Calorimetric Study and Modeling of Molecular Mobility in Amorphous

Oct 30, 2007 - ... Pharmaceutical Compounds Using a Modified Adam−Gibbs Approach .... Kirsten A. Graeser , James E. Patterson , J. Axel Zeitler , Ke...
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J. Phys. Chem. B 2007, 111, 13243-13252

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Calorimetric Study and Modeling of Molecular Mobility in Amorphous Organic Pharmaceutical Compounds Using a Modified Adam-Gibbs Approach Chen Mao,† Sai Prasanth Chamarthy, and Rodolfo Pinal* Department of Industrial and Physical Pharmacy, Purdue UniVersity, 575 Stadium Mall DriVe, West Lafayette, Indiana 47907 ReceiVed: April 2, 2007; In Final Form: August 20, 2007

The purpose of this study is to provide a quantitative characterization of the thermal behavior of amorphous organic pharmaceutical compounds across their glass transition temperature, and to assess their molecular mobility as a function of temperature and time by combining theoretical simulations with experimental measurements using differential scanning calorimetry. A computational approach built on the Boltzmann superposition principle of nonexponential decay and the Adam-Gibbs theory of entropic-dependent structural relaxation is presented. The heat capacities of the crystalline and amorphous forms are incorporated into the simulation in order to accurately assess the entropic fictive temperature as functions of temperature and time under any arbitrary set of experimental conditions. Using this method, we evaluated properties of the glass former, D and T0, and the nonexponentiality index β, for amorphous salicin, felodipine, and nifedipine, by fitting the simulated glass transition profile with the experimentally determined heat capacity across the glass transition region. From this fit, the evolution of the relaxation time of the model compounds following any thermal cycle, including heating, cooling, and isothermal holds can then be estimated a priori. This study reveals the profound and inextricable effect of thermal history on the molecular mobility of the amorphous materials, and the ability of the glass to undergo fast changes in its molecular motions over an aging process even at low temperatures.

Introduction The study of the glass transition and structural relaxation in amorphous systems has long been established on ceramics and polymeric materials. The theoretical and phenomenological aspects of structural relaxation in amorphous systems have been presented in detail in some excellent reviews.1-5 In recent years, the relevance of the amorphous state in small-molecule organic materials (M.W. < ∼1000 Dalton) of pharmaceutical interest has been increasingly recognized. Studies on this class of amorphous materials were initially triggered by the advancement of technology, such as freeze or spray drying, that typically employ amorphous sugars (such as trehalose) as stabilizing agents to process labile biomolecules.6-8 Investigations on nonsugar amorphous solids, particularly those deemed as active pharmaceutical ingredients (APIs), have been stimulated recently by an increasing interest within the pharmaceutical industry to deliver poorly water-soluble compounds in the amorphous state. When prepared in their glassy, higher free-energy form, such molecules exhibit greater apparent solubility and faster dissolution than those in crystalline form.9 It is evident that when amorphous compounds are delivered orally, their solubility advantage could translate into enhanced bioavailablity, especially under conditions where the dissolution in the gastrointestinal tract is the rate-limiting step for absorption.10 The development of every pharmaceutical product in the amorphous state presents the challenge of obtaining optimal processing and storage conditions. Crystallization during the * Corresponding author. E-mail: [email protected]. Ph. (765)496-6247. † Current Address: Discovery Support and Pharmaceutical Chemistry, S7-F2, Schering-Plough Research Institute, Summit, NJ 07901.

shelf life of the product is a major concern in pharmaceuticals. Another important consideration is the faster chemical degradation of drugs in the amorphous state in comparison with the crystalline form. It is possible for molecular organic glasses to be produced with undetected crystal nuclei that will grow upon heating.5 However, the higher degree of molecular motion in the glass, relative to crystalline form, is considered one of the main reasons for the accelerated crystallization11-13 and chemical degradation14,15 in amorphous pharmaceutical compounds, when subjected to prolonged and/or stressed storage conditions. The model presented here focuses on systems where molecular mobility can lead to nucleation in the amorphous material. The timescales of molecular mobility are typically evaluated indirectly by measuring the kinetics of structural relaxation following an environmental perturbation. Relaxations of different origins, including enthalpy relaxation,16-19 volume relaxation,16 dielectric relaxation,20-22 and spin-spin and spinlattice relaxation of proton and 13C nuclei,11,23 have been employed for this purpose. For low-molecular-weight pharmaceutical compounds, molecular mobility determination through thermal techniques, such as differential scanning calorimetry (DSC) or temperaturemodulated DSC, are favored because the enthalpy of reversible thermal events can be readily measured from heat flow under isobaric conditions. From such measurements, changes in properties related to the physical and chemical stability (heat capacity, entropy, and free energy change) can be obtained. Experimental approaches have been continuously refined, reflecting a continuing effort to incorporate the history dependence and the non-Arrhenius character of structural relaxation, in order to arrive to more reliable estimates of molecular mobility. A considerable number of amorphous pharmaceutical

10.1021/jp072577+ CCC: $37.00 © 2007 American Chemical Society Published on Web 10/30/2007

