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Scaling laws and size effects for amorphous crystallization kinetics: Constraints imposed by nucleation and growth specificities
Accepted Manuscript Scaling laws and size effects for amorphous crystallization kinetics: Constraints imposed by nucleation and growth specificities Marc Descamps, Jean-François Willart PII: DOI: Reference:
S0378-5173(18)30141-8 https://doi.org/10.1016/j.ijpharm.2018.03.001 IJP 17344
To appear in:
International Journal of Pharmaceutics
Received Date: Revised Date: Accepted Date:
22 December 2017 1 March 2018 2 March 2018
Please cite this article as: M. Descamps, J-F. Willart, Scaling laws and size effects for amorphous crystallization kinetics: Constraints imposed by nucleation and growth specificities, International Journal of Pharmaceutics (2018), doi: https://doi.org/10.1016/j.ijpharm.2018.03.001
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Scaling laws and size effects for amorphous crystallization kinetics: Constraints imposed by nucleation and growth specificities
Marc Descamps†*, Jean-François Willart†
†
Université de Lille, CNRS UMR 8207 – UMET – Unité Matériaux et Transformations, F-59000, Lille – France. *corresponding author: [email protected] (33) 03 20 43 49 79
Abstract. In the present paper we review different aspects of the crystallization of amorphous compounds in relation to specificities of the nucleation and growth rates. Its main purpose is: i) to underline the interest of a scaling analysis of recrystallization kinetics to identify similarities or disparities of experimental kinetic regimes. ii) to highlight the intrinsic link between the nucleation rate and growth rate with a temperature dependent characteristic transformation time (T), and a characteristic size (T). The consequences on the influence of the sample size on kinetics of crystallization is considered. The significance of size effect and confinement for amorphous stabilization in the pharmaceutical sciences is discussed. Keywords: Amorphous state, crystallization, nucleation and growth, scaling, size effect.
Graphical Abstract 1.0 L=600 L=100
L=z=75
0.8
L=50
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X(t)
L=40
0.4 L=25
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1. INTRODUCTION Recrystallization of amorphous forms is the major cause of their instability. This impacts the stability during a storage but it is also a key element of the mastery of amorphization during a desired formulation in this state. Independently of structural and microstructural modifications, the exothermic nature of the recrystallization process can also influence chemical stability. In the pharmaceutical literature, discussions on this topic try to find primarily a relationship between loss of stability and molecular mobility. These investigations often focus on a narrow temperature range. Often this concerns the search for a similarity between an activation energy controlling mobility and a supposed activation energy controlling the recrystallization[1]. The molecular mobility in the amorphous state has several facets corresponding to various relaxation processes: i) The non-Arrhenian relaxation which leads to times of the order of 100 - 1000s at Tg and is responsible for the gel manifesting at the glass transition. ii) The set of secondary relaxations called , , etc. , (of inter or intra-molecular origin), reflecting localized rapid motions. They are especially manifest in the glassy state. For the molecular materials, all these processes have very different activations energies. And thus, according to studies, various links with an assumed crystallization activation energy have been proposed. This leads to a variety of proposals that are not general in scope. In fact, there is no reason that a simple consideration of mobility can account for the entire mechanism. There are many other possible contributions to the kinetics of crystallization. The reason is that the overall process results from the combination of a dual mechanism of nucleation and growth. These two processes have different temperature evolutions. Both have a maximum rate for a certain degree of undercooling. For both, the existence of a maximum rate is due to the competition between the thermodynamic driving force which prevails for moderate undercooling and the molecular slowing down for larger undercooling. But these maxima are not at the same temperature. This is because the structural difference between the liquid and the crystal does not occur in the same way for nucleation and growth. The difference in temperature localizations of the two processes implies that they are not influenced to the same degree by the molecular mobility; and even more so their combination. On the other hand, for moderate undercooling, it is clear that the increases in nucleation rate and growth rate are in no way a consequence of the dynamics. Moreover, nucleation can occur in a homogeneous or heterogeneous manner with very different rates at the same temperature. Whatever the effect of the control factors, an inherent aspect that differentiates nucleation and growth is the dimension of their physical expressions: Nucleation rate [N] = L-3 t-1 Growth rate [V] = L t-1 This makes it possible to implement a scaling analysis of experimental kinetics of crystallization which can be very useful in a search for interpretation of the mechanisms involved. It can helps to identify the temperature domains where the same mechanism intervenes to give the same physical law. Moreover, the scaling analysis makes it possible to identify a characteristic time and a characteristic length for the global mechanism. This allows capturing a small size effect inherent in the process.
In the present paper we review different aspects of the crystallization of amorphous compounds in relation to specificities of the nucleation and growth rates (designated respectively by N and V). Its main purpose is to underline the interest of a scaling analysis of kinetics and to insist on an intrinsic link between the effects of reduced size and the key parameters of crystallization. The paper includes three sections: The first two summarize briefly - with some new perspectives and illustrations - considerations on crystallizations from the amorphous state, presented in a previous article [2]. These considerations are essential for the development of a new mode of taking into account the size effects which is presented in the third part. In the first section, we recall the expression of the classical law of Avrami expressing the kinetics of recrystallization [3, 4]. We use this to show the interest of a scaling analysis to identify similarities, or, disparities of experimental kinetic regimes. The second section is devoted to the physical properties which drive the temperature evolutions of N and V. The question of the localization of their respective maxima is especially discussed. This is used to identify the best vitrification possibilities and to clarify the reasons for the Oswald rule of stage [5, 6]. The third part of this review considers the intrinsic link between N and V with temperature dependent characteristic transformation time (T), and characteristic size (T). The consequence on the influence of the sample size on kinetics of crystallization is considered. This clarifies the role of sample size on the effective life time of metastability of an amorphous compound and may have a practical impact on amorphization processes and the manipulation of amorphous solids 2. SCALING ANALYSIS OF GROWTH CURVES. So called growth curves express the time (t) evolution of the fraction of amorphous material X(T,t) which is crystallized in the course of an isothermal process at temperature T. The isothermal methods which give access to X(t) are in principle all those which can discern, quantitatively, between the amorphous state and the crystalline state. In practice, it is most often the DSC thermal analysis (used in isothermal mode) and the in-situ X-ray diffraction. The determination of all X(T,t) curves at different temperatures T
If we suppose that a liquid is brought instantaneously to a given temperature T, lower than Tm, nanocrystals are produced at the constant nucleation rate N(T), per unit volume. Even if N is assumed to be constant in time, the nucleation rate decreases effectively as the amorphous volume, free for crystallization, also decreases. Once formed, the grain grows isotropically, with a constant velocity V(T). As it grows it starts to impinge upon other grains and eventually stops growing [4]. These two sources of slowing down are to be taken into account in the evaluation of X(t). This is done most directly by calculating the probability for a point P of the amorphous sample not to be engulfed at a time t, which is identical to 1-X(t). The probability obeys the following Poisson law: t 1-X(t) = exp(-N 0 f(t') dt') (1) where f(t') is the volume centred on P, in which nucleation should not have appeared before t' in order to be sure that P is not reached by a grain at time t. If we consider the isotropic growth in a 3-dim space, this lead to the classical Avrami law[7]: X(t) = 1 - exp[- (4/3) (t/t0)4]
(2)
which is a universal function of the time scale: t0 = (N V3)-1/4
(3)
If nucleation and growth occur simultaneously and N and V have the particular characteristics given above, then a universal transformation law of type (2) is expected. X(t) takes the form of an S-shaped curve that is almost symmetrical with respect to the half-transformation point (that occurs at t0.5 t0). At different temperatures, and therefore for different values of N and V, X(t) can then be rescaled on a single curve when plotted against the reduced time t/t0.5. 2.2 Scaling analysis of the growth curves A scaling analysis of isothermal growth curves allows to locate the temperature domains for which an Avrami hypothesis is possibly applicable. It is also very useful for identifying kinetic behaviors associated, on the contrary, with evolutions of N and V that do not fit into the initial hypotheses. An example of such a scaling analysis is given in Figure 1. Figure 1a, reports different curves X(T, t), obtained by isothermal DSC, on l-arabitol[8, 9], which has been quenched rapidly from the liquid state to different temperatures between the melting temperature of the most stable phase Tm = 101°C and the glass transition T g -14°C. These curves all have an S-shaped appearance whose rapid evolution is more or less time-delayed as a function of temperature. From this observation alone, it is difficult to discern possible kinetic regime changes. Figure 1b shows the curves X(T, t) corresponding to those of Figure 3 but plotted against a rescaled time (t / t0.5). For the lowest temperatures (15°C. T 40 ° C), the scaled curves are fairly well superimposed to show that a scaling, of the Avrami type, is well satisfied. This suggests that in this temperature range, the recrystallization kinetics probably corresponds to a nucleation and growth process, and obey the same law. The scaled curve shows that the kinetics recorded at the highest temperatures (45 ° C and 50 ° C) strongly deviate from the symmetrical universal sigmoidal shape.
In particular, the growth curve measured at 50 ° C shows a very abrupt evolution preceded by a long waiting time. Such an evolution can absolutely not be represented by an Avrami type curve, unless to replace in (2) the exponent 4 with an exponent value greater than 10, which is totally unrealistic. The scaling analysis thus reveals changes in kinetic regime when the melting temperature Tm is approached. The nucleation rate becomes so low and the rate of growth is so great that the two mechanisms do not occur simultaneously. The consequence of the absence of universal behavior is that it would be totally illusory to try to interpret the temperature evolution of the whole of the growth curves shown in Figure 1. by the only change of a relaxation time or the search for a single activation energy. In paragraph 4 it will be seen that a scaling analysis also makes it possible to identify how a confinement can induce modifications of the kinetic laws themselves, in addition to a modification of the time scale of the crystallization.
