Polymorphism metastable equilibrium

The concepts of metastable equilibrium and suspended transformations are extremely important in any practical understanding of polymorphism. Metastable equilibrium can be defined as a state which will exist for some time period without change, even though a more stable state does exist (Findlay et al. 1951 Zemike 1955). This is distinct from unstable equilibrium that results in spontaneous and instantaneous change. A suspended transformation is one that should occur on the basis of the thermodynamic considerations but does not occur because of kinetic factors. Suspended transformations must and do result in metastable states. Without metastable equilibrium and suspended transformations, polymorphism would not be as significant an issue as it is. 

In the real world, the transition or equilibrium point between two enantiotropic polymorphs is often not directly observed because of suspended transformations in connection with metastable equilibrium. Measurement of various physical properties at different temperatures can be related to the relative free energy of the polymorphs, and the forms, can be classified as enantiotropic or monotropic. 

Closely allied with the concepts of metastable equilibrium and suspended transformations is Ostwald s rule (Ostwald s step rule or law of successive reactions). Essentially Ostwald s rule states that in all processes it is not the most stable state with the least amount of free energy that is initially obtained but the least stable state lying nearest to the original state in free energy (Ostwald 1897). It is easy to see how this rule and the concept of suspended transformations can explain the production of a metastable polymorph through crystallization from a melt or solution. 

As the alkyl chains assume an ordered arrangement with weak intermolecular forces, the thermal liberation of rotational freedom around the chains takes place at a relatively low temperature. Molecular motion within the chain increases gradually as the temperature increases until, at characteristic temperatures, there is a considerable increase in the molecular motion, causing the formation of various polymorphs. Polymorphic crystals may be defined as crystals that are formed from the same molecule and have the same composition but are different in crystal structure. During a phase transition the crystal that exists at low temperatures may be tfansformed on heating into a different structure. Two different kinds of polymorphs exist equilibrium and metastable [14,15]. A form that has a range of temperature over which it is stable with respect to other polymorphs is said to be an equilibrium polymorph. An equilibrium polymorph exhibits thermodynamically reversible isothermal phase fiansitions. Metastable polymorphs are kinetically stable states whose existence depends on the presence of a kinetic barrier to the attainment of equilibrium polymorphs. The fiansformation of a metastable polymorph to the corresponding equilibrium polymorph is an irreversible process. 

Crystallization, by definition, implies that the initial structure be a glass, followed by the nucleation and growth of a crystalline phase, be it the equilibrium one or a metastable phase. The process is a first-order transformation and involves atomic diffusion, or at least atomic shuttles. Types of crystallization reactions that occur include polymorphous crystallization, which is a composition invariant transformation such as that in Fe-B, and eutectic crystallization, T, in FeNiPB glass, where line lamellae of iron-nickel austenite and mclastable (FeNiJj PB phases grow cooperatively. 

A systematic study of salting-out precipitation is carried out to obtain the operational limits within which this precipitation method can be applied for the production of fines (mean particle size <10 xm) with acceptable quality and productivity. The model substances glycine and sodium chloride are salted-out from their aqueous solutions by using ethanol as antisolvent. The main operational parameter is the initial supersaturation of the solutions. It is shown that the smallest particles can be produced at the limits of the metastability domain determined by three optional process parameters the initial solution concentration, the equilibrium solubility and the operational time. The product quality (crystallinity, polymorphic states, aggregation) and productivity considerably change with the operational conditions. 

At temperatures below the main transition, a basic equilibrium stracture is the subgel (crystalline) Lc phase. Its formation usually requires prolonged low-temperature incubation. In addition to the Lc phase, many intermediate stable, metastable, and transient lamellar gel structures are adopted by different lipid classes—with perpendicular or tilted chains with respect to the bilayer plane, with fully interdigitated, partially interdigitated, or noninterdigitated chains, rippled bilayers with various ripple periods, and so forth. (Fig. 1). Several polymorphic phase transitions between these structures have been reported. Well-known examples of polymorphic transitions are the subtransition (Lc- L ) and the pretransition (Lp/- Fp/) in phosphatidylcholines (33). Recently, a polymorphic transition that included rapid, reversible transformation of the usual gel phase into a metastable, more ordered gel phase with orthorhombic hydrocarbon chain-packing (so-called Y-transition) was reported to represent a common pathway of the bilayer transformation into a subgel (crystalline) Lc phase (62). 

