**3. Thermodynamic vs. kinetic factors: formation and transformation of polymorphs**

A brief summary of fundamental thermodynamics principles is required prior to understand their relevance on *polymorphism*. Despite ideal systems are not viable, the definition of boundaries within this phenomenon based on simple laws could be achieved with a sufficient accuracy to be useful. It is essential to stress that the result of any structural change will be reflected as a change on the properties of the system. Since energy changes are empirically measurable, they are a pragmatic way to face *polymorphism* and led these energy differences to be a useful descriptor. The identification of the relative intensities within the different forms of energy displays whether exchanges of energy between two polymorphs will occur. Remarkably, this allows to induce energetic modifications, by an external stimulus, only on a selected form of energy to establish relationships in a linear manner.

The quantity of energy a system exchanges with environs is named *enthalphy* and is referred to as *H*. Taking a chemical reaction as the case study, the character of the energetics involved in it, is represented as the *enthalpy of reaction,* which is the difference in enthalpies between products and reactants. This concept is useful to determine the direction in which energy will flow but such information is not conclusive to establish *spontaneity*. Hence the introduction of the notion of *entropy* (*S*) is essential. It is a way to represent the organization of a system approached as the degree of disorder and it is measured in terms of entropy changes of the system. Using both terms, *Gibbs free energy* (*G*) provides further insights into *spontaneity* at constant temperature and pressure.

To understand the relative stability of the different polymorphs within a given system as well as their transformations, the energy/temperature diagrams (E/T) introduced by Buerger in 1951 [55] are the most valuable approach, being capable to encapsulate much data in a single representation. It is based on the *G* equation:

$$\mathbf{G} = \mathbf{H} - \mathbf{T} \mathbf{S} \tag{1}$$

As a general representation, the diagram on **Figure 2a** displays how these terms evolve at increasing temperature. Since the term *TS* is more significantly affected by temperature than *H*, its effect on *G* becomes intense as the temperature of the system raises and thus, *G* of the system tends to decrease. The energy of a transformation undergone within a polymorphic system, under conditions of constant temperature and pressure, are defined by:

$$
\Delta \mathbf{G} = \Delta \mathbf{H} - \mathbf{T} \Delta \mathbf{S} \tag{2}
$$

Within a series of polymorphic forms their crystal lattices are unique *per se*, so despite their *G* at a certain temperature could be equal (*isoenergetic* forms), the values of *H* and *S* will be different (**Figure 2b**). If isoenergetic forms are present, there is a crossing point between the two *G* value curves in the E/T diagram, named *transition point* (*t.p.*), which to be useful is to be placed below the *melting point* (*m.p.*). In such a scenario, if the formation of these polymorphs is allowed by kinetic factors, both forms coexist and can be formed as *concomitant polymorphs* (*vide infra in Section 5*).

For instance, considering a case study as the one represented in **Figure 2b**, containing two polymorphic forms (*dimorphic*) named as *A* and *B*, a brief analysis of this E/T diagram can provide some useful data. Below the *t.p.* (blue region) form *A* is the most thermodynamically stable as its lower *G* value states. Within this region, its transformation into form B is related to an increase in *H* and it is defined as an *endothermic* transition. At the *t.p.* both forms have the same *G* (ΔG = 0), which applied to Eq. (2) is set that ΔH = TΔS, where ΔH = HB-HA and ΔS = SB-SA. All these equations allow to quantify the entropic change of a polymorphic transformation once the enthalpic variation , ( ) ∆*HtB A* <sup>↔</sup> is experimentally determined (see Section 4.1). Above the *t.p.* (green region), within this range of temperature and before the change of state at the *m.p.*, form *B* is the thermodynamically stable and therefore, transformation from *A* to *B* is associated with a decrease in *H* (*exothermic* process). After defining these borders, one can infer the proper conditions in which the formation of one polymorph is favored instead of the other.

Considering that theoretical thermodynamic relations applied to systems undergoing phase transformation are bounded by experimental data, more empirical concepts were developed to better represent these changes. Consequently, the terms *Enantiotropism* and *Monotropism* were defined to meet this requirement. The former refers to those systems presenting a reversible transformation before the *m.p*. when heated above or cooled below the *t.p.* Contrarily, *Monotropism* sets that only one polymorph is the more thermodynamically stable over the entire range

#### **Figure 2.**

*(a) Progression of the terms G, S, TS and H at increasing temperature. (b) Representation of the evolution of H and G variables in a dimorphic system (polymorphs A and B) at increasing temperature.*

*Polymorphism and Supramolecular Isomerism: The Impasse of Coordination Polymers DOI: http://dx.doi.org/10.5772/intechopen.96930*

of temperatures until the *m.p*. Both examples are illustrated in **Figure 3a** and **b**, respectively.

