Polymorphism phase rule

The isolation of crystalline products having mixed polymorphic compositions (often referred to as concomitant polymorphism) remains a topic of interest, even though the phase rule predicts that a system at equilibrium consisting two components (solvent + solute) and three phases (solution + Form I + Form II) is uni variant. Hence, for crystallizations performed at a fixed pressure (typically atmospheric) the system becomes nonvariant and genuine equilibrium can exist at only one temperature. Therefore, concomitant products must be obtained under nonequilibrium conditions. Flexibility in molecular conformation was attributed to the concomitant polymorphs of a spirobicyclic dione [34] and of 3-acetylcoumarin [35],... 

Polymorphic structures of molecular crystals are different phases of a particular molecular entity. To understand the formation of those phases and relationships between them we make use of the classic tools of the Phase Rule, and of thermodynamics and kinetics. In this chapter we will review the thermodynamics in the context of its relevance to polymorphism and explore a number of areas in which it has proved useful in understanding the relationship between polymorphs and polymorphic behaviour. This will be followed by a summary of the role of kinetic factors in detecting the growth of polymorphic forms. We will then provide some guidelines for presenting and comparing the structural aspects of different polymorphic structures, with particular emphasis on those that are dominated by hydrogen bonds. 

There are significant flaws in Ostwald s conclusion that led to his rule. For instance, when a crystallization experiment yields only a single form there is the question of whether it contradicts the Rule or whether the material is simply not polymorphic. Moreover, there is no way of answering this question. However, a sufficient number of cases of successively crystallizing polymorphic forms have been observed to warrant considering the principles behind Ostwald s Rule as guidelines for understanding the phenomenon of the successive crystallization of different polymorphic phases. 

Brittain, H. G. (1999c). Application of the phase rule to the characterization of polymorphic systems. In Drugs in the pharmaceutical sciences (ed. J. Sarbrick), Vol. 95 Polymorphism in pharmaceutical solids (ed. H. G. Brittain), pp. 35-72. Marcel Dekker, New York. [30]... 

This difference is further reinforced by application of the phase mle to the equihbrium between two strictly defined polymorphs of a compound or the equihbrium between a compound and a corresponding solvate of that compound. In the former case, there is only one component (in the phase rule sense—the compound). There are two phases (the two polymorphs) and, therefore, there is only one degree of freedom for equihbrium between two polymorphs by application of the phase rule equation... 

Other well-known authors have made similar statements, a. F. Findlay, who authored the classic book The Phase Rule noted in that [polymorphism] is now recognized as a very frequent occurrence indeed. [34] Buerger and Bloom stated, "... polymorphism is an inherent property of the solid state and it fails to appear only under special conditions. [35] Similarly, Sirota in 1982 [Polymorphism is now believed to be characteristic of all substances, its actual non-occurrence arising from the fact that a polymorphic transition lies above the melting point of the substance or in the area of as yet unattainable values of external equilibrium factors or other conditions providing for the transition. [36]... 

EquiUbrium between Solid, Liquid and Vapour. The Triple Point.—From the Phase Rule, F = n + 2 — r, it follows that when oaie component is present in three coexisting phases, the system is invariant. Such a system can exist in stable equilibrium only at one definite temperature and one definite pressure. This definite temperature and pressure at which three phases coexist in equilibrium, as an invariant system, is called a triple point. Although the commonest triple point in a one-component system is the triple point, solid, liquid, vapour (S—L—V), other triple points are also possible when, as in the case of ice, sulphur, and other substances, polymorphic forms occur. Whether or not all the triple points can be experimentally realised will, of course, depend on circumstances. We shall, in the first place, consider the triple point S—L—Y. 

Compared to crystalline materials, the production and handling of amorphous substances are subject to serious complexities. Whereas the formation of crystalline materials can be described in terms of the phase rule, and solid-solid transformations (polymorphism) are well characterised in terms of pressure and temperature, this is not the case for glassy preparations that, in terms of phase behaviour, are classified as unstable . Their apparent stability derives from their very slow relaxations towards equilibrium states. Furthermore, where crystal structures are described by atomic or ionic coordinates in space, that which is not possible for amorphous materials, by definition, lack long-range order. Structurally, therefore, positions and orientations of molecules in a glass can only be described in terms of atomic or molecular distribution functions, which change over time the rates of such changes are defined by time correlation functions (relaxation times). 

Application of the Phase Rule to the Characterization of Polymorphic Systems... 

The following discussion of the phase rule, and its application to systems of polymorphic interest, has primarily been distilled from the several classic accounts published in the first half of this century [2-8]. It may be noted in passing that one of the most serious disagreements in the history of physical chemistry was between the proponents of computational thermodynamics and those interested in the more qualitative phase rule. Ultimately the school of exact calculations prevailed... 

