Exploring the Crystallization Tendency of Glass-Forming Liquid

Article pubs.acs.org/crystal

Exploring the Crystallization Tendency of Glass-Forming Liquid Indomethacin in the T−p Plane by Finding Different Iso-Invariant Points Karolina Adrjanowicz,*,†,§ Kajetan Koperwas,†,§ Grzegorz Szklarz,†,§ Magdalena Tarnacka,†,§ and Marian Paluch†,§ †

Institute of Physics, University of Silesia, ulica Uniwersytecka 4, 40-007 Katowice, Poland SMCEBI, ulica 75 Pulku Piechoty 1a, 41-500 Chorzow, Poland

§

S Supporting Information *

ABSTRACT: In this work, we examine in detail the crystallization tendency of the model glass-forming liquid indomethacin at varying thermodynamic conditions. To do that, we combine experimental results with predictions of the classical theories of nucleation and growth. We have considered changes in the crystallization rate k along iso-invariant curves located differently in the T−p phase diagram of the studied supercooled liquid. We include for the first time temperature and pressure conditions along which the thermodynamic driving force toward crystallization, commonly discussed as the difference in the chemical potential of the liquid/crystalline phases, Δμ, is expected to remain constant. Although the increase in pressure fosters crystallization (due to an increasing overlap of the nucleation and growth rates maxima), we show that when moving along certain thermodynamic paths the crystallization rate remains unchanged, or it is only slightly affected by the density increase. Therefore, by studying the kinetics of crystallization in the T−p phase space, we were able to provide information on the vast importance to get a better understanding of the crystallization behavior of the glass-forming materials and methods that can be used to tune this process.

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INTRODUCTION

significant deviations between theoretical predictions and experimentally measured temperature dependences of the crystallization rate have been reported in many cases. In agreement with the classical view, crystallization is a twostep process that involves the formation of an embryonic nucleus and then its subsequent growth to the macroscopic dimension. When decreasing the temperature of the melt, the rates of the crystal growth and nucleation go respectively through the maxima of the different magnitude and location with respect to each other. The overlapping zone between both processes determines good (or bad) glass-forming ability of the sample on cooling. When nucleation and growth curves are well-separated crystallization can be easily omitted on cooling.

A liquid is called “glass-forming” when it can avoid crystallization on cooling and become a disordered solid called a glass. Controlling crystallization and glass-forming abilities are of the vast importance in the science and technology to obtain materials of the desired characteristics and properties.1−4 However, it is a longstanding scientific problem to understand what makes a liquid vitrify or crystallize on cooling.5,6 For these reasons, knowledge of the fundamental principles governing the crystallization/glass-forming tendencies of various materials is an essential step. Crystallization is a complex process affected by various factors at the same time. A fundamental guiding picture of the crystal formation and description of its relation to the glass formation phenomenon come from the classical theories of nucleation and growth.7−10 This has remained substantially solid over the last century, although it is well-known that © 2016 American Chemical Society

Received: August 16, 2016 Revised: October 14, 2016 Published: October 17, 2016 7000

DOI: 10.1021/acs.cgd.6b01215 Cryst. Growth Des. 2016, 16, 7000−7010

Crystal Growth & Design

Article

pressure dependencies of the thermodynamic driving force and the specific interfacial energy.18 In the simplest case of the spherical nuclei and with assumption of the normal growth mechanism, we can directly relate Δμ (in J/g) to the thermodynamic barrier for nucleation (in J)

But if both curves considerably overlap, the crystallization process will be favored.11,12 The temperature dependences of the nucleation rate I (the number of nuclei formed per unit of time per unit of volume) and crystal growth rate U (the increase of the characteristic crystal size per unit of time) can be expressed by the classical expressions10,13−15

W *(T , p) =

⎛ W *(T , p) ⎞ ⎛ ΔG D(T , p) ⎞ I(T , p) = C1 exp⎜ − ⎟ ⎟ exp⎜ − kBT ⎠ kBT ⎠ ⎝ ⎝ ⎡ ⎛ ΔG(T , p) ⎞⎤ ⎛ ΔE(T , p) ⎞ U (T , p) = C2⎢1 − exp⎜ − ⎟ ⎟⎥ exp⎜ − ⎢⎣ kBT ⎠⎦⎥ kBT ⎠ ⎝ ⎝ (2)

where W* is the thermodynamic barrier to nucleation, ΔG is the thermodynamic driving force for crystallization (i.e., the difference between the free energies of liquid and crystalline phases), while ΔGD and ΔE are the kinetic barriers to nucleation and crystal growth, respectively. The kinetic term is often introduced by considering the selfdiffusion coefficient D (related to the viscosity η, via Stokes− Einstein relation) or the time of the molecular relaxation τα. Both dynamic quantities are experimentally (directly) accessible, at least for a significant number of the low-molecular weight glass-forming liquids. Assuming that the kinetic term describing nucleation and crystal growth rates can be equivalently expressed via the diffusion coefficient or the relaxation time, we can employ the same kinetic parameters/or their activation energies explicitly in relations 1 and 2. However, the reader should be also aware that in many cases, D is not correlated uniquely with η as given by the Stokes− Einstein relation. We will discuss this issue in the latter part of this paper. Thermodynamic aspects involve a description of the thermodynamic driving force of crystallization given by the difference in the chemical potentials between liquid/crystal phases (Δμ) and the crystal−liquid interface energy (σ). The free energy difference ΔG (in eq 2) can be as a matter of fact replaced by Δμ. Hence, one can assume that σ has no explicit influence on the growth rate. As there is no direct method allowing one to measure both thermodynamic parameters (Δμ and σ), we need to rely on the approximate equations derived from the classical consideration. This is a very problematic issue for specific interfacial energy which causes grave problems for the theory of crystallization processes (for the thermodynamic driving force at least some conventional methods exist to do that). The general expressions allowing to obtain the temperature and pressure dependences of Δμ and σ can be given as15−17

