Diversity Amidst Similarity: A Multidisciplinary Approach to Phase

Perhaps the best-known example of polymorphism among the proteins is lyzozyme. Here one finds at least six different space groups with variations intr...
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CRYSTAL GROWTH & DESIGN 2004 VOL. 4, NO. 6 1419-1429

Review Diversity Amidst Similarity: A Multidisciplinary Approach to Phase Relationships, Solvates, and Polymorphs Frank H. Herbstein* Department of Chemistry, Technion-Israel Institute of Technology, Haifa, Israel 32000 Received July 1, 2004

ABSTRACT: Description of the red and yellow polymorphs of thallium picrate provides the introduction to this review paper that attempts to set its subject within the framework of classical physical chemistry without losing the connection to the burgeoning activity in academic and industrial laboratories based on more limited sources of information. After a formal definition of polymorphism and a thrust at current trends in nomenclature, the relation of hydrates (solvates) and polymorphs is illustrated in terms of the binary sodium sulfate-water phase diagram, with high-throughput crystallization techniques seen as an important auxiliary when complete phase diagrams are not available. The fundamental thermodynamics is introduced through the polymorphism of tin and, to a lesser degree, adamantane, and this is then connected to energy-temperature diagrams. The polymorphism of the elements (about one-third do not have ambient-pressure polymorphs) is then surveyed, with arrangements of spheres having a predominant (but not exclusive) role. The structural variety found among the elements is a qualitative (but not quantitative) reflection of that in the overall population of crystalline materials. This is followed by a brief look at the polymorphism of inorganics, organics, and biomaterials. First-order enantiotropic phase transformations relate many polymorph pairs, and their mechanism is discussed in terms of Mnyukh’s nucleation and growth approach. We close by suggesting that monotropic systems, very important in practice, deserve more attention than they have received. The review is (mostly) restricted to systems at ambient pressure. Introduction Studies of polymorphism and associated phenomena have never been so popular.1-4 This is not only because the scientific problems, both experimental and conceptual, are fascinating but also because of a realization that different polymorphs have different properties, favorable or not in a particular context, and that this has made the subject of considerable practical, and hence financial, importance, especially in the pharmaceutical industry. This review is based on a considerably revised version of the introductory lecture that I gave at the 2004 Erice Summer School.4 Despite the change of medium, I have tried to retain some of the features of a lecture. I summarize the basic issues, as I see them, in the experimental study of polymorphism with the aim of setting this topic in the broader embrace of classical physical chemistry. The most important modern source on the polymorphism (essentially only of organics) is undoubtedly Bernstein’s book;1 the preferred brief in* E-mail: [email protected].

troduction is still McCrone’s classic article,5 full of wisdom and pithy one-liners. In my choice of references, I have deliberately favored the classical and well established. I introduce the subject rather informally through a description of the polymorphism of thallium picrate, followed by a formal definition of “polymorphism” and a consideration of nomenclature. The physical chemistry aspects are then exemplified by the temperature-composition phase diagram of the binary sodium sulfate-water system, including as it does both hydrates and polymorphs. Having dealt with the relations among solvates and polymorphs from solutions, I next proceed to view polymorphism as a phenomenon, with two aspectssI leave consideration of the relative stabilities of the two phases to others6,7 but consider briefly the modes of transformations between the phases. The experimental information needed for a full description of a polymorphic system is then set out in terms of the very few systems where our knowledge approaches the desired level. Included here is a discussion of the polymorphism of tin, which reviews the thermodynamic aspects, and

10.1021/cg030081l CCC: $27.50 © 2004 American Chemical Society Published on Web 10/15/2004

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references to “adamantane as a polymorphic system”, which has been discussed elsewhere;8,9 in my view adamantane is the most completely studied of the polymorphic systems. These two systems also illustrate an important practical featuresthe two polymorphs of tin can be prepared separately (these are “concomitant polymorphs” in the nomenclature of Bernstein, Davey & Henck10), and hence the properties of each can be measured separately, whereas the high-temperature polymorph of adamantane is not stable below 208 K, nor the low-temperature polymorph above this temperature; these could be called “sequential polymorphs”. A survey of polymorphism among the elements, inorganics, organics, and biomolecules shows resemblances and differences. Diverse kinds of polymorphic systems are then outlined9 for a series of organic examples, moving from a situation in which the chemical nature of the structural entity hardly changes from one polymorph to the other of a pair to the situation in which there are clear and obvious differences in chemical states of the members of the polymorph pair. (Gavezzotti & Filippini11 suggested the term “cluster” for a group of related polymorphs; the simplest cluster is a pair.) I then return to the classical distinction between enantiotropic and monotropic systems, drawing attention to the views of Mnyukh12 on first-order phase transformations. Finally, we briefly consider the rather neglected topic of monotropic systems. The two variables governing polymorphism are temperature and pressure; unless stated otherwise, the present review is restricted to atmospheric pressure. High pressure polymorphism is a very important and rapidly developing field but requires separate treatment at present, a distinction that is likely to disappear in the future. In addition to reviewing the fundamentals, we provide connections to two methods (energy-temperature diagrams; high throughput crystallization techniques) of limited compass in fundamental terms, but that nevertheless provide routes to vital information not always obtainable by more conventional methods. Polymorphs of Thallium(I) Picrate Tl(I) picrate was first described in 1866 by Kuhlmann and, separately, by Bo¨tiger as an explosive substance crystallizing as yellow needles or plates. Two years later, Cloiseaux & Lamy reported that there was a second, red crystal form. [We follow Bernstein (ref 1; see p 46) in using “[crystal] form”; “modification” has also been used.] Rabe13 in 1901 showed that, at neutral pH, yellow needles appeared first, transforming over a few days in contact with water into red prisms. Rabe also measured the solubilities of the two forms in water and methanol and concluded, “Das Thallopikrat tritt in zwei physikalisch isomeren Formen auf, deren Umwaldungspunkt bei 46 °C liegt.” These results were confirmed by Cohen & Moesveld;14 Cohen15 reported that ∆V () Vyellow - Vred) was about 6% with the transformation temperature at 46 °C. (Cohen used 150 g of each polymorph; as thallium is poisonous and picrate potentially explosive, it is unlikely that such an experiment would be sanctioned by a modern safety committee.) The crystal structures of both polymorphs have been reported;16,17 in both phases there are stacks of anions with inter-

