Physical Stability and Relaxation of Amorphous Indomethacin

The R- and β-relaxation processes in amorphous indomethacin have been studied by ... In the glass transition region, the effective activation energy ...
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J. Phys. Chem. B 2005, 109, 18637-18644

18637

Physical Stability and Relaxation of Amorphous Indomethacin Sergey Vyazovkin* and Ion Dranca Department of Chemistry, UniVersity of Alabama at Birmingham, 901 South 14th Street, Birmingham, Alabama 35294 ReceiVed: June 5, 2005; In Final Form: August 15, 2005

The R- and β-relaxation processes in amorphous indomethacin have been studied by using differential scanning calorimetry. The β-process has been detected as a small endothermic peak that emerges before the glass transition step when reheating samples previously annealed in the temperature region -20 to +5 °C. The activation energy of the β-process is ∼57 kJ mol-1, and shows an increase with increasing temperature as it approaches the glass transition region. In the glass transition region, the effective activation energy of relaxation decreases with increasing temperature from 320 to 160 kJ mol-1. Heat capacity measurements have allowed for the evaluation of the cooperatively rearranging region in terms of the linear size (3.4 nm) and the number of molecules (90). The β-relaxation fades below -30 °C, which provides a practical estimate for the lower temperature limit of physical instability in indomethacin. It is demonstrated experimentally that nucleation of indomethacin takes place in the temperature region of the β-relaxation.

Introduction Obtaining crystalline pharmaceuticals in an amorphous form in certain cases leads to a significant increase in the bioavailability. A fundamental problem associated with solid amorphous drugs is that they are inherently unstable because of a strong thermodynamic drive toward crystallization. A great deal of studies have been concerned with evaluating the physical stability of amorphous drugs. Most of these studies have been focused on the dynamics of the glass transition (R-relaxation), because this is the process that on its heating transforms an amorphous system from a low- to high-mobility state, in which crystallization occurs very quickly. It is known that the molecular mobility associated with the R-relaxation fades away rapidly as the temperature drops below the glass transition temperature, Tg. Nevertheless, slow crystallization of indomethacin has been reported by several workers1-4 at temperatures as low as 20 °C, which is about 25 °C below the respective Tg value. On the other hand, relaxation in amorphous indomethacin is still detectable as much as 47 °C below Tg.5 It seems to be generally thought that at temperatures 50 °C below Tg, the molecular mobility should be too negligible to cause any significant relaxation and, therefore, crystallization. This is certainly true for the R-relaxation that requires a cooperative motion of multiple molecules and therefore has a huge energy barrier to it. However, local noncooperative motion has a significantly smaller energy barrier and continues to occur at lower temperatures, giving rise to the β-process (Figure 1A). Although the β-relaxations had been well known for polymers, Johari and Goldstein6 discovered similar processes in simple inorganic glasses, and suggested these to be a universal feature of the glassy state. The β-relaxation processes are frequently termed the Johari-Goldstein process.7 Although the importance of low-temperature mobility has been emphasized,5,8 there have been very few systematic studies concerned with detecting the β-relaxations and measuring their kinetics in pharmaceutical glasses. Some important work in that area has been recently initiated by using the technique of thermally stimulated depolarization current (TSDC).9-11 Dif* To whom correspondence [email protected].

should

be

addressed.

E-mail:

Figure 1. Schematic representation of temperature dependencies for relaxation times (A) and effective activation energies (B). (A) The R-process follows the VTF/WLF dependence, and the β-process follows the Arrhenius dependence. (B) The dashed line shows ideal trends predicted by the VTF/WLF (R-process) and Arrhenius (β-process) equations; points represent actual variation that deviates from the ideal when approaching the mixed R-β region (hatched area).

ferential scanning calorimetry (DSC) can also be used to probe the β-relaxations, as was shown for a large variety of polymers.12 However, DSC is used for this purpose more rarely than other techniques such as mechanical and dielectric spectroscopy.13,14 This is partially because the respective thermal effects are small and hard to detect in regular DSC runs. The effects are enhanced on annealing, during which the glassy state relaxes, losing its enthalpy. On reheating, the lost enthalpy is recovered in the form of an endothermic annealing peak that emerges before the main glass transition step. The effect was originally found by Illers15 in reheating polymer samples annealed at temperatures well below Tg. The enthalpy associated with the annealing peak increases with increasing annealing time. The use of long annealing times (up to 500 h) was initiated by Chen16 in his extensive studies of sub-Tg relaxations of metallic and inorganic glasses. The annealing times can be reduced to less than 1 h by cooling the liquids at very fast cooling rates that produce the glassy state, which is further removed from equilibrium and, therefore, relaxes at a much faster rate. This approach was exercised by Bershtein and Egorov12 in their work on polymers. Chen as well as Bershtein and Egorov arrived at the conclusion that the annealing peaks are associated with the partial relaxation