∫

1/3 where K0 and g0 are constants, m ≅ V2/3 m NA , σ0 comes from the Skapski−Turnbull formula

σ0 = γ0ΔSmTm/m0

m

0

p

ΔV (Tm(0), p) dp

(6)

where γ0 ≈ 0.4−0.6 is the dimensionless Stefan coefficient. The above equation connects specific surface energy and enthalpy of melt crystallization. And, in the simplest case, we can assume that σ is temperature (and pressure independent). But, with such approximation, drastic discrepancies between the magnitudes of the experimentally measured and calculated nucleation rates were reported.19 For that reason, temperature and pressure dependence of the specific surface energy needs to be introduced. Equation 5 provides one of the possible solutions to that problem. So far, it has been tested by us for some glass-forming liquids. However, in the literature, one can also find some other expressions allowing to calculate σ(T,P) dependence.20 Nevertheless, in each case obtained values should be considered with caution. Ideally, predicted rates of crystallization should be compared with experimental ones. In addition to pressure and temperature, one can also change the composition of a given system. The dimensionality of the phase space of a given substance is defined by the Gibbs phase rule. Therefore, the above consideration is valid only for onecomponent systems. As resulting from eqs 1 and 2, the overall crystallization is a complex phenomenon affected by dynamic and thermodynamic parameters, highly variable and difficult to control. On lowering the temperature, the thermodynamic driving force for crystallization increases, but at the same time the molecular movements slow down. Therefore, the overall crystallization/ glass-forming tendencies of liquids depend on the complex interplay between both key elements. Apart from that, it is additionally modulated by the liquid/crystal interface energy.7,10,50,21−23 By varying only the temperature it is enormously difficult to disentangle and determine the individual contribution coming from the kinetic and thermodynamic factors. However, this can be achieved by changing not only the temperature but also pressure. Introducing pressure as another thermodynamic variable can be rationalized by the fact that the phase diagram for each substance is two-, not onedimensional. Therefore, both parameters are needed to explore

∫T (0) ΔS(T , p) dT

∫p

(4)

∫

T

+

3Δμ(T , p)2

where Vcry is the specific volume of the crystallizing phase. Temperature and pressure changes of the specific volume (1/ density) for liquid and crystalline phases can be obtained directly from PVT measurements. For the liquid/crystal interface energy we can use the following expression (in J/cm2)15−17 T g σ(T , P) = σ0 − 0 ΔS(T , P) dT m0 Tm(0) P K − 0 ΔV (Tm(0), P) dP m0 P0 (5)

(1)

Δμ(T , p) = −

16πVcry 2σ(T , p)3

(3)

where ΔV = Vliq − Vcry (in cm3/g) and ΔS = Sliq − Scry (in J/ (K·g)) are the differences in the specific volumes and entropies of the liquid and crystalline states, respectively. Here, we wish to mention that very recently Schmelzer and co-workers have proposed new, generalized equations for the temperature and 7001



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the entire phase space and obtain complete information on the crystallization behavior of the glass-forming liquids. Hydrostatic pressure is an essential thermodynamic variable that controls molecular spacing. This has a significant impact on the molecular dynamics of the glass-forming liquids, which we know from some experimental studies.24−28 So we could also expect the important role of pressure on the crystallization behavior of the glass-forming liquids. In the T−p phase diagram, we can easily find paths of the constant pressure (isobars), temperature (isotherm), or density (isochore). But for a liquid in the supercooled regime where the molecular motions change by many orders of magnitude, this can also be extended to (T, p) points with constant αrelaxation time (isochrone). Moreover, keeping in mind that the crystallization process can be observed only below the melting temperature, we can also find in the T−p space state points with the same contribution coming from the thermodynamic driving force (Δμ = const.). While moving along such iso-invariant paths, the time scale of the molecular motions and thermodynamic driving forces toward crystallization are individually under control. This gives a unique opportunity to probe their actual impact on the crystallization tendencies of liquids and understand the complex interrelationship between dynamics and thermodynamics. Recently, we have pioneered crystallization studies under isochronal conditions, i.e., at different combinations of temperature and pressure characterized by the same time scale of global/cooperative mobility.16,17,29,30 While moving along the isochrone, changes in the crystallization rate can be exclusively explained by the variation of the thermodynamic factor. However, this has surprisingly resulted in almost temperature and pressure invariant behavior of the thermodynamic driving force and a relatively small effect of the density changes on the crystallization rate k.30 The motivation and scope for the present work are to explore some of our recent findings and recognize the contribution coming from the kinetic factor once the thermodynamic driving force of crystallization remains constant. For that, we have made an attempt to locate in the T−p phase space of the supercooled liquid so-called “iso-Δμ” conditions and determine their position with respect to isochrones and some other iso-invariant conditions. This includes constant crystallization rate (iso-k), pressure (isobar), temperature (isotherm), and density (isochore) conditions. To do that, experimental data and predictions of the classical theories of nucleation and growth were employed to investigate the crystallization phenomenon at varying thermodynamic conditions. The studied sample is indomethacin, a model glassforming substance showing crystallization tendencies above and below the glass transition. The crystallization behavior of indomethacin has been intensively investigated at atmospheric pressure31−35 and partially at elevated pressure (even by some of us when introducing the isochronal crystallization approach).16,36−38 Now, we have enriched this knowledge by identifying in the phase diagram of the supercooled indomethacin various iso-invariant lines and determine their position regarding each other and the maxima of the nucleation and growth rate curves. Obtained results provide new insight into the fundamental aspects of the crystallization phenomenon by taking a full advantage of using different combinations of the temperature and pressure.