Review

posed cations, but the detailed arrangements differ (for example, the shortest Tl+...O distance is 0.2 Å less in the yellow polymorph than in the red). This pair presents several issues that recur in studies of polymorphism. The crystallization first of the less stable polymorph, followed by its transformation to the stable modification aided by the presence of water, is an example of Ostwald’s Rule of Successive Reactions18 (there is no sensible transformation in the dry state under ambient conditions). The difference between the thermodynamic transformation temperature (from solubility measurements) and that obtained from differential scanning calorimetry is found in many other systems, as is the hysteresis, the yellow modification not reverting to red on cooling. The crystal structures are unmistakably different, but there is only a small “chemical” difference between them. There is not yet a satisfactory explanation for the color difference. Background Fundamentals Formal Definition of Polymorphism. “Polymorphism” comes from the Greek (polus ) many; morph ) shape). The Concise Oxford Dictionary (COD) 1960 defines it as “multiform esp. (Nat. Hist., Biol.) varying in individuals, passing through successive variations. Physical sciences: the occurrence of different crystalline forms of the same chemical material”. Our definition (Box 1) is a minor variation on that of the COD.

Two emendations are required: What is “occurrence”, and how does this depend on the environment (temperature, pressure, fields of varying kinds)? What is “same”? Occurrence answers the question whether the individual polymorphs of a cluster can be “held in one’s hand” under ambient conditions. Bernstein, Davey & Henck10 have discussed many examples as well as the conditions governing the formation of “concomitant polymorphs”. A simple, and classical, example is tin. Gray (diamond) tin is stable below 286 K (see below) and metallic tin above this temperature. The transformation between them is relatively slow, and hence both polymorphs can be obtained in bulk amounts, and properties such as heat capacity and cell dimensions can be measured separately for each over a wide range of temperatures. Thallium picrate also belongs in this group. A counter example is adamantane, which transforms at 208 K to a high-temperature cubic (plastic) phase not sustainable below this temperature, nor is the low-temperature tetragonal phase sustainable above it; this is “sequential polymorphism”. Perylene and pyrene are also example and its counter; the optical properties of both phases of perylene have been measured at very low temperatures19 but not those of pyrene.

Review

What is “same”? Here one has no doubts that adamantane and pyrene are the same chemical entity in both phases, but what of perylene, with structural differences between the polymorphs, and thallium picrate, with small differences in closest metal-oxygen distances or, more seriously, of tin, in which the two polymorphs have different physical properties in accordance with their different crystal structures? My recommendation is to operate a broad and tolerant church, such that both the tin pair and even graphite/diamond can be accommodated within its walls. Melting is the transformation of a crystalline phase into a fluid phase; both are condensed phases. Similarly, polymorphism can be thought of as the transformation of one crystalline phase into a second crystalline phase; again both are condensed phases, but a polymorphic transformation is a more complex process than melting because the daughter phase is neither fluid nor isotropic. Nomenclature. We now turn to the thorny issue of nomenclature (Box 2).

There are two aspects here, first of principle and second of practice. The terms “pseudopolymorphism” and “solvate” have something of a current vogue, especially in the pharmaceutical literature, and even in legal documents. The results of crystallization experiments, which can include (true) polymorphs, hydrates and/or solvates, and amorphous products, are often referred to collectively as pseudopolymorphs. A better blanket term could be “solid forms” (suggested by Professor Joel Bernstein) which should then be differentiated into appropriate subgroups. Pseudopolymorphs is an undesirable term [McCrone5 (p 753) writes “By this term, which is convenient to use here but which should never be allowed to come into general use, is meant a variety of phenomena sometimes confused with polymorphism.”] because it lumps together a number of different types of solids under a single heading, suggesting in misleading fashion a connection to polymorphism. The issues are carefully discussed in a recent paper describing the crystal structures of five new mixed-stack π-π* donor-acceptor molecular compounds of substituted benzenes with sym-trinitrobenzene.20 It is unfortunate that this paper was entitled “Five new pseudopolymorphs of sym-trinitrobenzene” and that the first sentence of the abstract contained the words “TNB was recrystallized....to yield five new solvates” (my italics). It is true that the molecular compounds examined contain solvent in addition to TNB, but their structure (as well as their chemistry) shows that they belong to a structure type known since 1851 and subsequently delineated in detail;21 there is no

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Figure 1. The sodium sulfate-water phase diagram adapted from Glasstone’s Figure 192.26 Temperatures are in degrees Celsius, and the temperature scale is distorted. The polymorphism of anhydrous sodium sulfate is incompletely described in the diagram.