X (t)
time (min)
Figure 1.a Temperature dependency of the time evolution of the fractional volume X(t) of recrystallized L-arabitol in phase I. The different temperatures are that reached after a quench from the melt (obtainedfrom DSC isothermal measurements). (From[2] with permission)
Figure 1.b (L-Arabitol) The scaled curves of X(t) (corresponding to those shown in Figure 1a) plotted in terms of the scaled time t/t0.5. (From
[2]
with permission)
3. TEMPERATURE SEPARATION OF N(T) AND V(T) MAXIMA Another consequence of the above kinetic curve analysis is that N(T) and V(T) have maxima at very different temperatures: Indeed at 50 ° C the step like behavior of the growth curve is interpreted by the fact that it is necessary to wait about 7 hours before one nucleus appears in the DSC sample. The crystallization then corresponds to the very rapid crystal growth from this single nucleus. At 50°C, the value of the nucleation rate is very small while it is large for the growth rate. At lower temperatures both events occur simultaneously in the crystallization process. 3.1. Experimental identifications The previous analysis suggests that the maxima of N (T) and V (T) are very distant in temperature. This separation can be easily demonstrated by a simple systematic DSC analysis. It consists of recording heating scans of the sample previously melted then quenched at different temperatures Ta between Tm and Tg, and finally annealed isothermally at Ta for a duration ta. Figure 2 shows such curves for l-arabitol[2]. For Ta >35°C and Ta <-10°C, virtually no exothermic sign of crystallization is observable during reheating. On the other hand, for annealing temperatures between 65°C and 30°C, significant exothermic recrystallizations are observed. They have their maxima located between 60°C and 95°C, i.e. a little below the melting endotherm. This temperature range is clearly distinct from that of annealing at T a. These observations and the analysis of the shape of the growth curves presented above make it possible to conclude that the observed exotherms correspond to the growth of crystals from nuclei formed at a lower temperature. The consequence of this is that N (T) and V (T) have both a maximum and that these maxima are very separated. The preferential growth zone is situated slightly below Tm. The preferential nucleation zone is located at a lower temperature, a little above Tg [10]. Since the enthalpy of recrystallization increases with the number of nuclei formed, it can be estimated that the maximum of N(T) is located near 5 ° C. Figure 2 shows the approximate location of the two zones.. When cooling a liquid, the system first crosses the temperature zone where V is maximum and then the temperature zone where N is maximum. In the first zone, no crystallization occurs because no nucleus has been formed. In the second temperature zone nucleation occurs, but without growth, so that the crystallized fraction is tiny. The fact that N and V both have a maximum (NMax and VMax, ) is basically the reason why it is possible to reach a glassy state, by a sufficiently fast liquid quenching. This possibility is amplified by a greater separation of the maxima.
Vitesse de croissance Vitesse de nucléation (unités arbitraires)
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-20
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-20
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Figure 2 (L-Arabitol) DSC curves recorded upon heating (heating rate of 2 K/min) for samples undercooled at temperatures Ta ranging from −83◦C to 82◦C and annealed for 60 min at this temperature. ( adapted from[2] with permission)
3.2 Drivers for the N(T) and V(T) separate evolutions The essential ingredients of the temperature evolutions of N(T) and V(T) are well captured by the elementary theories which lead to the following expressions [7, 11-14]: N(T) f0(T) exp( - g*/ RT)
(4)
V(T) f0(T) [1- exp(-G/RT)]
(5)
g* is the nucleation barrier : g* 3-d 3/G2 and is the interface free energy between a crystalline nucleus and the parent amorphous medium. G (T) is the Gibbs free energy difference between the metastable liquid and stable crystal states. G (T) is the thermodynamic driving force for crystallization. f0 is an "attempt frequency" for the addition of molecules from the metastable melt to the stable crystal across the interface At Tm, the liquid and crystal states are in thermodynamic equilibrium. The thermodynamic driving force for crystallization is zero as well as N(Tm) and V(Tm). During the cooling of the liquid, N(T) and V(T) increase initially with the degree of undercooling and the progressive increase of G (T). In the domain of relatively high temperatures, the molecular mobility is large enough not to oppose the effect of the thermodynamic driving force. For larger undercoolings, the molecular mobility (f0(T)), decreases considerably (ie, increasing viscosity) approaching Tg. It eventually offsets the effect of the thermodynamic force, and slows the nucleation and growth kinetics. As a result, N(T) and V(T) both have a maximum for a certain temperature. The competition of thermodynamic and dynamic effects is reflected in the competition between, respectively, the second term and the first term of the right-hand side of the equations (4) and (5) giving respectively N (T) and V (T). If the thermodynamic driving force and the molecular mobility compete similarly for N(T) and V(T) and induce maxima, these two factors alone can not justify the noticeable differences in their temperature positioning. The greater or lesser temperature separation is mainly due to the influence of the surface free energy between a crystallite and the parent amorphous medium[2]. plays a fundamental role in the value of N because of the important surface effects related to the initial formation of nanocrystalline nuclei. plays only a marginal role in V(T). 3.3 Influence on nucleation of crystalline and amorphous structural similarities. Nucleation is the precursor event of crystallization. If has a null value or is small, nucleation is easier and nucleation rate has more chance of being large. is a free energy, and, as such, has a fundamental entropic contribution. Figure 3 gives an illustration of the molecular origin of , during the crystallization of a molecular liquid. The value of decreases as the entropy difference between the bulk amorphous state and crystalline state, S (bulk), is smaller i.e. if there is a structural proximity between them[15-17]. Moreover, its value is as lower as the entropy difference between the core of the amorphous phase and the interfacial zone is weak i.e. if the liquid has little to organize at the interface with the crystal. Or says otherwise if S (bulk) is closer to S (interface). This helps to rationalize Ostwald's postulate that the first nuclei to form are often those of a metastable phase rather than of the more stable phase. Obviously that can not be related to the thermodynamic driving force which is smaller for the metastable phase. Furthermore, when we compare kinetics at the same temperature, that cannot be either explained by a difference of molecular mobility in the amorphous phase. In fact the metastable phase gives the lowest melting entropy and highest crystalline entropy (closer to that of the amorphous state) as illustrated in Figure 5a. Consequently the metastable phase opposes the lowest interfacial free energy and is therefore easier to nucleate.
Interf. LIQUIDE
CRYSTAL
TM S (interf.) (interf)
Hc
Hl TMSl TM S (bulk) = HM
TMSc
Figure 3 Diagram showing the structural origin of the interface free energy . is as lower as the molecular arrangement in the bulk of the liquid is close to that of the crystal and/or the organization of the liquid at the interface is similar to that existing in the bulk of the liquid.
There are two types of situations that require little reorganization at the interface: - When the local structure of the liquid has strong similarities with the long-range structure of the crystal[15]. - When the crystal phase is highly disordered. The disorder can be static or dynamic[18-20]. Moreover, for a given case, both situations can coexist We present two examples that illustrate these situations: l-arabitol l-arabitol has at least two crystalline polymorphs[8, 9]. The variety II is metastable and melts at lower temperature than the stable variety I (Figure 4a). Variety II is the first stage of the recrystallization from the amorphous state after a deep quenching of the melt. Raman scattering allows to show clearly that the origin of this preferred nucleation is associated with structural similarities between the amorphous state and variety II [21]. Figure 4b. allows to compare the low frequency Raman susceptibilities, recorded at the same temperature, of varieties I and II as well as the amorphous state. In the latter case the susceptibility is characterized by a broad and asymmetric band which is the image of the vibrational state density (VDOS) of the amorphous state. The maximum intensity of the VDOS, located at about 40-45 cm-1, corresponds quite exactly with the low-frequency, sharp and intense phonon peak of the metastable variety II. By cons there is no phonon peak of the stable form I, which is detected below 50cm-1. This indicates that there is a strong reminiscence of the long-range order of variety II in the local order of the undercooled liquid.
Figure 4a DSC heating scans showing the melting endotherms of the stable (I) and metastable (II) phases of L-Arabitol (From[2] with permission)
Figure 4b Low frequency Raman spectra of crystalline forms I and II and the undercooled liquid state of L-arabitol at 265 K. (From[21] with permission)
RS Ibuprofen The second example is RS ibuprofen [23, 24]. Under deep quenching of the liquid the RS ibuprofen preferentially crystallizes towards a metastable phase II. The crystallization of the stable ordered racemic phase I is kinetically much slower. The thermodynamic relationships of phases I, II and the amorphous state are summarized in the Gibbs diagram of Figure 5a. The low melting entropy value of phase II compared to that of phase I suggests that phase II exhibits a high degree of disorder
that allows easy adaptation of phase II nuclei to the amorphous matrix. This appears clearly in the confrontation of Raman spectra. The Raman susceptibilities (recorded at 190K) of the crystalline phases I and II are compared to that of the glassy state on Figure 5b. It appears clearly that the spectra of the glassy state and phase II are nearly superimposed. They are typical of disordered molecular systems. The similarity of the vibrational density of state (VDOS) spectrum of glassy ibuprofen and the ''()-spectrum of phase II, clearly reflects the existence of a significant structural disorder in phase II. It also shows that the local molecular arrangement which exists in glassy ibuprofen is the reflect of the long range order of crystalline phase II.