13244 J. Phys. Chem. B, Vol. 111, No. 46, 2007 compounds11,16,17,24 have been studied through enthalpy recovery experiments. In such studies, samples are allowed to anneal below the glass transition temperature (Tg) for various lengths of time, and the corresponding loss of enthalpy is measured upon heating the sample above Tg using DSC. A single relaxation time constant, τ, and the stretch parameter, β, can then be calculated by fitting the enthalpy versus annealing time to the empirical Kohlrausch-Williams-Watts (KWW) equation. An improved approach for estimating the temperature dependence of relaxation times was later proposed by Shamblin et al.25 The approach, based on the Adam-Gibbs theory, accounts for the effect of heat capacity (Cp) differences between crystalline and glass forms. This method can be extended to account for the time evolution of the relaxation time during annealing by tracking the change in fictive temperature (Tf) under isothermal conditions.25,26 Recently, the use of temperature-modulated DSC was proposed to study the molecular mobility of amorphous pharmaceutical solids through specific heat spectroscopy analysis.21,27,28 In these studies, an amorphous sample was excited with low-frequency temperature modulation (0.1-0.01 Hz). The temperature at which the relaxation time is in resonance with the modulation frequency can be determined from the phase angle peak. The fragility of the material is obtained from a plot of the non-Arrhenius temperature dependence of relaxation time at Tg and the relaxation times at different temperatures and annealing times are calculated from the Adam-Gibbs theory. The reliable estimation of molecular mobility in amorphous pharmaceutical solids requires a precise description of structural relaxation with temperature and time, for which the following features should be addressed: (1) the relaxation is nonlinear and (2) the relaxation is nonexponential. For molecular pharmaceutical compounds in the glassy state, none of the available calorimetric approaches is capable of accounting for both characteristics of relaxation. Such methods employ either the KWW equation, which addresses the nonexponentiality alone, or the nonlinear form of the Adam-Gibbs equation, which addresses the nonlinearity alone. In addition, the effect of thermal history remains unaccounted for in these methods. It is evident that improvement in the thermal characterization of amorphous pharmaceutical solids rests on developing methods capable of integrating the nonlinear and nonexponential nature of structural relaxation with the thermal history of the material, that is, with the structural changes taking place by the very act of creating the glass. In principle, this is possible by making use of all of the information encoded in a DSC scan, which records enthalpy resulting from all structural changes associated with the particular path leading to the current state of glass, starting from the equilibrium supercooled liquid. The objective of this work is to present a characterization method that accurately simulates the structural relaxation response of amorphous materials when subjected to different thermal treatments. The method presented is based on a model encompassing all essential elements inherent to the structural relaxation in glasses. By combining theoretical simulations with careful DSC measurements, the critical parameters associated with the relaxation properties of amorphous pharmaceutical compounds can be determined. The obtained parameters can then be used in the model to assess, a priori, the molecular mobility (structural relaxation time) as a function of temperature and time under different experimental conditions. Method Development Relaxation Model. The structural relaxation kinetics of amorphous systems produced upon cooling an equilibrium

Mao et al. supercooled liquid was modeled elegantly by DeBolt et al.,29 based on a phenomenological nonexponential decay function (often termed the KWW equation). Specifically, the process of cooling/heating at the rate of q is replaced with a series of differential temperature jumps, dT, followed by isothermal holds, dt ) dT/q. Because the duration of the isothermal holds, dt, is very short, relaxation kinetics within each individual holding step can be represented accurately by the linear form of the KWW equation. Therefore, the entire path of relaxation can be expressed as a Boltzmann superposition of all steps responding to changes in temperature and time

Tf ) T 0 +

∫TT dT′{1 - exp[- (∫TT′ dTqτ′′) ]} β

0

(1)

where Tf is the fictive temperature of the amorphous system at temperature T, T 0 is the starting temperature from which the supercooled liquid is cooled in order to form the glass, β is the nonexponentiality index with a value between 0 and 1, and τ is the characteristic structural relaxation time during isothermal hold at temperature T. The fictive temperature is the temperature at which a nonequilibrium system like a glass, containing excess thermodynamic properties such as enthalpy and entropy, would need to be in order for it to be in equilibrium in relation to such thermodynamic properties.30 The significance of Tf is discussed in detail below. The nonexponential nature and the thermal history dependence of relaxation are represented in eq 1 by a β value less than unity and by the integration that always starts from T 0, respectively. The nonlinear character of molecular relaxation is most frequently treated using the Narayanaswamy-Moynihan (NM) expression:29,31-34

τ ) A exp

[

x∆h* (1 - x)∆h + RT RTf

]

*

(2)

where A, x, and ∆h*are constants, and R is the ideal gas constant. Although numerical integration of eqs 1 and 2 was shown to follow actual enthalpy relaxation reasonably well, the validity of such an approach has come into question because of a number of limitations.35 The most-significant limitation is that the NM expression predicts an Arrhenius temperature dependence of relaxation time for the supercooled liquid. This prediction conflicts with the behavior observed in liquids, for which the Vogel-Tamman-Fulcher (VTF) expression provides a closer description of the temperature dependence of relaxation time:

τ ) A exp

(

)

B T - T0

(3)

This limitation is critical because a simulation of structural relaxation needs to start from the equilibrium liquid. Errors in initial estimates of τ carried over from the liquid region can lead to inaccurate subsequent predictions of the relaxation kinetics around Tg. One way to circumvent this limitation is to adopt the Scherer-Hodge approach,36 which replaces the NM expression with the Adam-Gibbs-Vogel (AGV) equation:

τ ) A exp

(

B T(1 - T0/Tf)

)

(4)

The near-equivalency of the NM and AGV equations was demonstrated by Hodge.35 In the equilibrium supercooled liquid state, where Tf ) T, eq 4 is numerically indistinguishable from the VTF expression so that relaxation can be modeled without

Calorimetric Study and Modeling

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loss of physical consistency, independent of the physical state in which the material exists across the temperature range. Equation 4 is often expressed in an alternative form to include Angell’s fragility parameter, D:37

τ ) τ0 exp

(

DT0

T(1 - T0/Tf)

)

(5)

Equation 5 suggests that relaxation in the nonlinear (glass) regime is partially controlled by the properties of the glass former (D and T0). Therefore, by combining eqs 1 and 5, it is possible to relate all of the physical attributes leading to the observed glass transition behavior, even if their molecular origins remain largely unknown. These attributes are the nonexponentiality index β, the properties of the glass former D and T0, and the configurational entropy through Tf (discussed in the following section). Fictive Temperature of Organic Amorphous Compounds. The Adam-Gibbs theory ascribes the kinetics of relaxation to the configurational entropy (Sc), which can be expressed in notation consistent with the AGV expression (eq 4) in terms of the fictive temperature (Tf)36