profit in subsuming them in the amorphous phrase “solvate”, which we discuss below. Now what about practice? How do we designate the different members of a cluster of polymorphssA, B, C, ... R, β, γ, ... I, II, III, and in what order? The chaotic situation has been well described (see endnote 35 in ref 10). Many solutions have been suggested (Appendix 1), an early one from McCrone5 (p 736), and through a report of a Committee of the International Union of Crystallography; I have even made a contribution of my own.9,22 The polymorph community generally treats these well-intentioned suggestions with disdain, leaving its practitioners to sort out the confusion for themselves. Crystallization from Binary Systems. Materials are usually prepared from solution, leading to formation, under appropriate conditions, of different crystalline modifications of different compositions. What are these conditions, and what are the different crystalline modifications? For simplicity, we consider a binary (twocomponent) system. The treatment is classical physical chemistry, developed more than 100 years ago. The essential tool is the Phase Rule, due to J. Willard Gibbs and the great names associated with the particular aspects considered here are H. W. B. Roozeboom, F. A. H. Schreinemakers, G. Tammann, J. H. van der Waals, and J. H. van’t Hoff. Among classical standard texts in the general area of the Phase Rule, Phase Diagrams, and Phase Transformations are those by Alexander Findlay (updated by A. N. Campbell and N. O. Smith, 1951)23 and John E. Ricci (1951; Dover edition 1966),24 to which should be added a recent text by Mats Hillert (1998).25 A classical example is the sodium sulfate (Glauber’s salt)-water system. I reproduce (in somewhat abbreviated form and with insertion of comments) the description given in Glasstone’s Textbook of Physical Chemistry26 under the heading of “Hydrates with Incongruent Melting Points” (see pp 773-775 and 786-788). A similar description is given by Findlay.

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Review

A case which has been well studied since GayLussac (1819) first observed the unusual behaviour, is the system sodium sulfate-water, the phase diagram being depicted schematically in (our) Fig. 1 (Glasstone’s Fig. 192). Along curve AB the liquid phase is in equilibrium with ice [this shows the depression of the freezing point of ice because of dissolved sodium sulfate in the aqueous solution], whereas along BC the solid phase is Na2SO4‚10H2O; the latter may thus be regarded as the solubility curve of the decahydrate. Before the curve BC reaches a maximum, however, an incongruent melting point, or transition point, is reached at C, when [ortho]rhombic crystals of anhydrous sodium sulfate commence to separate. At C the decahydrate and [ortho]rhombic anhydrous salts are in equilibrium with saturated solution, and the condensed system is invariant [from the Phase Rule F ) C- P +2, C ) 2 (Na2SO4 and H2O), P ) 3 and so F ) 2 - 3 + 2 ) 1. This degree of freedom is used up by setting the pressure to atmospheric, and hence the temperature must have a specific value]. This transition temperature is so definite at atmospheric pressure, viz. 32.383 °C, that it has been suggested as a fixed point in thermometry. The curve CD gives the compositions of solutions in equilibrium with [ortho]rhombic anhydrous salt, and is the solubility curve of this substance. [Glasstone next discusses the behavior of the system at higher temperatures and pressures, but this is omitted because it is not relevant here; however, one should note that the curve CD initially has a negative slopesin other words, the solubility of the anhydrous orthorhombic crystals decreases with increasing temperature. This applies even more strikingly for the monoclinic polymorph]. If a solution saturated with respect to anhydrous [ortho]rhombic sodium sulfate is cooled along DC, then when the transition point C is reached the decahydrate should commence to deposit; sometimes, however, this does not happen, the anhydrous form remains metastable, and the composition of the solution continues along DC beyond C until F is reached, at 23.465(4) °C, when the heptahydrate Na2SO4‚7H2O separates. This substance is always metastable in the presence of the decahydrate, and it can only be formed if Na2SO4‚10H2O is completely absent. In these circumstances, the metastable curves BG and GF, the latter being the solubility curve of the heptahydrate, can be realized, G being a metastable eutectic. The point F is the transition at which heptahydrate and anhydrous salt are in metastable equilibrium with solution [note the analogy with point C]. The fact that the metastable solubility curves GF and FC are found to the right of the stable curve BC shows that at a given temperature the metastable forms are more soluble than a stable form. The addition of a small crystal of decahydrate [a seed] will cause the excess salt to be precipitated in that form, and the composition will fall onto the line BC.

...that a metastable form is more soluble than a stable form represents a rule of universal applicability, which can be derived from thermodynamic considerations. The metastable form can be converted into the stable form as a result of a spontaneous change, but the reverse never happens [compare with Ostwald’s rule]. [Now follows a discussion of the chemical potentials, which we omit as too detailed for this review]. In addition to the curves BG, GF and FC, it is possible, on account of suspended transformation, for the solubility curve BC of the decahydrate to be continued into the metastable region CH, and that of the heptahydrate along FJ. In each case the metastable solutions are supersaturated with respect to the stable solutions at the same temperature. The dependence of the solubility on the nature of the solid phase in contact with the solution is evident, and hence in stating solubility data it is important always to define the exact nature of the salt with which the solution is saturated. The principal general conclusions from Glasstone’s treatment are summarized in Box 3.