10.1021/jp052985i CCC: $30.25 © 2005 American Chemical Society Published on Web 09/13/2005

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Vyazovkin and Dranca

of the glassy state that occurs via the local noncooperative motion or the β-process. This relaxation represents unfreezing the faster part of the overall relaxation time distribution, the slower part of which is the cooperative R-process. A common molecular origin of the R- and β-relaxation processes has been advocated widely in the literature.7,12,17-19 Although both processes contribute to the relaxation of the glassy state, their respective contributions change with temperature (Figure 1A). On the higher-temperature side of the glass transition region, relaxation is driven predominantly by the cooperative R-process, whose relaxation time typically obeys a non-Arrhenius temperature dependence such as that given by the Vogel-TammanFulcher (VTF) and/or Williams-Landel-Ferry (WLF) equations.7 At temperatures well below Tg, relaxation occurs via the noncooperative β-process, the relaxation time of which demonstrates an Arrhenius temperature dependence. Because of the temperature dependence of the relative contributions of the Rand β-process, the effective activation energy of relaxation defined as

d ln τ dT-1

E)R

(1)

also demonstrates a characteristic temperature dependence (points in Figure 1B). This type of dependence was originally predicted by Fox and Flory,20 and somewhat later was observed experimentally by McLoughlin and Tobolsky.21 From the standpoint of the physical stability of amorphous pharmaceuticals, one would be mostly interested in the physical aging that occurs at temperatures well below Tg, i.e., in the region in which relaxation is either predominately or exclusively driven by the β-process. Nemilov and Johari22 have recently discussed the importance of the β-process in the long-term subTg aging, and specifically stressed that the latter process has a lower activation energy and occurs significantly faster than predicted from the R-relaxation kinetics. For this reason, R-relaxation studies may provide very limited insight into the problem of the stability of amorphous pharmaceuticals. In this paper, we use DSC to detect the β-process, to determine temperature variations in the effective activation energy throughout the β-and R-relaxation processes, and to evaluate the size of cooperatively rearranging regions at the glass transition in indomethacin. The latter has been chosen as a model compound because it has been extensively studied, and therefore, a great deal of data on its physical properties and thermal behavior are available in the literature. In particular, the β-process in indomethacin has recently been detected in low-temperature TSDC studies.9 Although dielectric23,24 and mechanical25 spectroscopy have also been applied to the relaxation of indomethacin, the studies have been focused on the temperature region of the R-relaxation, and the β-process has not been reported. To the best of our knowledge, the present paper provides the first report on the application of DSC to studying β-relaxation in pharmaceutical glasses. Experimental Section Indomethacin (1-(p-chlorobenzoyl)-5-methoxy-2-methylindole-3-acetic acid) was purchased from MP Biomedicals, LLC (catalog number 190217, lot number 9331E) and used without further purification. The melting point of the sample was 160 °C as measured by DSC. About 22 mg of crystalline indomethacin was placed in 40 µL Al pans and turned into the glassy state by being melted in a furnace at 170 °C and quenched in liquid nitrogen. The thermal stability of indomethacin in that