Sc )

∫TT

f

2

( )

∆Cp dT T

Figure 1. Depiction of alternative definitions of the fictive temperature, Tf. Scheme 1: Using glass as the configurational ground state, Tf is temperature-independent and equivalent to Tg. Scheme 2: Crystal form as the configurational ground state. Tf is temperature-dependent. Because the heat capacity of glass is greater than that of the crystal, reducing temperature in the glass region leads to a loss of configurational entropy, as well as a drop on the entropic Tf.

as shown in Figure 1 (Scheme 2). The relaxation model (eq 1) can therefore be modified to incorporate this refinement

(6)

where ∆Cp is the configurational heat capacity of the liquid, and T2 is the configurational ground-state temperature, which is shown to be physically equivalent to T0 in eq 5, based on a hyperbolic temperature dependence of heat capacity (Cp).38 Tf in the AGV equation refers specifically to the temperature where the equilibrium liquid has the same configurational entropy, Sc, as the (nonequilibrium) glass under study. The quantity Sc is regarded as the amount of entropy in an amorphous phase exceeding that of the crystalline form at the same temperature. This quantity is obtained from the heat capacity difference between the liquid and crystalline phases (configurational heat capacity, ∆Cp). However, when studying polymeric or ceramic materials, identifying a reference crystalline form is often not possible. Thus, a common practice has been to approximate ∆Cp in eq 6 as the heat capacity difference between the liquid and glass forms. Such an approximation implicitly assumes that the heat capacities (and hence the configurational properties) of the glass and crystal are the same. But this presents a serious problem, as shown graphically in Figure 1 (Scheme 1): a direct consequence of the above approximation is a vanishing ∆Cp below Tg. Under these circumstances, both Tf and Sc become temperature-independent,39 which is an unrealistic situation. In the case of small-molecule, pharmaceutical organic compounds, stable crystalline forms are often obtained. The ability to characterize crystalline forms permits a more detailed description of the relaxation behavior of pharmaceutical glasses. The measured heat capacity of crystalline forms are always smaller than those of the glassy state.25,40 Because the vibrational properties of the inherent structure of the crystal and glass forms are similar,41 the excess heat capacity of the glass over the crystal can be expected to have a configurational component. Consequently, it is reasonable to expect that these materials have higher configurational entropy in the glassy state than in the crystal and that such differences may be too significant to be ignored for an accurate description of their structural relaxation. From these considerations, a nonzero ∆Cp below Tg leads to the more realistic temperature-dependent Sc and Tf for the glass,

Tf ) T 0 +

∫T

T 0

∫T

T

[

0

(Cgp - Cxp) (Clp - Cxp)

1-

dT ′ +

]{

(Cgp - Cxp) (Clp

-

Cxp)

[ (∫ dTqτ′′) ]} (7)

dT′ 1 - exp -

T

β

T′

where Cgp and Cxp and Clp are the heat capacity of the glass, crystalline, and liquid forms, respectively. This modification yields a relaxation model that is entirely consistent with the AGV equation (eq 5). Numerical integration of eqs 5 and 7 (see below for details) gives a detailed description of the structural relaxation of glasses within the framework of the Adam-Gibbs theory. This approach is believed to offer an improved treatment of structural relaxation and molecular mobility in pharmaceutical glasses, compared to the traditional methodology based on the NM equation (i.e., numerical integration of eqs 1 and 2), for the following reasons: (1) The AGV equation correctly predicts the VTF temperature dependence for the liquid, rather than the Arrhenius behavior predicted by the NM expression. (2) The stable crystalline form is taken into consideration for obtaining the configurational properties. (3) Because every parameter in eqs 5 and 7 bears an explicit physical interpretation, it is possible to assess, a priori, how material properties and environmental conditions control the relaxation behavior and molecular mobility in a pharmaceutical glass. It should be pointed out that in the treatment presented here the crystal represents the configurational ground state. Consequently, eq 7 bears the implicit assumption that the entropy difference between the liquid (or the glass) and the crystal is entirely configurational. Goldstein has shown that the configurational difference contributes only partially to the observed excess entropy in liquids and that part of the excess heat capacity is attributed to vibrational degrees of freedom.42 In this regard, Angell considers that the reason that the Adam-Gibbs theory has worked so well is that the total excess entropy and its configurational component vary proportionally with temperature.5 Vibrational contributions to the excess entropy notwithstanding, another important consideration is that the entropy

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Mao et al.

difference between liquid and glass cannot be exclusively vibrational or it would result in the unrealistic situation depicted in Scheme 1 of Figure 1. The difference in heat capacity between glass and crystal must have a configurational component. Only when the experimentally obtained glass corresponds to the “ideal glass” the structural entropy and (and hence the Cp) of the crystal and glass are the same. Numerical Simulation and Fitting Procedure. The structural relaxation of an amorphous material can be described in terms of the entropic Tf by combining eqs 5 and 7 through numerical integration, expressed, respectively, in the following forms:

Tf,m ) T +

∑ ∆T j)1

m

∑

(

∆Tj 1 -

j)1

|

(Cgp - Cxp)

m

0

j

(Clp

(Cgp

-

-

+

| ){ [ ∑ ( ) ]} 1 - exp -

k)j

∆Tk

β

qkτk

(8)

τk ) τ0 exp(W); W)

(

DT0

k

T0+

∑ n)1

)[ (

∆Tn 1 - T0/ Tf,k-1 + ∆Tk

| )]

(9)

