T0 Tg ) (ln 10)(1 −

T0

2 Tg )

m = 56 (2)

ΔHm

(4)

Here, ΔHm is the enthalpy of fusion and ΔCp is the heat capacity change at glass transition. The m calculated from eq 4 has been referred to as “thermodynamic fragility” as opposed to kinetic fragility that is measured using DES and DSC. A reasonable correlation between the thermodynamic and kinetic fragility for 50 nonpolymeric glass-forming liquids was shown, suggesting a thermodynamic basis for fragility.24 Even though it is claimed that the calorimetrically determined kinetic fragility (using the scanning rate dependence of fictive temperature method) shows good agreement with kinetic fragility determined using DES or other methods, an evaluation of the existing literature data for pharmaceutical compounds shows major discrepancies that need to be addressed.20 Specifically, there is lack of a systematic comparison between the fragility values obtained using different methods such as DSC and DES and within the same methods itself (depending upon the experimental parameters used). This in turn has led to a wide range of fragility values being reported for the same compound.5,10,11,25−29 This scatter in the reported values makes rank ordering glass forming liquids based on fragility meaningless. Furthermore, it is unclear how the socalled “thermodynamic” fragility compares with kinetic fragility calculated using DSC or DES for small organic molecules. Finally, it is also unclear whether the fragility (calculated by any or all of the aforementioned methods) is correlated to properties of pharmaceutical interest such as crystallization potential in the supercooled liquid region. The objective of this study is not only to compare the fragility values obtained by calorimetry and dielectric spectroscopy, but also to address the discrepancy that exists in the literature for the same. We have measured fragility values for 10 organic small molecules of pharmaceutical interest using DES and a modified DSC method that directly determines fragility. The data obtained are compared with the existing literature data. We have also analyzed the different parameters used for these calculations to highlight how they contribute to variability in determining fragility. Finally discussion is provided on the utility of measuring fragility for pharmaceutical materials.

2 10)mmin

(ln m − mmin

Tg ΔCp

(3)

Here, mmin represents the relaxation of an unrestricted material. By using τ = 100 s (relaxation time estimate at Tg) and τ0 = 10−14 s, the value of mmin can be approximated as 16. Kinetic fragility can also be measured using differential scanning calorimetry (DSC). A survey of the pharmaceutical literature reveals DSC as the most popular method to determine fragility owing to its widespread availability in research laboratories and ease of use.10−14 Calorimetric fragility is determined using three different methods described briefly as follows. (a) Extrapolation of configurational entropy to zero at Kauzmann temperature (Tk): in this method, the Kauzmann temperature Tk is determined from calorimetrically measurable parameters such as configurational heat capacity change at Tg, enthalpy of fusion, melting temperature, and glass transition temperature. Tk is considered equivalent to T0 and is used in the VTF equation (eq 1) to determine the strength parameter D.10,15,16 (b) Calculation of activation enthalpy of structural relaxation using the glass transition width: here, the temperature width of the glass transition (extrapolated onset and extrapolated offset temperatures following enthalpy relaxation at Tg) has been used to calculate fragility. The product of the apparent activation enthalpy for viscosity (ΔEη) and glass transition width are related to a constant that is empirically determined to be 5 ± 0.5 based on ΔEη data obtained for 22 “strong” glass formers.17 Considering ΔETg and ΔEη as equivalent, ΔETg is thus obtained using the same empirical relationship. Fragility “m” is estimated by normalizing the apparent activation energy’s temperature dependence at Tg. (c) Use of scanning rate dependency of Tg: the heating/ cooling rate dependency of Tg is utilized in this calorimetric method to determine the activation enthalpy of structural relaxation (ΔETg) at Tg.18 Once the apparent activation energy is estimated, m and consequently D are determined in the same manner as described for method b. A modification of this method for direct determination of m from Tg measurements has also been reported in literature.19,20 This method utilizes the fictive temperature Tf (temperature where the glass and extrapolated supercooled liquid have the same specific volume/enthalpy) that can be precisely defined from the heat capacity versus temperature curve.21−23 In this paper, we have implemented a modified version of this method for kinetic fragility determination using DSC, the details of which are outlined in the Methods section. In addition to kinetic fragility measured using DSC and DES, Angell et al. have proposed an empirical correlation between fragility and thermodynamic parameters, which is shown in eq 4:24

