Structural transformations of D-mannitol induced by in situ milling

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Structural transformations of D-mannitol induced by in situ milling using real time powder synchrotron radiation diffraction Pauline Martinetto, Pierre Bordet, Marc Descamps, Emeline Dudognon, William Pagnoux, and Jean-Francois Willart Cryst. Growth Des., Just Accepted Manuscript • DOI: 10.1021/acs.cgd.7b01283 • Publication Date (Web): 06 Oct 2017 Downloaded from http://pubs.acs.org on October 10, 2017

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Crystal Growth & Design

Structural transformations of D-mannitol induced by in situ milling using real time powder synchrotron radiation diffraction Pauline Martinetto1,2,*, Pierre Bordet1,2, , Marc Descamps3, Emeline Dudognon3, William Pagnoux1,2,3, Jean-François Willart3 1 Univ. Grenoble Alpes, Inst NEEL, F-38000 Grenoble, France 2 CNRS, Inst NEEL, F-38000 Grenoble, France 3 UMET, Univ. Lille 1, F-59655 Villeneuve d’Ascq, France ABSTRACT We have investigated the solid-state transformation of anhydrous β-D-mannitol upon milling using synchrotron X-ray powder diffraction. The 1720 diffraction patterns, collected in real time during the operation of the mill (6h), have been analyzed by sequential Rietveld refinements. From this analysis we confirm that a polymorphic conversion has occurred from the form β toward the form α. Moreover, the quantification and the microstructural analysis of both polymorphs have allowed obtaining new results: (i) the kinetic of the transformation is characterized by a sigmoidal shape (ii) little microstructural evolution occurs during the solidstate transformation, (iii) no intermediate amorphous phase, preceding the transformation toward the metastable crystalline α phase, can be detected. A comparison in depth of the two crystal structures is given and two models are proposed in order to explain this direct conversion β → α.

*corresponding author: Pauline Martinetto, Institut Néel, CNRS, BP166 38042 Grenoble Cedex 9, France e-mail: [email protected] tel : +33 (0)4 76 88 74 14

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Structural transformations of D-mannitol induced by in situ milling using real time powder synchrotron radiation diffraction Pauline Martinetto1,2,*, Pierre Bordet1,2, , Marc Descamps3, Emeline Dudognon3, William Pagnoux1,2,3, Jean-François Willart3 1

Univ. Grenoble Alpes, Inst NEEL, F-38000 Grenoble, France 2

3

CNRS, Inst NEEL, F-38000 Grenoble, France

UMET, Univ. Lille 1, F-59655 Villeneuve d’Ascq, France

*To whom correspondence should be addressed. E-mail: (P.M.) [email protected] KEYWORDS: D-mannitol; polymorphism; in situ oscillatory milling; powder diffraction; Rietveld refinements; synchrotron.

ABSTRACT We have investigated the solid-state transformation of anhydrous β-D-mannitol upon milling using synchrotron X-ray powder diffraction. The 1720 diffraction patterns, collected in real time during the operation of the mill (6h), have been analyzed by sequential Rietveld refinements. From this analysis we confirm that a polymorphic conversion has occurred from the form β toward the form α. Moreover, the quantification and the microstructural analysis of both polymorphs have allowed obtaining new results: (i) the kinetic of the transformation is characterized by a sigmoidal shape (ii) little microstructural evolution occurs during the solidstate transformation, (iii) no intermediate amorphous phase, preceding the transformation toward the metastable crystalline α phase, can be detected. A comparison in depth of the two crystal structures is given and two models are proposed in order to explain this direct conversion β → α. 2 ACS Paragon Plus Environment

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I. INTRODUCTION

Milling of molecular organic materials, frequently used in the pharmaceutical industry to reduce the size of the particles, is known to also induce solid-state transformations 1,2,3. It can be an alternative route to amorphization of crystalline powders compared to more conventional ones, such as supercooling of liquids or concentrating noncrystallizing solutes 4,5

. It can also lead to a crystal-crystal transformation between different polymorphic forms

6,7,8

. Previous studies indicate that amorphization is generally observed when milling is

performed far below the glass transition temperature of the compound (Tg), while polymorphic transformations mainly occur when milling above Tg