Other systems with a hydrate (or solvate) crystallizing together with two enantiotropically related polymorphs will show a similar phase diagram, the details differing. Among examples of pharmaceutical interest are the ampicillin-water and erythromycin-water systems. Few complete parent molecule-solvent phase diagrams have been determined (perhaps this should be “published”), even for drug systems of immediate commercial importance. There is currently considerable interest in determining at least the solid forms (polymorphs, hydrates, solvate, amorphous solids) that occur in such systems. This is being done by “high-throughput crystallization” techniques requiring specially developed computer-controlled assemblies for producing and analyzing large numbers of samples while keeping the overall amount of material within sustainable limits. For example, in a recent high-throughput crystallization study of Ritonavir, a total of five forms was found, including both known polymorphs and three forms not encountered previously. There were over 2000 crystallization attempts.27 Solvates. Solvates are crystalline materials containing a parent component (the solute, such as a drug molecule) and molecules of the solvent from which the solvate is crystallized. The important point here is that

Review

Figure 2. Thermodynamics of the tin polymorphs (grey ) diamond ) R; white ) metallic ) β). Reproduced with permission from ref 28. Copyright 1951 Institute of Metals.

both components are part of the same crystal structure; this differs from the situation in which solvent is located (zeolitically) in cracks or on internal surfaces of crystals of the parent material. Solvates are a subset of the more general class of “host-guest intermolecular complexes”, where the “host” is the parent component and the “guest” is not necessarily the solvent. In host-guest inclusion complexes, the host molecules are (almost) invariably ordered with the often-disordered guest molecules located among the host molecules in cavities, channels (“tunnel” is a more correct but less popular term) or sheets. “Disorder” in this context means that the guest in one unit cell will have one position and/or orientation and a different position and/or orientation in the next, with the numbers of both positions and orientations limited. Crystal structure analysis produces a model of the arrangement in the crystal averaged over space and time, and so we see a superposition of positions and/or orientations and an averaging of motions. When the solvent is water, it is customary to refer to the “solvate” as a “hydrate”; the amount of molecular water compared to one molecule of host is denoted by terms such as “hemihydrate”, “monohydrate”, “dihydrate”, etc. “Anhydrate” is actually a redundant term (a pleonasm) but is used to emphasize that water (or solvent) is not incorporated in the crystal structure. Water molecules are generally (not always) hydrogen bonded to the host. “Solvent” molecules are sometimes hydrogen bonded to the host framework, and sometimes not; sometimes they occupy ordered positions and sometimes not. When the solvent molecules are only loosely linked to the framework structure, the amount of solvent may not reach its maximal value (such as the 1:1 composition often assumed) and solvent may be lost to the atmosphere on standing even at room temperature, leading to decomposition of the solvate.

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In broad terms, there are two crystallographic possibilities. If the crystal structure of the solvate is the same as that of the anhydrous (nonsolvated) parent material, then one has a “primary solid solution” of solvent in host material. Alternatively, the solvate can have a crystal structure different from that of the anhydrous parent. The solvent can be ordered or disordered as noted above. The crystal structure determines the nomenclature, not the other way around. The crystal structure of the solvate depends on the nature of host and guest; there are so many possibilities that generalization does not seem possible. One could argue that any binary adduct in which one of the components is liquid under ambient conditions is a “solvate” but such a broad definition reduces its usefulness to homeopathic dimensions. Thermodynamics of Tin in its Diamond and Metallic Phases. A classical example is provided by tin, for which most of the required experimental measurements have been reported.8 In particular, the specific heats of the two polymorphs have been measured (on separate samples) over the temperature range 0-300 K. The enthalpy function and the entropy are given by

H(T) - H(0) )

∫0T CpdT and S(T) ) ∫0TCpd ln T

are calculated from the Cp - T and Cp - ln T curves. The value of ∆G for the reaction gray tin w white tin is positive at 0 K and is reduced by T∆S as the temperature rises until it reaches zero at Tc ) 286 K. The individual values of ∆H and T∆S are more informative than ∆G. To a first approximation, ∆H does not vary with temperature, and its positive value shows that the bonding in diamond tin is stronger than that in metallic tin. T∆S is linear with T and positive, showing that ∆S is also constant to a first approximation. Here the entropy of metallic tin is greater than that of diamond tin, presumably due to the larger thermal vibrations in the former. It is the increase in temperature that gives rise to the phase transition. Polymorphic clusters should always be described by a similar group of curves showing the balance between enthalpy and entropy factors as a function of temperature, but the required information is generally lacking. Adamantane is perhaps the only other polymorphic cluster studied to a similar level.8,9 When calorimetric measurements are not available (the usual situation), energy-temperature (E/T) diagrams, introduced by M. J. Buerger,29 provide a useful, if partial, substitute. An E/T diagram describes the temperature dependence of the Gibbs free energy and enthalpy of the two phases in the polymorphic system; solubility and melting point data are used for their construction.30 The Clapeyron-Clausius EquationsRelating Measured Transformation Parameters. There are a number of parameters of a polymorphic transformation that can be cross-checked through the ClapeyronClausius [due to B. P. E. Clapeyron (1834) and extended by R. Clausius (1850)] equation dP/dT ) ∆H/(Tc ∆V). dP/dT is obtained from the phase diagram. Tc is often obtained by direct measurement from a DSC trace, but misleading values can be obtained. We illustrate through