Article

EXPERIMENTAL SECTION

Materials and Methods. Examined Material. The crystalline form of indomethacin (as γ-polymorph) was supplied from SigmaAldrich and used as received. The supercooled liquid state of indomethacin was obtained by quenching of the melt below the glass transition temperature (lit. value Tg = 316 K as obtained for rapidly quenched glass upon heating with 10 K/min39). During reheating of the glassy sample and holding it for several hours at around 363 K a cold-crystallization event can be observed. At atmospheric pressure supercooled indomethacin recrystallizes either as α-form or a mixture of α (ρα‑form = 1.40 cm3/g, Tm = 428 K) and γ (ργ‑form = 1.38 cm3/g, Tm = 434 K) forms.33 Yoshioka et al. have demonstrated that by varying the cooling rate one can obtain predominantly α-form (fast cooling) or γ-form (slow cooling).31 However, on increased pressure, the α-form of indomethacin is expected to be thermodynamically more favorable.36 This we have also observed for our samples crystallized under compression. The thermal analysis of indomethacin samples obtained at varying thermodynamic conditions can be found in the Supporting Information. Preparation of Samples for High Pressure Studies. Samples for crystallization studies were prepared by placing a crystalline material on a stainless steel electrode (10 mm diameter) and heating above the melting temperature of the γ-polymorph. To ensure complete melting we held it for 5 min at 440 K. Then, the liquid was covered with another electrode of the same diameter to yield a layer of ca. 0.08 mm thickness provided by Teflon spacer and cooled to room temperature on a chilled brass block. The capacitor was inserted between a Teflon ring, wrapped with a few layers of Teflon tape and latex to separate it from the pressure transmitting silicon oil. Following Crystallization Progress. Crystallization kinetics studies were carried out by using dielectric spectroscopy. The basic principle of using a dielectric technique to study crystallization is based on monitoring changes in the static permittivity with time. A drop of the dielectric signal signifies, in this case, lowering the number of the actively reorientating dipoles as the liquid/crystal transformation proceeds. By following these changes, it is possible to follow the crystallization kinetics of glass-forming liquids at the real-time of measurements. To do that, obtained data were expressed as normalized quantities ε′N(t) = (ε′initial − ε′(t))/(ε′initial − ε′final) and then fitted with the use of the Avrami equation,40,41 ε′N(t) = 1 − exp(−ktn); k is an experimentally measured crystallization rate constant being a function of nucleation and crystal growth rates (k ≈ IUn−1), while n is te Avrami parameter containing information about the shape of the growing crystallites. From that, we see that the crystallization rate k obtained from the dielectric measurements contains, in principle, combined information about nucleation and growth rates (so the overall crystallization progress). By resting only on the dielectric crystallization data, we cannot however distinguish their individual contributions. Dielectric Measurements on Increased Pressure. To study the crystallization kinetics of supercooled indomethacin at varying temperature and pressure conditions, we have utilized a high-pressure system with MP5 micropump (Unipress, Institute of High-Pressure Physics, Warsaw, Poland) and external pressure control. The pressure was controlled with the tolerance better than 2 MPa by an automatic pressure pump. The temperature was controlled with a precision better than 0.1 K by the environmental chamber (Weiss Umwelttechnik GmbH). For dielectric measurements, we have utilized an impedance analyzer (Alpha-A Analyzer, Novocontrol GmbH, Montabaur, Germany). The real and imaginary parts of the complex permittivity were measured every 600 s within the same frequency range (0.5 Hz to 3 MHz). Crystallization measurements at each given (T, p) point were repeated 2−3 times to ensure good reproducibility. Finding Different Iso-Invariant Points. In this study, we have considered a number of iso-invariants which includes iso-thermodynamic driving force (Δμ = const.), isochoric (Vliq = const.), isocrystallization constant rate (k = const.), isobaric (p = const.), isothermal (T = const.), and isochronal (τα = const.) conditions. The results of the crystallization kinetic studies along isobars (p = 10 and 7002