(Clp - Cxp) Tk

where Cgp, Cxp, and Clp are the heat capacities of the glass, crystal, and liquid forms. This numerical analysis is designed to represent any heating or cooling process, from an initial temperature T 0, as a series of sequential steps, each consisting of a small but finite temperature jump of magnitude ∆T, followed by a short isothermal hold. The initial temperature T 0 is necessarily a temperature where the material exists as the equilibrium liquid. The duration of each individual hold can be calculated from the size of the temperature jump (∆T) and the heating/cooling rate, q. The numerical operation is an iterative procedure where the fictive temperature at the end of the mth step, Tf,m, is evaluated in eq 8 by accounting for the effect of the temperature jumps (the second term on the right-hand side of the equation) and the effect of time during isothermal holds (third term of the equation). The relaxation time at the end of the kth step, τk, is obtained from the numerical form of the AGV equation (eq 9) using the fictive temperature at the end of the kth temperature jump, right before the onset of the isothermal hold. This model can simulate a typical DSC experiment where a supercooled liquid is cooled across the glass transition. The model can accommodate subsequent multistep cooling/heating cycles at any rate. The foregoing discussion shows that for a given thermal treatment, the response of an amorphous material can be modeled adequately according to the Adam-Gibbs theory if the following parameters are known: (1) the liquid parameters D (fragility) and T0; (2) the nonexponentiality index β; (3) the heat capacity of the glass, crystal, and liquid (Cgp, Cxp, and Clp, respectively); and (4) the pre-exponential factor τ0. For organic pharmaceutical compounds, τ0 is commonly taken as being in the order of the life time of atomic vibrations (10-14 s).2,43 Because the heat capacities of the different forms are measurable by DSC, D, T0, and β are the three unknowns and can be obtained through a fitting procedure of the fictive

f

(10)

)

(C (

p

)|

- Cxp

Clp - Cxp

)|

T

(11)

Tf

where Cp is the heat capacity of the sample observed across the glass transition region. Simulated values of dTf/dT are obtained from eq 8 as follows:

|

dTf dT

(Cgp - Cxp)

∫TT CxpdT ′

where H(T) and Hl(Tf) represent the apparent and equilibrium values of enthalpy at temperature T, respectively. Note that eq 10 has been modified from its original form30,44 by replacing Cgp with Cxp, as a result of choosing the crystalline form, rather than the glass, as the configurational ground state. Differentiating eq 10 allows fictive temperatures to be determined from heat capacity measurements

dT m

(Clp - Cxp) Tj

H(T) ) Hl (Tf) -

dTf

Cxp) Tj

Cxp)

temperature over a DSC scan. Tf can be defined as follows (Figure 1, Scheme 2)

m

)

(Tf,m - Tf,m-1) ∆Tm

(12)

Any given amorphous compound can be characterized if the parameters D, T0, and β are known. These parameters can be obtained from the best fit of dTf/dT (eq 12) to the experimental values (eq 11) over the glass transition region (during a reheating process because of the endothermic overshoot observed in a heating scan). These parameters are then used in the simulation model (eqs 8 and 9), from which the relaxation time, τ, of the material can be predicted as a function of time and temperature, under any arbitrary set of experimental conditions. In essence, the model presented here consists of two parts. The first part (eqs 8, 9, 11 and 12) is a material characterization step, where the model is used to obtain the glass-forming parameters D, T0, and β of the material from the experimentally determined heat capacities across the glass transition. In this step, the computational cost can be reduced without incurring in large error by approximating Tf with T (the experimental temperature) in eq 11. This is possible because in the characterization step the fit is limited to the upper temperature range of the glass transition (through the endothermic overshoot) where Tf approaches T. In addition, organic pharmaceutical glasses typically exhibit small variation in Clp - Cxp within this narrow temperature range. It should be pointed out that the approximation in question is a computational simplification restricted to the high end of the Tg during the characterization step. The temperature dependence of ∆Cp is actually more complex than any simple functionality may anticipate (although the best empirical assessment suggests that it is a hyperbolic function)45 and the heat capacity difference becomes quite significant at lower temperatures. The second part of the presented model (eqs 8 and 9) is a predictive step where the glass-forming parameters obtained in the characterization step are used to predict the behavior of the material under different experimental conditions. A synopsis of this methodology is presented graphically in Figure 2. It should be pointed out that Tf in eq 12 is of entropic nature, whereas Tf in eq 11 is derived from enthalpic relaxation. The underlying assumption here is that the enthalpy- and entropybased fictive temperatures are the same, that is, that the degree of relaxation of the sample is the same, whether we are looking at it from an enthalpy or from an entropy point of view. In

Calorimetric Study and Modeling

Figure 2. Synopsis of application of the methodology for molecular mobility determination presented in this study.