Figure 5a Gibbs diagram showing the situations of the stable phase (I) and metastable phase (II) of Racemic RS ibuprofen (From[2] with permission)
Figure 5b Low-frequency (5–200 cm−1) Raman susceptibility in the under-cooled (or glassy) state and crystalline phases I and II of Racemic RS ibuprofen (From.[25] with permission)
3.4. Practical consequences for the formation and stability of amorphous states. The crystallization of an undercooled liquid requires both crystal nucleation and growth processes to occur. A Time-Temperature-Transformation (TTT) plot (Figure 6) is useful for describing the combination of the two factors and their specific velocities N and V. The set of TTT curves is build from growth curves such as those in Figure 1a. Such a diagram has a nose at temperature where the conjunction of N and V leads to the fastest crystallization. The position of the nose is generally different from that of Nmax and Vmax. The diagram can be used to determine the critical cooling rate (measured in K / s) that must be imposed on a liquid, to quickly pass the nose, and to succeed in vitrifying the compound without significant crystallization. The temperature zone of the nose is also a domain of maximum instability for the amorphous compound. However, caution must be exercised when using a TTT chart. Indeed, even if the vitrification is successful, without detectable crystallization, nanoscale nuclei could be trapped during rapid cooling. This is because, at low temperatures, the nucleation rate can be significant, while the growth rate is negligible. The presence of crystalline nuclei can be a source of instability of the amorphous compound, because they have a seeding effect, and can generate, on heating, a massive crystallization by the effect of growth. It is for this reason that the crystallization of an amorphous compound is more often observed during a reheating than during the cooling of the liquid. To avoid the formation of these nuclei it is necessary to use an even faster cooling rate. Determinations of the temperature localization of the nose of the TTT diagram, as well as Nmax and Vmax, are decisive factors in the strategy for forming a compound in the amorphous state and for controlling its stability. T
LIQUID
Tm
G
COOLIN
GROWTH ALONE
VMax
NUCLEATION And Growth
Tn 1%
NMax
50%
99%
NUCLEATION ALONE
Tg GLASS log (time)
Figure 6 Time-Temperature-Transformation (TTT) curve for crystallization (nucleation + growth), showing the critical cooling rate (black curve) which is necessary to vitrify the compound. Because nucleation is a faster process at low temperature a fast cooling may not avoid nucleation. Much faster cooling is required to obtain a non-nucleated glass, as shown by the red curve at left. Tm is the melting temperature, Tg is the glass-transition temperature. An experimental study which illustrates [2][26] this situation is found in reference .
4. EFFECT OF SAMPLE SIZE ON CRYSTALLIZATION KINETICS 4.1. Characteristic time, Characteristic size: dimensional analysis. By a simple dimensional analysis[27, 28] of the nucleation rate N ([N] = L-3t-1) and linear growth rate V ([V] = Lt-1) we can highlight that the crystallization process is specified by (for a 3-dim growth): - a characteristic time t0 = (N V3)-1/4 (6) 1/4 - a characteristic size = (V/N) (7) Specifying the values of t0 and , which are both temperature dependent, is equivalent to give the values of N and V. The expression of the kinetic law, X(t), of a nucleation and growth transformation is fundamentally dependent on the position of the sample size (L) with respect to . This is illustrated schematically in Figure 7. Such sample size influence is intrinsic to the crystallization mechanism and is not related to a chemical phenomenon or a change in N or V values. It depends only on the relative importance of the growth rate with respect to the nucleation rate. Sample size modification has effect on the time scale of the crystallization process but also on the expression of the kinetic law itself. In particular, a dramatic slowing down of actual kinetics is inevitably expected at small sample sizes.
1.0 L=600 L=100
L=z=75
0.8
L=50
0.6
X(t)
L=40
0.4 L=25
0.2
L=15 L=10 L=5
0.0 0
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200
300
Time (arb. Time units)
Figure 7 2-Dimensional simulation of the time evolution of the transformed fraction X(t) for different grain sizes (from above to below ). The values of the nucleation and growth rates are fixed and [27, 29] correspond to = 75 and t0 = 75.(From with permission)
4.2. Growth laws as a function of sample size. In the limiting cases of the very large and very small sample sizes, one expects two very different expressions for the growth laws [27, 28] L >> . We are in the case of the Avrami's law presented above. t0 intervenes in the expression of the Avrami equation, to give a dimensionless term in the exponential: X(t) = 1 - exp[- (4/3) (t/t0)4]
(8)
However does not intervene in this equation, nor does the size (L) of the sample. This is so because the Avrami equation is only valid for samples of significantly larger size than . Under these conditions, the Avrami expression is a sigmoidal universal function.
L << . We consider a sample consisting of a set of small cells of size L. Each cell is immediately crystallized as soon as it is the site of a nucleation event. The crystallized fraction of the global sample coincides with the fraction of "nucleated" cells. The evolution of the crystallized fraction is calculated by: dX = (1 - X) N L 3 dt -->X(t) = 1 - exp(-t/tn)
(9)
X(t) is exponential with the characteristic time tn = (L3 N-1) which is now highly size dependent. Figure 7 shows a general outlook of the influence of sample size on the evolution of the kinetic law[28]. It results from a simple 2-dimensional simulation of a nucleation and growth transformation with fixed values of N and V. Different simulations were made for a large sample being subdivided in equivalent square cells whose linear size L has been varied. In this simulation the value of L varied from 5 lattice units (l.u.) to 600 l.u. N et V were chosen so that = 75 lattice units and t0 = 75 time units. The expressions of and t0 are adapted from those given by (6) and (7), for a 2-dim system. For large values of L we observe that X(t) is sigmoidal. The shape of the curve is then only weakly dependent on L. When L decreases the shape of X(t) turns into an exponential whose relaxation time becomes very dependent on L. A marked change of behavior is noted when L crosses the value of the characteristic size .. The results of an exact analytical calculations have been reported in ref.[27]. It shows that, in the general case, a scaling relation involving a reduced time (t/t0) and a reduced size (L/) is applicable. Figure 8 shows that the effect determining the form of X(t) is related to the relative position of t/t0 with respect to L/. is the length which determines the crossover from the Avrami regime (L>>) to the size dependent exponential regime (L>>). In the latter case the transformation satisfies a scaling relation which involves the ratio L/. Experimentally the global variation of X(t) always marks a more or less clear transition from one situation to another. The awareness of this influence of the sample size on a growth curve should make it possible to better understand the influence of a formulation process on the stability of an amorphous compound with respect to crystallization.
1 1−
𝜋 𝑡 4 − 𝑒 3 𝑡 𝑎0
crossover
t/ta0
0 0
L/
1
1 1−
1−
𝜋 𝑡 4 − 𝑒 3 𝑡0𝑎
4𝜋 𝐿 3 𝑡 − 𝑒 3 𝜁 𝑡 𝑎0
crossover
t/ta0
0 0
1
1 crossover
1−
𝜋 𝑡 4 − 𝑒 3 𝑡0𝑎
1−
4𝜋 𝐿 3 𝑡 − 𝑒 3 𝜁 𝑡0𝑎
t/ta0
0
0
L/ 1
Figure 8 Typical behaviour of the growth curves X(t) plotted vs rescaled time (t/t0)for different values of (L/.) (adapted [27] with permission)
4.3. Relevance of the characteristic sizes and times in the analysis of the effect of confinement. The dimensional considerations presented above help to gain more understanding into the consequences of sample size reduction on the kinetics of crystallization and in turn on the amorphization possibilities and amorphous state stabilities. More generally this is applicable to phase transformations managed by a nucleation and growth mechanism. There are basically two types of confinement processes:
- The confinements imposed extrinsically by spreading the active compounds in a mesoporous carrier -The confinements resulting intrinsically from the formulation of multicomponent systems. We present below two representative illustrations. Extrinsic size effect: mesoporous carriers. The concept of mesoporous carriers has been recently developed in the pharmaceutical field to stabilize amorphous drugs. Porous excipients such as porous silicon or silicates materials (for example Neusilin or Florite [30-32] ) are used more and more frequently for delivery of amorphous drugs. The pharmaceutical molecules are then adsorbed and spatially constrained in the nanometer-sized pores of the carrier material. Significant changes in crystallization kinetics are then observed experimentally. The nanometric confinement of fenofibrate, studied by G. Szklarz et al., is particularly illustrative of these effects [33]. Fenofibrate was confined in cylindrical nanopores of diameter ranging between 18 nm and 100 nm. Isothermal crystallization curves were obtained from the time evolution of the static dielectric permitivity. Figure 9 corresponds to isothermal recordings made at 298 K (56 K below Tm). It clearly shows an evolution of an Avrami type sigmoidal behavior for the bulk sample to a pseudo exponential behavior when the pore diameter decreases. This also results in a gradual increase in the characteristic crystallization time. This type of evolution is consistent with that theoretically predicted for a size reduction below . Figure 10 shows the evolution of the crystallization rate k ( t0-n in the Avrami equation) as a function of temperature, for a bulk sample and different samples confined in the nanopores. The position of the maximum of the crystallization rate, and therefore the minimum of the crystallization time, corresponds to the nose in a TTT diagram (Figure 6). It can be noted that this maximum shifts to lower temperatures as the pore diameter decreases. At the same time the crystallization rates measured at the maximum decrease. Moreover, it can be seen that the confinement effect is considerable at temperatures close to Tm, but that the effect of nanometric confinement is much less pronounced for deep quenching (The curves of the k(T) diagrams are getting closer). The overall influence of the confinement on the modification of the TTT-type diagrams of Figure 10 can be easily understood if it is noted that, near Tm, the growth rates (V) are large whereas the nucleation rates (N) are in general weak. According to formula (7), this leads to a large value of , and therefore to a very pronounced size effect. Approaching Tg, N increases sharply and V decreases. As a result decreases, and the confinement effect is less pronounced, which results in the k(T) curves corresponding to the different pore sizes being closer to one another. This progressive modification is therefore also at the origin of the general sliding of the k(T) curves towards the low temperatures. It is therefore not necessary to imagine changes in the nucleation mechanisms to explain the evolutions measured.
