Toward Better Understanding Crystallization of Supercooled Liquids

Communication pubs.acs.org/crystal

Toward Better Understanding Crystallization of Supercooled Liquids under Compression: Isochronal Crystallization Kinetics Approach Karolina Adrjanowicz,*,†,‡ Andrzej Grzybowski,‡ Katarzyna Grzybowska,‡ Jürgen Pionteck,§ and Marian Paluch‡ †

NanoBioMedical Centre, Adam Mickiewicz University, ul. Umultowska 85, 61-614 Poznan, Poland Institute of Physics, University of Silesia, ul. Uniwersytecka 4, 40-007 Katowice, Poland § Leibniz Institute of Polymer Research Dresden, Hohe Str. 6, D-01069 Dresden, Germany ‡

S Supporting Information *

ABSTRACT: In this paper, we present dielectric studies on the effect of different thermodynamic conditions on the physical stability of van der Waals glass-forming material of pharmaceutical interest, indomethacin. By maintaining isochronal condition during measurements, we were able to control the kinetic factor of the crystallization process and untangle purely thermodynamic effects on crystallization from kinetic ones. This cannot be achieved by any other experimental attempt performed at atmospheric pressure. Along with experimental studies, crystallization of supercooled indomethacin under pressure was described theoretically. We have demonstrated within the studied pressure range (0.1−220 MPa) that one should expect an increase of thermodynamic driving force, decrease in melt/crystal interface energy, and critical nuclei size. Therefore, an experimentally observed increase in the overall crystallization rate under isochronal conditions can be exclusively rationalized as due to variations of the thermodynamic factor.

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and geophysics.12−14 Compression techniques were also found out to be extremely useful upon determining new polymorphic forms.15 However, despite significant progress made in the past, the effect of pressure on the physical stability of glass-forming liquids is not thoroughly understood. It is also very important to stress that high pressure crystallization at varying temperature has been investigated to a much lesser extent. We presume that this might be due to experimental problems arising when using a high pressure. Crystallization of glass-forming melts under pressure is a very attractive scientific topic because on increased pressure it is possible to produce materials with extraordinary properties, sometimes not accessible by any other techniques. As an example, one can mention some polymers, for which crystallization induced by high pressure produces a material with high tacticity, crystallinity, and, consequently, improved mechanical and thermal properties.16−18 On the other hand, crystallization of some small molecular weight liquids performed upon increased pressure significantly slows down (or is being suppressed), which might open up a new route in the preparation and manufacture of extraordinary stable amorphous materials.19,20 Given above, contradictory examples clearly show that the impact of pressure on crystallization abilities of various materials is not universal, which points out a

rystallization is one of the oldest and probably the most commonly investigated processes, particularly because of many important implications in chemistry and material science. Although nucleation and crystal growth rate have been studied for several generations, up to now, in many aspects, it remains one of the most enigmatic and intriguing phenomena. In recent years, a lot of effort has been put toward studying crystallization phenomenon of organic supercooled liquids and glasses because of their considerable interest in the food and pharmaceutical industry.1−3 Spontaneous ability of amorphous pharmaceuticals toward recrystallization is the main challenge while working with these materials. Thus, most of the research interests concentrate on thorough identification and understanding critical factors that influence the crystallization process,4−6 which should help to develop highly stable glassy systems. An extremely interesting aspect associated with the crystallization phenomenon of glass-forming liquids is the impact of hydrostatic pressure on its progress. We all know that pressure is an important thermodynamic variable of obvious theoretical but also practical importance. Indeed, from experimental studies we know that pressure affects molecular packing and thermodynamic properties of investigated materials and, thus, might be a very promising parameter that governs crystallization.7,8 In the field of pressure-induced crystallization, a lot of work has been performed already for inorganic materials.9−11 Nucleation and crystal growth under pressure had been intensively investigated in the past, particularly in mineralogy © XXXX American Chemical Society

Received: August 22, 2013 Revised: October 10, 2013

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dx.doi.org/10.1021/cg401274p | Cryst. Growth Des. XXXX, XXX, XXX−XXX