practice, this approach results from the fact that enthapy-based Tf can be expressed conveniently in terms of heat capacities alone. Although it is still unclear whether enthalpic and entropic relaxation are coupled in pharmaceutical glasses,46 the approach used here uses small temperature intervals, making integrals of ∆Cp and ∆Cp/T essentially proportional so that Tf evaluated from either path is expected to be very similar. Indeed, this assumption embodies a very common practice used to apply the AdamGibbs expression in enthalpy relaxation studies. Materials and Methods Materials. Crystalline forms of salicin and felodipine were purchased from Sigma (St. Louis, MO). Crystalline nifedipine was obtained from Hawkins, Inc. (Minneapolis, MN). All compounds were obtained by selecting the highest available grade and used as received. The amorphous forms of all model compounds were prepared in the DSC by quench cooling the melt. The lack of crystallinity was confirmed by the complete absence of melting endotherms at their corresponding melting temperatures. Differential Scanning Calorimetry. Calorimetric experiments were performed using a Perkin-Elmer DSC 7 differential scanning calorimeter (Norwalk, CT), equipped with a refrigerated cooling accessory. Samples (5-10 mg) were analyzed in hermetically sealed aluminum pans under dry nitrogen purge at 50 mL/min. The cell constant and temperature calibration were conducted using indium and zinc as standards. The thermal history of the amorphous materials was standardized by heating the samples to 10 °C above their Tg, followed by an isothermal hold for 3 min before the starting of the designated thermal cycle. Isothermal Annealing. The DSC thermal cycle used in this study is presented graphically in Figure 3 (where the specific values for salicin are shown). When studying the effect of aging on amorphous systems, it is important to make sure that the thermal history is controlled adequately. The thermal cycle used here consists of three main parts: (a) creation of the fresh glass and heating through the Tg, (b) recreation of the fresh glass for annealing, and (c) aging of the glass and subsequent heating through its Tg. The amorphous sample was (a1) held at 80 °C for 4 min in order to standardize the thermal history, (a2) cooled to -20 °C, and immediately (a3) heated back to 80 °C, both at 10 °C/min. The thermogram from 25 to 80 °C (a3) during the heating step shows the glass transition of the fresh (unaged) glass. The sample was then (b1) held at 80 °C for 4 min, (b2) cooled back to -20 °C, and immediately (b3) heated back to 25 °C, both at 10 °C/min. The sample was then (c1) annealed at 25 °C for different lengths of time ranging from 1 to 16 h. After the annealing time had elapsed, the aged sample was (c2)

J. Phys. Chem. B, Vol. 111, No. 46, 2007 13247

Figure 3. Schematic representation of the experimental procedure for isothermal annealing. The double lines represent the steps of which the thermograms are integrated. (a) creation of the fresh glass and glass transition measurement, (b) placement of the glass at the annealing temperature, (c) isothermal annealing and measurement of the glass transiton of the aged glass. See text for detailed explanation of the experimental procedure.

cooled to -20 °C and (c3) heated immediately back to 80 °C, both at 10 °C/min. The thermograms from 25 to 80 °C (c3) during the final heating were compared with those predicted by the simulation with the exact same thermal history (a1 through c3). This process was adopted because the history of a glass can only be standardized, never erased, since the very act of creating the glass bears its own history. Steps a1 and a2 in Figure 3 are the creation of fresh glass under controlled conditions, starting from the liquid. Step a3 is the glass transition measurement of the fresh glass. Steps b1 and b2 are the recreation of the fresh glass under the exact same conditions. The difference is that the glass is now brought to the annealing temperature (b3) and left to age isothermally (c1) instead of immediately heating it through the glass transition again. Step c3 is the glass transition measurement of the aged sample. In this procedure, steps a3 and c3 are analogous, with the only difference between the two being the aging of the sample. Heat Capacity Measurements. All DSC data presented in this report are heat capacity measurements, not heat flow. The constant-pressure heat capacities of the amorphous and crystalline forms of the model compounds were measured using DSC in accordance with the ASTM method (E1269-04). Sapphire was used as the heat-capacity standard. The Cp measurements involve three DSC runs: a baseline was obtained by heating the empty sample pan and a reference pan of equivalent weight through the temperature range of interest, bracketed by isothermal hold steps to establish equilibrium. The sapphire standard was placed in the sample pan and subjected to the same temperature program as the baseline. A third identical run was conducted with the sample. The heat capacity of the sample was obtained by referencing the baseline-corrected, weighted heat-flow data with the NIST reported values of sapphire heat capacity. Data used in the study were averaged from a minimum of three replicates. Numerical Simulation. A computer program coded on C Sharp was developed to carry out the fitting and numerical simulations. An interface was built on Microsoft.Net Framework 2.0, allowing the user to adjust the parameters of the DSC simulation and investigate the relaxation behavior under different experimental conditions. Because the accurate simulation results rely largely on choosing a sufficiently small value for the temperature increment, ∆T, this parameter is also made adjustable in order to find a cutoff value below which the simulation results remain unchanged. In our calculations, the value of

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Mao et al. TABLE 1: Parameters of Amorphous Salicin, Felodipine, and Nifedipine Obtained by Fitting the Simulated dTf /dT Data to the Experimental Profiles salicin felodipine nifedipine

D

T0 (°C)

β

Tg (°C)a

7.7 8.3 8.1

-3.4 -14.9 -13.3

0.61 0.62 0.66

52.8 43.5 44.0

a Tg values are calculated using the VTF equation, τ(Tg) ) τ0 exp(DT0 /(Tg - T0)), assuming the relaxation time, τ, at Tg is 100 s.

Figure 4. Heat capacities of crystalline and amorphous felodipine and the corresponding dTf/dT curve for amorphous form calculated from eq 11.

Figure 5. Results of fitting the dTf/dT plot for amorphous salicin, felodipine, and nifedipine during reheating at 10 K/min over the glass transition region. Symbols represent the experimental data, and the solid lines represent the best-fit from the simulation model.

0.2 K for ∆T is chosen because it is small enough to provide reliable simulation results while remaining computationally inexpensive. Results Characterization of Amorphous Compounds. The experimental heat capacity and corresponding dTf/dT profiles for felodipine are shown in Figure 4. The experimentally obtained dTf/dT profiles for the three compounds are presented in Figure 5. For all materials tested, the glass transition profile is characterized by an endothermic peak during a heating scan. This is due to the kinetic process of materials regaining the enthalpy from the glass to the supercooled liquid state as the relaxation time of the material catches up with the experimental time scale. The height and width of the endothermic peaks differ with each compound, reflecting the difference in their properties because all materials were subjected to identical thermal treatments. It is also noteworthy that the values of dTf/dT are always grater than zero, even at low temperatures, supporting the preceding argument on the persistence of the configurational entropy in the glassy state. Figure 5 also shows the simulated dTf/dT profiles, as solid lines, for the three compounds. In all three cases, the fitted dTf/dT curves agree very well with the experimental data throughout the glass transition region. At lower temperatures,