Figure 9 Isothermal (at 298K) time evolution of the normalized dielectric permittivity ('N) for fenofibrate confined in nanoporous alumina of different pore diameter. (from G. Szklarz et al.[33] with permission). We can notice the change from sigmoïdal Avrami to exponental type of evolution closely similar to that shown in Figure 7.
Figure 10 Temperature evolution of the crystallization rate k for fenofibrate confined in nanoporous [33]
material of different pore diameters. (from G. Szklarz et al. with permission). The maxima of the crystallization rate are the mirror images of the minimum of the crystallization time corresponding to the nose of the TTT diagram, such as that shown in Figure 6.
Intrinsic size effect: Amorphous Solid Dispersions It is known that by producing amorphous solid dispersions (ASD) of a polymer and an API it is possible to inhibit the crystallization of the API [34, 35]. This is the case, for example, for ASDs of PVP with piroxicam[36] or indomethacin[37] (IMC). In the case of ASDs prepared by solvent evaporation or HME, it has been shown that the formation of hydrogen bonds between the polymer and the small molecule certainly plays an important role in blocking crystallization. However, co-grinding also makes it possible to form ASDs. In the case of PVP / Indomethacin (IMC) and PVP / piroxicam, it has been shown by A. Hedoux et al.[22], for example, that the kinetics of crystallization of the small molecule is very considerably slowed down even at low PVP concentration (10%) (cf Figure 11). Figure 11 shows, moreover, that the shape of the crystallization curves changes from a sigmoidal behavior for bulk IMC, to a pseudo exponential behavior, almost without significant induction time, when the PVP concentration increases. Figure 12 b shows that when the PVP ratio increases up to 50% the value of Tg of the ASD is however practically unchanged and remains close to that of pure IMC (no antiplastifying effect). In addition, the high frequency (Figure 12 a) and low frequency Raman spectra are similar to that of the pure API for concentrations up to 50% which indicates the absence of noticeable interaction between the two components. This suggests that direct interactions between IMC and PVP do not play an important role in reducing drug crystallization. In this case, the polymer would rather have a partitioning effect within the dispersion, the PVP accumulating around nano / micro particles of IMC. Its stabilizing role would then come from a confinement effect of the type described above in 4.1. The presence of very low levels of the low-molecular mass additives has often a significant plasticizing effect on the pharmaceutical glasses. On the other hand adding small amounts of the high-molecular mass additives often had a minimum antiplasticizing effect[38, 39]. This suggests that self-confinement stabilization of the type described above is quite common.
Figure 11 Time evolution of the crystallized volume fraction of - IMC in the absence and presence [22]
of PVP (Reprinted from reference
, with permission)
a)
b)
Figure 12 a : High frequency Raman spectra (in the 1500 - 1800 cm-1 range) of solid amorphous IMC- PVP mixtures prepared by cryogrinding. (Reprinted from [22], with permission)
b)
Figure 12 b : Glass transition temperatures of IMC–PVP cryoground mixtures plotted vs. the mass fraction of PVP: The measured Tg values. The prediction of the Gordon–Taylor equation [36] (Reprinted from , with permission)
5. CONCLUDING REMARKS 5.1. Impact on the analysis of parameters managing the stability of an amorphous state. Crystallization of an amorphous product has a practical significance to the development of pharmaceuticals and biopharmaceuticals. It has a significant impact on formulation and storage conditions and consequently on bioavailability [40, 41]. The practical lifetime of metastability - or even instability for T
5.2. Significance of size effect and confinement for amorphous stabilization The changes in crystallization kinetics when a pharmaceutical molecule is confined in the nanometer-sized pores of a carrier material are usually attributed to two main reasons: i) A reduction in the molecular mobility of active molecules in the confined space. This is indeed possible if the pore sizes are extremely small and if the interactions of the molecules with the carrier are very important at the interface, (This then leads to an increase in Tg). However, size reduction can have a reverse effect and increase molecular mobility. This is due to an actual cleavage of the domains where the molecules rearrange in a cooperative manner (CRR zones) [44]. One could then think that the increase in mobility - which leads to a decrease in Tg - could lead to an acceleration of crystallization kinetics. But this is contrary to what is observed because the intrinsic size effect associated with the nucleation and growth mechanism is prevalent. ii) A pore size close or below the critical nucleation size rc* at the storage temperature. That is expected to minimize crystallization influence. Near Tg or below Tg the sizes predicted by classical nucleation theories are however extremely small. For the fenofibrate example, cited above, the critical nucleation size was calculated [33] using a conventional nucleation model. It varies between 20nm (at Tm-10K) and 3nm (at Tm-70K). Note that the critical nucleation size r c * is always significantly smaller than the pore diameter in the majority of situations. The slowing down of the kinetics when the pore size decreases cannot be attributed to a competition between rc * and the pore size that would block the nucleation mechanism. The two arguments mentioned above are only conceivable for extremely nano-sized pores. Confinement effects can however be observed even for micro size pores. In that case an interpretation based on a competition of the pore size with the critical size is to be considered because this last one is not necessarily nanometric and may intervene as soon as one has a nucleation/growth mechanism. The greater or lesser separation between NMax and VMax obviously affects the temperature evolution of the characteristic size . It therefore affects the sensitivity of a given compound to size reduction effects.
Acknowledgments The authors are very grateful to the ANR (The French National Research Agency) for their financial support (N° ANR-14-CE16-0025-01/02). They also thank the colleagues of UMET: E. Dudognon, F. Danède, P. Derollez, A. Hédoux, Y. Guinet, L. Paccou, and Professor J. Siepmann (Lille pharmacy faculty) for a fruitful long term collaboration. We thank Karolina Adrjanowicz and Marian Paluch for a very interesting discussion about fenofibrate.
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K. Grzybowska, K. Adrjanowicz, and M. Paluch, in Disordered Pharmaceutical Materials (Wiley-VCH Verlag, 2016), p. 301. M. Descamps and E. Dudognon, Crystallization from the amorphous state: Nucleation-growth decoupling, polymorphism interplay, and the role of interfaces, Journal of Pharmaceutical Sciences 103, 2615 (2014) M. Avrami, Kinetics of Phase Change. I General Theory, The Journal of Chemical Physics 7, 1103 (1939) J. Christian, The Theory of Transformations in Metals and Alloys (Pergamon, Oxford UK, 1975). W. Ostwald, Studien über die Bildung und Umwandlung fester Körper, Zeitschrift fur Physik 22, 289 (1897) T. Threlfall, Structural and Thermodynamic Explanations of Ostwald's Rule, Organic Process Research and Development 7, 1017 (2003) D. A. Porter, K. E. Easterling, and M. Sherif, Phase Transformations in Metals and Alloys, Third Edition (Revised Reprint) (Taylor & Francis, 2009). L. Carpentier, K. Filali Rharrassi, P. Derollez, and Y. Guinet, Crystallization and polymorphism of l-arabitol, Thermochimica Acta 556, 63 (2013) L. Carpentier, S. Desprez, and M. Descamps, Crystallization and glass properties of pentitols : Xylitol, adonitol arabitol, Journal of Thermal Analysis 73, 577 (2003) V. Andronis and G. Zografi, Crystal nucleation and growth of indomethacin polymorphs from the amorphous state, Journal-of-non-crystalline-solids 271, 236 (2000) K. F. Kelton, Crystal nucleation in liquids and glasses, Solid State Physics 45, 75 (1991) P. G. Debenedetti, Metastable Liquids - Concepts and Principles (Princeton University Press, 1996). I. Gutzow, Kinetics of crystallization processes in glass forming melts, Journal of Crystal Growth 48, 589 (1980) I. Gutzow, The mechanism of crystal growth in glass forming systems, Journal of Crystal Growth 42, 15 (1977) D. Turnbull, in Solid State Physics - Advances in Research and Applications, 1956), Vol. 3, p. 225. F. Spaepen, A structural model for the solid-liquid interface in monatomic systems, Acta Metallurgica 23, 729 (1975) D. W. Oxtoby and A. D. J. Haymet, A molecular theory of the solid-liquid interface. II. Study of bcc crystal-melt interfaces, The Journal of Chemical Physics 76, 6262 (1981) J. N. Sherwood, The Plastically Crystalline State, Chichester, New York, Brisban, Toronto, 1979). H. Suga and S. Seki, Thermodynamic investigation on glassy states of pure simple compounds, Journal of Non Crystalline Solids 16, 171 (1974) M. Descamps, C. Caucheteux, G. Odou, and J. L. Sauvajol, Local molecular order in the glassy crystalline phase of cyanoadamantane: diffuse X-ray scattering analysis, Journal de Physique (Lettres) 45, 719 (1984) Y. Guinet, L. Carpentier, L. Paccou, P. Derollez, and A. Hédoux, Analysis of stochastic crystallization in micron-sized droplets of undercooled liquid l-arabitol, Carbohydrate Research 435, 76 (2016)