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growth rates. On the other hand, within the same time, exponential terms describing the kinetic factor are responsible for slowing or even halting the crystallization progress as viscosity (relaxation times) increases rapidly on decreasing temperature/increasing pressure of the vitreous liquid. The interplay between both factors gives back the outcome of the overall crystallization process. Thus, depending on the magnitude of the kinetic and thermodynamic factors, one of them might get the dominance or nullify the influence of another one. As a result, crystallization of supercooled liquid will slow down or accelerate. From that, one can easily conclude that in the way toward a better understanding of crystallization phenomenon, it is necessary to put considerably more attention into controlling the kinetic and thermodynamic factors as well as the interplay between them. This can be achieved only by varying in wide temperature and pressure ranges, which we are capable of doing in our lab. Our current work is devoted to a new approach of studying crystallization that is based on keeping constant the structural relaxation time (so-called “isochronal” condition). The advantage of isochronal kinetics experiments stems from the fact that by keeping the same structural relaxation time while crystallizing indomethacin under different (T, p) combinations, we restrain its global molecular motions and, as a consequence, have the unique possibility to control the kinetic factor of the crystallization process (the second terms in eqs 1−3 are constant). It is worth stressing that this cannot be achieved by any other experimental attempt performed at atmospheric pressure. Under this condition, a major contribution to the change of nucleation and crystal growth rates of indomethacin under pressure comes exclusively from variations of the thermodynamic factor. Therefore, by analyzing isochronal crystallization kinetics, we were able to attain for the first time disentanglement thermodynamics effects on the crystallization from kinetic ones. To introduce our new approach, we have followed the crystallization progress under the pressure of selected van der Waals glass-forming liquid, indomethacin (the chemical structure of which is presented in the inset of Figure 1). The investigated material is a very important pharmaceutical substance, often termed as a modeled amorphous drug to study recrystallization phenomenon from the supercooled and glassy states.27 Some previous studies on increased pressure have demonstrated that compression has a significant impact on the physical stability of crystalline and amorphous forms of indomethacin. For example, the glassy state of indomethacin compressed at 43.7 MPa was found to recrystallize more easily.28 In agreement with the breakdown of the Ostwald rule of stages under hydrostatic pressure,29 indomethacin crystallized from ethanol slurry was reported to transform to the more dense α-form, being thermodynamically more favored on increased pressure.30 It is worth stressing that identification of the isochronal condition by controlling the global molecular mobility of a given system (so therefore kinetic factor of the crystallization process) is valid only when the diffusion coefficient is coupled to the viscosity. In the case of the vitreous liquids in the vicinity of the glass transition, we typically observe violation of the Stokes−Einstein equation. However, for investigated material we can presume, based on literature data,31 that at selected isochronal conditions, self-diffusion and viscosity should agree rather well (assuming as well that their relation does not change with pressure).

need of more detailed studies on the interplay between crystallization and glass formation. Knowledge on that matter seems to be a key point in many theoretical considerations and practical applications. The first attempt at describing high-pressure crystallization kinetics of inorganic glasses were made by Turnbull and coworkers.9−11,21,22 However, even at that time, the authors were fully aware they did not have a complete understanding of the crystallization process under pressure. A more elaborate theoretical study on the crystallization kinetics of glass-forming materials under pressure was provided only in the 1990s by Gutzow and co-workers23−25 who have presented a detailed analysis on the influence of increased hydrostatic pressure on crystallization of glass-forming melts and have provided some means to estimate basic parameters characterizing thermodynamic and kinetic factors of crystallization on increased pressure. In order to obtain complete information about the overall crystallization rate, it is necessary to describe three basic parameters characterizing the crystallization progress within the classical theory [i.e., steady-state nucleation rate (Ist), the time lag (τ#), and crystal growth rate (g)]. In a very basic assumption, these parameters at varying thermodynamic conditions (T, p) can be expressed as follows. Steady-State Nucleation Rate ⎡ ΔW (T , p) + ΔGD(T , p) ⎤ Ist(T , p) = A exp⎢ − ⎥ ⎣ ⎦ kT