temperature region was tested by heating a sample from 30 to 180 °C at 20 °C min-1 in a Mettler-Toledo TGA/SDTA851e thermobalance that did not detect any appreciable mass loss in the melted sample. For the glass transition (R-relaxation) studies, the glassy samples of indomethacin were heated to ∼40 °C above their glass transition temperature and held at this temperature for 10 min to erase thermal history. The samples were then cooled to ∼40 °C below the glass transition temperature at the rates of 10, 15, 20, 25, and 30 °C min-1. Immediately after completion of the cooling segment, the samples were heated at a rate with an absolute value equal to the rate of the preceding cooling. For the annealing (β-relaxation) measurements, the samples quenched in liquid nitrogen were quickly transferred from it into the DSC cell maintained at -40 °C. After a short period of stabilization at this temperature, the samples were heated to an annealing temperature, Ta, and held at it for 30 or 180 min. The annealing temperatures were -30, -20, -10, 0, 5, 10, and 20 °C. After the completion of the annealing segment, we cooled the samples to -40 °C and immediately heated them to above Tg. The heating rates were 10, 15, 20, 25, and 30 °C min-1. The heat capacity was measured on an ∼27 mg sample by using a standard procedure that is precise to 1%.26 A sapphire sample of 41.48 mg was used as the calibrant. The temperature program used for the heat capacity measurements involved 5 min of isothermal hold at 0 °C, followed by heating at 10 °C min-1 to 90 °C, and another 5 min of isothermal hold at the final temperature. All the aforementioned measurements we conducted by using a Mettler-Toledo heat flux DSC 822e in the atmosphere of nitrogen flow (80 mL min-1). The temperature and heat flow calibration were performed by using an indium standard. Long-term annealing was performed in two freezers, kept at 5 and -10 °C. Melted, quenched samples (∼10 mg) were sealed in 40 µL Al pans, placed in the freezers, and annealed there for approximately 1 month. During this period, some of the samples were removed and subjected to heating DSC scans. To avoid any significant heating above the annealing temperature, we quickly transferred the samples from the freezer to liquid nitrogen, from which they were transferred directly into the DSC cell that was pre-cooled to -40 °C. After a short moment of equilibration at this temperature, the samples were heated at 10 °C min-1 from 0 to 200 °C. Results and Discussion Glass Transition or r-Relaxation. In DSC, the glass transition shows up as a heat-capacity step. (Figure 2). An increase in the heating rate q causes this step to shift to a higher temperature. This effect is used in the popular method by Moynihan et al.27 for evaluating the effective activation energy, E, from the slope of the ln q versus Tg-1 plots. In their original work, Moynihan et al.27 used three different estimates of Tg (the extrapolated onset, the inflection point, and the maximum of the enthalpy overshoot), and did not observe any significant difference in E. However, the value of E is most frequently evaluated by using only one estimate of Tg, such as the midpoint Tg or the limiting fictive temperature. The resulting single E value is then commonly substituted in the Tool28-Narayanaswamy29-Moynihan30,31 (TNM) model, which includes three more fitting parameters (the preexponential factor, the nonlinearity parameter, and the stretch exponent). A considerable limitation32,33 of the TNM model is that it predicts the Arrhenius temperature dependence for the relaxation time of a liquid that contradicts the typically observed VTF and/or WLF depend-

Stability and Relaxation of Amorphous Indomethacin

J. Phys. Chem. B, Vol. 109, No. 39, 2005 18639

Figure 2. DSC curves obtained from the heating of indomethacin at 10 °C min-1 immediately after quenching (“no aging”) and after annealing for 180 min at different temperatures, Ta. Arrows show the location of the annealing effect. The midpoint Tg is ∼46 °C for the non-aged sample.

encies. Consequently, Kovacs et al.32 stressed that the application of the TNM model is limited “only to systems in which viscosity obeys an Arrhenius dependence within and above the glass transition range”. Such systems later became known as strong glass-forming liquids as opposed to the fragile liquids that demonstrate the VTF/WLF behavior.34 The degree of deviation from the Arrhenius behavior is characterized by the dynamic fragility, m. The original work by Moynihan et al.27 reports the E values for B2O3 and As2Se3, whose dynamic fragilities are, respectively, 3235 and 37.36 This qualifies the respective liquids as very strong, which explains why no significant variation in the activation energy was observed when using the three different definitions of Tg. On the other hand, for the glass transition of sorbitol (m ) 9335), Angell et al.37 obtained a significantly larger E when Tg was determined as the onset temperature than when Tg was estimated as the temperature of the heat capacity peak. A similar effect was reported by Hancock et al.38 for a number of pharmaceutical glasses. Note that the applications of the TNM model with the forced-constant E value tend to demonstrate39-42 a temperature dependence of the stretch exponent parameter for fragile systems that is likely to be a form of manifestation of a variation in the effective activation energy. Recently we reported43 on a correlation between the dynamic fragility and the variability in E for the glass transition. To be able to detect this effect, we proposed43,44 using an advanced isoconversional method.45,46 Compared to the popular methods by Flynn and Wall47 and by Ozawa,48 this method has two important advantages. First, it can treat the kinetics that occur under arbitrary temperature variations, T(t), which allows for the accounting of self-heating/cooling. For a set of n experiments performed under different heating programs, Ti(t), the activation energy is evaluated at any given R by finding ER, which minimizes the function n

Φ(ER) )

n

J[ER,Ti(tR)]