9-13

. However, in general

the final state of the compound still remains unpredictable and physical parameters which drive the transformation are not yet clearly identified and understood. This article is focused on the case of polymorphic conversions induced by milling. Literature reports many cases of molecular organic materials for which a conversion, usually from the stable polymorph to a metastable one, is observed 14. Two different behaviors upon milling have been described: -

if the milling experiments are performed at temperatures very close to Tg, an intermediate amorphous phase which precedes the transformation toward the metastable crystalline phase has been clearly identified

15-17

. Nucleation may take place in this

intermediate state and a new crystal phase may be generated upon growth from these nuclei. -

if the molecular compounds are milled at temperatures substantially higher (e.g. several tens of degrees) than their Tg’s, the evidence of a transient amorphous phase is not detectable 18 or highly questionable 8. One proposed explanation could be the very short



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lifetime of the amorphous state (fast kinetics of crystallization), which would make it very difficult to detect experimentally. The purpose of this paper is to obtain new insights into the solid-state transformation of anhydrous D-mannitol (called mannitol in the following) upon milling. Mannitol, C6H14O6, is an acyclic sugar alcohol produced by various plants, algae and fungi, mainly used in the pharmaceutical industry as an excipient in the formulation of tablets and granulated powders. It is one of the classic examples of a compound which crystallizes in several polymorphs with different physical properties. After some confusion in the literature over the years, it is now proven that only three polymorphs (labelled δ, α, β) exist, with increasing stability at room temperature 19,20. Crystal structures of these three polymorphs have been solved and described respectively in 20,21 (δ) ; 20,22 (α) ; 20,23 (β). Previous investigations 24 have shown that milling at room temperature of the forms δ and β (respectively the less stable and the most stable polymorph) under the same conditions induces a transformation toward form α of intermediate stability. In both cases, the form α obtained by milling slowly reverses toward the stable crystalline form β within a few days after the end of the milling treatment. With a Tg = 13°C 25,24,9, mannitol can be viewed as a model compound to investigate the mechanism of solid state transformations in molecular organic materials. Here, we studied in situ the milling of the polymorph β, at room temperature i.e. some degrees above Tg, by using synchrotron X-ray powder diffraction in order to follow in realtime the kinetics of transformation and clarify whether it is possible to detect a transient intermediate amorphization stage.

II. EXPERIMENTAL SECTION

1. Laboratory DSC measurements of samples milled in a planetary mill 4 ACS Paragon Plus Environment

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Crystalline mannitol (purity ≥ 99.0 %) was purchased from Fluka and was used without any further purification. The commercial compound is the stable polymorph β. Ball-milling was performed in a high energy planetary mill (Pulverizette 7 − Fritsch) at room temperature. We used ZrO2 milling jars of 45 cm3 with seven balls (Ø = 1.5 cm) of the same material. One gram of β-mannitol was placed in the planetary mill, corresponding to a ball/sample weight ratio of 75:1. The rotation speed of the solar disk was set to 400 rpm. We took care to alternate milling periods (typically 20 min) with pause periods (typically 10 min) in order to limit the overheating of the sample. A set of samples with increasing milling times (15 min, 1h, 10h) was prepared. Each sample taken out from the mill was immediately submitted to a characterization by Differential Scanning Calorimetry (DSC). DSC measurements were performed with a TA Instruments Discovery calorimeter from room temperature up to above the melting point of β-mannitol using approximately 3-5 mg of sample. All the experiments were performed with a heating rate of 5 °C/min. The sample was placed in an open aluminium pan (container with no cover) and was flushed with highly pure nitrogen gas (50ml/min). Temperatures and enthalpies were calibrated using Indium at the same heating rate and the same environmental conditions as the experiments.