(1)

where ΔW and ΔGD denote thermodynamic and kinetic barrier to nucleation, respectively. ΔW is the work required to form critical nuclei, whereas ΔGD is often discussed in terms of an effective diffusion coefficient (D), related to the viscosity (η) via the Stokes−Einstein relation (D ∼ 1/η). Nucleation Time Lag ⎛ σ(T , p) ⎞ ⎡ ΔGD(T , p) ⎤ ⎟exp⎢ − τ#(T , p) = ⎜B · 2 ⎥ ⎦ kT ⎝ Δμ (T , p) ⎠ ⎣

(2)

where Δμ is thermodynamic force for crystallization and σ is melt/crystal interface energy Growth Rate ⎡ ⎛ ΔG(T , p) ⎞⎤ ⎛ ΔE(T , p) ⎞ g (T , p) = C ⎢1 − exp⎜ − ⎟⎥exp⎜ − ⎟ ⎝ ⎠⎦ ⎝ ⎠ kT kT ⎣ (3)

where ΔE and ΔG are the kinetic and thermodynamic barriers to the crystal growth. Analogously, as in the case of ΔGD, the kinetic barrier to crystal growth ΔE can be discussed in terms of the diffusion coefficient, associated with viscosity. In addition, the value of the thermodynamic barrier ΔG can replaced by Δμ.26 Although a detailed analysis Ist(T,p), τ#(T,p), and g(T,p) dependences is beyond the scope of this paper, it is very important to note that each eq 1−3 is actually composed of two parts. The first terms of the above equations describe the thermodynamic factor of crystallization, whereas their second terms refer to kinetic factors. Both factors have, in fact, the opposite effect on the crystallization kinetics. The exponential terms related to the thermodynamic factor act in such way that upon decreasing temperature/increasing pressure of a supercooled liquid, they are responsible for increasing nucleation and B



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counterpart.35 On that basis, the glass transition temperatures at isobaric conditions were determined (Tg = T at which τα = 100 s). As it turned out, indomethacin reveals crystallization abilities in the vicinity of the glass transition; however, isochronal crystallization conditions were carefully selected (τα ≅ 7.8 × 10−4s) to avoid impractically long induction times. Dielectric measurements performed at elevated pressure were combined with pressure−volume−temperature (PVT) measurements and heat capacity studies that were necessary to estimate selected thermodynamic variables under pressure. The PVT data for indomethacin are shown in the inset of Figure 1. In Figure 2, panels a and b, we present the evolution of real and imaginary part of the complex dielectric permittivity during crystallization under isochronal conditions (τα ≅ 7.8 × 10−4 s), respectively. From these data, the normalized dielectric constant ε′N for each (T, p) pair was determined and plotted versus crystallization time as presented in Figure 2c. These experimental results were fit to the Avrami equation,36 given in the following form

Figure 1. Isobaric dependences of structural relaxation time and volume. Structural relaxation times obtained during isobaric measurements. Solid lines are temperature VFT fits. To guide the eye, dotted horizontal lines are also shown. They indicate the position of isochronal crystallization points with respect to the glass transition temperature at respective isobaric conditions. The inset shows isobaric PVT data for indomethacin.

α(t ) = 1 − exp( −kt n)

(4)

where k is crystallization rate constant and n is the Avrami exponents. Herein, it is very instructive to explain that experimentally determined values of the rate coefficient (k) contain important information about nucleation and crystal growth rates. In accordance with the Kolmogorov−Johnson− Mehl−Avrami (JMAK) model, the volume fraction of a growing crystalline can be described as37,38

It is worth noting that before isochronal crystallization kinetics experiments were performed, it was necessary to perform the whole set of additional measurements, to determine the most important features in the relaxation dynamics and physical properties of investigated materials under the examined pressure and temperature range. In the first step, we have studied molecular dynamics of indomethacin at ambient and elevated pressure. From dielectric measurements, we have determined structural relaxation times as a function of temperature and pressure (Figure 1)41. In order to describe the temperature and pressure dependences of structural relaxation times, we have used the VFT32−34 equation and its pressure