∑ ∑ i)1 j*i J[E ,T (t )] R

(2)

j R

where

J[ER,Ti(tR)] ≡

∫t t

R

[ ]

exp

R-∆R

-ER

RTi(t)

dt

(3)

The second advantage originates from numerical integration over small time regions (eq 3), which eliminates a systematic

Figure 3. Variation of the activation energy for R-relaxation with the extent of relaxation.

error46 produced by the Flynn and Wall and Ozawa methods when ER varies extensively with R. In eq 3, R varies from ∆R to 1 - ∆R with a step ∆R ) m-1, where m is the number of intervals selected for analysis. The integral J in eq 3 is estimated numerically by employing the trapezoid rule. The minimization is performed iteratively for each R to establish the dependence of ER on R. The conversion R is determined from DSC data (Figure 2) as the normalized heat capacity49 as follows

CNp )

(Cp - Cpg)|T (Cpe - Cpg)|T

≡R

(4)

where Cp is the current heat capacity, and Cpg and Cpe are the glassy and equilibrium heat capacity, respectively. As the values of Cpg and Cpe depend on temperature, they need to be extrapolated into the temperature regions of the glass transition. By applying the isoconversional method to the R versus T data for different heating rates, we obtain the ER dependence shown in Figure 3. The effective activation energy decreases from 320 to 160 kJ mol-1. A decrease in E with the transition from the glassy to liquid state has also been reported by Hancock et al.,38 who estimated E for the onset, midpoint, and offset of the transition, and obtained respective values of 385, 263, and 190 kJ mol-1. The single values of E reported in the literature5,9,50,51 span the range 180-380 kJ mol-1. The ER versus R plots have been converted into ER versus T plots by replacing R with the temperature that is estimated as an average of the temperatures corresponding to this R at different heating rates. The resulting plot is presented in Figure 4. A decrease in the effective activation energy with increasing temperature is typical for the glass transition. We have recently observed similar effects for polystyrene and polystyrene composite44 as well as for poly(ethylene terephthalate) (PET) and boron oxide (B2O3).43 The decrease in E has been found43 to correlate with the dynamic fragility of glasses, being the largest for a fragile glass (PET) and the smallest for a strong glass (B2O3). Indomethacin provides an intermediate case showing a 2 times decrease in E per 10 °C. In PET the decrease is more than 2 times per 7 °C, and in B2O3 it is less than 1.5 times per 57 °C.43 The decrease in E can be explained in terms of cooperative molecular motion. The glassy state has a small amount of free volume that permits only the local motion (i.e., the β-process) (Figure 5) that persists well below the glass transition temperature. As the temperature increases and approaches the region of the glass transition, the molecular motion becomes more intense, and the free volume rises, initiating the R-process. This

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Figure 4. Variation of the activation energies for the R- and β-relaxation processes with the average temperature of the process. Numbers by the points represent annealing temperatures. The dashed line shows the value of the activation energy for the β-relaxation according to the correlation E ) 24RTg.

Figure 6. Temperature dependence of the heat capacity for indomethacin. The heat capacity at the glass transition (319 K) changes from 1.19 to 1.56 J K-1 g-1. ∆T is determined as the temperature interval in which Cp changes from 16 to 84% of the total ∆Cp.

Size of Cooperatively Rearranging Regions. Glasses are heterogeneous systems that are characterized by density fluctuations. The lower-density areas form the islands of increased mobility or so-called7 Glarum defects that, in the simplest form, are holes.53 According to Donth,7 the average distance between the mobility islands, ξ, is determined by the volume of the cooperatively rearranging region, VR, that can be estimated from calorimetric data as

ξ3 ≡ V R )

kBTg2∆(CV-1) F(δT)2

(6)

where F is the density (1.31 g cm-3 for amorphous indomethacin3), δT is the mean temperature fluctuation, and CV is the isochoric heat capacity. The value of ∆(CV-1) is determined as -1 -1 ∆(C-1 V ) ) CVg - CV l

Figure 5. Schematic presentation of the cooperative R-process and noncooperative β-process occurring in a cooperatively rearranging region of size ξ. The open area in the middle represents a mobility island.

process needs a great degree of cooperativity between the molecules, which gives rise to a large energy barrier represented by a large value of E at the initial stages of the transition (Figures 1B and 4). As the free volume grows with increasing temperature, the molecular packing becomes looser, allowing molecules to relax more independently (i.e., with a smaller degree of cooperativity). As a result, the energetic constraints relax, and the effective activation energy drops. Note that a fall of E is consistent with both the WLF and VTF equations (Figure 1B). A decrease is also predicted by the Adam-Gibbs equation52

τ ) A exp

( ) z*∆µ kBT

(5)

where kB is the Boltzmann constant, ∆µ is the activation energy per particle, and z* is the number of particles that rearrange cooperatively. In eq 5, z* is inversely proportional to the configurational entropy that rises with T, so that both z* and the effective activation energy (i.e., z*∆µ) decrease with increasing T.