The incorporation of energetics associated with the liquid state into the E/T diagram, enthalpy (*H*liq) and free energy (*G*liq), ease to parse their behavior and classification. If the crossing points between the *G*liq curve and the *G* curves of both *A* and *B* are after the *t.p.*, they are said to be *enantiotropes*. The absence of crossing points between *A* and *B* before their *m.p.* classify them as *monotropes*. The terms ΔHf,*B* and ΔHf,*A* are the energy that *B* or *A* require to change into liquid state, respectively.

Burger and Ramberger [56] in 1979, pioneered in the development of many experimental rules to empirically establish the nature of this relationship between polymorphic forms and their applicability was exemplified with 113 substances [57]. All giving fundamental knowledge to determine the proper conditions to control their formation: Heat-of-transition, Heat-of-fusion, Entropy-of-fusion, Phase transformation reversibility, Enthalpy of sublimation, Heat-capacity, Infrared, Solubility and Density rules. Among them, the most applied are Heat-of-transition and Heat-of-fusion rules. The former being based on using Differential Scanning Calorimetry (DSC) to identify the endothermic or exothermic character of the transition. If this rule cannot be applied, Heat-of-fusion rule, which rely on the determination of the *m.p.*, states that the relation is *enantiotropic* when the higher melting polymorph has the lower enthalpy of fusion while *monotropic* systems are defined by the higher melting polymorph having the higher enthalpy of fusion. Detailed explanation on the application of them could be found in Burger publication [57] or in the more recent Brittain's review [58].

Crystallization is the process related with nucleation and growth of a crystal structure. This process is, in principle, directed by thermodynamic factors that tend to reach the structure with a lower lattice energy. However, the crucial stage is supersaturation which is determined by kinetics, in particular, by the rate of nucleation and thus, the first structure to be formed is the one with preferred nucleation. Therefore, a metastable form can grow despite being unfavored by thermodynamic factors. Subsequently, this form can be converted into a more thermodynamically stable by solution or solid state-mediated phase transformation. This successive phase change was identified and proposed as the *Law of States* by Ostwald [13].

To avoid common mistakes in ascribing the nature of the *nucleation* process, Mullin [59] divided it into *primary*, in which any crystalline matter is directing the process, and *secondary*, pertaining to the circumstances of nucleation generated in the vicinity of crystals, previously nucleated or intentionally placed, a methodology known as *seeding* (*vide infra* in Section 4.2). Furthermore, secondary nucleation can be classified as *homogeneous* if it is spontaneous or *heterogeneous* if it is induced by foreign particles.

**Figure 3.** *E/T diagrams of (a) enantiotropic and (b) monotropic dimorphic systems.*

#### *Crystallization and Applications*

Solid-state phase transformations are usually promoted by an external stimulus in the form of mechanical work or temperature. They are thought to be related with intrinsic defects, whether coming from the original structure or being caused by mechanical stress, which commence and propagate the formation of the new phase. The most studied transformations are those based on order–disorder changes, mainly promoted by temperature variations. *Disordering* processes arise from increasing temperature whereas *ordering* processed are observed at decreasing temperature. An excellent analysis of such phenomena with the most relevant examples were summarized by Dunitz and Bernstein [60].

The reversibility of these changes could not be evident or even accessible since *hysteresis* plays a crucial role. It is the lagging of the transition behavior respect to the applied stimulus. Therefore, it is possible to need heating or cooling beyond the *t.p.* to let the phase change occurs and even with the sufficient high degree of *hysteresis* is conceivable to avoid transformation [61]. Although ideal reversibility is often desired, control of hysteresis [62] also leads to unique properties [63], often advantageous to the application of these materials, only achievable by a precedent transformation. *Hysteresis* phenomenon is associated to *structural fatigue* [64–66], which means the rise of structural changes in the crystalline material as dislocations as well as to a different nucleation with different energy barriers from one to another [67]. Further concerns regarding the basis of this topic can be addressed reading Flanagan's publication [68].

Overall, since crystallization is a competitive process between minimizing lattice energy (thermodynamic) and reaching supersaturation (kinetic), the achievement of non-minimal energetically stable forms allows to the transformation into lower energetic forms after reaching the activation barrier.