α(t ) = 1 − exp[−aIss(t )g n − 1(t )(t − t ′)n − 1dt ′]

(5)

where a is a shape factor. By assuming that nucleation and growth rates do not change throughout the whole crystallization process, eq 5 simplifies to the Avrami form (eq 4), where the rate constant k = Issgn−1. It is clearly seen in Figure 2c that the crystallization of supercooled indomethacin proceeds faster with increasing

Figure 2. Crystallization progress under pressure followed by dielectric spectroscopy. The evolution of the (a) real and (b) imaginary parts of the complex dielectric susceptibility during crystallization at T = 343 K and p = 10 MPa. Data were measured every 600 s. (c) Normalized dielectric constant as a function of time for crystallization carried out at three different T, p combinations at the same τα. The solid lines represents Avrami fits. C



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Figure 3. Activation energy and activation volume of the crystallization process under pressure. (a) Temperature and (b) pressure dependences of the crystallization constant rate k determined from Avrami fits on the basis of time-dependent crystallization experiments performed at constant pressures (p = 10 and 75 MPa) and temperatures (T = 368 and 391 K), respectively. Solid lines represent Arrhenius and its pressure counterpart fits to the experimental data. The activation energies and activation volumes are given as well.

For indomethacin, values of ΔVm = Vliq − Vcry (= 0.082 cm3/ g), ΔSm = Sliq − Scry (= 0.253 J K−1 g−1), and ΔHm (=110 J g−1) at ambient pressure were taken directly from PVT measurements and literature.40 Melting temperatures of indomethacin at investigated pressures (at which crystallization experiments were performed) were calculated from the integrated Clausius− Clapeyron equation under the assumption of a constant ratio ΔVm/ΔSm

compression under isochronal conditions. In order to confirm the lower physical stability of supercooled indomethacin under pressure, we have also monitored its crystallization process under isobaric and isothermal conditions. The crystallization rate, k, extracted from Avrami fits was plotted versus the reciprocal of temperature [Figure 3b] and pressure [Figure 3b inset]. The activation barriers of the overall crystallization process Ea at 10 and 75 MPa were estimated from the Arrhenius equation, whereas the activation volume of the crystallization process Va at 368 and 391 K was calculated from the volume activation law.39 We found that Ea and Va decrease with pressure from 139 kJ/mol at 10 MPa to 128 kJ/mol at 75 MPa and from 89.6 cm3/mol at 368 K to 74.5 cm3/mol at 391 K, respectively. These experimental results clearly show that compression facilitates crystallization of indomethacin within the examined pressure range. It has been noted that by performing isochronal crystallization experiments under hydrostatic pressure, we are able to modify the interplay between thermodynamic and kinetic factors as well as their influence on nucleation and growth rates. In addition, by keeping the same macroscopic structural relaxation time, we have the unique possibility to control the kinetics of the crystallization process. This might help in our understanding of crystallization phenomenon of supercooled liquids because, regardless of thermodynamic conditions in which the liquid is kept, the influence of kinetic factors on crystallization remains identical. Thus, a major contribution to the change of the overall crystallization kinetics with pressure solely has a thermodynamic origin. In this work, we have made an attempt to characterize thermodynamic effects induced by pressure on the crystallization of supercooled indomethacin. We have considered herein several aspects related with the effect of pressure on the undercooling, thermodynamic driving force of crystallization, crystal/liquid interfacial energy, and critical radius of the crystal nuclei. The first three parameters were estimated based on the theoretical scheme provided by Gutzow and co-workers.23

⎛ ΔV ⎞ Tm(p) − Tm(p0 ) ≅ ⎜ m ⎟(p − p0 ) ⎝ ΔSm ⎠

(6)

On the basis of the melting temperatures (Tm) and the crystallization temperatures (Tc) at investigated pressures, we have determined the changes of undercooling as ΔT = Tm(p) − Tc(p)

(7)