(7)

where CVg and CVl are, respectively, the values of the glassy and liquid CV extrapolated to Tg (Figure 6). The difference between the isobaric and isochoric heat capacities can be neglected,7 so that CV is replaced with Cp, which is accessible by calorimetric measurements. Hempel et al.54 showed that one can account for the difference as follows

∆(CV-1) ) (0.74 ( 0.22)∆(Cp-1)

(8)

For the glass transition measured on heating of the samples, δT is estimated as

δT )

∆T 2.5

(9)

where ∆T is the temperature interval in which Cp changes from 16 to 84% of the total ∆Cp step at Tg54 (Figure 6). Figure 6 shows a Cp versus T curve that is an average of three independent measurements. The obtained Cp data agree very well with the earlier measurements.8 The application of eqs 6-9 to the Cp data (Figure 6) yields the volume of the cooperatively rearranging region of 40.5 nm3. The characteristic length of the cooperatively rearranging region (Figure 5) or the average distance between the mobility islands7 estimated as ξ ) (VR)1/3 is 3.4 nm. The value is comparable to that estimated for sorbitol, 3.6 nm.54 The number of molecules involved in

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J. Phys. Chem. B, Vol. 109, No. 39, 2005 18641

the cooperatively rearranging region is determined by eq 107

NR )

RTg∆(C-1 V ) M(δT)2

(10)

where R is the gas constant and M ) 357.8 g mol-1 is the molecular weight of indomethacin. The resulting value is ∼90 molecules. Sub-Tg Annealing and β-Relaxation. In addition to the aforementioned limitations, the TNM model has not been successful55,56 in describing annealing at temperatures significantly below Tg. Its quantitative application is generally limited to ∼20 °C below Tg.57 It should, however, be mentioned that the TNM model is capable of predicting the annealing peaks, as shown by Hodge and Berens,58 who applied the model to fit the main R-relaxation process simultaneously with the annealing peak. In their procedure, they first estimated the activation energy and preexponential factor for the R-relaxation event, and then varied the nonlinearity parameter and the stretch exponent to fit simultaneously both the R-relaxation and the annealing peak. This procedure does not obviously allow for individually estimating the activation energy for the annealing process should it be different from that of the R-process. Yet, it is known22 that the activation energy for annealing at temperatures far below Tg is significantly smaller than that for the R-process. A similar result was produced by Chen16 and by Bershtein and Egorov,12 who proposed to evaluate the effective activation energy of annealing from the shift in the annealing peak with the heating rate as follows

E ) -R

d ln q dTp-1

(11)

where Tp is the peak temperature. Although this approach has been used widely by Chen et al. and by Bershtein et al. and become an established routine, it should be kept in mind that the value of Tp depends generally on the thermal history of the glass, so that the resulting value of E represents the temperature coefficient of relaxation that may not be identical to the intrinsic activation energy. On the other hand, annealing at temperatures far below Tg predominantly relaxes the faster part of the relaxation spectrum associated with the β-process. Because the position of the Arrhenius plot for the β-process is known7 to be virtually independent of free-volume variations, one may also expect the resulting E value not to be critically dependent on the variation in the glass structure resulting from different thermal histories. A strong support for the validity of the method comes from the extensive work by Bershtein and Egorov, who demonstrated12 experimentally for a number of polymers that the E values obtained by eq 11 agree well with the activation energies of the β-relaxations as determined from dynamic mechanical and dielectric experiments. Since a good agreement was also accomplished in our previous work44 on polystyrene systems, we continue to apply this method in the present study. Unlike the R-relaxation that requires a great degree of cooperativity from nearby molecules, the β-relaxation is associated with a local noncooperative motion that is easily initiated in the area of mobility islands (Figure 5). The β-relaxation is a thermally activated process, and can be initiated only above certain temperatures. Annealing of a glass above those temperatures causes a loss of the enthalpy that can be recovered on reheating, and is typically detected in the form of a broad and shallow endothermic peak that starts to emerge immediately above Ta. Usually the annealing peaks are easily