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the example of N-anilinophthalimide.31,32 The solubilities of the two phases were measured in a number of solvents as a function of temperature31 and show that Tc ) 283 K. DSC traces32 showed an endotherm on heating at 401 K (∆Htr ) 1.62 kJ/mol) and melting at 457 K (∆Hfus ) 26.9 kJ/mol). ∆V is obtained from cell dimensions measured as a function of temperature for both polymorphs. Cell dimensions measured under standard conditions are not particularly precise and ∆V is the difference between two large numbers. Its standard uncertainty is correspondingly large. As the P-T phase diagram has not been determined for N-anilinophthalimide, we cannot cross-check via the Clapeyron-Clausius equation but this can be done for adamantane, where the measured (phase diagram) value of dP/ dT is 51 bar/K to be compared with the value calculated from ∆H, Tc, and ∆V as 43.5 bar/K. When the complete P-T phase diagram is lacking, the value of (dP/dT) at Tc can be calculated from the more accessible parameters ∆H, ∆V, and Tc. This has been done for perylene and pyrene.22

Polymorphism of Materials We briefly survey the polymorphism of different types of material under the more-or-less traditional divisions of elements, inorganics, organics, and biomaterials (proteins and nucleic acid oligomers); however, a more logical classification would be structure based. In very broad terms, this would give a division of the field into “molecular” and “network” areas, leading to a comparison of the polymorphism of octasulfur and aromatic hydrocarbons in the molecular category, while the wurtzite-zinc blende pair would go together with R- and β-quartz, and the ice polymorphs into the network category. We have not followed this route here because, although polymorphism of organics has been recently and comprehensively surveyed by Bernstein,1 there does not appear to be a comparable discussion of inorganics; furthermore, studies of polymorphism of biomaterials are still in swaddling clothes. I believe that the correct way to approach the structural behavior of the solid state is in terms (for unary systems) of the two-dimensional P-T phase diagram. This should produce a considerable economy in the discussion of the relations between analogous phases of different materials (consider as an example carbon, silicon, germanium, and tin, all with the diamond structure). Higher pressures lead us into a new world,33

Review

fascinating and important but not for us heresone has to set boundaries. Polymorphism of the Elements. The term “allotropy” is sometimes used: the Concise Oxford Dictionary (1960) defines it as followssGreek. Allo other; allotropy is “[the] variation of physical properties without change of substance”. There do not seem to be strong reasons for keeping a second name, a view also held by McCrone5 (p 728). The hundred-odd crystalline elements (the data come from Elmsley34) compose a microcosm of structural types that is remarkably representative of the macrocosm of crystalline substancessthere are structures based on the packing arrangements of individual atoms and of molecules, and a variety of network types. Although the resemblance between sample (elements) and population (all crystalline materials) is qualitative and not quantitative, the structural variety of the elements can be condensed into a number of cohorts to give an illuminating picture that can be carried over to a discussion of the population of all crystalline materials. Cohort I. The elements for which polymorphism at ambient pressure has not been reported. Group 1: K, Rb, Cs; Group 2: Mg, Ba; Group 5: V, Nb, Ta; Group 6: Cr, Mo, W; Group 7: Re; Group 8: Ru, Os; Group 9: Rh, Ir; Group 10: Ni, Pd, Pt; Group 11: Cu, Ag, Au; Group 12: Zn, Cd; Group 13: Al, In; Group 14: C, Si, Ge, Pb; Group 15: Bi; Group 16: Te; Group 17: Br2, I2; Group 18: He, Ar, Kr, Xe. (C and He are special cases). Thus, about one-third of the elements (37 in the list above) do not show polymorphism at ambient pressure. Why? I do not know whether the question has ever been posed. Remember Conan Doyle’s The Hound of the Baskervilles? Sherlock Holmes asked why the dogs did not bark. Cohort II. The elements where polymorphism involves interconversion of the two simplest sphere packings fcc f hcp (or the inverse process). Transformation temperatures (K) are given in brackets. Group 1: Li (LT), Na (5); Group 2: Sr (506); Group 9: Co (690); Group 15: N2 (35); Group 18: Ne (3). Actinoids: Am hcp f fcc (1347); Cm hcp f fcc (1550); Bk hcp f fcc (1523); Cf hcp f fcc (1173); Es hcp f fcc (1133). Cohort III. The elements where polymorphism involves conversion from coordination number 12 (the two simplest sphere packings fcc or hcp) to CN 14 (bcc, 8 + 6). A number of related examples have been included here for convenience. Group 2: Be (1123), Ca; Group 3: Sc (1223); Group 4: Ti (1155), Zr (1135), Hf (?, 2033); Group 13: Tl (503); Group 18: Ne (3). Lanthanoids: La hcp f fcc (583) fbcc (1137); Ce fcc, hcp f fcc (441), bcc; Pr hcp f bcc (1065); Nd hcp f bcc (1135); Sm rhombohedral f bcc (1190); Gd hcp f bcc (1535); Tb hcp f orthorhombic (200); hcp f bcc (1590); Dy orthorhombic f hcp (86); Er hcp f bcc (1640); Yb fcc f bcc (1073). Actinoids: Th fcc f bcc (1673); Pa tetragonal f bcc (1440). Cohort IV. The “molecular” elements. Group 16: O2 monoclinic f rhombohedral (23.8) f cubic (43.8); S8 orthorhombic f monoclinic (369); Se8 monoclinic f monoclinic. Group 17: F2 monoclinic f cubic (45.6); Cl2 tetragonal f orthorhombic (100).