2. In-situ synchrotron powder diffraction during oscillatory milling The experiment was carried out at the ID15B beamline of the ESRF. We took advantage of the very high energy (λ=0.14259 Å) and highly collimated x-ray beam to perform real time diffraction data collection while the powder sample was submitted to milling at room temperature within the vessel of a modified oscillating Retsch MM400 ball mill26,27. About 0.6 g of commercial mannitol was placed in the Perspex® vessel with a single ZrO2 ball (Ø = 10 mm, ball/sample weight ratio of 9:1) and the oscillation frequency was set to 30 Hz. The mill was operated continuously for 6h with 5 min interruptions after each hour of milling. The 5 ACS Paragon Plus Environment


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collection was then extended for 1h40’ after the mill was stopped. The x-ray beam (size 0.4x0.4 mm2) was aligned across the vessel so as to maximize the amount of sample powder allowed to diffract. The diffracted signal was collected with a 10s/image exposure time (plus a ≈2.4s dead-time between two consecutive images) on an amorphous Si flat panel PerkinElmer detector located at 1.25 m from the sample and was azimuthally integrated using Fit2D28. The scattering signal from the empty vessel (i.e. measured without sample powder and zirconia ball) was then subtracted and the diffraction patterns were normalized to compensate for the evolution of the direct beam intensity. The Instrument Resolution Function was determined from the measurement of a CeO2 standard with the mill ball removed. The evolution of the sample microstructure was obtained by sequential Rietveld refinements using the FP_Suite29. The diffraction signal from the ball was found to consist of two different ZrO2 phases with tetragonal (S.G.: P42/nmc) and monoclinic (S.G.: P21/c) symmetries, which were treated with LeBail refinement, whereas the mannitol phases (β and α) were treated with the Rietveld method.

III. RESULTS

1. Calorimetry


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Figure 1. Heating DSC curves (5°C/min) of β-mannitol samples recorded after increasing milling times. Four samples were prepared by milling crystalline β-mannitol powder during increasing times from 0 min to 10 hours and the obtained samples were characterized by DSC. The corresponding DSC curves, displayed on figure 1, are very similar whatever the milling time and show one endothermic peak located at about 165 °C, reflecting the melting of the β and/or α crystalline phase. Both polymorphs have very close melting points and enthalpies of melting: respectively 166.5°C / 53.5 kJ mol−1 for the form β and 166°C / 52.1 kJ mol−1 for the form α, which prevents the formal identification of the crystalline form

19

from this

measurement alone. Changes of the melting temperatures and enthalpies when the milling time is increased are too weak to be assigned either to the conversion from β form to α form 7 ACS Paragon Plus Environment


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or to microstructural modifications (decrease of the crystallite size as usually observed under milling). Moreover, no event characteristic of a glass transition (and thus of the presence of an amorphous fraction) can be detected. In conclusion, DSC traces are not able to provide significant observations to allow understanding the modifications induced in milled crystalline β-mannitol.

2. Real time powder synchotron radiation diffraction During the whole in situ milling experiment, 2100 diffraction patterns have been recorded: 1720 diffraction patterns allow monitoring the 6 hours of mill and 380 more are available to follow the evolution of the powder after the stop of the mill (for about 1h40’). a. In situ milling: phase identification The evolution of the β-mannitol diffraction patterns during in-situ milling can be seen on Figure 2 after normalization and subtraction of the signal from the empty Perspex® milling vessel. A crystal-crystal transformation is clearly observed to take place around pattern #1000, corresponding to about 3h30 of milling time.


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Figure 2. Evolution of the 1720 β-mannitol diffraction patterns recorded at the ID15B-ESRF beam line (λ=0.14259 Å) during in-situ oscillatory milling. A 2D surface plot has been chosen in order to qualitatively show the evolution of the phases. It displays the low 2θ domain, free of the diffraction signal of the mill ball.

The first diffraction pattern and the last one, obtained just before the mill was stopped after 6 hours of in situ milling, have been further analysed and the corresponding Rietveld refinement results are shown figure 3. The first diagram confirms that the powder was pure β-mannitol. In the final Rietveld refinement, the scale factor, cell parameters, instrumental zero point, an overall thermal parameter and a parameter describing broadening effects of particle size origin were refined. The crystal structure was considered as fixed and described with positions taken from reference 23. β-mannitol is still present in the last pattern but in very weak amount (only its scale factor is then refined). The major phase is α-mannitol and the 9 ACS Paragon Plus Environment


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Rietveld refinements were conducted as previously described for the β form by using reference 22 (refinement of the scale factor, cell parameters, instrumental zero point, an overall thermal parameter and size broadening parameter). The instrumental zero point refined to the same -0.006° value as for the first pattern, and this parameter was fixed at this value in the following sequential refinements.