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EXPERIMENTAL SECTION Materials. Ten compounds (crystalline organic small molecules) were chosen for this study and were melt- quenched to obtain the corresponding amorphous forms. These compounds, used as received, are as follows: carbamezepine, celecoxib, droperidol, indomethacin, nifedipine, pimozide, ritonavir (Sigma-Aldrich, St. Louis, MO), itraconazole (Spectrum Chemical Mfg Corp., Gardena, CA), loratadine, and felodipine (TCI America, Portland, OR). Approximately 500 mg of each compound was heated beyond its melting point, and the melt was rapidly quenched in liquid nitrogen for vitrification. The resultant glass was characterized by PXRD (powder X-ray diffraction), DSC, and thermogravimetric analysis (TGA). These ex-situ melt-quenched samples were used for fragility determination using DES. For DSC runs, samples were melt-quenched in situ in the DSC pans. Melting points and midpoint Tg values of these ten compounds are tabulated in Table 1. Methods. Differential Scanning Calorimetry (DSC). Approximately 10−15 mg of powder sample was analyzed using a Discovery differential scanning calorimeter (TA B

DOI: 10.1021/acs.molpharmaceut.7b01068 Mol. Pharmaceutics XXXX, XXX, XXX−XXX

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of 20, 10, 5, 2, 0.5, and 0.25 °C/min, with a constant heating rate of 2 °C/min up to Tg + 30 °C for each cooling rate. In the heating scan, a modulation of ±1 °C (amplitude) and a period of 60 s were applied. The 2 °C/min cooling scan rate was chosen as the standard scan. Only the heating scan data were collected on the instrument. Experiments were conducted in triplicate for each sample. Because of fast crystallization in carbamazepine at slow cooling rates, the heating scan range was reduced to Tg + 15 or 20 °C. Each modulated DSC experiment yielded six total heat flow curves, which could be separated into reversing and nonreversing heat flows to obtain Tg and ΔHE (enthalpy associated with recovery of structural relaxation), respectively. ΔCp and Tf were obtained at the glass transition event from the reversing heat flow trace of the standard scan. ΔH increased progressively with slower cooling scans since slow cooling provided greater residence time for the sample and allowed for increased structural relaxation. For each nonstandard cooling scan, the ΔH(Q) was obtained from the difference in ΔH between the standard scan and the said nonstandard scan. These ΔH(Q) values were then used to determine the Tf at every nonstandard cooling rate. Finally, eq 5 was used to plot log (Qs/Q) versus Tfs/Tf to obtain m from either the slope or the intercept. In the method proposed in ref 20, DSC experiments were conducted in the standard mode where a single heat flow signal is obtained from which Tg and the corresponding ΔH at the glass transition are calculated. Since these two events are not separated, there is a potential for incorrect estimation of ΔH values. Use of modulation enables the complete separation of reversing (Tg) and nonreversing events (ΔH), thereby resulting in improved baselines and greater reproducibility in measured fragility values. Dielectric Spectroscopy (DES). A broadband dielectric spectrometer Novocontrol Alpha-A high performance frequency analyzer (Novocontrol Technologies, Montabaur, Germany) was used for isothermal dielectric analysis (isothermal frequency sweeps) in the temperature range 0− 120 °C (covering Tg ± 30 °C) and frequency range of 10−2− 106 Hz. The powder sample (melt quenched) was packed between two 20 mm diameter gold plated copper electrodes using a polytetrafluoroethylene (PTFE) ring as a spacer (1 mm thick, 59.69 mm2 in area and with 1.036 pF capacitance). DES measurements were corrected for stray capacitance, spacer capacitance, and edge compensation. The Havriliak−Negami (HN) function (eq 6) was used to fit the dielectric data to obtain the relaxation time τ (from the peak of the dielectric loss spectrum) and obtain the shape parameters αHN and βHN:30

Table 1. Melting Point and Glass Transition Temperature of Model Compounds Selected for the Present Studya compound

Tm (°C)

Tg (°C)

Tg/Tm

carbamazepine (form 1) celecoxib droperidol felodipine indomethacin itraconazole (form 1) loratadine nifedipine pimozide ritonavir

187 163 148 147 161 168 136 173 222 126

53 57 29 44 44 58 37 45 58 49

0.71 0.76 0.72 0.75 0.73 0.75 0.76 0.71 0.67 0.81

a

Melting point peak temperatures (Tm), glass transition temperature midpoints (Tg), determined at 10 °C/min heating rate, and the ratio of Tg to Tm are reported.

Instruments, New Castle, DE) equipped with a refrigerated cooling accessory. Samples were packed in hermetically crimped aluminum pans (Tzero) and experiments were conducted under dry nitrogen purge. The instrument was calibrated using sapphire (heat capacity) and indium (temperature and cell constant). The data were analyzed using commercial software (Universal Analysis 2000, version 4.7A, TA Instruments). Two types of experiments were performed, which are as follows. Baseline Scan. Here, samples were melt quenched in situ by equilibrating them at 10 °C beyond their melting points (Table 1) followed by equilibration at 0 °C. These melt quenched samples were then heated to the melting temperatures of their crystalline counterpart at 10 °C/min. Direct determination of Kinetic Fragility Using Modulated DSC. In this method originally proposed by Robertson et al. and also reported by Li-Min Wang et al., the sample is cycled through different cooling rates to experience different thermal histories but with the same heating rate.19,20 This concept utilizes a standard scan (where the heating and cooling rates are the same) and a standard fictive temperature (Tfs), as determined by Moynihan’s method.18,21 The Tf values for different cooling rates (−Q) are then assessed using an enthalpy differing procedure, caused by changes in thermal history of the sample in the cooling cycle. Essentially, this accounts for the enthalpy difference in the heating scan caused by the standard cooling scan (where the cooling and heating rates are the same) and the cooling scans at different rates, albeit with the same heating scan rate as the standard scan. Details of this method and relevant equations are outlined by Li-Min Wang et al. Ultimately, “m” is obtained either from the slope or intercept (identical in value) from the logarithmic plot of Q/Qs versus Tfs/Tf as shown in eq 5: ⎛Q ⎞ Ts log⎜⎜ ⎟⎟ = m − m f Tf ⎝Qs ⎠