Review

Cohort V. Miscellaneous examples of polymorphism, some of considerable complexity Group 7: Mn bcc f tetragonal (973) f fcc (1352) f bcc (1413). Group 8: Fe bcc f fcc (1183) f bcc (1663). Group 12: Hg tetragonal f rhombohedral (79). Group 13: B tetragonal, rhombohedral, rhombohedral, orthorhombic, monoclinic (or trigonal), hexagonal; Ga orthorhombic, orthorhombic, orthorhombic. Group 14: Sn diamond f metallic (286.4). Group 15: P4 cubic (white), cubic (red), monoclinic (purple), orthorhombic (black); As rhombohedral (metallic) f hexagonal (yellow) (501); Sb rhombohedral (grey), cubic (grey), hexagonal (metallic). Actinoids: U orthorhombic f tetragonal (941) f bcc (1047); Np orthorhombic f tetragonal (551) f bcc (850); Pu monoclinic f monoclinic (395) f orthorhombic (473) f fcc (583) f tetragonal (725) f bcc (753). Transformation of one arrangement of close packed layers of spheres to the other arrangement (i.e., fcc f hcp or conversely) is found in 10 elements, of which half come from the actinoid group. Transformation of CN 12 (i.e., one of the close packings fcc or hcp) to bcc (coordination 8 + 6) is found for some 20 elements and is thus the most common polymorphic transformation among the elements. The sphere packing data for all the elements are consolidated in Table 1. Transformation is obviously entropy-driven, and one explanation is that the bcc structure (not close packed) has a higher vibrational entropy. A large spread in Tc values for similar types of transformation is found in Cohorts II and III. If one makes the approximation that ∆H for various polymorph pairs are roughly similar, then the spread in the values of Tc derives from a spread in entropy differences. As Tc ) ∆H/∆S then, for given ∆H, the smaller ∆S the higher Tc. The elements having the largest numbers of polymorphs are B and Pu (both six); it is also intriguing that U and Np have the same sequence of polymorphs. Polymorphism of Inorganics. Here I shall only mention, without details, two examples. One is NaCl, which transforms to the CsCl structure at 613 K. The other is titanium oxide TiO2, which exists in the three modifications rutile, brookite, and anatase. These are described by Evans35 as follows: in rutile each Ti ion is octahedrally coordinated by oxygen, but the coordinating octahedra are not perfectly regular because the two shared edges are slightly shorter than the remaining ten. In brookite and anatase, the coordination about Ti is again octahedral ...but the structures differ in the way in which the octahedra are united: in brookite three and in anatase four of the twelve edges are shared. [Following Pauling’s Third Rule for ionic structures] we should expect rutile to be the most stable and anatase the least stable of the three modifications. Polymorphism of Organics. There are at least some thousands of examples. Among the polycyclic aromatics, benzene, naphthalene, anthracene, and tetracene have polymorphs only under nonambient conditions. Pyrene has an enantiotropic transformation at 110 K, while perylene has an enantiotropic transformation at about 500 K.22 The polymorphs are concomitant for perylene but sequential for pyrene.

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The different types of polymorphism found among 15 (hopefully representative) organics have been discussed in some detail,9 the guiding principle being the degree of chemical resemblance between the chemical molecules as they occur in the two different polymorphs. At one extreme, the chemical molecules in the two polymorphs of 1,2-4,5-tetrabromobenzene have negligible difference, while at the other extreme the mixed stack structure of red semiconducting {TMTSF-TCNQ} (TMTSF is 4,4′,5,5′-tetramethyl-2,2′-bis-1,3-diselenole; TCNQ is tetracyanoquinodimethane) differs strikingly in physical properties from the black conducting segregated stack structure with the same composition. Pyrene and perylene are intermediate examples; both pyrene polymorphs have face-to-face dimers, while only the E-II(β) high-temperature perylene polymorph has this arrangement. Conformational polymorphs fit next into the intermediate group, followed by polymorphs with different hydrogen-bonding arrangements. These comparisons are essentially qualitative and subjective, but there is growing interest in more quantitative approaches to comparing molecular arrangements in polymorphs of similar chemical type (and other groups). A comprehensive background is given by Dziubek and Katrusiak36 who also compare organic (e.g., resorcinol) and inorganic (e.g., ice, erbium pentaphosphate) polymorph clusters. A parallel development is that of Hirshfeld surfaces by Spackman and co-workers;37,38 here color plots of intermolecular contacts displayed graphically make possible comparison of packing interactions in polymorph clusters. Polymorphism of Proteins. When considering the polymorphism of proteins it becomes essential (not merely desirable) to define “chemical entity” and “same” in some appropriate manner. Perhaps the best-known example of polymorphism among the proteins is lyzozyme. Here one finds at least six different space groups with variations introduced by differences in water content, nature, and amount of anions, and differences in packing arrangements. The situation, summarized by Vaney et al.,39 is nothing if not complicated. A simpler situation is found in jack bean canavalin, where the protein has a single source (Table 2). We quote from Ko et al.40: “Three of the crystals have as asymmetric unit a single subunit of canavalin with Mr ) 47000. The orthorhombic form has the entire trimeric molecule of Mr ) 140000 as its asymmetric unit. All crystals are obtained under similar crystallization conditions.... Hexagonal crystals are obtained with NaCl concentration 2-4% (w/v) and temperature around 277 K. ...Crystals will dissolve and reappear if the temperature exceeds 283 K.... The various polymorphs can interconvert through dissolution and regrowth...with orthorhombic crystals generally the final and presumably the most stable form to appear.” This quotation shows that the authors appreciate the need to consider the thermodynamics (the relative stabilities) of the various polymorphs, but they have only been able to take tentative steps to this end. What are the “chemical entities”? The ratio of the volumes for the orthorhombic and the rhombohedral cells is 2736800/ 44910 ) 6.10; for cubic and hexagonal we find 1191000/ 784049 ) 1.52. However, orthorhombic and cubic give 2736800/1191000 ) 2.30. Thus, we may conclude that

