**4. Crystal growth mechanisms**

Once the nucleation step of crystallization is underway, more solute molecules can become incorporated into these nuclei and crystal growth can begin. Generally, crystal growth is considered to consist of two steps in series, namely volume diffusion and surface integration. Volume diffusion consists of the diffusion of solute molecules from the solution, through the boundary layer surrounding the crystal, to its surface. Surface integration can be split into three subprocesses, starting with the desolvation of a solute molecule unto the surface of the crystal, followed by the transfer of this molecule from its point of arrival across the crystal's surface. The latter is referred to as surface diffusion. Finally, the desolvated molecule reaches an energetically favorable location (kink site) on the crystal's surface and becomes incorporated into the crystal lattice [13]. The steps in crystal growth described above are illustrated in **Figure 3**.

In this section, we will discuss the different crystal growth mechanisms, their theoretical basis, and examples of how they are used in pharmaceutical sciences. The topic of surface energy models is quite extensive and will not be covered in detail in this section. However, it plays an important role in how different crystal growth rates can lead to different crystal habits and is, therefore, a relevant topic in solid-state pharmaceutical investigations, as it influences downstream processes, such as dissolution, powder flow, milling, granulation, and compaction. Therefore, we will start this section with a discussion on the Gibbs–Curie–Wulff theorem and then look at a popular surface energy model, namely the Bravais–Friedel–Donnay–Harker (BFDH) model.

Also, in this section, we will not discuss adsorption layer or diffusion-reaction models but will instead focus on the more popular power law, birth-and-spread, and screw dislocation models, and show how information regarding the rate-limiting steps in crystal growth can be inferred from the power-law model.

#### **Figure 3.**

*A schematic representation of crystal growth. The steps are (1) the diffusion of a molecule from the solution to the crystal surface, (2) desolvation of the solute molecule unto the crystal surface, (3) transfer across the surface, and (4) incorporation of the molecule into the lattice. The numbering represents—(a) the terrace, (b) the step, and (c) the kink site. Reproduced from ref. [21] with permission from Springer Nature.*

#### **4.1 Surface energy models**

The Gibbs–Curie–Wulff theorem originated when Gibbs postulated that, at equilibrium, a crystal should take on a form such that the product of its surface area and the surface-free energy is minimized. Later, Curie proposed that there exists a direct proportionality between the normal growth rates of crystal faces and surface-free energy. Wulff later stated that, at equilibrium, there exists a central point (the Wulff point) within a crystal such that the distances of the crystal faces from this point are proportional to the specific surface-free energies of those faces [22].

Li and coworkers [22] investigated whether organic crystals, with their diverse molecular shapes, noncovalent interactions, and hydrogen bonding, would still follow the Gibbs–Curie–Wulff theorem. Using a heteroacene molecule as crystal former, they found that not only did the shape and shape evolution of their equilibrium crystals follow the Gibbs–Curie–Wulff theorem, but that the packing of the molecules inside the crystals also reflected the principle of minimizing surface-free energy.

A crystal lattice can be thought of as a repeating 3D pattern consisting of a set of points such that each point has identical surroundings. A lattice has three spatial dimensions (*a*, *b*,*c*), also referred to as the lattice constants. In crystallography, Miller indices are used to express a crystallographic plane in terms of a 3D lattice using the notation (*hkl*). The physical meaning of Miller indices (*hkl*) is a set of parallel crystal facets with equivalent interplanar spacing [23]. The interplanar spacing is defined as the distance between two adjacent parallel planes (facets) with the same Miller indices, denoted as *dhkl*. So, for a cubic lattice system, where the lattice constant is simply *a*, the interplanar spacing is defined as:

$$d\_{hkl} = \frac{a}{\sqrt{h^2 + k^2 + l^2}}\tag{13}$$

And for a more complex tetragonal system [23, 24]:

$$d\_{hkl} = \frac{1}{\sqrt{\left(\frac{h}{a}\right)^2 + \left(\frac{k}{b}\right)^2 + \left(\frac{l}{c}\right)^2}}\tag{14}$$

The BFDH model attempts to predict the crystal habit based on the growth rates of the crystal facets. Concretely, the BFDH model states that the growth rate of a facet is inversely proportional to that facet's interplanar spacing [21, 25]. Unlike the other crystal growth models that will be discussed later in this chapter, the BFDH model does not have a mathematical representation. Instead, predictions of crystal morphology based on the BFDH model are obtained from computer modeling software, such as Mercury 2020.1, Cerius<sup>2</sup> , or Materials Studio.