ε*(ω) = ε∞ +

Δε (1 + (iωτHN)αHN )βHN

(6)

In the equation above, ω is the angular frequency, ε*(ω) is the complex dielectric permittivity consisting of real (ε′) and imaginary (ε′′) components, and Δε is the dielectric strength, which is equal to εs − ε∞, where εs represents the low frequency limit of ε′(ω) with ω → 0 and ε∞ represents the high frequency limit of the same, with ω → ∞. The shape parameters αHN and βHN account for the symmetric (αHN) and nonsymmetric (βHN) peak broadening and their values lie between 0 and 1.31 Since a considerable contribution was obtained from conductivity at low frequencies at higher temperature, the conductivity component σ0/iεsω was added to the HN equation where σ0 is the dc conductivity.

(5)

In our work, calorimetric m was determined using the modified version of the method described above, with the modification being the application of modulation to the heating scans. The crystalline samples were melt quenched in situ by heating them 10 degrees above their melting points, followed by equilibration to temperature at least 30 degrees below the Tg and then reheating to 30 degrees above Tg. After this cycle, the sample was then cooled to Tg − 30 °C at different cooling rates C

DOI: 10.1021/acs.molpharmaceut.7b01068 Mol. Pharmaceutics XXXX, XXX, XXX−XXX

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Molecular Pharmaceutics The distribution of relaxation times was determined using the Kohlrausch distribution parameter, βk, which was estimated from the dielectric shape parameters αHN and βHN using the following relationship:32 βk1.23 = αHNβHN

(7)

The value of βk lies between 0 and 1. A value of 1 implies a single relaxation mode (Debye relaxation), while smaller values denote greater spread of relaxation times.33−36 The half-width parameter for the α-relaxation spectra was also determined, which is another indicator of distribution of relaxation times. A half-width value of 1.14 decades of frequency denotes a single, Debye-type relaxation process.37 Determination of Kinetic Fragility Using DES. The logarithm of the relaxation time, that is, τ, obtained by the H−N fit at each temperature, was plotted against the temperature (K) and the VTF equation was fit to this data, using at least five points on the curve. The fitting was done using WinFit software, v.3.5 (Novocontrol Technologies). The strength parameter D and temperature of zero mobility, T0, were obtained from this fit. Two values of τ0 (relaxation time of the unrestricted material) were used for the VTF fit, which are 10−14 and 10−17 s.29,38 By using τ = 100 s, the dielectric Tg was calculated using the VTF equation.33 The values of D alone and those of D, Tg (dielectric), and T0 were used to calculate kinetic fragility using eqs 3 and 2, respectively. Powder X-ray Diffractometry (PXRD). Melt quenched samples prepared ex-situ were analyzed using PANalytical Empyrean powder X-ray diffractometer (PANalytical Inc., Almelo, Netherlands). The powder sample was packed in a zero-background silicon holder and run in reflection mode (Bragg−Brentano configuration). The instrument was equipped a Cu Kα source with tube voltage and current of 45 kV and 40 mA, respectively. Data were collected at ambient temperature from 3.0 to 40.0° 2θ using a step size of 0.0263°, with a revolution speed of 8 s. The incident beam path was equipped with a 0.02° solar slit, a fixed 1° antiscatter slit, a fixed incident beam mask of 10 mm, and a programmable divergence slit in automatic mode. A beam knife for linear detectors was used. The diffracted beam was equipped with a 0.02° solar slit, a programmable antiscatter slit in automatic mode, and a nickel K-β filter. A PIXcel 1D detector was used in the scanning line detector (1D) mode. Data were analyzed using commercial software (JADE, version 9, Materials Data Inc., Livermore, CA).

Figure 1. PXRD patterns of melt quenched samples of the ten model compounds.