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Figure 3. Rietveld refinement results and corresponding zoom in inserts (experimental pattern: data points, calculated pattern: grey full line, difference: black full line): top) at the start of the mill. Tick marks are from top to bottom: β-mannitol20,23, ZrO2 tetragonal phase, ZrO2 monoclinic phase, Rwp =5.61 % Rp =3.82 %, bottom) after 6 hours of in-situ milling. Tick marks are from top to bottom: β-mannitol20,23, α-mannitol20;22, ZrO2 tetragonal phase, ZrO2 monoclinic phase, Rwp = 5.29 %.

The solid-state transformation can be clearly visualized through the inspection of the two Bragg peaks located at 1.25° and 1.35° and respectively attributed to α form and β form (figure 4): the intensity of the first peak increases while that of the second peak decreases, which means that α form appears while β form disappears. It is difficult to precisely determine the time when the polymorph α appeared (because it is first present as an ultraminor phase) but this phase is significantly present from the pattern 500, corresponding to a time t ≈ 1h44.



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Figure 4. Transformation of the β-mannitol form (Bragg peak at 1.35°) toward α form (Bragg

peak at 1.25°) induced by in situ oscillatory milling

Moreover, no noticeable evolution of the cell parameters can be observed during the whole experiment, i.e. the Bragg peak positions of the mannitol phases remain unchanged within their whole domain of existence. Broadening of the Bragg peaks does not evolve either, except during the first minutes of milling for the initial β form. This broadening has been quantitatively analysed by using sequential Rietveld refinements (see § c, figure 6). Lastly, no significant evolution of the diffuse scattering halo is observed: this signal has been attributed to self-scattering from the mannitol compound because it is identical for all patterns whatever 12 ACS Paragon Plus Environment

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the crystalline phase. A similar signal has been already observed for molecular organic compounds studied under the same conditions 5.

b. Evolution at rest after the stop of mill

Once the mill stopped, the collection of powder diffraction patterns has been extended for 1h40’. They are displayed on Figure 5. No sign of reconversion of the form α toward the stable crystalline form β, nor modification of position or width of Bragg reflexions can be observed until the end of the experiment.

Figure 5. Evolution of the 380 α-mannitol (obtained by milling) diffraction patterns recorded at the ID15B-ESRF beam line after the end of the milling. 13 ACS Paragon Plus Environment


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c. Sequential Rietveld refinements: evolution of phase proportion and coherent domain size

Despite the strong contamination from the ZrO2 ball diffraction signal, the scale factor and the size broadening parameter of the β-mannitol and α-mannitol phase fractions could be followed throughout the 1720 patterns representing 6h of milling by sequential Rietveld refinements. After attempting to take into account in the refinements the effect of microstrains on the Bragg reflexion width, the size broadening effect was found to dominate and the best results were obtained using a single isotropic size broadening parameter, which was sufficient to satisfactorily fit the patterns. The refinements were carried out between 2θ = 0.5° and 7.5°. The background was described by linear interpolation of a set of 2-theta points with refined intensities. The instrumental zero point was fixed at -0.006° and the overall temperature factors at the values obtained for β-mannitol and α-mannitol by refining respectively the first diagram and the last one. The starting structure models were adopted from literature 23,22. The scale factor, the lattice parameters and the isotropic size broadening parameters were refined for each phase only for data sets where the corresponding phase represents to more than 20 wt % of the sample. Otherwise, these last parameters were kept fixed for the minor phase and only its scale factor was refined. Figure 6 represents the evolution of the phase fractions, obtained from the variations of the scale factors, and coherent domain sizes of the phases, obtained from the isotropic size broadening parameters, as function of milling time.


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Figure 6. Top): Crystalline phase proportion and coherent domain size of β-mannitol and αmannitol obtained by sequential Rietveld refinement for in-situ oscillatory milling at ID15B 15 ACS Paragon Plus Environment


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(ESRF, Grenoble). Domain size is shown only in the range where the isotropic size broadening parameters have been refined. Error bars (black vertical bars) are values of standard deviations corresponding to the numerical refinement uncertainties given by the FP_Suite. Bottom): Corresponding zoom for the first 80 minutes.