In a discussion of polymorphic systems, one would encounter the vapor and liquid phases of the compound under study as separate phases. In addition, each polymorph would constitute a separate phase. Once the general rule is deduced and stated, the phase rule can be used to deduce the conditions under which these forms can be in an equilibrium condition. 

Another way to state this is that with an increase in the number of phases at equilibrium, the condition of the system must become more defined and less variable. Thus for polymorphic systems where one can encounter additional solid-state phases, the constraints imposed by the phase rule can be exploited to obtain a greater understanding of the equilibria involved. 

Setting aside any consideration of solvate species or considerations of chemical reaction, systems of polymorphic interest consist of only one component. The complete phase diagram of a polymorphic system would provide the boundary conditions for the vapor state, the liquid phase, and for each and every true polymorph possible. From the phase rule, it is concluded that the maximum amount of variance (two degrees of freedom) is only possible when the component is present in a single phase. All systems of one component can therefore be perfectly defined by assigning values to a maximum of two variable factors. However, this bivariant system is not of interest to our discussion. 

When a single component is in equilibrium between two phases, the phase rule predicts that it must be a univariant system exhibiting only one degree of freedom. It is worthwhile to consider several univariant possibilities, since the most complicated phase diagram of a polymorphic system can be broken down into its component univariant systems. The phase rule applies equally to all of these systems, and all need to be understood for the entire phase diagram to be most useful. 

According to the definition of Section LA, each solid-state polymorphic form constitutes a separate phase of the component. The phase rule can be used to predict the conditions under which each form can coexist, either along or in the presence of the liquid or vapor phases. One immediate deduction is that since no stable equilibrium can exist when four phases are simultaneously present, two polymorphic forms cannot be in equilibrium with each other and be in equilibrium with both their solid and vapor phases. When the two crystalline forms (denoted S, and Sj) are in equilibrium with each other, then the two triple points (Sj-Sj-V and S,-S2-L) become exceedingly important. 

Nevertheless, the phase rule is extremely useful for yielding a physical understanding of polymorphic systems and for providing a physical interpretation of phase transformation phenomena. Its greatest power is in its ability to rule out the existence of simultaneous multiple equilibria that violate its fundamental equation, permitting more quantitative investigations to focus on the possible aspects of such systems. 

This book represents an attempt to summarize the major issues pertaining to the pharmaceutical aspects of polymorphism, as well as the effects of solvate formation. The book is subdivided into five main sections, the first consisting of two chapters defining the phenomenon. Chapter 1 presents the theory and origin of polymorphism in solids, and Chapter 2 examines the phase characteristics of polymorphic systems in the systematic manner permitted by the Phase Rule. The second section covers the crystallographic considerations that define different polymorphic and solvate species, with Chapters 3 and 4 providing detailed summaries of the structural aspects. 

A simple classification scheme of solids is given in Fig. 7.1. In order to differentiate between the types of solids, we have to consider the Gibbs phase rule, which is discussed in any physical chemistry textbook. The basic question is whether the solid substance consists of only one chemical entity (component) or more than one. Usually the component is one molecular unit, with only covalent bonded atoms. However, a component can also consist of more constituents if their concentration cannot be varied independently. An example of this is a salt. The hydrochloride salt of a base must be regarded as a one-component system as long as the acid and the base are present in a stoichiometric ratio. A deficiency of hydrochloric acid results in a mixture of the salt and the free base, which behave as two completely different substances (i.e. two different systems). Polymorphic forms, the glassy state, or the melt of the base (or the salt) are considered as different phases within such a system (a phase is defined as the portion of a system that itself is homogeneous in composition but physically distinguishable from other phases). When the base (or salt) is dissolved in a solvent, a new system is obtained this is also tme when a solvent is part of the crystal lattice, as in the case of a solvate. Thus, each solvate represents a different multicomponent system of a compound, whereas, polymorphic forms are different phases. The variables in the solvate are the kind of solvate (hydrate. 

Tricalcium silicate is the most important component of Portland cement clinker. Its share as a rule is overpassing 55% and reactivity with water has the decisive effect on paste hardening. At room temperature CjS is triclinic. The structures of different polymorphic phases of tricalcium silicate was determined by Jeffery [99], The basis was the rhombohedral pseudo-stracture with hexagonal unit cell [99] ... 

All crystals of one given substance, which may exhibit different habits, have identical physical properties. On the other hand, the different polymorphs of a given substance, which may also differ in habit, will exhibit different physical properties density, hardness, melting point, solubility, reactivity, thermal properties, optical and electrical behaviour, etc. Each polymorph constitutes a separate phase of the given substance, in the Gibbs phase rule sense, whereas crystals of different habit constitute the same phase. Polymorphs may transform in the solid state, but crystals of different habit cannot. 