Figure 2. DSC thermograms of in situ melt quenched samples that show crystallization in the heating cycle. Itraconazole is also included to demonstrate the mesophase transitions after Tg, although it does not undergo crystallization upon heating. The glass transition temperatures are indicated by (−). The crystallization exotherms are indicated by ∗, and the melting endotherms are indicated by #. The inset figure shows the mesophase transitions in amorphous itraconazole near the Tg.

heating cycle of carbamazepine, droperidol, nifedipine, and pimozide as is evident from the crystallization exotherms and melting endotherms observed in the heating cycle. Determination of Kinetic Fragility Using DSC. Figure 3 shows representative modulated DSC thermograms of in situ melt quenched celecoxib. Six thermal traces, resulting from six different cooling rates (constant heating rate), are shown in the figure. The total heat flow, reversing heat flow (depicting glass transition event), and nonreversing heat flow (representing enthalpy associated with recovery of structural relaxation) are included in the Figure. Figure 4 is a representative plot of log (Qs/Q) versus Tfs/Tf obtained from the thermal parameters in Figure 3, as per method detailed in the previous section. The calorimetric fragility “m” was obtained from the slope or intercept value of the plot. The calorimetric fragility (m, DSC) of the model compounds is shown in Table 2 along with the literature values wherever available. For comparison, the kinetic fragility values obtained using DES, where the strength parameter D is converted to m via eq 3 are also included. In case of Itraconazole, due to appearance of mesophase

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RESULTS Baseline Characterization of Melt-Quenched Samples. Figures 1 and 2 show the PXRD patterns and DSC thermograms of the melt quenched samples of the model compounds, respectively. The amorphous halos shown in the PXRD overlay suggest that glass formation occurs upon melt quenching. The differences in the shapes of the PXRD peaks of these amorphous samples may be attributed to contributions from different populations of local structures or localized molecular order (short-range) existing in the glassy state.39 In Figure 2, thermograms of only those compounds have been included, which show crystallization in the heating cycle (heating rate of 10 °C/min) after melt-quenching the samples in situ. Itraconazole does not crystallize upon heating, but a few endotherms are observed following Tg that have been attributed to mesophase transitions.40 Crystallization is observed in the D

DOI: 10.1021/acs.molpharmaceut.7b01068 Mol. Pharmaceutics XXXX, XXX, XXX−XXX

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Determination of Kinetic Fragility Using DES. Figure 5 shows a representative DES plot for Celecoxib where the

Figure 3. DSC thermograms generated to determine fragility for a representative sample (here, celecoxib). Application of modulation enabled a complete separation of the total heat flow event in to reversing (step transition or Tg) and nonreversing events (enthalpy associated with recovery of structural relaxation).

Figure 5. Dielectric spectra of celecoxib acquired at different temperatures over 10−2−106 Hz frequency range, depicting α or primary relaxations. The solid lines represent the Havreliak−Negami fit to the data. For clarity, only selected temperatures are shown.

dielectric loss (ε′′) is plotted against the frequency. αrelaxations also known as primary relaxations, responsible for global mobility, were observed in the supercooled region and showed a progressive shift toward higher frequencies with an increase in temperature. The solid line is the Havriliak−Negami fit to the DES data to obtain the relaxation time, τ from the spectral peaks. Figure 6 shows the plot of log τ versus

Figure 4. Representative plot of log (Qs/Q) versus Tfs/Tf obtained from the thermal data shown in Figure 3. The value of “m” was obtained either from the slope or the intercept. The average values of the fictive temperature ratios are plotted with the standard deviation being within the data points shown in the plot.

Table 2. Kinetic Fragility (DSC and DES) Values Data Obtained from Our Research along with Those Obtained from Literaturea compound

m (kinetic, DSC) mean ± SD

m (kinetic, DES)b

carbamazepine celecoxib

96 ± 1 124 ± 1

84 132

±5 ±0 ±7

85 109 115 117 100 109 95 87

droperidol felodipine indomethacin itraconazole loratadine nifedipine pimozide ritonavir

102 115 112 N/A 93 142 121 105

± ± ± ±

5 3 4 11

literature data 3513 104,5 85,11 11041 10811 6611 98,5 5811 64.2,26 73111 8411 78.6,26 3311 17011 107.3,25 12711

Figure 6. Representative plot showing the VTF fit (solid line) to the relaxation time, τ versus temperature (K) plot. At least five data points were used to generate the VTF plot for each model compound. The zero mobility temperature T0 (287 K) and strength parameter D (5) were obtained from the VTF fit to the data using τ = 10−14 s.

temperature (K), the solid line being the VTF fit to the data. Table 3 shows a comparison of the strength parameter D, obtained experimentally from DES, using two different values of τ0. Literature data have been included for comparison of D values of the different model compounds, wherever available. Additionally, a comparison of calorimetric and dielectric Tg values, obtained from our research, is included in Table 3. Dielectric Tg was calculated from the VTF equation assuming τ(Tg) = 100 s. A comparison of the kinetic fragility values obtained using DSC and DES (included in Table 2) is shown in Figure 7. The

a

Literature values are of kinetic fragility (DSC and DES) determined using various methods discussed previously (refs in superscript). b Calculated using eq 3, with mmin = 16.