In Figure 4, we demonstrate the pressure dependences of Tm and the glass transition temperature (Tg) for indomethacin. The behavior of the glass transition temperature with pressure was taken from literature data.41 Along with the curves of Tm(p) and Tg(p) in Figure 4, we have shown isochronal crystallization points (⧫), from which we have determined the undercooling ΔT (eq 7). From the results collected in Table 1, we get a very important finding that with increasing pressure, the undercooling ΔT also increases. Therefore, isochronal crystallization conditions do not guarantee that we are still in the same place with respect to the melting temperature. Let us move now to the difference in the chemical potentials of the liquid/crystal phases (Δμ), which is a thermodynamic driving force of crystallization. dμ = − ΔSm dT + ΔVm dp

(8)

By integrating from T = Tm (p = 0) and p = p0 = 0, Gutzow and co-workers get the following formula for the chemical potential under different combinations of temperature and pressure (T, p) D



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ature dependence of σ(T,po) for the γ-crystal form of indomethacin at ambient pressure follows the relation23 σ(T , po ) ≅ 0.85 + 0.06T (mJ × m−2)

(12)

By assuming constant values of Ko ≈ 0.55 and go ≈ 0.4 parameters (for details we refer the readers to the Gutzow and co-workers paper23), we are able to calculate the liquid/crystal interface energy under pressure. These results are collected in Table 1, and show that σ(T,p) systematically decreases with pressure. Then, the thermodynamic barrier for two-dimensional nucleation ΔW under increased pressure can be estimated in the following way:23 ΔW (T , p) ≈

rc(p) = T

∫T (0) ΔSm(T , p0 ) dT ∫p

p

ΔVm(Tm , p) dp

(9)

0

It is worth stressing that Tm(0) in the integration limits of the entropy term is set to act as a reference point. In fact, crystallization experiments for indomethacin under pressure were performed only below that point, which was related mostly to our equipment limitations. After assuming temperature and pressure independent constant values of ΔV and ΔS and exploiting eq 6, eq 9 can be rewritten as Δμ(T , p) ≅ ΔSm[Tm(p) − T ]

(10)

We have calculated the difference in the chemical potential of coexisting phases for three (T, p) conditions at which the indomethacin sample was crystallized. These results are also summarized in Table 1 and indicate that in the studied pressure range, there is an increase in the thermodynamic driving force of crystallization. When considering thermodynamic aspects of crystallization under elevated pressure, we should also analyze the behavior of the specific surface energy parameter σ, which can be estimated from the following formula proposed by Gutzow and coworkers σ(T , p) ≈ σ(T , po )[1 − (Ko/γo)Tm(p)]

(13)

4σ(T , p)Tm ΔT ΔHmρc

(14)

where ΔT denotes undercooling (i.e., the difference between the melting and crystallization temperatures at a certain pressure). The density of the crystalline state at the investigated pressures was determined from the PVT measurements (Table 1), whereas the pressure dependences of Tm and σ(T,p) were calculated using Clausius−Clapeyron formula (eqs 6 and 11), respectively. As it turned out, the critical nucleus size for indomethacin should decrease with pressure from 2.65 nm at 10 MPa to 2.25 nm at 220 MPa. In order to summarize our data, we have plotted pressure dependences of Δμ, σ, and rc in Figure 5. It is illustrated that within the studied pressure range there is an increase in the thermodynamic driving force of crystallization (approximately 20% growth of Δμ) and a decrease in the melt/crystal interface energy (approximately 10% drop in σo) as well as in critical nuclei size of 15%. As Δμ and σ are directly involved in equations defining thermodynamic barriers of nucleation and crystal growth, we can expect that their variations with compression portray in some way adequate changes in the thermodynamic factor of the crystallization process under constant influence of the kinetic factor. Taking into account all results presented above, it becomes evident that the increase in nucleation and crystal growth rates of indomethacin under pressure determined upon isochronal measurements must have purely thermodynamic grounds. In summary, this paper provides highly desirable data on the effect of pressure on crystallization of the glass-forming van der Walls liquid, indomethacin. By considering only isochronal conditions, we were able to untangle purely thermodynamic effects on crystallization from kinetic ones. This cannot be achieved by any other experimental attempts performed at

m

+

3Δμ(T , p)2

We expect it to decrease, as the [σ(T,p)3]/[Δμ(T,p)2] ratio decreases with compression. Finally, we have also made an attempt to determine for indomethacin, the pressure dependence of the critical nucleus size, rc, which can be calculated from the following equation