Figure 7. DSC curves obtained from the heating of indomethacin after annealing for 30 min at different temperatures, Ta. The solid curve represents heating at 20 °C min-1; all other curves correspond to 10 °C min-1. Arrows show the location of the annealing peak.

obtained by annealing a glassy sample in the temperature region of ∼0.8Tg and higher. Initially we performed a series of 30 min annealing runs at temperatures, Ta, from -30 to 20 °C in 10 °C increments. The annealing peaks were detected for Ta ) -20, -10, and 0 °C (Figure 7). Although the peaks are broad and shallow, they can be immediately recognized when comparing a DSC curve for an annealed sample (Figures 2 and 7) with that for a non-aged sample (Figure 2), which demonstrates a practically straight baseline before the glass transition step. The fact that the annealing peaks were detected for a rather narrow interval of Ta was surprising, as in our pervious work with polystyrene systems44 the effect was reliably detected for a Ta interval of more than 40 °C. To enhance the effect, we performed a series of 3 h runs at Ta ) -30, 5, 10, and 20 °C. No annealing peaks were reliably detected on reheating samples annealed at -30 °C, which indicates that below this temperature the β-relaxation ceases. Determining this low-temperature limit is important for evaluating the physical stability of a glassy material because, according to Struik,59 aging does not occur below the temperature region of the β-relaxation. The annealing effects became visible in samples annealed at 5, 10, and 20 °C (Figure 2). For a sample annealed at 5 °C, a shallow endothermic effect is observed at the initial stage of the glass transition. For a sample annealed at 10 °C, one can see an endothermic deflection that is strongly overlapped with the major glass transition step, which explains why we could not detect it after annealing for 30 min. Annealing at Ta ) 20 °C does not produce any sub-Tg effects. Instead, we observe an increased enthalpy overshoot that is typically observed in glasses aged not far below Tg. The aforementioned results indicate that in indomethacin, the location of the annealing peak is strongly dependent on the annealing temperature. This suggests that the underlying relaxation process has a rather low effective activation energy. For evaluating the E values, we used the samples annealed at -20, -10, and 0 °C. The values of Tp have been determined from DSC curves after subtracting a straight baseline. The samples annealed at Ta ) 5 and 10 °C (Figure 2) could not be used for kinetic evaluations because the annealing events were strongly overlapped with the glass transition step, which made it practically impossible to determine the values of Tp. Figure 7 provides an example of how the annealing peaks shift with the annealing temperature and heating rate. The plots of ln q against Tp-1 for each Ta are shown in Figure 8. It is seen (Figure 8) that the plots are linear, as is typical of β-relaxations, which normally obey the Arrhenius law (cf., Figure 1A).7 At the lower annealing temperatures (-20 and -10 °C), the plots are

18642 J. Phys. Chem. B, Vol. 109, No. 39, 2005

Vyazovkin and Dranca TABLE 1: Physical Parameters of the r- and γ-Forms of Indomethacin form

Tm (°C)

Hf (J g-1)

F (g cm-3)

Lf (J m-3)

σ (J m-2)

r* (nm)

R γ

155 161

91 110

1.40 1.38

1.27 × 108 1.52 × 108

0.017 0.027

0.7 0.9

for diffusion across the phase boundary, and ∆F* is the maximum free energy for the formation of the critical size nucleus. Equation 12 contains two exponential terms that have opposite effects on the nucleation rate. Because the ∆F* value decreases with decreasing temperature as Figure 8. Evaluating activation energies (eq 11) for the β-relaxation of indomethacin annealed for 30 min at different temperatures Ta.