the experimental dTf/dT curve starts to show slight positive deviation from the predictions. This is a natural consequence of approximating the fictive temperature with the actual temperature in eq 11. We should emphasize that the region of interest for the fit shown in Figure 5 is the high end of the Tg including the overshoot (the region where the above approximation applies). The fit is the characterization step of the model, and its purpose is to obtain the relaxation parameters, β, D, and T0 of the glass former. Subsequent predictive calculations use the actual fictive temperature. Parameters of the best fit for salicin, nifedipine, and felodipine are given in Table 1. All compounds exhibit D values ranging from 7 to 9, indicating that they are relatively fragile glass formers, as is typically the case for low-molecular-weight, organic compounds.47 The β values are also very close, suggesting that this class of compounds might share a similar degree of relaxation nonexponentiality. The Tg values in Table 1 are calculated using the VTF equation at Tg: τ(Tg) ) τ0 exp(DT0/(Tg - T0)) based on the common assumption that τ(Tg) ≈ 100 s.48 These calculated Tg values fit well within the onset phase of the observed glass transition region, affirming the robustness of the model in predicting the relaxation of amorphous materials in response to the given thermal treatments. Relaxation Time in Amorphous Compounds. Unlike the empirical NM expression (eq 2), fitting within the Adam-Gibbs framework yields the parameters D, T0, and β. These parameters determine the behavior of the glass-forming liquid as well as the relaxation in the glass. Using these parameters, we can evaluate the relaxation time of an amorphous compound as a function of temperature for any given thermal treatment, based on eqs 8 and 9. Figure 6 shows the relaxation time as function of temperature for amorphous salicin, felodipine, and nifedipine when they are cooled into glass from the liquid state at 10 K/min. During cooling, the change in relaxation time over the glass transition region follows a path with a smooth crossover between the liquid and glass regimes. The fragility of the compounds can also be readily identified from Figure 6 because fragility is, by definition, the non-Arrhenius temperature dependence of relaxation time as temperature approaches Tg. In fact, Angell’s dynamic fragility index m48,49

m)

d ln τ d(Tg/T)

|

(13)

Tg

in Figure 6 is the slope the relaxation time curve for each compound at their corresponding Tg. The magnitude of the slopes of the different lines in Figure 6 agrees with the corresponding fragility values. Discussion Effect of Nonlinearity and Nonexponentiality. Because molecular motion is thought to be one of the major factors leading to destabilization upon the processing and storage of amorphous drugs, there has been considerable interest in assessing structural relaxation in pharmaceutical materials in

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Figure 6. Left: change in the estimated relaxation time for amorphous salicin, felodipine, and nifedipine through the glass transition region following cooling the liquid at 10 K/min. Right: a magnified view of the temperature dependence of relaxation times above Tg. The difference in slope at Tg corresponds to the difference in fragility among compounds.

the glassy state. However, existing approaches for evaluating molecular mobility do not fully account for the combined effects of the nonlinear and nonexponential nature of structural relaxation. Relaxation time of a glass is often estimated employing the AGV equation (eq 5) with the liquid parameters, D and T0, determined from thermal, dielectric, or mechanical measurements.43,50 For the fictive temperature, a frequently used methodology, proposed by Shamblin et al., involves evaluating the temperature dependence of the fictive temperature entirely from the observed heat capacity of the crystalline, glassy, and liquid forms. Specifically, by assuming a hyperbolic temperature dependence of heat capacities, the entropic fictive temperature can be obtained using the following expressions:25

1 γCp 1 - γCp ) + Tf Tg T γCp )

Clp Clp

-

|

Cgp Cxp Tg

(14)

(15)

The relaxation time of a glass can therefore be obtained from eq 5 using the fictive temperature obtained from eqs 14 and 15, based on heat capacity measurements. This method is denoted “AGV method” in the discussion that follows. This approach is experimentally convenient but still bears two inherent limitations: (1) It neglects the effect of the nonexponential nature of relaxation (i.e., the β parameter) on molecular mobility. (2) It implicitly assumes that molecular mobility is independent of the method of preparation and thermal history of the glass. In fact, because the formation and performance of glasses largely rely on the relative timescales of the structural relaxation and the temperature perturbation, the thermal history could have a profound effect on the molecular mobility in pharmaceutical glasses. This point is discussed in greater detail below. The effect of relaxation nonexponentiality, β, on molecular mobility is illustrated in Figure 7, where the relaxation time of amorphous salicin, using the AGV method,25 is presented as a function of temperature. The salicin data used to generate the plot are listed in Table 1. The same salicin data (including β) are then employed to obtain relaxation time versus temperature

Figure 7. Comparison of the relaxation times of amorphous salicin evaluated using the proposed method and those evaluated using the AGV method.

based on the model presented here (eqs 8 and 9), following cooling the material from the liquid at 10 K/min. To illustrate the effect of β, curves for salicin with different hypothetical values of β are also shown. Figure 7 shows that in the temperature region not far away from Tg (within ∼30 K below Tg, T/Tg ) 0.9 in the case of salicin) the AGV method and the method proposed here give comparable results. However, the curvature (non-Arrhenius) profiles are not the same and predictions by the two methods diverge at lower temperatures. In addition, the model proposed here (eqs 8 and 9) predicts VTF behavior at the higher temperatures where the liquid is formed. The degree of relaxation nonexponentiality also contributes to the molecular mobility of glasses. The effect of β becomes more pronounced as the material is brought to lower temperatures. When cooled below Tg, a less-exponential relaxation leads to the material exhibiting greater relaxation times. A change in β by 0.1 unit is sufficient to generate significant variation, especially when molecular mobility is evaluated at temperatures farther away from Tg. One very important aspect of amorphous systems is the significant role that thermal history has on their observed behavior. Predictive methods based on configurational entropy

13250 J. Phys. Chem. B, Vol. 111, No. 46, 2007

Figure 8. Estimated relaxation in cool-heat cycles, with different cooling rates, for amorphous salicin. The reheating rate is the same, 10 K/min, in all cases.