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75 MPa), isotherms (T = 368 and 391 K), and isochrone (log10(τα/s) ≅ −3.1) were taken from the previous paper.16 The isochronal conditions include the following (T, p) pairs: (340 K, 0.1 MPa), (343 K, 10 MPa), (368 K, 100 MPa), and (391 K, 220 MPa). An additional point from the atmospheric pressure (T = 340 K, p = 0.1 MPa) was measured for the purpose of the present study. The properties of the supercooled liquid indomethacin at this condition (Vliq = 0.77503 cm3/g, log(τα/s) ≅ −3.1, Δμ = 14 J/g, log10(k/s−1) = −5.2) will be considered as a starting point for determining considered iso-invariant conditions. Isochoric (T, p) points having a constant value of the specific volume of the liquid phase (Vliq = 0.77503 cm3/g) were obtained by using PVT data parametrized with the use of the equation of state (EOS) for supercooled liquids (see ref 16 for details). This includes the following combinations of temperature and pressure: (340 K, 0.1 MPa), (346.5 K, 10 MPa), (379 K, 60 MPa), (404 K, 100 MPa), and (459 K, 200 MPa). To determine thermodynamic conditions along which the crystallization constant rate remains the same (log10(k/s−1) = −5.2) we have used isobaric and isothermal crystallization data. This gives the following (T, p) combinations: (340 K, 0.1 MPa), (341.6 K, 10 MPa), (354.6 K, 75 MPa), (368 K, 137 MPa), and (391 K, 241 MPa). As already mentioned, the thermodynamic driving force is difficult to approach from the experimental point of view. In contrast to the αrelaxation time, Δμ cannot be measured directly. At most, it can be estimated from eq 3 by using the specific heat capacity and volumetric data. However, this requires making some additional assumptions, especially since we have limited access to the calorimetric data on increased pressure. The procedure allowing estimation of the most probable changes of the thermodynamic driving force of crystallization on increased pressure can be found elsewhere.17,30 On the basis of obtained dependences we have generated a number of (T, p) points along which Δμ should remain constant, and it equals Δμ ≅ 14 J/g: (340 K, 0.1 MPa), (T = 344 K, p = 10 MPa), (T = 353 K, p = 30 MPa), (T = 371 K, p = 90 MPa), (T = 389 K, p = 150 MPa), and (T = 402.6 K, p = 200 MPa). The desired thermodynamic conditions were reached by starting from p = 0.1 MPa and T = 340 K. Then, we have systematically increased the temperature and pressure, however, always remaining in the supercooled liquid regime. Each time crystallization studies were performed for freshly prepared indomethacin samples of the same history of preparation. After complete recrystallization of indomethacin on increased pressure, the temperature was decreased first. Afterward, the pressure was released. Crystalline materials obtained in this way were taken out for calorimetric measurements to verify which polymorphic form of indomethacin is favorable under different thermodynamic conditions. Obtaining Overall Crystallization Curves at Atmospheric Pressure. For indomethacin, the curve describing the overall rates of crystallization was obtained by performing a series of time-dependent crystallization measurements at atmospheric pressure with the use of Novocontrol Alpha Analyzer Concept 80 (frequency range from 1 × 10−2 to 3 × 106 Hz). The sample was placed between two stainless steel electrodes (diameter: 10 mm, gap: 0.08 mm), mounted inside a cryostat, and maintained under dry nitrogen gas flow. The temperature was controlled by Quatro Cryosystem using a nitrogen gas cryostat, with a stability better than 0.1 K. Crystallization was followed by analyzing changes in the static permittivity ε′ in the frequency region not affected by the electrode polarization. We explore the range of temperature from 340 K up to 413 K. The samples for crystallization kinetics studies were prepared by following the same protocol as the ones which were crystallized on increased pressure. The melted compound was first cooled to room temperature and waited for some time to active the nucleation process. Then the temperature was raised to the final crystallization temperature, Tc.

some important features related to the molecular mobility aspect and its relation to the crystallization process. By its nature, crystallization is driven by thermodynamics. However, molecular movements are also essential to proceed with the crystal formation. Therefore, it is vital to recognize which dynamic quantity determines the rate of the molecular rearrangement relevant to the crystallization process. The classical nucleation and growth pictures do so by introducing molecular diffusivity through the self-diffusion coefficient D as an essential component required to align the molecules into the crystalline lattice. Owing to the limited range of the diffusivity data, the self-diffusion coefficient is often replaced by the viscosity (η), or the α-relaxation time (τα) describing the molecular reorientation in the supercooled liquids. In general, the dynamic properties of the liquid between the melting and the glass-transition temperatures have a nonlinear character and change much faster than predicted from the simple temperature activated Arrhenius law. This behavior can be captured by, e.g., the Vogel−Fulcher−Tammann equation. The choice of D, η or either τα should provide similar information regarding the molecular mobility which affects the crystallization, on the condition that D is coupled to η (and τα) as provided by Stokes−Einstein expression. However, this becomes far more complicated for deeply supercooled liquids, because the temperature dependences of D and η (similarly D and τα) do not follow each other. In such case, identifying whether translational or reorientational movements represent the effect of the molecular mobility on the crystallization progress is an essential step for providing a proper description of the kinetic factor. For indomethacin, Ediger et al. have demonstrated that close to the glass transition the temperature dependence of D changes weaker than the structural relaxation time (or viscosity) and scales as τ−0.76 .42 In such case, the Stokes− α Einstein equation is expected to hold only far above the glass transition temperature Tg, i.e., T > 1.12 Tg. We expect that for the considered starting conditions (0.1 MPa, 340 K), the Dτα product shows only a weak deviation from the constancy predicted by the Stokes−Einstein formalism. However, an open question that remains is the relationship between both quantities on increased pressure. As we assumed, in the studied range of pressures (up to 200 MPa) the relation between D-η (and D-τα) remains unchanged. This can be rationalized by experimental evidence reported for simple molecular liquids (except water).43,44 Because of the decoupling phenomenon, an important question arises: what type of the molecular motions controls the crystallization process in the supercooled liquid state of indomethacin? Ediger42 and Yu45 have reported a strong correlation between the crystal growth rate and the molecular motions responsible for self-diffusion of indomethacin over a wide range of temperature. This suggests that the translational diffusion seems to play a significant role in controlling the rate of crystal growth. However, does it mean that they play the dominant role over the reorientational movements in controlling the time scale of crystallization? To test this supposition, we have investigated changes in the rate of crystallization for supercooled indomethacin over a wide temperature range. The crystallization rate k, as obtained by treating the dielectric crystallization data with the use of Avrami equation, contains combined information about the rates of nucleation and crystal growth. This we have explained in the Experimental Section. Figure 1a illustrates the temperature

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RESULTS AND DISCUSSION Molecular Mobility Aspect and the Crystallization Rate. Before examining the crystallization behavior of indomethacin in the T−p plane, we wish to focus a bit on 7003