Figure 4. Dependence of the melting point and glass transition temperature on pressure. Pressure dependences of melting Tm(p) and glass transition temperature Tg(p) are denoted by ○ and ■, respectively. ⧫ signify crystallization temperatures, Tc(p) for τα= const. The dependence Tg(p) for indomethacin was taken from ref 41.

Δμ(T , p) = −

16πσ(T , p)3 Vc 2

(11)

where σ(T,po) is the liquid/crystal interface energy at ambient pressure and Ko and γo are parameters introduced by the Gutzow formalism. The experimentally determined temper-

Table 1. Calculated Values of Thermodynamic Parameters As Well As Density of Supercooled Indomethacin under Pressure pressure (MPa)

Tm (K)

Tc (K)

ΔT (K)

Δμ (kJ/mol)

σ (mJ/m2)

ρc (g/cm3)

rc (nm)

10 100 220

434 466.36 505.25

340 368 391

94 98.4 114.25

8.52 8.95 10.4

21.2 20.7 19.6

1.347 1.374 1.398

2.65 2.60 2.25

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atmospheric pressure. We conjecture that this seems to be a very interesting and universal way toward studying and understating crystallization of glass-forming liquids under pressure. Our novel experimental studies were also supported by the theoretical description given in terms of Gutzow et al. formalism. As demonstrated, within the studied pressure range, we should expect the increase in the nucleation and crystal growth rates purely due to thermodynamic reasons. In addition, we can assume that theoretical studies given in terms of Gutzow formalism work reasonably well in the case of indomethacin, though require some assumptions (especially ΔVm, ΔSm, and ΔHm). On the other hand, it should also be pointed out that in our current studies, we have considered relatively narrow ranges of temperature and pressure (due to specific equipment limitations). In the future, it is highly desirable to extend the applied pressure range and verify if the same pattern of behavior can be applied to describe crystallization of glass-forming liquids above the reference point, at higher pressure regimes. In fact, our current studies open a new perspective for much more interesting studies to be carried out in the future.

ASSOCIATED CONTENT

S Supporting Information *

Experimental and theoretical procedures considering crystallization studies under pressure are provided. This material is available free of charge via the Internet at http://pubs.acs.org.

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REFERENCES

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Figure 5. Thermodynamic factor of crystallization process under pressure. Pressure dependences of the thermodynamic driving force of crystallization Δμ (main panel), melt/crystal interface energy σ (inset on the left), and critical nuclei size rc (inset on the right).

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS K.A., K.G., and M.P. are deeply thankful for the financial support received from the National Science Centre (DEC2012/05/B/NZ7/03233, UMO-2012/05/B/NZ7/03233). K.A. acknowledges financial assistance from POKL and START (FNP) programs. F



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(36) Avrami, M. J. Chem. Phys. 1939, 17, 1103. (37) Gutzow, I.; Schmelzer, J. The Vitreous State: Thermodynamics, Structure, Rheology and Crystallization; Springer: Berlin, 1995. (38) Gutzow, I.; Schmelzer, D. A. J. Non-Cryst. Solids 1997, 219, 1− 16. (39) Forsman, H.; Anderson, P.; Bäckström, G. J. Chem. Soc., Faraday Trans. 1986, 2, 857. (40) Andronis, V.; Zografi, G. J. Non-Cryst. Solids 2000, 271, 236− 248. (41) Wojnarowska, Z.; Adrjanowicz, K.; Wlodarczyk, P.; Kaminska, E.; Kaminski, K.; Grzybowska, K.; Wrzalik, R.; Paluch, M.; Ngai, K. L. J. Phys. Chem. B 2009, 113, 12536−12545.

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Toward Better Understanding Crystallization of Supercooled Liquids

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