Mao et al. preparation will contribute significantly to its performance by altering its initial (and, consequently, the evolution of) molecular mobility. Any characterization method neglecting these factors may be insufficient in providing an accurate estimation of the molecular mobility of the glasses. The study of the effect of thermal history and experimental conditions has many practical implications. For instance, from Figure 9, pharmaceutical glasses produced by rapid cooling could be stabilized by slowly heating the material to a temperature below Tg, such that during the heating process the molecular mobility of the glass could be made to decrease, despite the increasing temperature. Effect of Annealing. In addition to the effect of temperature, changes in relaxation time during annealing (aging) at a fixed temperature are of particular interest with amorphous pharmaceutical products because the kinetics of molecular mobility may have a profound effect on their physical and chemical stability during storage.26 The evolution of τ as function of time can be assessed by adding an annealing process into the cooling/heating cycle modeled with eqs 8 and 9 at any given temperature, following the computational approach proposed by Hodge and Berens.32 Specifically, in order to track changes in τ as a function of time at a given temperature during the cycle, the term ∆Tk/ qk in eq 8 is replaced with tk, representing a subinterval of the annealing time. The evolution of τ of an amorphous material subjected to a thermal cycle consisting of isothermal holds can therefore be obtained as follows: (1) Calculate Tf and τ using eqs 8 and 9 for a total of mA steps, until the annealing temperature is reached. (2) The calculation of Tf during annealing (assuming a total of mB isothermal steps over time t) is incorporated into eq 8 after the completion of step 1. Because the temperature is held constant during annealing, the summation over temperature (j) is stopped, while the summation over time (k) continues through the annealing process (replacing ∆Tk/qk with tk). Equation 8 and 9 are therefore modified to give mA

Tf,m ) T + 0

Figure 9. Predicted relaxation time as a function of temperature for amorphous salicin when the equilibrium liquid is cooled at 100 K/min and subsequently heated back above Tg at 0.1 K/min.

alone cannot account for the effect of thermal history on molecular mobility. The effect of thermal history and experimental conditions is illustrated for the case of salicin in Figures 8 and 9. The rate of cooling the liquid clearly has significant influence on the molecular mobility of the resulting glass. Figure 8 shows that an increase in cooling rate by an order of magnitude could lead to a corresponding decrease in relaxation time of comparable scale in the resulting glass. The hysteresis of molecular mobility between cooling and reheating is also evident, even when the cooling and reheating rates are equal. At the molecular structural level, heating and cooling are never the reverse of each other in glassy materials, and this lack of symmetry carries important implications. Because of the relatively low molecular mobility in glasses, materials reheated from the glassy state respond sluggishly to the rise in temperature and continue to relax toward a more-stable structure during heating, resulting in a greater relaxation time, at the same temperature, than that of the glass prepared by cooling the liquid. An extreme example of this condition is observed in Figure 9, where equilibrium liquid salicin undergoes rapid cooling to form the glass, followed by slow reheating. Depending on the experimental conditions, the molecular mobility of the salicin glass from heating and cooling could differ by over an order of magnitude at the same temperature. This indicates that for any particular glassy material, the thermal history and method of

mA

∑ j)1

(

∆Tj ∑ j)1

∆Tj 1 -

(Clp - Cxp)

(Cgp - Cxp) (Clp - Cxp)

τk ) τ0 exp

| | ){ [ ( ) ]}

(Cgp - Cxp)

[(

+

Tj

1 - exp -

Tj

tk

k)mA

τk

∑

DT0 mA

T0+

)

β

m

∑1 ∆T (1 - T0/Tf,k-1)

]

(16)

(17)

where

tk ) t1/mB tk ) t

(k-mA)/mB

-t

(k-mA-1)/mB

if k ) mA + 1 if k > mA + 1

Equations 17 and 18 apply only during annealing (i.e., when mA < m emA + mB). To accurately describe the change in τ during annealing, we divided the duration of isothermal hold into a large number of subintervals (mB is typically set to be above 200) that are even-logarithmically spaced32,51 so that within each subinterval the relaxation time does not change significantly and can be represented appropriately by a single value. (3) Calculations based on eqs 8 and 9 are resumed after the isothermal hold is completed, that is, when m > mA + mB, to describe the cooling/heating cycle following the annealing step.

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J. Phys. Chem. B, Vol. 111, No. 46, 2007 13251

Figure 11. Change in the estimated τ with time during annealing at 25 °C for amorphous salicin prepared by cooling the equilibrium liquid at different cooling rates.

Figure 10. Experimental and simulated DSC thermograms showing the enthalpy recovery of amorphous salicin aged at 25 °C (T/Tg ) 0.92), as a function of aging time. Aging time varied between 0 and 16 h.