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Table 1. Compilation of Sphere-Packing Transformations for Various Polymorphs of the Elements transformation

Group 1

Group 2

Group 3

Group 4

Close Packed Layers to bcc Be, Ca (fcc, bcc, hcp), Sr Sc, Y, Lu Ti, Zr, Hf (to “cubic”)

hcp to bcc

Na, Tl (503)

fcc to bcc

Li, Fe (1663)

hcp to fcc

Group 18., Ne (3)

lanthanoids

actinoids

Pr, Nd, Sm (rhombohedral to bcc) Gd (1535) Tb Er (1640) Th

Close Packed Layers to Close Packed Layers Am (1347) Cm (1550) Bk (1523) Cf (1173) Es (1133)

fcc to hcp

Group 9 Co

Table 2. Crystallographic Data (Å, deg, Å3) for Four Forms of Jack Bean Canavalin40

a chemical entity of one kind is found in the pair of orthorhombic and rhombohedral cells, while a different chemical entity is found in the pair of hexagonal and

cubic cells. Each of these groupings could be a polymorph pair. Clearly polymorphism among the proteins has much to offer as a research topic (probably including a considerable amount of frustration). The situation regarding polymorphism of nucleic acid oligomers is not very different from that of the proteins. There are many tantalizing hints, but incomplete information makes a survey premature. Relation between Polymorphism and Phase Transformations. The fundamental distinction is between enantiotropic and monotropic systems, but one

Figure 3. P-T phase diagram of sulfur (S8) reproduced from Glasstone’s Figure 103 (ref 26) with permission. Copyright 1947 Van Nostrand. The solid lines represent stable phase boundaries and the broken lines are metastable boundaries. Triple point F (orthorhombic, monoclinic, liquid) is at 151 °C and 1200 atm, while triple point C (monoclinic, liquid, vapor) is at ≈119.5 °C and a low pressure; thus, the lines BF and CF are actually essentially vertical, and C can be taken as the 1 bar stable melting point of the monoclinic phase, while the metastable 1 bar melting point of the orthorhombic phase (E) is at 114.5 °C. The orthorhombic to monoclinic transition point (the datum of principal interest in the present context) is at 95.5 °C.

Figure 4. Putative P-T diagram for a monotropic system. The solid lines represent stable phase boundaries and the broken lines are metastable boundaries. The phase boundaries in the high-pressure region have been corrected from McCrone’s original presentation and now show solid-phase II (“the metastable phase” at low pressure) as appearing as a stable phase in the high-pressure region (shaded). The 1 bar isobar can be taken as a horizontal line in the middle of the diagram and shows the melting behavior of the two phases as described in the text. The stability region of solid-phase I is shown by the region with horizontal lines.

crystal modification

a

orthorhombic 136.5 rhombohedral 83.0 hexagonal cubic

126.35 106.0

b

c

150.3 133.4

β

Vcell

space group

2736800 C2221 111.1 44910 R3 (pseudo R32) 51.64 784049 P63 1191000 P213

Z 8 1 2 4

Review

should remember McCrone’s warning5 (p 734) “...that these two terms, enantiotropic and monotropic, are dangerous words indeed.” As noted earlier, in an enantiotropic system there is at least one phase transformation between 0 K and the melting point. In a monotropic system, the two phases melt (in general at different temperatures) without showing intermediate transformations. “Concomitant polymorphs” may be enantiotropes or monotropes, but “sequential polymorphs” must be enantiotropes. A clear discussion of enantiotropy and monotropy was given by Glasstone26 (see pp 468-471), who pointed out that many real systems, such as phosphorus and silica, show both phenomena. Here we shall illustrate the two facets separately. Enantiotropic Systems. As an illustration, we use the schematic P-T phase diagram of sulfur (S8) taken from Figure 103 of Glasstone.26 [It seems possible (from a comparison of the diagrams) that Figure 2a of McCrone5 (which is the same as Figure 2.7a of Bernstein1) was based on this sulfur phase diagram.] The schematic nature of the diagram arises from the need to accommodate a wide range of pressures and a small range of temperatures in the same diagram. Some examples are given in the caption. The 1 bar isobar will be a horizontal line roughly in the middle of the diagram. Proceeding from left to right one passes from the region where the orthorhombic phase is stable across the transition line to the region where the monoclinic phase is stable and then to melting of the monoclinic phase. At higher pressures only orthorhombic sulfur is stable; it is surmised that the large crystals of orthorhombic sulfur found in nature were formed by cooling of liquid sulfur in the high-pressure, single (solid) phase region. A phase diagram of this kind is essential for the understanding of an enantiotropic system, preferably in conjunction with a binary x-T diagram such as that in Figure 1. Monotropic Polymorphs. These are often obtained as the metastable first crystals from a crystallization from solution (cf. Ostwald’s Rule of Stages); the metastable (designated II) and stable (designated I) phases melt independently, II at a lower temperature than I, and there is no interconversion during heating. What is the appropriate form of the P-T phase diagram, and where does the metastable phase appear in this diagram? A suggestion can be found in Ricci’s “The Phase Rule and Heterogeneous Equilibrium” (1966 Dover edition referenced here, the book having been first published in 1951).24 Ricci draws attention to the resemblance between the P-T phase diagram where a high-pressure phase appears, and that for a monotropic system, but in a somewhat reticent fashions“The highpressure polymorph Sh of Figure 2.12 may conceivably appear as a monotropic form at low pressure (at the vapor pressure of the system), while the monotropic form S’ of Figure 2.15 may appear as a stable form in the condensed system under high P.” We illustrate this by using the phase diagram for a monotropic system given by McCrone5 (his Figure 2b) reproduced by Bernstein1 as his Figure 2.7b. This is a putative diagram, but, as far as I know, an actual diagram for a real substance has not yet been reported. Another quotation from McCrone5 (p 734) is appropriate: ”It is not necessarily true that the melting point