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A. Hédoux, Y. Guinet, and M. Descamps, The contribution of Raman spectroscopy to the analysis of phase transformations in pharmaceutical compounds, International Journal of Pharmaceutics 417, 17 (2011) E. Dudognon, N. T. Correia, F. Danède, and M. Descamps, Solid-solid transformation in racemic Ibuprofen, Pharmaceutical Research 30, 81 (2013) E. Dudognon, F. Danède, M. Descamps, and N. T. Correia, Evidence for a New Crystalline Phase of Racemic Ibuprofen, Pharmaceutical Research 25, 2853 (2008) A. Hédoux, Y. Guinet, P. Derollez, E. Dudognon, and N. T. Correia, Raman spectroscopy of racemic ibuprofen: Evidence of molecular disorder in phase II, International Journal of Pharmaceutics 421, 45 (2011) M. Descamps and J.-F. Willart, in Disordered Pharmaceutical Materials (Wiley-VCH Verlag GmbH & Co. KGaA, 2016), p. 1. O. Delcourt, M. Descamps, and H. J. Hilhorst, Size effect in a nucleation and growth transformation, Ferroelectrics 124, 109 (1991) M. Descamps, J. F. Willart, O. Delcourt, and M. Bertault, Extrinsic and intrinsic size effects on the phase tranformation and metastbility of a glassy crystal, Thermochimica Acta 266, 49 (1995) M. Descamps, J. F. Willart, and O. Delcourt, Intrinsic instability and nonergodic behaviour of a glassy crystal, AIP Conference Proceedings. no. 256, 71 (1992) S. Sharma, P. Sher, S. Badve, and A. P. Pawar, Adsorption of meloxicam on porous calcium silicate: Characterization and tablet formulation, AAPS PharmSciTech 6, E618 (2005) C. Sander and P. Holm, Porous Magnesium Aluminometasilicate Tablets as Carrier of a Cyclosporine Self-Emulsifying Formulation, AAPS PharmSciTech 10, 1388 (2009) P. Grobelny, I. Kazakevich, D. Zhang, and R. Bogner, Amorphization of itraconazole by inorganic pharmaceutical excipients: comparison of excipients and processing methods, Pharmaceutical Development and Technology 20, 118 (2015) G. Szklarz, K. Adrjanowicz, M. Tarnacka, J. Pionteck, and M. Paluch, ConfinementInduced Changes in the Glassy Dynamics and Crystallization Behavior of Supercooled Fenofibrate, The Journal of Physical Chemistry C 122, 1384 (2018) T. Vasconcelos, B. Sarmento, and P. Costa, Solid dispersions as strategy to improve oral bioavailability of poor water soluble drugs, Drug Discovery Today 12, 1068 (2007) S. Qi, W. J. McAuley, Z. Yang, and P. Tipduangta, Physical stabilization of lowmolecular-weight amorphous drugs in the solid state: A material science approach, Therapeutic Delivery 5, 817 (2014) T. P. Shakhtshneider, F. Danède, F. Capet, J. F. Willart, M. Descamps, S. A. Myz, E. V. Boldyreva, and V. V. Boldyrev, Grinding of drugs with pharmaceutical excipients at cryogenic temperatures : PPPart I. Cryogenic grinding of piroxicampolyvinylpyrrolidone mixtures, Journal of Thermal Analysis and Calorimetry 89, 699 (2007) T. P. Shakhtshneider, F. Danède, F. Capet, J. F. Willart, M. Descamps, L. Paccou, E. V. Surov, E. V. Boldyreva, and V. V. Boldyrev, Grinding of drugs with pharmaceutical excipients at cryogenic temperatures, Journal of Thermal Analysis and Calorimetry 89, 709 (2007) B. C. Hancock and G. Zografi, Characteristics and significance of the amorphous state in pharmaceutical systems, Journal of Pharmaceutical Sciences 86, 1 (1997) S. L. Shamblin, E. Y. Huang, and G. Zografi, The effects of co-lyophilized polymeric additives on the glass transition temperature and crystallization of amorphous sucrose, Journal of Thermal Analysis and Calorimetry 47, 1567 (1996)
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Graphical Abstract 1.0 L=600 L=100
L=z=75
0.8
L=50
0.6
X(t)
L=40
0.4 L=25
0.2
L=15 L=10 L=5
0.0 0
100
200
Time (arb. Time units)
300
S0378-5173(18)30141-8 https://doi.org/10.1016/j.ijpharm.2018.03.001 IJP 17344
To appear in:
International Journal of Pharmaceutics
Received Date: Revised Date: Accepted Date:
22 December 2017 1 March 2018 2 March 2018
Please cite this article as: M. Descamps, J-F. Willart, Scaling laws and size effects for amorphous crystallization kinetics: Constraints imposed by nucleation and growth specificities, International Journal of Pharmaceutics (2018), doi: https://doi.org/10.1016/j.ijpharm.2018.03.001
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Scaling laws and size effects for amorphous crystallization kinetics: Constraints imposed by nucleation and growth specificities
Marc Descamps†*, Jean-François Willart†
†
Université de Lille, CNRS UMR 8207 – UMET – Unité Matériaux et Transformations, F-59000, Lille – France. *corresponding author: [email protected] (33) 03 20 43 49 79
Abstract. In the present paper we review different aspects of the crystallization of amorphous compounds in relation to specificities of the nucleation and growth rates. Its main purpose is: i) to underline the interest of a scaling analysis of recrystallization kinetics to identify similarities or disparities of experimental kinetic regimes. ii) to highlight the intrinsic link between the nucleation rate and growth rate with a temperature dependent characteristic transformation time (T), and a characteristic size (T). The consequences on the influence of the sample size on kinetics of crystallization is considered. The significance of size effect and confinement for amorphous stabilization in the pharmaceutical sciences is discussed. Keywords: Amorphous state, crystallization, nucleation and growth, scaling, size effect.
Graphical Abstract 1.0 L=600 L=100
L=z=75
0.8
L=50
0.6
X(t)
L=40
0.4 L=25
0.2
L=15 L=10 L=5
0.0 0
100
200
Time (arb. Time units)
300
1. INTRODUCTION Recrystallization of amorphous forms is the major cause of their instability. This impacts the stability during a storage but it is also a key element of the mastery of amorphization during a desired formulation in this state. Independently of structural and microstructural modifications, the exothermic nature of the recrystallization process can also influence chemical stability. In the pharmaceutical literature, discussions on this topic try to find primarily a relationship between loss of stability and molecular mobility. These investigations often focus on a narrow temperature range. Often this concerns the search for a similarity between an activation energy controlling mobility and a supposed activation energy controlling the recrystallization[1]. The molecular mobility in the amorphous state has several facets corresponding to various relaxation processes: i) The non-Arrhenian relaxation which leads to times of the order of 100 - 1000s at Tg and is responsible for the gel manifesting at the glass transition. ii) The set of secondary relaxations called , , etc. , (of inter or intra-molecular origin), reflecting localized rapid motions. They are especially manifest in the glassy state. For the molecular materials, all these processes have very different activations energies. And thus, according to studies, various links with an assumed crystallization activation energy have been proposed. This leads to a variety of proposals that are not general in scope. In fact, there is no reason that a simple consideration of mobility can account for the entire mechanism. There are many other possible contributions to the kinetics of crystallization. The reason is that the overall process results from the combination of a dual mechanism of nucleation and growth. These two processes have different temperature evolutions. Both have a maximum rate for a certain degree of undercooling. For both, the existence of a maximum rate is due to the competition between the thermodynamic driving force which prevails for moderate undercooling and the molecular slowing down for larger undercooling. But these maxima are not at the same temperature. This is because the structural difference between the liquid and the crystal does not occur in the same way for nucleation and growth. The difference in temperature localizations of the two processes implies that they are not influenced to the same degree by the molecular mobility; and even more so their combination. On the other hand, for moderate undercooling, it is clear that the increases in nucleation rate and growth rate are in no way a consequence of the dynamics. Moreover, nucleation can occur in a homogeneous or heterogeneous manner with very different rates at the same temperature. Whatever the effect of the control factors, an inherent aspect that differentiates nucleation and growth is the dimension of their physical expressions: Nucleation rate [N] = L-3 t-1 Growth rate [V] = L t-1 This makes it possible to implement a scaling analysis of experimental kinetics of crystallization which can be very useful in a search for interpretation of the mechanisms involved. It can helps to identify the temperature domains where the same mechanism intervenes to give the same physical law. Moreover, the scaling analysis makes it possible to identify a characteristic time and a characteristic length for the global mechanism. This allows capturing a small size effect inherent in the process.