With Gibbs free-energy data from a single reference source (why one source ), calculate the difference in the free energies of diamond and graphite (polymorphs of pure carbon) at 25°C. What is the value of the equilibrium constant of the reaction between them Which phase is stable and which metastable at 25°C and 1 bar total pressure Discuss. 

The solubility of the most stable crystal form in a polymorphic system is termed the equilibrium solubility. While the measurement of equilibrium solubility at a given temperature is a routine practice in pharmaceutical research (Grant and Brittain, 1995), evaluation of the solubility of a metastable polymorph is frequently more complicated owing to the tendency of metastable forms to undergo a phase transformation to the more stable polymorph in the medium of measurement. It is therefore prudent to include a determination of the phase at the completion of any solubility measurement to verify exactly which polymorphic form has been the subject of the measurement. 

As discussed earlier in the Thermodynamics section of this chapter, the difference in free energy between solid phases (i.e. polymorphs and solvates) is directly proportional to their relative solubilities (Equation 2). Therefore, a saturated solution of a less stable (more soluble) phase is super-saturated with respect to a more stable (less soluble) phase. Since super-saturated solutions are metastable, a more stable phase will eventually crystallize in order to establish equilibrium and remove supersaturation. According to Ostwald s (1897) Law... 

The melting curve of sulfur with respect to metastability and experimental procedures has been critically analyzed by Vezzoli and Walsh [194]. The authors also discussed the previously reported phase boundaries of the sohd high-pressure allotropes in the range up to 4 GPa and from room temperature up to about 670 K [132]. However, the structures of these high-pressure polymorphs are unknown and their probability to exist under equilibrium conditions is still awaiting confirmation. 

Figure 3.29 shows the dissolution profile obtained during a solubility experiment for the amorphous form of a development compound. The initial dissolution of the compound is high but falls as the compound crystallizes and the equilibrium reaches that of a crystalline form of the compound. By measuring the solubility of polymorphs, the thermodynamic quantities involved in the transition from a metastable to a stable polymorph can be calculated. Experimentally, the solubilities of the polymorphs are determined at various temperatures, and then the log of the solubility is plotted against the reciprocal of the temperature (the van t Hoff method). A straight line results (the problem of non-linearity has been dealt with by... 

Univariant Systems.—Equilibrium between liquid and vapour. Vaporisation curve. Upper limit of vaporisation curve. Theorems of van t Hoff and of Le Chatelier. The Clausius-Clapeyron equation. Presence of complex molecules. Equilibrium between solid and vapour. Sublimation curve. Equilibrium between solid and liquid. Curve of fusion. Equilibrium between solid, liquid, and vapour. The triple point. Complexity of the solid state. Theory of allotropy. Bivariant systems. Changes at the triple point. Polymorphism. Triple point Sj—Sg— V. Transition point. Transition curve. Enantiotropy and monotropy. Enantiotropy combined with monotropy. Suspended transformation. Metastable equilibria. Pressure-temperature relations between stable and metastable forms. Velocity of transformation of metastable systems. Metastability in metals produced by mechanical stress. Law of successive reactions. 

Conversions of a metastable phase into a more stable phase may include the transformation of one polymorphic phase into another, the solvation of an anhydrous phase, the desolvation of a solvate phase, the transformation of an amorphous phase into a crystalline anhydrate or solvate phase, the degradation of a crystalline anhydrate or solvate phase to an amorphous phase, or in the case of digoxin, the conversion of imperfect (less crystalline, more amorphous) crystals with a high density of defects into more perfect (more crystalline) crystals with a lower density of defects. While it is straightforward to determine the equilibrium solubility of a phase that is stable with respect to conversion, the measurement of solubilities of metastable phases that are susceptible to conversion is not a trivial matter. 

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