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Figure 5. Optical microscopy (Mnyukh12) of the growth of daughter high-temperature phase from parent low-temperature phase of p-dichlorobenzene (the temperature of transformation is around 300 K). There is no rational orientation relation between daughter and parent, as is shown by the appearance of a number of differently orientated daughter crystals. However, the locus of nucleation is at a fixed point in the crystal, and it is possible to cycle through the transformation many times with the daughter phase returning to the same state each time the temperature is raised above Tc.

and transition curves intersect at high pressures, but the possibility exists....in this case the enantiotropic system becomes a monotropic system at high pressures and vice versa.” (italics added). The first of these situations is shown in McCrone’s5 Figure 2a and the second in our corrected version of his Figure 2b (our Figure 4). First-Order Enantiotropic Phase Transformations. There are two descriptions of the transformation process in an enantiotropic system. Mnyukh12 states that the daughter phase is nucleated at particular imperfections (groups of vacancies) in the parent crystal; homogeneous nucleation is not considered possible. Two situations are distinguished: (1) When the structures of the two phases differ, there is no orientation relation between the two phases (Figure 5); matter is transferred from irrational planes of the parent phase to rational planes of the daughter phase. (2) When there are close resemblances between the structures of the two phases, there is an epitaxial relation between the two phases and the transformation takes place at an interface between them (Figure 6). Mnyukh provides an explanation of the dependence of transformation rate on temperature for both heating and cooling regimens. A somewhat different description was developed by Ubbelohde.41,42 Ubbelohde considers that the low and high temperature phases have their usual structures in temperature ranges far removed from the vicinity of the transformation, but that there is an

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Figure 6. Tetrathiofulvalene-chloranilsthe epitaxial relation at 70 K between the neutral N phase (stable above ≈81 K) and the ionic I phase (stable below ≈81 K) and their coexistence as observed by optical microscopy with unpolarized light. This is adapted from Figure 6 of Buron-Le Cointe et al.43

intermediate region where the system has a hybrid nature, possibly consisting of coexisting domains of the two phases. Some Concluding Thoughts The experimental study of polymorphism rests on two complementary foundationssstructure and thermodynamics. The techniques of crystal structure determination have advanced so dramatically over the past decade that one can be confident that the structure of any new polymorph can be determined even though only micrograms of material may be available. Automation of the experimental measuring process is matched by automated solution of the (crystallographic) Phase Problem; computerized databases are used to place the structural results in their appropriate context. Single studies of the individual phases at, say, room temperature can now be easily extended to studies of the dependence of properties such as cell dimensions and atomic displacement parameters on temperature and even on pressure. Individual crystal structure determinations at small temperature intervals can be a valuable complement to the more usual cell dimension measurements in the study of phase transformations.44 The importance of thermodynamic measurements complementary to structure determinations has been illustrated in this review by three examplessthe binary phase diagram of Figure 1, determined from solubility measurements, the temperature dependence of ∆H and ∆S (Figure 2) derived from heat capacity-temperature measurements and the P-T phase diagrams (Figures 3 and 4). All three types of measurements currently require appreciable amounts of material (but note that heat capacities of {[TTF][TCNQ]} have been measured at low temperatures (≈40 K) using milligram samples45,46), and the measurements themselves are both difficult and tedious; the results, despite their fundamental character, are hardly dramatic. Small wonder that the first two of these techniques are becoming lost arts, unbeloved of the funding agencies. Automation and miniaturization need to be introduced. Remedial programs in these areas should handsomely repay the efforts involved and the funds invested. Structural chemists studying polymorphism could enhance their diffraction studies by including observations under the polarizing microscope of phase transformations and studying them by DSC. For this, only small amounts of material are requiredsthe profit would come from defining particular systems as enan-

Review

tiotropic or monotropic. This is a small but necessary step on the way to a full suite of thermodynamic measurementsssome redress of the imbalance between Dominant Structure and Neglected Interrelationships is overdue. Thus, the coming generation of investigators may be able to mitigate a rather pessimistic conclusion of Richardson, Yang, Novotny-Bregger and Dunitz47 that “the experimental study of polymorphic transitions is very difficult. The phenomena are complex and seldom reproducible, and our understanding of them leaves much to be desired.” Appendix. Nomenclature of Polymorphs The “Report of an IUCr Working Group on Phase Transition Nomenclature” provides background information and a proposal for a “Six-field phase-transition nomenclature”.48 This proposal has the virtue of completeness, but the deficiencies of not distinguishing enantiotropic and monotropic systems, and of using only inorganic substances as illustrative examples. A more compact nomenclature for convenient designation within a text has been proposed.9,22 This revises (and revives) an earlier proposal of McCrone5 (pp 736-737) and incorporates part of the information included in the Working Group proposal. A shortened version is given here: Polymorphs of a compound are designated as follows: Enantiotropic System: Name (of Compound) [EI (