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Figure 1. (a) Behavior of the crystallization rate k as a function of temperature determined for indomethacin based on the crystallization kinetic studies carried out at atmospheric pressure. The solid line denotes the best fit to the experimental data which was achieved by using exponential linear combination function (k = p1 exp−T/p2 + p3 + p4T with p1 = −867, p2 = 478, p3 = 700, p4 = −0.8). The k(T) dependence from the highlighted range was used to compare with the corresponding τα(T) and D(T) dependences. The crystallization rate k plotted as a function of (b) the αrelaxation time and (c) the self-diffusion coefficient. Solid lines with the slopes 0.39 and 0.53 indicate that the temperature dependences of τα and D change at least two times faster than the crystallization rate. The dashed line in panel (a) shows probable evolution of the crystallization rate for supercooled indomethacin assuming that its temperature dependence in the highlighted region is perfectly coupled to the self-diffusion coefficient. The self-diffusion data for indomethacin were taken from ref 45, while the α-relaxation times are from ref 46.

evolution of k as obtained for indomethacin at 0.1 MPa. It is interesting to note that within the temperature range covering almost 100 K the rate of crystallization changes only within two decades. Moreover, the activation energies of the crystallization process on both sides of the crystallization curve maximum have turned out to be very similar. In the next step, we have compared the temperature dependences of the experimentally measured crystallization rate k and the corresponding τα or D dependencies. To do that we have taken data from the same temperature range and plot them versus each other in a log−log plot, as presented in Figure 1b,c. The temperature evolution of both dynamic observables was taken from the literature.42,46 If τα(T) (or D(T)) dependence tracks alone changes in the rate of crystallization, a straight line with slope = 1 should be obtained. However, the results presented in Figure 1b,c indicate that neither the molecular mobility reflected in the α-relaxation time (slope 0.39) nor the self-diffusion coefficient (slope 0.53) can predict alone changes in the crystallization rate in of supercooled indomethacin. It is clear that D(T) and τα(T) dependences change at least two times faster than k. The dashed lines visualize the relationship between the crystallization rate and the corresponding dynamic quantities assuming that their temperature dependences match each other. In the literature, it has been speculated that molecular mobility determines the rate of crystallization in glass-forming liquids.47,48 However, our finding shows that this does not seem to be essentially true for the studied sample. At first sight, the results presented in Figure 1b,c are inconsistent with the literature data showing a strong correlation between the crystal growth rate and the molecular motions (particularly the self-diffusion coefficient).42,45 However, we would like to draw the reader’s attention to the fact

that the growth rate is measured by tracking on the macroscopic level the distance from the center of a single cluster to its edges and then averaging over the rates obtained for some other crystals of the same polymorph. On the other hand, the crystallization rate obtained from the time-dependent dielectric measurements gathers information about the overall crystallization progress from the entire sample volume. This can essentially include more than one crystallizing centers/clusters, or different polymorphic forms that grow at the same time. Therefore, the rate of crystallization obtained from the experimental study is faster than that predicted assuming that the crystallization is perfectly coupled with the translational diffusion (or α-relaxation). We also wish to note that at a fixed pressure the crystallization process is affected by a number of factors, not only the molecular mobility; k is defined by all quantities entering the growth rate and the nucleation rate: driving force, surface energy, and kinetic coefficients. This explains why the temperature dependence of the overall crystallization rate does not behave in accordance with the changes of the self-diffusion coefficient (or α-relaxation time). Crystallization in T−p Plane. Figure 2 demonstrates the location of the different iso-invariant points in the phase diagram of the supercooled indomethacin. This includes various temperature and pressure combinations along which the αrelaxation time (iso-τα, log(τα/s) ≅ −3.1), thermodynamic driving force of crystallization (iso-Δμ, Δμ = 14 J/g), crystallization constant rate (iso-k, log10(k/s−1) = −5.2), and specific volume of the liquid phase (isochore, Vliq = 0.77503 cm3/g) are expected to be constant. For the sake of clarity, isobaric and isothermal lines are not shown. As can be seen, all considered iso-curves start from the same (T, p) point, but with increasing the temperature and pressure their evolution in T−p space becomes much different. Interestingly, we note that iso-k 7004



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curves, we have also considered changes in Δμ along some other iso-invariant points that include isochoric (V = const.), isochronal (τα = const.), and iso-crystallization constant rate (k = const.) conditions. The starting condition in each case is T = 340 K and p = 0.1 MPa at which the supercooled liquid state of indomethacin has the following properties Vliq ≅ 0.77503 cm3/ g, log10(τα/s) ≅ −3.1, log10(k/s−1) = −5.2. By looking at the results presented in Figure 3, it is evident that the thermodynamic driving forces toward crystallization vary differently along considered iso-conditions. The most significant drop off is expected when increasing both T, p to maintain the same density of the supercooled liquid. This suggests lowering the contribution coming from the thermodynamic driving force of crystallization when moving along isochoric conditions. On the other hand, when keeping a constant time scale of the molecular movement upon crystallization progress (and crystallization rate) Δμ is expected to increase. However, it should be noted that these changes are relatively small especially when compared to, e.g., isobaric or isothermal dependences. Interestingly, we found that the temperature and pressure evolution of Δμ have very similar characteristics for iso-(τα) and iso-(k) conditions. From Figure 3 one can also see that when moving in T−p plane it is possible to vary with the temperature and pressure in such way to retain the same Δμ (open stars in Figure 3). Along with iso-(Δμ) lines the crystallization behavior of the supercooled liquid is expected to be driven primarily by the molecular mobility factor. This will be tested for indomethacin in the latter part of this paper using various combinations of (T, p) to maintain the same contribution of Δμ, Δμ ≅ 14(J/g) selected to match the starting point from the atmospheric pressure. To investigate the crystallization tendency of indomethacin at varying thermodynamic conditions we have followed the kinetics of crystallization with the use of the dielectric spectroscopy. In Figure 4a we present obtained evolution of the crystallization constant rate as a function of pressure. The