In the present paper we review different aspects of the crystallization of amorphous compounds in relation to specificities of the nucleation and growth rates (designated respectively by N and V). Its main purpose is to underline the interest of a scaling analysis of kinetics and to insist on an intrinsic link between the effects of reduced size and the key parameters of crystallization. The paper includes three sections: The first two summarize briefly - with some new perspectives and illustrations - considerations on crystallizations from the amorphous state, presented in a previous article [2]. These considerations are essential for the development of a new mode of taking into account the size effects which is presented in the third part. In the first section, we recall the expression of the classical law of Avrami expressing the kinetics of recrystallization [3, 4]. We use this to show the interest of a scaling analysis to identify similarities, or, disparities of experimental kinetic regimes. The second section is devoted to the physical properties which drive the temperature evolutions of N and V. The question of the localization of their respective maxima is especially discussed. This is used to identify the best vitrification possibilities and to clarify the reasons for the Oswald rule of stage [5, 6]. The third part of this review considers the intrinsic link between N and V with temperature dependent characteristic transformation time (T), and characteristic size (T). The consequence on the influence of the sample size on kinetics of crystallization is considered. This clarifies the role of sample size on the effective life time of metastability of an amorphous compound and may have a practical impact on amorphization processes and the manipulation of amorphous solids 2. SCALING ANALYSIS OF GROWTH CURVES. So called growth curves express the time (t) evolution of the fraction of amorphous material X(T,t) which is crystallized in the course of an isothermal process at temperature T. The isothermal methods which give access to X(t) are in principle all those which can discern, quantitatively, between the amorphous state and the crystalline state. In practice, it is most often the DSC thermal analysis (used in isothermal mode) and the in-situ X-ray diffraction. The determination of all X(T,t) curves at different temperatures T
If we suppose that a liquid is brought instantaneously to a given temperature T, lower than Tm, nanocrystals are produced at the constant nucleation rate N(T), per unit volume. Even if N is assumed to be constant in time, the nucleation rate decreases effectively as the amorphous volume, free for crystallization, also decreases. Once formed, the grain grows isotropically, with a constant velocity V(T). As it grows it starts to impinge upon other grains and eventually stops growing [4]. These two sources of slowing down are to be taken into account in the evaluation of X(t). This is done most directly by calculating the probability for a point P of the amorphous sample not to be engulfed at a time t, which is identical to 1-X(t). The probability obeys the following Poisson law: t 1-X(t) = exp(-N 0 f(t') dt') (1) where f(t') is the volume centred on P, in which nucleation should not have appeared before t' in order to be sure that P is not reached by a grain at time t. If we consider the isotropic growth in a 3-dim space, this lead to the classical Avrami law[7]: X(t) = 1 - exp[- (4/3) (t/t0)4]
(2)
which is a universal function of the time scale: t0 = (N V3)-1/4
(3)
If nucleation and growth occur simultaneously and N and V have the particular characteristics given above, then a universal transformation law of type (2) is expected. X(t) takes the form of an S-shaped curve that is almost symmetrical with respect to the half-transformation point (that occurs at t0.5 t0). At different temperatures, and therefore for different values of N and V, X(t) can then be rescaled on a single curve when plotted against the reduced time t/t0.5. 2.2 Scaling analysis of the growth curves A scaling analysis of isothermal growth curves allows to locate the temperature domains for which an Avrami hypothesis is possibly applicable. It is also very useful for identifying kinetic behaviors associated, on the contrary, with evolutions of N and V that do not fit into the initial hypotheses. An example of such a scaling analysis is given in Figure 1. Figure 1a, reports different curves X(T, t), obtained by isothermal DSC, on l-arabitol[8, 9], which has been quenched rapidly from the liquid state to different temperatures between the melting temperature of the most stable phase Tm = 101°C and the glass transition T g -14°C. These curves all have an S-shaped appearance whose rapid evolution is more or less time-delayed as a function of temperature. From this observation alone, it is difficult to discern possible kinetic regime changes. Figure 1b shows the curves X(T, t) corresponding to those of Figure 3 but plotted against a rescaled time (t / t0.5). For the lowest temperatures (15°C. T 40 ° C), the scaled curves are fairly well superimposed to show that a scaling, of the Avrami type, is well satisfied. This suggests that in this temperature range, the recrystallization kinetics probably corresponds to a nucleation and growth process, and obey the same law. The scaled curve shows that the kinetics recorded at the highest temperatures (45 ° C and 50 ° C) strongly deviate from the symmetrical universal sigmoidal shape.
In particular, the growth curve measured at 50 ° C shows a very abrupt evolution preceded by a long waiting time. Such an evolution can absolutely not be represented by an Avrami type curve, unless to replace in (2) the exponent 4 with an exponent value greater than 10, which is totally unrealistic. The scaling analysis thus reveals changes in kinetic regime when the melting temperature Tm is approached. The nucleation rate becomes so low and the rate of growth is so great that the two mechanisms do not occur simultaneously. The consequence of the absence of universal behavior is that it would be totally illusory to try to interpret the temperature evolution of the whole of the growth curves shown in Figure 1. by the only change of a relaxation time or the search for a single activation energy. In paragraph 4 it will be seen that a scaling analysis also makes it possible to identify how a confinement can induce modifications of the kinetic laws themselves, in addition to a modification of the time scale of the crystallization.
X (t)
time (min)
Figure 1.a Temperature dependency of the time evolution of the fractional volume X(t) of recrystallized L-arabitol in phase I. The different temperatures are that reached after a quench from the melt (obtainedfrom DSC isothermal measurements). (From[2] with permission)
Figure 1.b (L-Arabitol) The scaled curves of X(t) (corresponding to those shown in Figure 1a) plotted in terms of the scaled time t/t0.5. (From
[2]
with permission)
3. TEMPERATURE SEPARATION OF N(T) AND V(T) MAXIMA Another consequence of the above kinetic curve analysis is that N(T) and V(T) have maxima at very different temperatures: Indeed at 50 ° C the step like behavior of the growth curve is interpreted by the fact that it is necessary to wait about 7 hours before one nucleus appears in the DSC sample. The crystallization then corresponds to the very rapid crystal growth from this single nucleus. At 50°C, the value of the nucleation rate is very small while it is large for the growth rate. At lower temperatures both events occur simultaneously in the crystallization process. 3.1. Experimental identifications The previous analysis suggests that the maxima of N (T) and V (T) are very distant in temperature. This separation can be easily demonstrated by a simple systematic DSC analysis. It consists of recording heating scans of the sample previously melted then quenched at different temperatures Ta between Tm and Tg, and finally annealed isothermally at Ta for a duration ta. Figure 2 shows such curves for l-arabitol[2]. For Ta >35°C and Ta <-10°C, virtually no exothermic sign of crystallization is observable during reheating. On the other hand, for annealing temperatures between 65°C and 30°C, significant exothermic recrystallizations are observed. They have their maxima located between 60°C and 95°C, i.e. a little below the melting endotherm. This temperature range is clearly distinct from that of annealing at T a. These observations and the analysis of the shape of the growth curves presented above make it possible to conclude that the observed exotherms correspond to the growth of crystals from nuclei formed at a lower temperature. The consequence of this is that N (T) and V (T) have both a maximum and that these maxima are very separated. The preferential growth zone is situated slightly below Tm. The preferential nucleation zone is located at a lower temperature, a little above Tg [10]. Since the enthalpy of recrystallization increases with the number of nuclei formed, it can be estimated that the maximum of N(T) is located near 5 ° C. Figure 2 shows the approximate location of the two zones.. When cooling a liquid, the system first crosses the temperature zone where V is maximum and then the temperature zone where N is maximum. In the first zone, no crystallization occurs because no nucleus has been formed. In the second temperature zone nucleation occurs, but without growth, so that the crystallized fraction is tiny. The fact that N and V both have a maximum (NMax and VMax, ) is basically the reason why it is possible to reach a glassy state, by a sufficiently fast liquid quenching. This possibility is amplified by a greater separation of the maxima.
Vitesse de croissance Vitesse de nucléation (unités arbitraires)
N
V
18 -40
-20
0
20
40
60
80
100
120
(W/g) Flow HeatFlux de chaleur (W)
16
14
12
10
8
6
Exo Exo.
4
Tg
2 -40
-20
0
Tm 20
40
60
80
100
120
Température (°C) Temperature ( C)
Figure 2 (L-Arabitol) DSC curves recorded upon heating (heating rate of 2 K/min) for samples undercooled at temperatures Ta ranging from −83◦C to 82◦C and annealed for 60 min at this temperature. ( adapted from[2] with permission)
3.2 Drivers for the N(T) and V(T) separate evolutions The essential ingredients of the temperature evolutions of N(T) and V(T) are well captured by the elementary theories which lead to the following expressions [7, 11-14]: N(T) f0(T) exp( - g*/ RT)
(4)
V(T) f0(T) [1- exp(-G/RT)]
(5)
g* is the nucleation barrier : g* 3-d 3/G2 and is the interface free energy between a crystalline nucleus and the parent amorphous medium. G (T) is the Gibbs free energy difference between the metastable liquid and stable crystal states. G (T) is the thermodynamic driving force for crystallization. f0 is an "attempt frequency" for the addition of molecules from the metastable melt to the stable crystal across the interface At Tm, the liquid and crystal states are in thermodynamic equilibrium. The thermodynamic driving force for crystallization is zero as well as N(Tm) and V(Tm). During the cooling of the liquid, N(T) and V(T) increase initially with the degree of undercooling and the progressive increase of G (T). In the domain of relatively high temperatures, the molecular mobility is large enough not to oppose the effect of the thermodynamic driving force. For larger undercoolings, the molecular mobility (f0(T)), decreases considerably (ie, increasing viscosity) approaching Tg. It eventually offsets the effect of the thermodynamic force, and slows the nucleation and growth kinetics. As a result, N(T) and V(T) both have a maximum for a certain temperature. The competition of thermodynamic and dynamic effects is reflected in the competition between, respectively, the second term and the first term of the right-hand side of the equations (4) and (5) giving respectively N (T) and V (T). If the thermodynamic driving force and the molecular mobility compete similarly for N(T) and V(T) and induce maxima, these two factors alone can not justify the noticeable differences in their temperature positioning. The greater or lesser temperature separation is mainly due to the influence of the surface free energy between a crystallite and the parent amorphous medium[2]. plays a fundamental role in the value of N because of the important surface effects related to the initial formation of nanocrystalline nuclei. plays only a marginal role in V(T). 3.3 Influence on nucleation of crystalline and amorphous structural similarities. Nucleation is the precursor event of crystallization. If has a null value or is small, nucleation is easier and nucleation rate has more chance of being large. is a free energy, and, as such, has a fundamental entropic contribution. Figure 3 gives an illustration of the molecular origin of , during the crystallization of a molecular liquid. The value of decreases as the entropy difference between the bulk amorphous state and crystalline state, S (bulk), is smaller i.e. if there is a structural proximity between them[15-17]. Moreover, its value is as lower as the entropy difference between the core of the amorphous phase and the interfacial zone is weak i.e. if the liquid has little to organize at the interface with the crystal. Or says otherwise if S (bulk) is closer to S (interface). This helps to rationalize Ostwald's postulate that the first nuclei to form are often those of a metastable phase rather than of the more stable phase. Obviously that can not be related to the thermodynamic driving force which is smaller for the metastable phase. Furthermore, when we compare kinetics at the same temperature, that cannot be either explained by a difference of molecular mobility in the amorphous phase. In fact the metastable phase gives the lowest melting entropy and highest crystalline entropy (closer to that of the amorphous state) as illustrated in Figure 5a. Consequently the metastable phase opposes the lowest interfacial free energy and is therefore easier to nucleate.