IV. DISCUSSION a. Experimental kinetics

These diffraction data, collected in real time during the operation of the mill, make it possible to analyze in depth the different steps of the microstructural and structural transformations of crystalline β-mannitol induced by milling. From these results, it can be deducted: 1) A polymorphic transformation has occurred from the form β toward the form α, as already observed by ex-situ experiments 24 and as expected. Indeed, previous characterization by DSC of amorphous mannitol, obtained by thermal quench of the liquid phase directly into liquid nitrogen (in order to avoid the very quick recrystallization of the liquid), has allowed to determine a glass transition located at Tg = 13°C

25,24,9

. Hence, milling the mannitol at room

temperature corresponds to a milling at about 10°C above its Tg, a situation which favors polymorphic transformations 3. During this solid-solid transformation, the α phase proportion evolves in three steps as function of milling time (figure 6): i) from t = 0 min to t = 15 min, the α phase forms in low concentration until it reaches about 2.5 wt %. ii) from t = 15 min to t = 60 min, the α phase proportion remains stable (at 2.5 wt %) iii) for t > 60 min, the α phase proportion increases rapidly so that the transformation is almost completed at t = 360 min (98 wt % of α phase). These three steps give the overall kinetic a sigmoidal shape.


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2) Little microstructural evolution occurs during the solid-state transformation. The microstructure of the β phase evolves only during the first 15 minutes of milling, with a decrease of its coherent domain size from 6000 Å to about 500 Å. This value corresponds to the coherent domain size of the formed α phase and it is worth noting that it then keeps constant whatever the nature of the phase and the transformed fraction.

3) No intermediate amorphous phase, preceding the transformation toward the metastable crystalline α phase, is detected. Recent work

30

reported similar results, suggesting there was

no amorphous regions and a very small amount of α mannitol present in powders milled for short periods of times (15 min and 30 min). In order to investigate further this issue, we have looked carefully at the evolution of the sum of the scale factors (provided by the sequential Rietveld refinements) as function of milling time. Indeed, because this sum is not required to be equal to 1 (unlike the sum of the phase fractions), its decrease may reveal the presence of an amorphous (or ill-ordered) phase not taken into account in the Rietveld refinement, where only crystalline phases are considered. Here, it remains constant within experimental precision all along the milling (figure 7), which means that the total amount of crystalline α and β phases remains constant and no amorphous phase formed at their expense is detected. We can conclude that no intermediate amorphous phase is formed upon milling or in a content so low that it could not be detected in our experiment estimated as < 1% from the refinement results.



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Figure 7. Evolution of the sum of the scale factors (provided by the sequential Rietveld

refinements) as function of milling time. Our study indicates that most of the conversion β (the thermodynamically stable form) → α (the metastable form) occurs without significant microstructural modifications and that the coherent domain size in the β phase is equal to ~500 Å. It is quite remarkable that the coherent domain size of the β phase remains constant throughout the transformation and the α phase also directly appears with the same size. This strongly suggests that the mechanism of transformation does not require a drastic reconstruction which would unavoidably lead to a loss of structural coherence and to microstructural reorganization. This may be understood by observing that the crystal structures of the polymorphs α and β share strong similarities. Indeed, the two polymorphs crystallize in the same orthorhombic space-group P212121 with four molecules in the unit cell of dimensions a = 5.5381(10), b= 8.580(2) and c = 16.795(5) Å (V = 798.0 Å3) for the β form 20,23 and a = 4.8653(10), b= 8.873(2) and c = 18.739(3) Å (V =


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809.0 Å3) for the α form

20,22

. Moreover, they present nearly identical molecular

conformations (figure 8a) (only differing by the OH groups orientations), with similar hydrogen-bonding schemes, described in details by Kim et al., 1968

22

(figures 8b and 8c).