transitions within 10−20 degrees of glass transition (Figure 2), calorimetric fragility could not be determined reliably. E

DOI: 10.1021/acs.molpharmaceut.7b01068 Mol. Pharmaceutics XXXX, XXX, XXX−XXX

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Table 3. Comparison of Strength Parameter D Obtained Experimentally from Dielectric Spectroscopy Using τ0 = 10−14 s or 10−17 sa compound

D (τ0 = 10−14 s)

D (τ0 = 10−17 s)

carbamazepine celecoxib droperidol felodipine indomethacin itraconazole loratadine nifedipine pimozide ritonavir

9 5 9 6 6 6 7 6 7 8

15 9 16 12 11 10 13 11 13 16

literature data 7,5 6.341 942 7,5 8.96 3.927 7.8,28 12.929

dielectric Tg (°C)

calorimetric Tg (°C)

47 56 25 44 41 52 33 40 56 34

52 57 29 44 44 55 37 45 58 49

a Literature DES data are included for comparison. A comparison of the calorimetric and dielectric Tg of model compounds is also included. Midpoint values of calorimetric Tg, at a scan rate of 10 °C/min, are reported. The dielectric Tg values are reported using τ0 = 10−14 s in the VTF equation.

Figure 8. Plot of “thermodynamic fragility” versus kinetic fragility measured using DSC. The “thermodynamic” m was calculated from eq 4.

Figure 7. Plot of calorimetric fragility (kinetic fragility, DSC) versus dielectric fragility (kinetic fragility, DES). The straight line is the diagonal shown for data visualization. A slope of 0.6 and R2 of 0.3 were obtained from a linear fit. Fragility measured using DES was calculated using the strength parameter D and mmin = 16 in eq 3 (Table 2). For the dielectric data, standard deviation of the fitting is shown. The average of triplicate measurements with corresponding standard deviations is shown for calorimetrically measured fragility.

where Tc is the crystallization temperature and Tm is the melting temperature. The authors had proposed the concept of reduced temperature to allow for the crystallization propensity of molecules in the supercooled region that have different Tg and Tm values to be compared. The strength parameter D and the reduced temperature data are tabulated in Table 4 for these four compounds. Even with this limited data set, a correlation was not found between D and the reduced temperature.

strength parameter D, obtained using DES, was converted to fragility (m) using eq 3, with τ0 = 10−14 s. Correlation of values obtained using these different techniques is poor for the compounds in this study. Similarly, for fragility values calculated using eq 2, with D, T0, and Tg derived from DES and τ = 10−14 or 10−17 s, no correlation was observed between dielectric and calorimetric fragility. In fact, the correlation was found to be worse. All dielectric fragility values, obtained using either eqs 2 or 3 and τ = 10−14 or 10−17 s, are included in the Supporting Information. The plot of “thermodynamic fragility” calculated using eq 4 versus the calorimetrically measured m (kinetic fragility, DSC) is shown in Figure 8. No obvious correlation was found between the two data sets. A correlation was also not observed between “thermodynamic fragility” and the m measured using DES (data included in Supporting Information). The parameters used in eq 4 are included in the Supporting Information as well. The crystallization propensity of the four compounds that showed crystallization upon heating was assessed by their reduced temperature ratio as proposed by Zhou et al.25 The reduced temperature is represented by (Tc − Tg)/(Tm − Tg),

Table 4. Crystallization Tendency and Strength Parameter (Determined Using DES) for Four Compounds That Crystallized from the Glassy State in DSC Heating Cycle compound

D

Tc − Tg/Tm − Tg

droperidol carbamazepine nifedpine pimozide

9 9 6 7

0.71 0.47 0.53 0.53

To determine if a correlation exists between the crystallization potential and the distribution of relaxation times, the Kohlrausch distribution parameter (βk) was calculated using eq 7. Table 5 shows the calculated values of βk for all the ten compounds at temperature ∼Tg + 10 K. The half width parameter, which is representative of the distribution of relaxation times, for the α-relaxation spectra is also included. F

DOI: 10.1021/acs.molpharmaceut.7b01068 Mol. Pharmaceutics XXXX, XXX, XXX−XXX

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Molecular Pharmaceutics Table 5. Calculated βk and Half Width Values for Ten Compounds at Temperature ≈ Tg + 10 K compound

βk (at Tg + 10 K)

half width (at Tg + 10 K) (decades)

carbamazepine droperidol nifedipine pimozide celecoxib felodipine loratadine indomethacin itraconazole ritonavir