Using this method, we can determine the evolution of τ and Tf during the isothermal annealing, as well as the subsequent thermal treatment. In other words, the relaxation of an amorphous organic compound following any arbitrary thermal treatment can be quantitatively determined by combining eqs 8 and 9 with eqs 16 and 17. The extent of relaxation during annealing may alter the glass transition profile observed upon reheating the aged samples. This effect can be assessed from the proposed model by calculating Cp using eq 11. The calculated Cp versus temperature profiles as the result of reheating the aged salicin samples are compared with the experimental DSC thermograms in order to test the accuracy of the simulation model. The detailed DSC thermal cycles are specified under the Materials and Methods section. The experimental and simulated DSC thermograms following the designated thermal treatment are shown in Figure 10, aged at 25 °C (T/Tg ) 0.92) for periods ranging between 0 and 16 h. The simulation properly predicts the observed shifts in the glass transition profiles of the aged samples, relative to the Tg of the fresh glass, as well as the entire enthalpy recovery process as function of annealing time. This suggests that the model is capable of describing a continuous relaxation process for an organic amorphous compound with a high level of confidence. The time dependence of relaxation time of amorphous salicin aged at 25 °C (T/Tg ) 0.92), calculated using the model, is shown in Figure 11. The figure shows the evolution of τ over a 24-hour period from the onset of annealing for glassy salicin produced by cooling the liquid at different cooling rates. At any given temperature, the material cooled at 100 K/min shows a much shorter relaxation time than the material cooled at 1 K/min. Having greater molecular mobility to start, the fastercooled material also exhibits greater change in τ during

annealing. This result agrees with previous studies indicating that changes in τ by several orders of magnitude are possible for some pharmaceutical materials within the time scale of the experiment.26,52 This situation in turn raises questions about the common practice of characterizing molecular mobility at a given temperature using the relaxation time averaged over the entire annealing process. The fast dynamics of molecular mobility observed under isothermal conditions render the concept of mean or characteristic relaxation time inadequate in many cases. Time is just as important of a dimension as temperature on the dynamics of molecular mobility in amorphous pharmaceuticals. Depending on how a particular sample came into being, the kinetics of molecular mobility of an amorphous compound could vary significantly, even if the storage temperature is held constant. The very act of creating a glass bears a thermal history, which imprints information on the material that significantly affects its subsequent thermal and relaxation behavior. Without such information, any account of the molecular mobility in amorphous pharmaceutical systems will be incomplete at best. Conclusions There is a general notion that the structural relaxation of an amorphous material can be understood by studying its behavior during the glass transition event. Published approaches include investigations on the width of the Tg53,54 or the scanning rate dependence of Tg.55,56 In this report, we present a modified Adam-Gibbs approach for the time and temperature dependence of the molecular mobility of low-molecular-weight, amorphous pharmaceutical compounds. In contrast to the original nonlinear Adam-Gibbs model,36 which takes the amorphous form as configurational ground state, our approach employs the crystalline forms to account for the configurational entropy. The AGV equation, rather than the more widely used NM expression, is used here to calculate the relaxation time in order to accord physical, conceptual, and mathematical continuity between the behavior of the liquid (VTF) and the behavior of the glass (AGV). The model has the ability to fully characterize an amorphous pharmaceutical material from one set of carefully collected heat-capacity data and to estimate, a priori, the time and temperature dependence of structural relaxation time under any arbitrary set of experimental conditions, including heating, cooling, and isothermal holds. The proposed model has no adjustable parameters; it is based on the premise that the parameters that characterize the relaxation

13252 J. Phys. Chem. B, Vol. 111, No. 46, 2007 behavior of the glass, β, D, and T0, are all encoded in the glass transition event. Fitting of the experimental heat capacity during the characterization step of the model is done in order to “extract” these parameters from the DSC data. These parameters, however, can all be experimentally obtained, independently of the model. In fact, the extent to which the extracted parameters match the independently determined values (and predict the DSC thermograms) is a measure of the applicability of the model to the particular system under study. Because there are no adjustable parameters, refinements of the model will not come from “tweaking” the values of the relaxation parameters of the glass former. Model refinements should come from properly dissecting the heat capacity terms into their configurational and vibrational components. The approach presented here integrates the nonexponentiality and nonlinearity of structural relaxation with the thermal history of the material. The model, which employs the entire glass transition profile, including the position, width, and overshoot, as well as the inherent effect of the measurement itself, is believed to be the closest quantitative description of the structural relaxation of amorphous pharmaceutical compounds among the thermal methods currently available. Acknowledgment. The financial support from the National Science Foundation, I/UCRC 000364-EEC, and the PurdueMichigan Joint Program on Supramolecular Assemblies of Pharmaceutical Solids is acknowledged. We thank an anonymous reviewer for very helpful comments and suggestions. References and Notes (1) Angell, C. A. Science 1995, 267, 1924. (2) Ediger, M. D.; Angell, C. A.; Nagel, S. R. J. Phys. Chem. 1996, 100, 13200. (3) Hodge, I. M. J. Non-Cryst. Solids 1994, 169, 211. (4) Scherer, G. W. J. Non-Cryst. Solids 1990, 123, 75. (5) Angell, C. A.; Green, J. L. Thermodynamics and Dynamics of Glassforming Systems. In Lyophilization of Biopharmaceuticals; Constantino, H. R., Pikal, M. J., Eds.; AAPS Press: Arlington, VA, 2004; Vol. 2, p 367. (6) Carpenter, J. F.; Pikal, M. J.; Chang, B. S.; Randolph, T. W. Pharm. Res. 1997, 14, 969. (7) Elversson, J.; Millqvist-Fureby, A. J. Pharm. Sci. 2005, 94, 2049. (8) Franks, F. Eur. J. Pharm. Biopharm. 1998, 45, 221. (9) Hancock, B. C.; Parks, M. Pharm. Res. 2000, 17, 397. (10) Fukuoka, E.; Makita, M.; Yamamura, S. Chem. Pharm. Bull. 1987, 35, 2943. (11) Aso, Y.; Yoshioka, S.; Kojima, S. J. Pharm. Sci. 2000, 89, 408. (12) Aso, Y.; Yoshioka, S.; Kojima, S. J. Pharm. Sci. 2004, 93, 384. (13) Zhou, D. L.; Zhang, G. G. Z.; Law, D.; Grant, D. J. W.; Schmitt, E. A. J. Pharm. Sci. 2002, 91, 1863. (14) Guo, Y. S.; Bryn, S. R.; Zografi, G. J. Pharm. Sci. 2000, 89, 128.

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