practically parallel, giving rise to the respective values of E ) 56.5 and 57.1 kJ mol-1, whereas at the larger Ta ) 0 °C, the value of E is somewhat greater: 74.8 kJ mol-1. Figure 4 presents a dependence of the effective activation energy of the β-relaxation on the average temperature that has been determined as the mean of the Tp values for different heating rates. The obtained results are in good agreement with the TSDC data by Correira et al.,9 who also observed an increase in E (from 66 to 85 kJ mol-1) for the relaxation of indomethacin samples polarized at -10 to 15 °C. The increase in E determined from the annealing peaks is consistent with the idea of the common molecular origin of the β- and R-processes. When we start by heating a glass at lower temperatures, we unfreeze predominantly the faster part of the relaxation time distribution, i.e., the part associated with the local molecular motion of individual molecules that is responsible for the β-process (Figure 5). As temperature rises, the local motion intensifies, engaging nearby molecules in the cooperative motion responsible for the R-processs that is associated with the slower part of the relaxation time distribution. Since the R-processs has a markedly larger activation energy, the total effective activation energy for the annealing process increases with increasing annealing temperature (see Figure 1B). This obviously suggests that the use of the lowest feasible annealing temperature would produce the closest estimate for the activation energy of the β-process. That means that our value of ∼57 kJ mol-1 obtained at Ta ) -20 °C should provide a reasonable estimate for the β-relaxation in indomethacin. This value agrees well with the value of 66 kJ mol-1 obtained by Correira et al.9 from TSDC data at their lowest Ta ) -10 °C. It also fits well into the empirical equation for the activation energy of the β-relaxation (E ) 24RTg) that was found by Kudlik et al.,60 and scrutinized extensively by Ngai and Capaccioli.61 For the experimental value of Tg ) 319 K (Figure 6), 24RTg ≈ 64 kJ mol-1. Implications for Crystallization. The processes of relaxation drive a nonequilibrium glassy system toward the thermodynamically stable crystalline state. Although crystallization should occur spontaneously below the melting point, the actual occurrence of the process in the glassy state is controlled by kinetic factors, regardless of the thermodynamic driving force.62 Crystallization is preceded by nucleation, whose rate is defined as63

I ) I0 exp

( ) ( ) -ED -∆F* exp kBT kBT

(12)

where I0 is the preexponential factor, ED is the activation energy

∆F* ∼

1 (Tm - T)2

(13)

the ∆F* exponential term (eq 12) increases, giving rise to an increasing nucleation rate. Once ∆F* falls below a certain limit, the rate of nucleation becomes controlled by diffusion, with a temperature dependence defined by the ED exponential term (eq 12). If ED is independent of temperature, the rate of nucleation should vary with temperature in accordance with the Arrhenius law. In reality, the rate of self-diffusion in the glass transition region is associated with the rate of the R-relaxation process, whose kinetics are not driven by a single relaxation time, having a single activation energy, but by a distribution of the times having different activation energies (cf., Figures 1 and 4). The situation is, however, simplified as the temperature drops markedly below Tg, and the rate of self-diffusion eventually becomes determined by the β-relaxation that can be characterized by a single activation energy. That is, when a glass is aged at the lower temperatures, the ED in eq 12 is likely to be similar to the activation energy of the β-relaxation. Nevertheless, the low energy barrier does not automatically mean a fast nucleation rate. Because the process occurs at low temperatures, the frequency of diffusion jumps over the energy barrier is significantly decreased. In other words, the nucleation rate is determined by the ED/T ratio, and can still be slow despite a low activation energy. Another important issue for nucleation is the formation of the critical size nucleus in the low-mobility environment. Spontaneous crystallization can start only after the formation of stable nuclei that reach the critical size determined as63

r* )

2σTm Lf∆T

(14)

where σ is the surface tension, Lf is the heat of fusion per volume unit, and ∆T ) Tm - T. In the temperature region of the β-relaxation, the formation of the critical nucleus could occur only via the local noncooperative motion of the individual molecules. The critical nucleus can form without engaging the R-process if its size is significantly smaller than the characteristic length of the cooperative motion, which is defined by the previously determined value of ξ. The critical nucleus size for indomethacin can be estimated by using the literature data.4 The values of Tm and σ for crystallization of amorphous indomethacin have been measured4 for the R- and γ-forms (Table 1). The values of Lf can be easily determined from the heats of fusion per mass unit and respective densities3 (Table 1). For instance, for the nucleation of amorphous indomethacin at -10 °C, substitution of the literature data in eq 14 yields r* ) 0.7 and 0.9 nm for the R- and γ-crystalline forms, respectively. Because these values are markedly smaller than the ξ ) 3.4 nm

Stability and Relaxation of Amorphous Indomethacin

Figure 9. DSC curves obtained from the heating of indomethacin at 10 °C min-1 after long-term annealing at -10 °C. Arrows show the location of the annealing effect.