0.42 0.47 0.52 0.47 0.59 0.57 0.61 0.54 NCa 0.46

3.78 2.21 2.01 2.02 1.80 1.62 1.70 1.88 NC NC

In general, the issue of limited cooling rates is a problem with DSC measurements of fragility. Regardless of the type of DSC instrumentation used, only about four decades of cooling rates can be accessed. For all compounds in our study, the colorimetric fragility was found to be >90. Thus, these compounds can be considered moderate to high fragility materials. For such materials, the curvature of the τ versus temperature (T) changes rapidly, and employing a narrow range of cooling rates may lead to errors in the estimation of fragility. In contrast, dielectric measurements span over 8−9 decades of frequencies and are thus more robust in determining fragility. Thus, the determination of kinetic fragility via spectroscopic methods may be considered more reliable than DSC estimates for small molecular organic glass formers. The type of method used will be irrelevant for strong glasses (m ≈ 16) as they show nearly an Arrhenius τ−T behavior. Variability in Kinetic Fragility Measured Using DES. In the previous section, we had argued for the use of spectroscopic methods to determine fragility. However, even the spectroscopic methods can produce variability in the measured strength parameter depending on the value of τ0, time scale of atomic vibrations, used in the VTF fit. Table 3 shows the values for the strength parameter D obtained in our work versus that in the literature. The value of τ0 appears to contribute significantly to the variability in D obtained from the VTF fit to dielectric data. τ0 values typically range from 10−11−10−17 s, with 10 −14 s and 10 −17 s being the most used in literature.3,5,28,29,38 Although we have used both values for the VTF fit to our relaxation data, the fitting can also be done without restricting the τ0 parameter to any predetermined value. In all these cases, the mean square error of fit will not change appreciably, but one may get considerably different values of D. Since it is evident that strength parameter calculation depends on assumptions in fitting, it is advisable to report τ0 values alongside those of D to facilitate any meaningful comparison of data. By using a τ0 value of 10−14 s, a good agreement was observed between dielectric and calorimetric Tg (c.f. Table 3). Tg values determined using 10−17 s are included in the Supporting Information. On the basis of this agreement, it may be more appropriate to fix the value of τ0 at 10−14 s for the VTF fit of τ−T data. “Thermodynamic” versus Kinetic Fragility. Angell et al. have proposed the empirical concept of thermodynamic fragility whereby m can be calculated from readily measurable quantities such as heat of fusion (ΔHm) and specific heat capacity change at glass transition (ΔCp) as shown in eq 4.24 They have shown that the relationship in eq 4 has also been derived from random first order transition theory (ROFT) bestowing a theoretical foundation to the empirical equation. Figure 8 shows the plot of the “thermodynamic fragility” versus kinetic fragility as measured by DSC. There is no correlation between the two fragility measures for our compounds, which is in contrast to the results reported by Angell et al. where they saw a good correlation between “thermodynamic fragility” (calculated using eq 4) with kinetic fragility for 54 nonpolymeric glass formers. There have also been previous studies where other measures such as heat capacity ratios have been used to define thermodynamic fragility.12,44 Huang and McKenna have shown that no correlation exists between thermodynamic (heat capacity ratios) and kinetic fragilities (determined spectroscopically) for their data set of both hydrogen bonding and non-hydrogen bonding nonpolymeric glass formers.

a

NC, not calculated (i) as mesophase transition follows glass transition in Itraconazole and (ii) due to considerable peak asymmetry in Ritonavir.

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DISCUSSION Comparison of Methods Used To Measure Fragility Using DSC. It is clear from the data presented in Table 2 that there is considerable variation in calorimetric fragility values depending on the method used to determine fragility. This variability is not surprising considering the inherent assumptions involved in the methods used. For example, in the calorimetric method of the extrapolation of configurational entropy to Tk, errors may arise since the configurational entropy is extrapolated over a wide temperature range from Tm to Tk.10 As Tk cannot be measured directly, it is calculated from thermodynamic parameters (ΔHm, ΔCpconf, Tg, and Tm) and used to determine D from the VTF equation by assuming Tk = T0. Such an assumption of equating a thermodynamic parameter with a kinetic one may not be valid.43 In addition, any nonconformational contribution to configurational entropy may affect the prediction of D value by this method.10 In the second calorimetric method of utilizing the Tg width to calculate fragility, a constant that has been derived empirically from 22 strong glass formers is used, and therefore, this method may not work for fragile glass formers.10 Finally, in the case of calculation of calorimetric fragility using the heating rate dependence of Tg, the use of Tg midpoint or onset can significantly influence the outcome.10 The method employed in this study is the only thermal analysis method that provides a direct estimation of calorimetric fragility. The additional use of modulation during the heating scan, as described in our work, helps to maintain baseline integrity and completely separate thermal events to reproducibly determine fragility. The use of this calorimetric technique results in less uncertainty in the measured fragility values.20 Comparison of Kinetic Fragility Measured Using DSC and DES. As shown in Figure 7, we did not find any correlation between kinetic fragility measured using DSC and DES. This result does not agree with the observations of Angell et al. and Robertson et al., who have reported a good correlation between DSC and DES-measured fragility for small molecular organics and polymers.19,20 It should be noted that the range of cooling rates used in this study (20−0.25 K/min) is lower than that used in Angell’s work (80 to 0.5 K/min) and in Robertson’s work (100 to 0.1 K/min) due to limitation of the DSC instrument used in this study.20 The limited range of heating rates employed in this study may have contributed the lack of correlation seen between DSC and DES-measured fragility. G