Figure 2. T−p phase diagram of supercooled indomethacin showing the location of the different iso-invariant points along which changes in the crystallization rate was considered.

and iso-(τα) curves are located close to each other. This suggests that keeping the same time scale of the molecular motions upon crystallization progress produces, to some extent, the same effect as controlling the rate of crystallization at varying thermodynamic conditions. Now, let us focus on the effect of pressure on the thermodynamic driving force of crystallization. By taking into account a number of the assumptions that requires calculating Δμ at different (T, p) conditions, obtained dependences serve only as an approximation of the most probable evolution of Δμ in the T−p space. In Figure 3 we present the evolution of Δμ at

Figure 3. Temperature dependence of the thermodynamic driving force toward crystallization for indomethacin as predicted for a few pressures from 0.1 to 200 MPa by using eq 3. Changes of Δμ along representative iso-invariant points are also shown.

various thermodynamic conditions calculated from eq 3. As can be seen, at a fixed pressure (i.e., at isobaric conditions) Δμ decreases with increasing the temperature and vanishes at the melting point (Δμ = 0). On the other hand, the increase of pressure shifts the entire Δμ(T) dependences toward higher temperatures. This signifies that while moving along an isotherm in the T−p phase diagram of the supercooled liquid the thermodynamic driving force toward crystallization is expected to increase with compression and should favor the crystal formation. Apart from the isobaric and isothermal

Figure 4. Experimentally measured crystallization constant rate for indomethacin plotted versus the corresponding pressure p (a) and density ρ (b). Pressure and density dependencies of k were studied along two isobars, two isotherms, isochrone, and for iso-Δμ points. 7005



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crystallization data spans the available phase space to demonstrate that k changes differently along considered isopaths. For example, the increase of temperature at a fixed pressure speeds up the crystallization progress, while compression of liquid at constant temperature slows it down. Using isobaric and isothermal crystallization data, it is also possible to find in the phase space (T, p) points with the same rate of crystallization. These points create in the phase space iso-crystallization constant rate (“iso-k”) curve. This is a clear signature that by varying with temperature and pressure, it is possible to tune the crystallization tendency of glass-forming liquids. Figure 4b shows the corresponding crystallization data plotted as a function density. Here, the density values were taken from the parametrized PVT data. It is very interesting to note that the experimentally observed behavior of the crystallization rate along iso-τα and iso-Δμ curves are very much alike. In both cases, we observe a slight increase of k with increasing pressure/density. It is also evident that the density exerts the weakest effect on the crystallization progress along iso-τα and iso-Δμ curves. Quite a good overlap of the crystallization data for iso-τα and iso-Δμ paths is a very surprising finding. Moreover, it indicates that controlling exclusively the time scale of the molecular mobility (as reflected by α-relaxation) or the thermodynamic driving force leads to almost the same crystallization behavior in the T−p plane. The straightforward implication of our finding is also that there must be some hidden link between the kinetic and thermodynamic driving forces. This explains why keeping one factor to remain constant upon crystallization progress is immediately sensed by the other one, and why controlling thermodynamics and dynamics independently eventually leads to the same crystallization behavior. To the best of our knowledge, this has never been reported experimentally for molecular glass-formers to date. At this stage, we cannot provide a detailed explanation why crystallization rates determined along iso-τα and iso-Δμ points change in the T− p space in a very similar manner. For that, detailed theoretical considerations and computer modeling are needed. We hope that our experimental results could be a great inspiration for that. Intuitively, when controlling the thermodynamic driving force, we expect that changes in the crystallization behavior of the investigated sample arise primarily from the molecular mobility factor. Indeed, for the considered (T, p) points located along iso-Δμ path we have observed a change of τα for almost two decades (it decreases when increasing the temperature and pressure to maintain constant Δμ). At first sight, this seems to be a very pronounced change over the investigated T−p range. On the other hand, we wish to mention that the α-relaxation time changes by many (7−9) orders of magnitude along isothermal or isobaric conditions when increasing pressure by ∼200 MPa or temperature by almost 120 K, respectively. By taking this experimental evidence into account, we believe that the changes in the crystallization rate along iso-Δμ are noticeable, but not considerable. In Figure 5 we have examined the correlation between αrelaxation time and the characteristic crystallization time τcryst (τcryst = 1/k) along iso-Δμ and isobaric paths. If both time scales are coupled to each other, we should get a straight line with slope 1. Such a scenario is expected when the thermodynamic factor remains constant so that the molecular motions are the dominant force responsible for the crystallization. As a matter of fact, controlling the entire

Figure 5. Main plot: Characteristic crystallization time τcryst plotted versus α-relaxation time τα as measured for indomethacin along iso-Δμ (Δμ = 14 J/g) and isobar (p = 10 MPa). Solid lines denote linear fit to the data. Inset: The dependence of the crystallization time plotted as a function of τα−0.76 which should describe the self-diffusion coefficient assuming that the decoupling ratio provided by Swallen and Ediger holds also at elevated pressure.