The definition of solubility permits the occurrence of a single solid phase which may be a pure anhydrous compound, a salt hydrate, a non-stoichiometric compound, or a solid mixture (or solid solution, or "mixed crystals"), and may be stable or metastable. As well, any number of solid phases consistent with the requirements of the phase rule may be present. Metastable solid phases are of widespread occurrence, and may appear as polymorphic (or allotropic) forms or crystal solvates whose rate of transition to more stable forms is very slow. Surface heterogeneity may also give rise to metastability, either when one solid precipitates on the surface of auiother, or if the size of the solid particles is sufficiently small that surface effects become important. In either case, the solid is not in stable equilibrium with the solution. See (21) for the modern formulation of the effect of particle size on solubility. The stability of a solid may also be affected by the atmosphere in which the system is equilibrated. 

As different polymorphs of a given substance represent different phases under a given set of experimental conditions, the well-known Gibbs s phase rule (1) is applicable to polymorphic systems... 

C = 1 for polymorphs because there is only one component in polymorphic systems. For a dimorphic system, P = 2 when both the polymorphs are in equilibrium, which means that one of the parameters can be varied. This implies that at constant pressure, the temperature at which the two polymorphs coexist can vary, which is defined as the transition point. The phase rule also implies that, since the system cannot have a negative number of degrees of freedom (F < 1), only a maximum of three polymorphs can coexist in equilibrium, and the set of conditions under which this occurs is defined as the triple point. 

The application of the phase rule to a dimorphic system suggests that, at a given pressure, phase transition from one polymorph to the other may occur by changing the temperature (F = 1). If such a phase transition is reversible, the two polymorphs are said to be enantiotropes and the energy of transition on heating is endothermic. When the phase transition is irreversible, the two polymorphs are termed as monotropes, in which case only one form is stable whatever the temperature, and the transformation of the metastable form to the stable one is exothermic. 

A single compound system exhibiting two solid states is univariant according to the Gibbs phase rule. Thus, at constant pressure, temperature is sufficient to define the solid state. The temperature at which both solid states (polymorphs) exist in equilibrium is called the transition temperature, T,. Essentially, there are two types of polymorphic transformations that can occur (a) reversible enan-tiotropic and (b) irreversible monotropic. The thermodynamic state of polymorphic transitions can be explained in terms of Gibbs free energy, G, which is given by the relation ... 

McCrone WC (1965) In Fox D, Labes MM, Weissemberg A (eds) Polymorphism in physics and chemistry of the organic solid state, vol 2. Interscience, New York, p 726 Findlay AF (1951) The phase rule and its applications, 9th edn. Dover, New York, pp 7-16 Buerger MJ, Bloom MC (1937) Z Kristallogr 96 182-200 Sirota NN (1982) Cryst Res Technol 17 661-691 Kuhnert-Brandstatter M (1965) Pure Appl Chem 10 133-144... 

During precipitate ageing, a gradual transformation of an initially precipitated metastable phase into a final crystalline form often occurs. The metastable phase may be an amorphous precipitate, a polymorph of the final material, a hydrated species or some system-contaminated substance (Mullin, 2001). In 1896, Ostwald promulgated his rule of stages which states that an unstable... 

Based on the reversibility of their phase transformation behavior, polymorphs can easily be classified as being either enantiotropic (interchange reversibly with temperature) or monotropic (irreversible phase transformation). Enantiotropic polymorphs are each characterized by phase stability over well-defined temperature ranges. In the monotropic system, one polymorph will be stable at all temperatures, and the other is only metastable. Ostwald formulated the rule of successive reactions, which states that the phase that will crystallize out of a melt will be the state that can be reached with the minimum loss of free... 

These compounds are almost all polymorphic, and high conductivity is generally restricted to the high-temperature form, usually designated a-. Other phases, designated (5- or 7-, are lower temperature phases that mostly have normal ionic conductivity. The phases (3-CuBr and (3-CuI are exceptions to this general rule. 

Metastable crystalline phases frequently crystallise to a more stable phase in accordance with Ostwald s rule of stages, and the more common types of phase transformation that occur in crystallising and precipitating systems include those between polymorphs and solvates. Transformations can occur in the solid state, particularly at temperatures near the melting point of the crystalline solid, and because of the intervention of a solvent. A stable phase has a lower solubility than a metastable phase, as indicated by the solubility curves in Figures 15.7a and 15.7/ for enantiotropic and monotropic systems respectively and,... 

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