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Molecular Pharmaceutics

tions.12 Fragility is measured at temperatures close to the glass transition and as such a correlation between fragility and crystallization in the supercooled liquid state is not expected especially if crystallization occurs at a much higher temperature than Tg. Crystallization is a complex phenomenon and depends on interplay of several factors such as molecular diffusion, nucleation, and molecular mobility. If mobility is considered the primary driver for crystallization in the supercooled liquid state, then an evaluation of relaxation behavior and distribution of relaxation times becomes more relevant. Using the Havreliak−Negami shape parameters, we have calculated the Kohlrausch distribution parameter βk, which indicates the spread of relaxation times (Table 5). βk was calculated at an arbitrarily chosen temperature of Tg + 10 K to (a) evaluate the distribution of relaxation times in the vicinity of glass transition temperature for the compounds used in this study and (b) normalize the temperature range since different compounds have different Tg. βk values range from 0 to 1, with an increase in the distribution of relaxation times as βk value decreases. The four compounds that show crystallization at T > Tg are also the ones with the smallest βk values, possibly indicating the influence of mobility on crystallization. The greater spread of relaxation times suggests the existence of faster relaxation modes that may facilitate molecular diffusion, nucleation, and hence crystallization. Ritonavir is the only exception among the noncrystallizing glass formers and shows βk values in the same range as those compounds that show crystallization. In case of ritonavir, the dielectric spectra showed relatively lower values for both αHN (0.78−0.83) and βHN (0.47−0.53) compared to other noncrystallizing glass formers (αHN > 0.9 and βHN > 0. 51) that resulted in a lower value of βk. The corresponding αrelaxation spectra showed narrower width with a highly skewed tail, depicting considerable asymmetry. It thus appears that although the distribution of relaxation times is broader for ritonavir compared to other noncrystallizing liquids at Tg + 10 K, molecular mobility is possibly not a dominant contributor to crystallization. An increased heterogeneity of relaxation times exists at glass transition, due to which the estimate τ = 100 s used to determine the dielectric Tg may not be valid for this compound. This may explain the lack of agreement between the calorimetric and dielectric Tg for ritonavir (Table 3). On the basis of our data, it appears that crystallization propensity in the supercooled liquid state shows a reasonable correlation with the Kohlrausch distribution parameter βk. However, this correlation needs to be considered as a preliminary finding and validated by expanding the data set to include more compounds. A similar correlation between amorphous stability and distribution of relaxation time has also been reported by Shamblin et al. where glasses with lower βk values showed decreased “shelf lives”, the shelf life being time taken for 10% of glass to relax completely upon storage.47 In addition to βk, the half width parameter determined for the dielectric loss spectra is also indicative of distribution of relaxation times. A half width value of 1.14 is indicative of a single Debye type relaxation and an increased deviation from this value indicates a wider spread of relaxation times, that is, existence of multiple relaxation modes. The half width parameter for the compounds studied is shown in Table 5. It follows the same trend as βk values, with the four crystallizing compounds showing the highest half width value, which suggests a wider distribution of relaxation times in the vicinity of Tg.