Interf. LIQUIDE
CRYSTAL
TM S (interf.) (interf)
Hc
Hl TMSl TM S (bulk) = HM
TMSc
Figure 3 Diagram showing the structural origin of the interface free energy . is as lower as the molecular arrangement in the bulk of the liquid is close to that of the crystal and/or the organization of the liquid at the interface is similar to that existing in the bulk of the liquid.
There are two types of situations that require little reorganization at the interface: - When the local structure of the liquid has strong similarities with the long-range structure of the crystal[15]. - When the crystal phase is highly disordered. The disorder can be static or dynamic[18-20]. Moreover, for a given case, both situations can coexist We present two examples that illustrate these situations: l-arabitol l-arabitol has at least two crystalline polymorphs[8, 9]. The variety II is metastable and melts at lower temperature than the stable variety I (Figure 4a). Variety II is the first stage of the recrystallization from the amorphous state after a deep quenching of the melt. Raman scattering allows to show clearly that the origin of this preferred nucleation is associated with structural similarities between the amorphous state and variety II [21]. Figure 4b. allows to compare the low frequency Raman susceptibilities, recorded at the same temperature, of varieties I and II as well as the amorphous state. In the latter case the susceptibility is characterized by a broad and asymmetric band which is the image of the vibrational state density (VDOS) of the amorphous state. The maximum intensity of the VDOS, located at about 40-45 cm-1, corresponds quite exactly with the low-frequency, sharp and intense phonon peak of the metastable variety II. By cons there is no phonon peak of the stable form I, which is detected below 50cm-1. This indicates that there is a strong reminiscence of the long-range order of variety II in the local order of the undercooled liquid.
Figure 4a DSC heating scans showing the melting endotherms of the stable (I) and metastable (II) phases of L-Arabitol (From[2] with permission)
Figure 4b Low frequency Raman spectra of crystalline forms I and II and the undercooled liquid state of L-arabitol at 265 K. (From[21] with permission)
RS Ibuprofen The second example is RS ibuprofen [23, 24]. Under deep quenching of the liquid the RS ibuprofen preferentially crystallizes towards a metastable phase II. The crystallization of the stable ordered racemic phase I is kinetically much slower. The thermodynamic relationships of phases I, II and the amorphous state are summarized in the Gibbs diagram of Figure 5a. The low melting entropy value of phase II compared to that of phase I suggests that phase II exhibits a high degree of disorder
that allows easy adaptation of phase II nuclei to the amorphous matrix. This appears clearly in the confrontation of Raman spectra. The Raman susceptibilities (recorded at 190K) of the crystalline phases I and II are compared to that of the glassy state on Figure 5b. It appears clearly that the spectra of the glassy state and phase II are nearly superimposed. They are typical of disordered molecular systems. The similarity of the vibrational density of state (VDOS) spectrum of glassy ibuprofen and the ''()-spectrum of phase II, clearly reflects the existence of a significant structural disorder in phase II. It also shows that the local molecular arrangement which exists in glassy ibuprofen is the reflect of the long range order of crystalline phase II.
Figure 5a Gibbs diagram showing the situations of the stable phase (I) and metastable phase (II) of Racemic RS ibuprofen (From[2] with permission)
Figure 5b Low-frequency (5–200 cm−1) Raman susceptibility in the under-cooled (or glassy) state and crystalline phases I and II of Racemic RS ibuprofen (From.[25] with permission)
3.4. Practical consequences for the formation and stability of amorphous states. The crystallization of an undercooled liquid requires both crystal nucleation and growth processes to occur. A Time-Temperature-Transformation (TTT) plot (Figure 6) is useful for describing the combination of the two factors and their specific velocities N and V. The set of TTT curves is build from growth curves such as those in Figure 1a. Such a diagram has a nose at temperature where the conjunction of N and V leads to the fastest crystallization. The position of the nose is generally different from that of Nmax and Vmax. The diagram can be used to determine the critical cooling rate (measured in K / s) that must be imposed on a liquid, to quickly pass the nose, and to succeed in vitrifying the compound without significant crystallization. The temperature zone of the nose is also a domain of maximum instability for the amorphous compound. However, caution must be exercised when using a TTT chart. Indeed, even if the vitrification is successful, without detectable crystallization, nanoscale nuclei could be trapped during rapid cooling. This is because, at low temperatures, the nucleation rate can be significant, while the growth rate is negligible. The presence of crystalline nuclei can be a source of instability of the amorphous compound, because they have a seeding effect, and can generate, on heating, a massive crystallization by the effect of growth. It is for this reason that the crystallization of an amorphous compound is more often observed during a reheating than during the cooling of the liquid. To avoid the formation of these nuclei it is necessary to use an even faster cooling rate. Determinations of the temperature localization of the nose of the TTT diagram, as well as Nmax and Vmax, are decisive factors in the strategy for forming a compound in the amorphous state and for controlling its stability. T
LIQUID
Tm
G
COOLIN
GROWTH ALONE
VMax
NUCLEATION And Growth
Tn 1%
NMax
50%
99%
NUCLEATION ALONE
Tg GLASS log (time)
Figure 6 Time-Temperature-Transformation (TTT) curve for crystallization (nucleation + growth), showing the critical cooling rate (black curve) which is necessary to vitrify the compound. Because nucleation is a faster process at low temperature a fast cooling may not avoid nucleation. Much faster cooling is required to obtain a non-nucleated glass, as shown by the red curve at left. Tm is the melting temperature, Tg is the glass-transition temperature. An experimental study which illustrates [2][26] this situation is found in reference .
4. EFFECT OF SAMPLE SIZE ON CRYSTALLIZATION KINETICS 4.1. Characteristic time, Characteristic size: dimensional analysis. By a simple dimensional analysis[27, 28] of the nucleation rate N ([N] = L-3t-1) and linear growth rate V ([V] = Lt-1) we can highlight that the crystallization process is specified by (for a 3-dim growth): - a characteristic time t0 = (N V3)-1/4 (6) 1/4 - a characteristic size = (V/N) (7) Specifying the values of t0 and , which are both temperature dependent, is equivalent to give the values of N and V. The expression of the kinetic law, X(t), of a nucleation and growth transformation is fundamentally dependent on the position of the sample size (L) with respect to . This is illustrated schematically in Figure 7. Such sample size influence is intrinsic to the crystallization mechanism and is not related to a chemical phenomenon or a change in N or V values. It depends only on the relative importance of the growth rate with respect to the nucleation rate. Sample size modification has effect on the time scale of the crystallization process but also on the expression of the kinetic law itself. In particular, a dramatic slowing down of actual kinetics is inevitably expected at small sample sizes.
1.0 L=600 L=100
L=z=75
0.8
L=50
0.6
X(t)
L=40
0.4 L=25
0.2
L=15 L=10 L=5
0.0 0
100
200
300
Time (arb. Time units)
Figure 7 2-Dimensional simulation of the time evolution of the transformed fraction X(t) for different grain sizes (from above to below ). The values of the nucleation and growth rates are fixed and [27, 29] correspond to = 75 and t0 = 75.(From with permission)
4.2. Growth laws as a function of sample size. In the limiting cases of the very large and very small sample sizes, one expects two very different expressions for the growth laws [27, 28] L >> . We are in the case of the Avrami's law presented above. t0 intervenes in the expression of the Avrami equation, to give a dimensionless term in the exponential: X(t) = 1 - exp[- (4/3) (t/t0)4]
(8)
However does not intervene in this equation, nor does the size (L) of the sample. This is so because the Avrami equation is only valid for samples of significantly larger size than . Under these conditions, the Avrami expression is a sigmoidal universal function.
L << . We consider a sample consisting of a set of small cells of size L. Each cell is immediately crystallized as soon as it is the site of a nucleation event. The crystallized fraction of the global sample coincides with the fraction of "nucleated" cells. The evolution of the crystallized fraction is calculated by: dX = (1 - X) N L 3 dt -->X(t) = 1 - exp(-t/tn)
(9)
X(t) is exponential with the characteristic time tn = (L3 N-1) which is now highly size dependent. Figure 7 shows a general outlook of the influence of sample size on the evolution of the kinetic law[28]. It results from a simple 2-dimensional simulation of a nucleation and growth transformation with fixed values of N and V. Different simulations were made for a large sample being subdivided in equivalent square cells whose linear size L has been varied. In this simulation the value of L varied from 5 lattice units (l.u.) to 600 l.u. N et V were chosen so that = 75 lattice units and t0 = 75 time units. The expressions of and t0 are adapted from those given by (6) and (7), for a 2-dim system. For large values of L we observe that X(t) is sigmoidal. The shape of the curve is then only weakly dependent on L. When L decreases the shape of X(t) turns into an exponential whose relaxation time becomes very dependent on L. A marked change of behavior is noted when L crosses the value of the characteristic size .. The results of an exact analytical calculations have been reported in ref.[27]. It shows that, in the general case, a scaling relation involving a reduced time (t/t0) and a reduced size (L/) is applicable. Figure 8 shows that the effect determining the form of X(t) is related to the relative position of t/t0 with respect to L/. is the length which determines the crossover from the Avrami regime (L>>) to the size dependent exponential regime (L>>). In the latter case the transformation satisfies a scaling relation which involves the ratio L/. Experimentally the global variation of X(t) always marks a more or less clear transition from one situation to another. The awareness of this influence of the sample size on a growth curve should make it possible to better understand the influence of a formulation process on the stability of an amorphous compound with respect to crystallization.