thermodynamic part of the crystallization process is practically impossible (at least in the experimental study) as this requires manipulation of Δμ and σ at the same time. However, their evolution in T−p space does not essentially follow each other. This is because Δμ is expected to increase with increasing pressure at fixed T/lowering the temperature at constant pressure, while σ decreases with increasing pressure/increases with lowering the temperature, respectively. In the present study, we have deliberately avoided examining σ, and consider only changes of Δμ as the fundamental thermodynamic driving force of crystallization. Figure 5 illustrates that the time scale of the crystallization changes two times slower than the αrelaxation time (slope = 0.5) for both iso-Δμ and isobaric paths. Additionally, in the inset of Figure 5 we have plotted the dependence of the crystallization time as a function of τα−0.76 which should describe the self-diffusion coefficient D assuming that the decoupling between both quantities on increased pressure can be still described by using the ratio provided by Swallen and Ediger.42 From that, we can presume that both dynamic quantities are loosely coupled to k and that the αrelaxation time/translational diffusion does not seem to be the dominant factor controlling crystallization in the supercooled liquid state. This is not surprising for isobaric conditions, as both key parameters governing crystallization (mobility and thermodynamics) vary when moving along this path, and we are not able to control their contribution. However, finding the same relationship between τcryst-τα quantities for iso-Δμ path is a very intriguing result. When we perform crystallization study along iso-Δμ conditions all the changes in the crystallization rate are expected to originate primary from the molecular mobility contribution. In such case, we could at least expect that the crystallization rate should change in accord with a time scale of the molecular movements. However, our experimental results indicate that this is not true; τα (or D) changes approximately two times faster than the characteristic crystallization rate. From a different perspective, this finding puts in doubt the validity of the general formalism used to describe the crystallization behavior of glass-forming liquids. Possibly, σ as another thermodynamic parameter should be taken into account to get a broad overlook on the contributions coming from the dynamics and thermodynamics in the crystallization phenomenon. 7006



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Figure 6. Shift of the nucleation (a) and crystal growth (b) rates curves for indomethacin with increasing pressure as predicted by the classical theory of nucleation and growth. The position of the different iso-invariant points regarding the maxima of the nucleation and growth is shown as well. (c) The probable location of the nucleation and growth rates maxima on increased pressure calculated for indomethacin. To make the comparison possible, obtained at different pressure curves were normalized with respect to the maximum values (U(T)/Umax (T) and N(T)/Nmax(T)).

Crystallization of Supercooled Indomethacin in T−p Plane from the Classical Viewpoint. The location and the magnitude of the nucleation and growth rates maxima are relevant to understand the crystallization tendency of the molecular liquid on cooling. In parallel, recognizing the effect of pressure on the nucleation and growth rates curves provides a complete picture of the crystallization behavior of the supercooled liquids in T−p space. At the same time, even at atmospheric pressure, it is not trivial to determine the entire temperature dependences of nucleation and growth rates. This is particularly very problematic for nucleation process, as the size of the critical cluster and the number of the particles involved in its formation (typically from few molecules up to hundreds/thousands of molecules) are sometimes too small to be observed by the conventional techniques probing the macroscopic signal of the sample.49 This becomes even more complicated on increased pressure, particularly because of the technical difficulties arising from the specificity of such experiments. Because of that reason, we have used the basic relations coming from the classical theories of nucleation and growth (eq 1 and eq 2) to predict the evolution of the nucleation and growth rates curves for indomethacin on increased pressure. It is well-known that the classical approach rests on numerous of the shortcomings (e.g., the critical nucleus is treated as macroscopic droplet that behaves in the same way as the bulk crystal regardless of its size) and often predicts

unreliable temperature dependences of the nucleation and growth rates. Nevertheless, it is still able to provide an elementary picture of the crystallization progress that has proven to be very useful for many systems.50−53 The position and the magnitude of the nucleation and growth rates maxima were estimated from the basic properties of indomethacin (such as temperature and pressure dependences of α-relaxation times, density, or specific heat capacity for liquid and crystalline phases) and following the general relations established within the classical theory.54,55 The results of our calculations are shown in Figure 6a,b. Since the melting temperature is typically expected to change (increase) with compression, we have plotted N(T) and U(T) dependences versus reduced temperature, i.e., Tm − T. Presenting obtained results in this way enable to compare the results coming from the different pressures. The change of the melting point was estimated from the Clausius−Clapeyron equation, which gives quite satisfactory agreement with the experimental results in the pressure range not exceeding 1 GPa.15,16 The evolution of the specific surface energy with the temperature and pressure was estimated from eq 5. To describe the temperature dependence of σ (T, p0) for the α-form of indomethacin we have used the relation provided by Zografi et al.34 The curves presented in Figure 6a demonstrate that the rate of nucleation is expected to magnify with increasing pressure. Also, its maximum shifts to higher temperatures, i.e., closer to 7007



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the melting temperature at a given pressure. On the other hand, the crystal growth rate seems to be less affected by the pressure effects. Assuming a normal growth mode, we have determined that the bell-shaped curves shown in Figure 6b shifts only a bit toward higher undercooling with compression. However, the magnitude of the growth rate remains almost the same. This is a clear sign that the changes in the nucleation rate on increased pressure seems to be the bottleneck in the high-pressure crystallization. On the other hand, the reader should be informed that the nucleation rate is strongly influenced by the value of the interfacial free energy, which we cannot measure directly, but only estimated making several assumptions. Variations of σ by 10% may change the value of the steadystate nucleation rate by 10 orders of magnitude, and the steadystate nucleation rate is an essential ingredient of the expression for the overall crystallization rate. In contrast, a small change in the viscosity or the thermodynamic driving force has a less pronounced influence on the nucleation rate. Except for the magnitudes of the nucleation and growth rate curves, their location with respect to each other is essential to recognize crystallization/glass-forming tendencies of liquids. Because of the entirely different time scales, the temperature dependences of both processes cannot be directly compared. This can be achieved by introducing reduced coordinates, as presented in Figure 6c. The zone of overlapping for N(T) and U(T) dependences is expected to expand with increasing pressure. This is a clear sign that, in general, the crystallization tendency of the glass-forming liquid increases with increasing pressure. We should also mention that this is consistent with many experimental results showing that under isobaric conditions compression facilitates the crystallization progress.16,56−58 As demonstrated above, the classical theory predicts increasing crystallization tendencies of indomethacin with increasing pressure. On the other hand, it does not essentially mean that the crystallization progress cannot be tuned by varying thermodynamic conditions, particularly by introducing different iso-invariant paths. As shown in Figure 6a,b, for iso(τα), iso-(k), and iso-(Δμ) curves we observe a shift of the considered iso-points toward higher undercooling, so that they progressively move away from the maxima of the nucleation and growth rates. This might have an important impact on the crystallization behavior of the investigated liquid at varying thermodynamic conditions. In contrast, when exploring the T− p space along isochoric conditions crystallization is expected to accelerate, as the isochoric crystallization points move toward the temperatures of the maxima of nucleation and growth rate. Using obtained N(T,p) and U(T,p) dependences and assuming that the crystallization process can be described reasonably well by the Johnson−Mehl−Avrami theory we have also predicted the overall crystallization rate for indomethacin using the following relation k theor(T , p) =