Both structures have the same number of hydrogen bonds per molecule and each hydroxyl group is involved in two bonds. In both unit cells, there is an infinite chain or spiral and a closed circuit of hydrogen bonds, which repeat after four links. The infinite chains involve only the oxygen atoms O1 and O2 at one end of the molecule, while the remainder form the closed circuits in the sequence O6, O5, O4, O3 (the donor direction of the closed circuit is reversed in the two structures.) As visible from figure 8b and c, the β → α transformation can be almost entirely described as a rotation of the molecules around the b-axis. This is accompanied by a lengthening of the c-parameter by 11.6% and a shortening of the aparameter by 12.1%, while the b-parameter remains almost unchanged (3.4% increase). Otherwise, the global topology of the structure remains unchanged, which explains why both compounds share the same space group symmetry. Hence, the conversion β → α can be viewed as a displacive transformation

14

where the crystal lattice is deformed but not broken

and which does not require major modifications at the molecular scale. This is also reflected in the results obtained by Burger et al., 1999

19

and Yu et al., 2005 31, which concluded that

only small energetic differences could be found between form β and α forms, as also manifested in the small differences between enthalpies of fusion and entropies of fusion (1.4 kJ mol−1 and 2 J mol−1K-1 respectively, which makes it possible to estimate the free-energy difference at 0.5 kJ mol−1). In the case where a solid state transformation by milling takes place from a

thermodynamically stable to a metastable form, the inversion of stability is found to occur below a critical coherent domain size. This can be qualitatively understood by considering that the forces exerted on the crystal structure result from the mill ball impacts (that can be considered as constant) and the effects of microstructure (surface forces, etc…) which



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increase when the size decreases. The effect of milling is first to reduce the size until it reaches a critical value where the force due to the impacts is strong enough to induce the phase transformation. With further processing, the whole milling energy is utilized for the phase transformation and the size remains constant. In the case of mannitol, given the small energetic difference between the two polymorphs, the stability inversion is achieved for a coherent domain size as large as 500 Å, a rather high value compared to those found in literature for other materials (for example: between 100 and 200 Å in metallic systems and 200 Å in sorbitol 8). As discussed for sorbitol, the polymorphic transformation in mannitol occurs before the spontaneous amorphization of the crystallites because it is triggered for a higher crystallite size.

(a)


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(b)

(c) Figure 8. (a) Overlay of the molecules from both forms: in black, mannitol form β and in grey, mannitol form α. Structural comparison of (b) form β and (c) form α viewed down the b axis, illustrating the similarity of the hydrogen-bonding schemes. Only the oxygen 21 ACS Paragon Plus Environment


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atoms are labelled for clarity.

b. Modelling of the transformation kinetics

Let us now attempt to a more difficult task: the modelling of the solid-solid transformation kinetics upon milling. Very few papers deal with this subject in the case of molecular solids because it may be difficult to quantify the transformed sample fraction and even more to determine to what extent the sample has evolved in the time lapse between milling and the actual characterization experiment – or even during the latter – and what does this evolution consist in. In this context, in situ characterization experiments such as the one presented here may be the key to the understanding of the complex mechanism at play during solid-state phase transformation in molecular solids. In order to model as precisely as possible the observed solid-state transformation kinetics, we have used two different models, which are presented below.

1) Model 1 developped by Delogu and Cocco 32,33

In a recent study performed in the same experimental conditions, we investigated the amorphization of trehalose under milling 5. Contrarily to the S-shape observed for the solid state transformation of mannitol (figure 6, top) the amorphization kinetics was found to follow an exponential evolution (see figure 7 in Bordet et al., 2016 5). Moreover, the appearance of a small amount of transformed phase followed by a plateau (figure 6, bottom) was not observed for trehalose. In order to explain the direct conversion of β-mannitol into αmannitol without strong modifications of the microstructure or formation of an intermediate amorphous phase, we have tried to use a model which takes into account the specificity of the milling process, consisting in the repetition of shocks. For this, we first used the approach proposed by Delogu and Cocco