Kawakami et al. have also reported lack of correlation between thermodynamic fragility and DSC measured kinetic fragility.12,44 Our own limited data set seems to support these findings. It is not surprising that there is a lack of correlation between the thermodynamic and kinetic fragilities due to the following reasons. (i) The empirical relationship shown in eq 4 has the implicit assumption that Tg/Tm = 0.67, which is definitely not the case for most of the molecules in this study (c.f. Table 1). In fact, this ratio is >0.7 for the majority of the compounds. Moreover, this ratio is currently at an average of 0.75 (3/4) for the new compounds emerging from discovery pipelines. (ii) The implicit assumption underlying thermodynamic fragility is that thermodynamic parameters alone can describe the kinetics of relaxation in supercooled liquids approaching glass transition as well as the glass transition event itself.45 The validity of this assumption is a topic of intense debate in the field. For example, Ngai has argued that although thermodynamics (degree of change of volume or entropy) does play a role in change of either η or τ as a function of temperature at T > Tg, the extent of this contribution will also depend on the chemical and physical structure of the molecule and will thus differ from one group of glass formers to another.45 Kinetic factors such as α-relaxation as well as time/temperature dependent distribution of this relaxation time are influenced by many-body relaxation dynamics and thus cannot be completely explained by thermodynamics alone.45,46 Similarly, glass transition cannot be viewed strictly as a thermodynamic parameter since there may be other underlying factors contributing to this phenomenon that are not thermodynamic in nature. While it is certainly attractive to calculate thermodynamic fragility from easily and readily measurable parameters, based on the data and the arguments presented, the use of “thermodynamic fragility” is questionable at best. Correlation of Kinetic Fragility with Pharmaceutical Properties of Interest. In our study, only four out of ten compounds crystallized and their strength parameters (D, τ = 10−14 s) tabulated against the reduced temperature are shown in Table 4. The reduced temperature has been used in literature to compare the crystallization propensity of compounds above Tg.25 There appears to be no correlation between these two parameters in our limited data set. There have been few publications in the pharmaceutical literature that have attempted to classify glass formers based on their fragility and understand the correlation between fragility and properties of pharmaceutical interest such as propensity for crystallization in supercooled and glassy states.11,12,26 For example, Baird et al. attempted to correlate glass stability parameters of several compounds to D and m values estimated using the configurational entropy extrapolation method via DSC. They could not find any correlation between calorimetric fragility and glass stability.11 Similarly, Graeser et al. could not correlate amorphous stability and D values calculated using the Adam− Gibbs equation. The authors also made the observation that pharmaceutically relevant molecules show a wide range of fragilities, and therefore, categorizing them as “moderately fragile” does not accurately reflect their relaxation behavior and crystallization tendency.26 Kawakami et al. showed a poor, albeit positive correlation between glass forming ability and calorimetric fragility calculated using the Tg temperature dependence method for five compounds. Thus, our data suggesting a lack of correlation between D and reduced crystallization temperature are in line with previous observaH

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Molecular Pharmaceutics Finally, it is also important to mention that amorphous pharmaceuticals are typically stored much below Tg (as amorphous glasses) to prevent crystallization. Therefore, a correlation between kinetic fragility and glass stability is not expected unless the glass is formed under experimental conditions similar to that used in fragility measurements. This condition is seldom met in normal pharmaceutical operations used to make glassy materials. Thus, structural relaxation times in the glassy state as well as nonexponentality of relaxation will depend on the method of preparation irrespective of the fragility of the equilibrium liquid. A more meaningful exercise would thus be to directly probe structural relaxation in the glassy state and determine its influence on physical stability as a function of time and temperature. In this context, measuring the relaxation times involved with Johari−Goldstein relaxation (β-relaxation) and trying to correlate that with the stability of amorphous glasses should be explored.48,49

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Phone: 650-225-2917. Fax: 650-225-6238. ORCID

Karthik Nagapudi: 0000-0003-4635-5109 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS P.C. and K.N. are grateful to Dr. Gyan Johari for in-depth discussions and guidance on the concept of fragility and relevant literature references. P.C. acknowledges initial DES training provided at the University of Minnesota in Prof. Raj Suryanarayanan’s group by Drs. Pinal Mistry and Subarna Samanta.

CONCLUSION Caution should be adopted in rank ordering organic small molecules on the basis of their fragility due to variability in existing literature data. A modified fictive temperature method employing modulation was implemented in this work from which kinetic fragility using DSC can be determined directly and reproducibly. The application of modulation provides a robust baseline and complete separation of thermal events and results in less uncertainty in the measured fragility values. However, the kinetic fragility determined using this method did not correlate to that measured using dielectric spectroscopy for the compounds investigated. As DES directly probes τ−T behavior, it may be considered a more appropriate method to determine kinetic fragility. The strength parameter, D, calculated from the VTF fit to τ− T data, is dependent on τ0 value used. It was found that when τ0 = 10−14 s was used, a good agreement between calorimetric and dielectric Tg was obtained suggesting that this value of τ0 is appropriate for fitting τ−T data to VTF equation. “Thermodynamic fragility” calculated using thermodynamic parameters showed no correlation to kinetic fragility calculated using DES or DSC. The use of thermodynamic fragility seems questionable as the theoretical basis for the equation, namely Tg/Tm = 0.67, is not met for many pharmaceutical compounds. Additionally, thermodynamic factors are not the sole contributors to either glass transition or relaxation in the supercooled liquid state. No correlation was observed between fragility and pharmaceutical properties of interest such as crystallization propensity or glass forming ability. However, the crystallization propensity in the supercooled liquid state shows a reasonable correlation with the Kohlrausch distribution parameter βk and half width. Thus, studying the α-relaxations above glass transition and the associated distribution of relaxation times is a more reliable approach to understanding crystallization propensity in the supercooled liquid state.

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equations; table comparing dielectric Tg calculated from DES data with T0 = 10−17 s and DSC Tg; table showing thermodynamic parameters from calorimetry and calculated thermodynamic fragility; figure showing correlation between thermodynamic fragility and kinetic fragility measured using DES (PDF)

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REFERENCES

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ASSOCIATED CONTENT

* Supporting Information S

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.molpharmaceut.7b01068. Table comparing kinetic fragility measured using DES and DSC calculated with different assumptions and I

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