g U (T , p)m I(T , p) m+1

Figure 7. Evolution of the overall crystallization rate with the temperature and pressure as predicted for indomethacin using general relations provided within the classical theories of nucleation and growth. Position of the considered iso-invariant points with respect to the theoretically predicted crystallization maxima is shown as well. Inset: Representative crystallization rate curves generated based on the classical expressions from the theory of nucleation and growth. The rate of crystallization is essentially expected to decrease with increasing pressure at T = const.. Red points indicate the most probable explanation way experimentally measured crystallization rate for supercooled indomethacin slows down with pressurization at a constant temperature. However, one should also note that a slightly different scenario will be observed if the starting point is located on the other side of the crystallization curve, i.e., at higher temperatures right above the peak maximum.

The classical expression for the crystallization progress predicts the increase of ktheor with increasing pressure and a shift of its maximum value toward higher temperatures. On the other hand, experimental (T, p) points along constant τα and Δμ-curves are expected to progressively move away from the maximum rate of crystallization. Regrettably, for (T, p) points located on the iso-k curve, the classical approach is not able to predict the constancy. By comparing experimentally measured (see Figure 1a) and numerically calculated crystallization rates we also get a clear discrepancy. This is a clear sign that it is not possible to describe perfectly well the crystallization behavior of the molecular glass-forming liquids by resting purely on the classical expressions. It is possibly due to the σ contribution which determination is still a very problematic procedure. In the inset of Figure 7 we also present temperature evolution of the crystallization rate for representative isobars that were normalized by the corresponding maximal values. From that, we clearly see that the predicted dependences shift toward higher temperatures with increasing pressure. This explains the experimental finding that the pressurization at constant temperature slows down the crystallization progress of supercooled indomethacin. From the above, we believe that the simplified picture of the crystallization process as given by the classical approach could still serve as a basic guidance allowing us to understand differences in the crystallization tendency of indomethacin as observed experimentally along various iso-curves.

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where g is a shape factor, m is the number of independent directions of growth (m + 1 = n, where n is Avrami parameter).40,54,59 For three-dimensional crystallization and spherical shapes of the crystallization sites we get g = 4π/3 and m + 1 = 4. Obtained temperature dependences of the crystallization rate ktheor for pressures from 0.1 to 200 MPa are presented in Figure 7.

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CONCLUSIONS To conclude, we show that in the T−p diagram of the supercooled indomethacin one can find various iso-invariant 7008



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ACKNOWLEDGMENTS K.A, K.K., G.S., and M.P. are grateful for the financial support from the National Science Centre within the framework of the OPUS project (Grant No. DEC 2014/15/B/ST3/00364).

paths along which a number of the parameters can be controlled. This includes α-relaxation time, the thermodynamic driving force of crystallization, crystallization constant rate, or the density of the supercooled liquid. Such iso-invariant points can be reached only by adapting thermodynamic variables such temperature and pressure. Therefore, by ably control of the thermodynamic conditions, it is possible to alter the crystallization/glass-forming propensity of the molecular liquids. We have also used predictions of the classical theory to rationalize crystallization behavior of indomethacin along different iso-invariant points. Obtained results have indicated that increasing the temperature and pressure points located along iso-curves evolve differently regarding the maxima of the nucleation and crystal growth. Nucleation and crystal growth rates curves are also expected to overlap on increased pressure progressively. This should promote the crystallization when increasing pressure. The experimental results have also demonstrated that the crystallization tendency of indomethacin along iso-τα and isoΔμ curves is very much alike. To the best of our knowledge, this is the first experimental evidence reported of such behavior so far. The very important implication of this finding is that the dynamic and thermodynamic factors of the crystallization are in some way connected. Fixing mobility is immediately sensed by the thermodynamic driving force and vice versa. Hence, they do not vary independently and cannot be separated from each other by any means. From the future perspective, it will be very interesting to recognize the contribution coming from the specific surface energy to the crystallization behavior in the T− p space, including different iso-paths. Finally, we have also shown that keeping approximately the same Δμ does not guarantee that the crystallization time couples very well with the time scale of the α-relaxation time (or the self-diffusion coefficient). This should be the case because along iso-Δμ conditions all the changes in the crystallization rate are expected to originate primarily from the molecular mobility contribution. As a matter of fact, the temperature behavior of both dynamic variables along the isoΔμ curve changes approximately two times faster than the characteristic crystallization rate. Thus, our experimental finding is worth to be given in the future in-depth theoretical consideration by the general formalism used to describe the crystallization phenomenon in supercooled liquids.

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REFERENCES

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.cgd.6b01215. The results of the calorimetric measurements that were performed to identify which polymorphic form of indomethacin is preferable on increased pressure (PDF)

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest. 7009



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Exploring the Crystallization Tendency of Glass-Forming Liquid

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