Figure 10. DSC curves obtained from the heating of indomethacin at 10 °C min-1 after long-term annealing at 5 °C. Inset shows a blowup of the section of the curve that is related to slow crystallization.

determined earlier, self-assembly of the critical nucleus may occur without engaging the cooperative R-motion. The question is whether the β-process alone can provide sufficient mass transport to secure nucleation. It certainly appears so, according to experiments of Oguni et al., who demonstrated that nucleation of o-terphenyl64 and toluene65 does occur via the β-process in the temperature region several degrees above and below Tg. Interestingly, Alie et al.66 have arrived at a similar conclusion when studying the crystallization of a drug 25-65 °C above its Tg. It may, therefore, be that nucleation is generally controlled by the local noncooperative motion below as well as above Tg. Nucleation is usually the slow step (induction period) that may significantly delay the growth of the crystalline phase, so that not detecting the crystal growth over an extended period cannot be taken as the absence of crystallization. For instance, aging of indomethacin at -20 and 4 °C reportedly does not4 result in any measurable crystallization for 6 months, which is not surprising, as even at 20 °C the induction period for indomethacin takes days.2,4 On the other hand, nucleation without any detectable crystallization has been reported67 on prolonged sub-Tg annealing of trehalose. Needless to say, nucleation is essential to the stability of amorphous systems, as prenucleated systems may crystallize at a significantly accelerated speed if the temperature is increased. To probe whether detectable nucleation occurs in indomethacin, we performed prolonged annealing tests at Ta ) -10 and 5 °C. The results of these tests are shown in Figures 9 and 10. The samples aged at -10 °C did not show any crystallization or melting events upon being heated (Figure 9). The only change seen in the DSC scans is a shift in the position of the annealing

J. Phys. Chem. B, Vol. 109, No. 39, 2005 18643 effect with an increasing annealing time. A similar effect was observed in long-term aging experiments by Chen.16 The effect is similar to that observed when increasing Ta (Figures 2 and 7) in the sense that the use of longer annealing times allows for the detection of increased contribution from the slower part of the overall relaxation time distribution that is associated with the R-process. As a result, the annealing peak shifts toward the main R-relaxation event, and ultimately turns into the regular enthalpy overshoot. This effect is clearly seen when comparing Figures 10 and 2. Annealing for 3 h at 5 °C yields an annealing peak right at the beginning of the main R-relaxation (Figure 2), whereas after 2, 10, and 22 days, we observe an increasing enthalpy overshoot. As seen from Figure 10, no crystallization or melting is observed either immediately after quenching the melt or after 2 days of annealing at 5 °C. However, we were able to detect reproducible crystallization and melting in heating scans performed on the sample annealed for 10 days. Even more pronounced effects were observed after 22 days of annealing. These results suggest that during annealing, the glassy samples undergo an active nucleation process. The formed nuclei initiate crystallization when the annealed samples are heated. The size of the melting peaks can give us an idea of the amount of the crystalline phase formed when the annealed samples are heated. The respective heats of melting are 1.2 and 7.4 J g-1 after annealing for 10 and 22 days, respectively. The measured heat of melting of crystalline indomethacin used in this study is 110.3 J g-1. Therefore the amount of the crystalline phase produced when samples are heated does not exceed 7%. It means that the process of nucleation in samples annealed at 5 °C is still in the early stages, as the amount of nuclei formed is insufficient to cause complete crystallization when samples are heated. The obtained data suggest that slow crystallization occurs at temperatures as low as 5 °C, i.e., at 0.87Tg or 41 °C below Tg. This appears to be the lowest temperature at which evidence of nucleation in indomethacin has been obtained so far. This experiment places us in the higher-temperature end of the β-relaxation region. Note that the experiments of Oguni et al. were conducted in the “mixed” R-β region (i.e., 0.94Tg or 7 °C below Tg for toluene,65 and 0.93Tg or 16 °C below Tg for o-terphenyl64). We plan to initiate a new series of annealing experiments on indomethacin at lower temperatures (0 and -5 °C) and continue the runs at -10 °C. The results of these studies will be reported when available. In the meantime, we conclude that nucleation studies in the temperature region of the β-relaxation of pharmaceutical glasses should be of both fundamental and practical importance. Conclusions DSC can be used for measuring the β-relaxations in pharmaceutical glasses. It also provides a convenient way for evaluating temperature variations of the effective activation throughout the R- and β-relaxations as well as for evaluating the sizes of cooperatively rearranging regions. The obtained results indicate that at temperatures as low as -20 °C, indomethacin undergoes β-relaxation driven by noncooperative molecular motion. Because the activation energy of the β-relaxation (∼57 kJ mol-1) is several times smaller than that of the R-relaxation, the former is relatively fast, and can contribute to the nucleation of indomethacin. Evidence of nucleation has been obtained for samples annealed at 5 °C. All in all, we believe that the β-relaxations should be paid more attention when one is studying the physical stability of amorphous pharmaceuticals. On the other hand, it does not seem likely that amorphous pharmaceuticals would demonstrate any measurable

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