1 1−
𝜋 𝑡 4 − 𝑒 3 𝑡 𝑎0
crossover
t/ta0
0 0
L/
1
1 1−
1−
𝜋 𝑡 4 − 𝑒 3 𝑡0𝑎
4𝜋 𝐿 3 𝑡 − 𝑒 3 𝜁 𝑡 𝑎0
crossover
t/ta0
0 0
1
1 crossover
1−
𝜋 𝑡 4 − 𝑒 3 𝑡0𝑎
1−
4𝜋 𝐿 3 𝑡 − 𝑒 3 𝜁 𝑡0𝑎
t/ta0
0
0
L/ 1
Figure 8 Typical behaviour of the growth curves X(t) plotted vs rescaled time (t/t0)for different values of (L/.) (adapted [27] with permission)
4.3. Relevance of the characteristic sizes and times in the analysis of the effect of confinement. The dimensional considerations presented above help to gain more understanding into the consequences of sample size reduction on the kinetics of crystallization and in turn on the amorphization possibilities and amorphous state stabilities. More generally this is applicable to phase transformations managed by a nucleation and growth mechanism. There are basically two types of confinement processes:
- The confinements imposed extrinsically by spreading the active compounds in a mesoporous carrier -The confinements resulting intrinsically from the formulation of multicomponent systems. We present below two representative illustrations. Extrinsic size effect: mesoporous carriers. The concept of mesoporous carriers has been recently developed in the pharmaceutical field to stabilize amorphous drugs. Porous excipients such as porous silicon or silicates materials (for example Neusilin or Florite [30-32] ) are used more and more frequently for delivery of amorphous drugs. The pharmaceutical molecules are then adsorbed and spatially constrained in the nanometer-sized pores of the carrier material. Significant changes in crystallization kinetics are then observed experimentally. The nanometric confinement of fenofibrate, studied by G. Szklarz et al., is particularly illustrative of these effects [33]. Fenofibrate was confined in cylindrical nanopores of diameter ranging between 18 nm and 100 nm. Isothermal crystallization curves were obtained from the time evolution of the static dielectric permitivity. Figure 9 corresponds to isothermal recordings made at 298 K (56 K below Tm). It clearly shows an evolution of an Avrami type sigmoidal behavior for the bulk sample to a pseudo exponential behavior when the pore diameter decreases. This also results in a gradual increase in the characteristic crystallization time. This type of evolution is consistent with that theoretically predicted for a size reduction below . Figure 10 shows the evolution of the crystallization rate k ( t0-n in the Avrami equation) as a function of temperature, for a bulk sample and different samples confined in the nanopores. The position of the maximum of the crystallization rate, and therefore the minimum of the crystallization time, corresponds to the nose in a TTT diagram (Figure 6). It can be noted that this maximum shifts to lower temperatures as the pore diameter decreases. At the same time the crystallization rates measured at the maximum decrease. Moreover, it can be seen that the confinement effect is considerable at temperatures close to Tm, but that the effect of nanometric confinement is much less pronounced for deep quenching (The curves of the k(T) diagrams are getting closer). The overall influence of the confinement on the modification of the TTT-type diagrams of Figure 10 can be easily understood if it is noted that, near Tm, the growth rates (V) are large whereas the nucleation rates (N) are in general weak. According to formula (7), this leads to a large value of , and therefore to a very pronounced size effect. Approaching Tg, N increases sharply and V decreases. As a result decreases, and the confinement effect is less pronounced, which results in the k(T) curves corresponding to the different pore sizes being closer to one another. This progressive modification is therefore also at the origin of the general sliding of the k(T) curves towards the low temperatures. It is therefore not necessary to imagine changes in the nucleation mechanisms to explain the evolutions measured.
Figure 9 Isothermal (at 298K) time evolution of the normalized dielectric permittivity ('N) for fenofibrate confined in nanoporous alumina of different pore diameter. (from G. Szklarz et al.[33] with permission). We can notice the change from sigmoïdal Avrami to exponental type of evolution closely similar to that shown in Figure 7.
Figure 10 Temperature evolution of the crystallization rate k for fenofibrate confined in nanoporous [33]
material of different pore diameters. (from G. Szklarz et al. with permission). The maxima of the crystallization rate are the mirror images of the minimum of the crystallization time corresponding to the nose of the TTT diagram, such as that shown in Figure 6.
Intrinsic size effect: Amorphous Solid Dispersions It is known that by producing amorphous solid dispersions (ASD) of a polymer and an API it is possible to inhibit the crystallization of the API [34, 35]. This is the case, for example, for ASDs of PVP with piroxicam[36] or indomethacin[37] (IMC). In the case of ASDs prepared by solvent evaporation or HME, it has been shown that the formation of hydrogen bonds between the polymer and the small molecule certainly plays an important role in blocking crystallization. However, co-grinding also makes it possible to form ASDs. In the case of PVP / Indomethacin (IMC) and PVP / piroxicam, it has been shown by A. Hedoux et al.[22], for example, that the kinetics of crystallization of the small molecule is very considerably slowed down even at low PVP concentration (10%) (cf Figure 11). Figure 11 shows, moreover, that the shape of the crystallization curves changes from a sigmoidal behavior for bulk IMC, to a pseudo exponential behavior, almost without significant induction time, when the PVP concentration increases. Figure 12 b shows that when the PVP ratio increases up to 50% the value of Tg of the ASD is however practically unchanged and remains close to that of pure IMC (no antiplastifying effect). In addition, the high frequency (Figure 12 a) and low frequency Raman spectra are similar to that of the pure API for concentrations up to 50% which indicates the absence of noticeable interaction between the two components. This suggests that direct interactions between IMC and PVP do not play an important role in reducing drug crystallization. In this case, the polymer would rather have a partitioning effect within the dispersion, the PVP accumulating around nano / micro particles of IMC. Its stabilizing role would then come from a confinement effect of the type described above in 4.1. The presence of very low levels of the low-molecular mass additives has often a significant plasticizing effect on the pharmaceutical glasses. On the other hand adding small amounts of the high-molecular mass additives often had a minimum antiplasticizing effect[38, 39]. This suggests that self-confinement stabilization of the type described above is quite common.
Figure 11 Time evolution of the crystallized volume fraction of - IMC in the absence and presence [22]
of PVP (Reprinted from reference
, with permission)
a)
b)
Figure 12 a : High frequency Raman spectra (in the 1500 - 1800 cm-1 range) of solid amorphous IMC- PVP mixtures prepared by cryogrinding. (Reprinted from [22], with permission)
b)
Figure 12 b : Glass transition temperatures of IMC–PVP cryoground mixtures plotted vs. the mass fraction of PVP: The measured Tg values. The prediction of the Gordon–Taylor equation [36] (Reprinted from , with permission)
5. CONCLUDING REMARKS 5.1. Impact on the analysis of parameters managing the stability of an amorphous state. Crystallization of an amorphous product has a practical significance to the development of pharmaceuticals and biopharmaceuticals. It has a significant impact on formulation and storage conditions and consequently on bioavailability [40, 41]. The practical lifetime of metastability - or even instability for T
5.2. Significance of size effect and confinement for amorphous stabilization The changes in crystallization kinetics when a pharmaceutical molecule is confined in the nanometer-sized pores of a carrier material are usually attributed to two main reasons: i) A reduction in the molecular mobility of active molecules in the confined space. This is indeed possible if the pore sizes are extremely small and if the interactions of the molecules with the carrier are very important at the interface, (This then leads to an increase in Tg). However, size reduction can have a reverse effect and increase molecular mobility. This is due to an actual cleavage of the domains where the molecules rearrange in a cooperative manner (CRR zones) [44]. One could then think that the increase in mobility - which leads to a decrease in Tg - could lead to an acceleration of crystallization kinetics. But this is contrary to what is observed because the intrinsic size effect associated with the nucleation and growth mechanism is prevalent. ii) A pore size close or below the critical nucleation size rc* at the storage temperature. That is expected to minimize crystallization influence. Near Tg or below Tg the sizes predicted by classical nucleation theories are however extremely small. For the fenofibrate example, cited above, the critical nucleation size was calculated [33] using a conventional nucleation model. It varies between 20nm (at Tm-10K) and 3nm (at Tm-70K). Note that the critical nucleation size r c * is always significantly smaller than the pore diameter in the majority of situations. The slowing down of the kinetics when the pore size decreases cannot be attributed to a competition between rc * and the pore size that would block the nucleation mechanism. The two arguments mentioned above are only conceivable for extremely nano-sized pores. Confinement effects can however be observed even for micro size pores. In that case an interpretation based on a competition of the pore size with the critical size is to be considered because this last one is not necessarily nanometric and may intervene as soon as one has a nucleation/growth mechanism. The greater or lesser separation between NMax and VMax obviously affects the temperature evolution of the characteristic size . It therefore affects the sensitivity of a given compound to size reduction effects.
Acknowledgments The authors are very grateful to the ANR (The French National Research Agency) for their financial support (N° ANR-14-CE16-0025-01/02). They also thank the colleagues of UMET: E. Dudognon, F. Danède, P. Derollez, A. Hédoux, Y. Guinet, L. Paccou, and Professor J. Siepmann (Lille pharmacy faculty) for a fruitful long term collaboration. We thank Karolina Adrjanowicz and Marian Paluch for a very interesting discussion about fenofibrate.
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Graphical Abstract 1.0 L=600 L=100
L=z=75
0.8
L=50
0.6
X(t)
L=40
0.4 L=25
0.2
L=15 L=10 L=5
0.0 0
100
200
Time (arb. Time units)
300