32,33

. Although this model was developed in the context of 22

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metallurgy, where structural and elastic properties are different and Tg’s have much higher values than observed for molecular compounds, it does not contain any explicit reference to a given mechanism of phase change and allows observing transitions with exponential or sigmoidal shapes. It is thus interesting to test this model and check its relevance for the case of molecular materials. This approach assumes that, during the ball milling treatment, the mechanical energy is transferred to the sample when grains are trapped between colliding milling tools. Under such processing conditions, the number of collisions experienced by the amount of sample trapped (or “loaded”) at each collision is the quantity to which refer the kinetics of any given transformation. Moreover, by assuming that the formation of the final (amorphous or polymorphic) phase requires that the loaded sample experiences critical loading conditions (namely a mechanical stress higher than a critical threshold value) at least j times, Delogu and Cocco have developed a mathematical formalism allowing to calculate the total volume fraction of the final phase at the nth collision. In this work, we applied the Delogu and Cocco model to the case of mannitol by choosing as the origin of time t= 15 min of milling. Before, we assume that the collisions are responsible of grain reduction in the βphase and that the low amount of α-phase formed is due to the conversion of coherent domains of size smaller than 500 Å already present in the initial powder. A satisfying agreement is found between the experimental kinetics and a model for which 15 critical loading events are needed to induce the transformation process (figure 9). Discrepancies between the experimental and calculated curves may be attributed to the fact that the model was developed for metallic systems and it does not take into account possible reverse structural transformations between two impacts. Nevertheless, the model correctly accounts for the conversion trends observed as a function of milling time, being the existence of a time lapse before the transformation process sets in and the S-shape evolution of the transformation kinetics. This model is also able to describe the differences between the two systems



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investigated by in situ milling in the same conditions: 50 % of β-trehalose is amorphized after 30min of milling whereas 3h30 is needed to transform 50% of β mannitol into α mannitol, meaning that amorphization of trehalose requires less of critical loading events than the polymorphic transformation of mannitol and proceeds with an exponential kinetics.

Figure 9. Evolution of the transformed fraction f as a function of milling time. Each kinetic has been rescaled by its half completion time t1/2 in order to compare experimental data obtained for mannitol with models calculated for different values of “critical loading events”, j, as proposed by Delogu and Cocco, 2007 33. NB: for the experimental kinetic, the origin of time was chosen at t= 15 min of milling (see text for details).


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However, this model is not entirely satisfactory. The appearance of a small amount of the α form immediately after the beginning of the milling process is not reproduced by the model and must thus be considered here as an extrinsic effect. Moreover, it is difficult to understand how the memory of previous shocks is recorded in the sample, since no change of the microstructure is observed during the transformation. Nevertheless, such memory must exist since it is the accumulation of successive shocks above some threshold which triggers the transformation. In order to avoid these difficulties, we propose here an alternative mechanism.

2) Model 2 developed in this work

The stationary stage, which is intercalated between two stages of increasing fraction of form α, is probably the most striking feature of the kinetic. In our new scenario, we consider this stage to correspond to a creation / annihilation process of form α due to a conversion β → α which would perfectly counter balance the conversion α → β. To test this possibility, we have developed a simple phenomenological model where each crystallite of the initial powder undergoes several (β → α; α → β) transformation cycles during the milling. Moreover, the phase transformations are made strongly dependent of the phase state of the surrounding crystallites. Representing a new approach with respect to the previous model, both features are justified by the very small structural and energetic differences between the two polymorphs. In this model, we consider an initial powder sample made of N grains which are themselves made of n3 crystallites in the form β. The crystallites inside a grain are arrayed on a n x n x n cubic lattice. The position of crystallites inside the grains does not change during milling so that a crystallite is in contact with the same six neighbors during all the simulation. This assumption is supported by the fact that the sample microstructure is not affected by the phase transformation. At each impact, one crystallite in one grain undergoes a phase transformation 25 ACS Paragon Plus Environment


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before the next impact. At this stage, the question of whether the transformation is direct or proceeds through an amorphization /recrystallization process is not relevant. The key feature of the model is that the transformation of one crystallite leads to a structural state which strongly depends on the structural state of the six neighboring crystallites. More precisely, the nature of the transformation is driven by the number (nα) of neighboring crystallites in the structural state α. nα≥3 will favor the transformation toward the form α while nα

Structural transformations of D-mannitol induced by in situ milling

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