Since the BFDH model does not explicitly take into account factors, such as molecular interactions but is instead concerned with the geometric aspects of the crystal, there have been instances where predictions made from the BFDH model were not in agreement with experimental observations. For example, while using the BFDH model to predict the morphologies of two carvedilol polymorphs, Prado and coworkers found that their BFDH model predicted thicker crystals of form II than were experimentally obtained and the model also predicted more faces for form III [26]. Similarly, Nichols and Frampton found that the BFDH model predicted morphologies for paracetamol polymorphic forms I and II that differed significantly from those observed experimentally [27]. Instead of using the BFDH model to predict morphologies, Turner and coworkers [28] used it to identify the important morphological faces of lovastatin crystals obtained from different solvents. They found that the calculated attachment energies of these important faces correlated reasonably well with the observed crystal morphologies.

#### **4.2 Power-law model**

The first mathematical model of crystal growth we will discuss is the power-law, or empirical, model. As the name suggests it is an empirical model, raised to some power, and as such it does not offer much insight into the actual growth mechanism. However, it is widely regarded for its flexibility and for being less complex and easier to use than the birth-and-spread or screw dislocation models.

Concretely, the power-law model expresses the growth rate ð Þ *GPWR* in terms of the degree of supersaturation as:

$$G\_{PWR} = k\_{\mathfrak{g}} (\ln \left( \sigma \right))^{\mathfrak{g}} \tag{15}$$

where the kinetics constant, *kg*, is expected to follow Arrhenius temperature dependence, and is defined as:

$$k\_{\rm g} = k\_{\rm g0} \exp\left(\frac{-E\_a}{RT}\right) \tag{16}$$

and where *g* is the temperature-independent growth order parameter, *kg*<sup>0</sup> is a pre-exponential factor, *Ea* is the activation energy and *σ*, *R*, and *T* have the same meanings as before [6].

We mentioned that the power-law model does not explicitly offer insights into the specific crystal growth mechanism. However, recently certain consistencies between the values of *g* and *Ea* and growth mechanisms have come to the fore. For instance, there is now evidence that for values of *Ea* ranging from 10–20 kJ/mol the ratelimiting step for crystal growth is volume diffusion, and for values ranging from 40–100 kJ/mol, surface integration [13, 17]. For values of *g* higher than unity, the rate-limiting step is also considered to be surface integration [13].

We also mentioned the flexibility of the power-law model. In a paper concerning the crystal growth kinetics of piracetam polymorphs, Soto and Rasmuson [13] substituted the supersaturation driving force in Eq. (15) with a different driving force, namely a mass balance expression based on Haüy's law, which states that one can assume a crystal's shape to be constant if nucleation, agglomeration, breakage and growth rate dispersion can all be assumed to be negligible. The driving force can then be expressed as Eq. (17):

$$\ln \sigma = \left( \Delta \mathbf{C}\_0 - \left[ \left( \frac{\overline{L}}{\overline{L}\_0} \right)^3 - \mathbf{1} \right] \frac{\mathcal{W}\_0}{\mathcal{M}} \right) \Big/ \mathbf{C}\_i^{\text{sat}} \tag{17}$$

where <sup>Δ</sup>*C*<sup>0</sup> <sup>¼</sup> *Ci* � *<sup>C</sup>sat <sup>i</sup>* , *L* is the crystal length at any instant, *L*<sup>0</sup> is the mean initial size of the seed crystals, *W*<sup>0</sup> is the mass of the seed crystals, *M* is the mass of solvent and *Ci* and *Csat <sup>i</sup>* have the same meanings as before [13]. Substituting back into Eq. (15) and combining with Eq. (16), they modeled the crystal growth rates using Eq. (18):

$$\mathbf{G}\_{\rm PWR} = \frac{d\mathbf{L}}{dt} = k\_{\rm g0} \exp\left(\frac{-E\_a}{RT}\right) \left[ \left( \Delta \mathbf{C}\_0 - \left[ \left( \frac{\overline{\mathbf{L}}}{\overline{\mathbf{L}}\_0} \right)^3 - \mathbf{1} \right] \frac{\mathbf{W}\_0}{\mathbf{M}} \right) \Big/ \mathbf{C}\_i^{\rm sat} \right]^{\rm \mathcal{S}} \tag{18}$$

where *dL=dt* is the rate of change of a crystal's characteristic linear dimension with time and all the other terms have the same meanings as before [13]. Using Eq. (18), they were able to accurately model the growth rates of their piracetam polymorphs and from the values of *g* and *Ea*, they were able to determine that surface integration was the rate-limiting step in their crystals' growth.

#### **4.3 Birth-and-spread model**

The birth-and-spread model (B+S) models surface, or two-dimensional, nucleation which is believed to occur at intermediate levels of supersaturation. Under such conditions, surface nuclei can form at the edges, corners, and even faces of crystals, creating new growth steps [13]. These surface nuclei then spread to create a new crystalline layer. The B+S model is presented here in Eq. (19)–(22):

$$G\_{B+S} = \tau\_1 (\sigma - 1)^{2/3} (\ln \left( \sigma \right))^{1/6} \exp \left( \frac{-\tau\_2}{T^2 \ln \left( \sigma \right)} \right) \tag{19}$$

$$
\pi\_1 = \left(\frac{16}{\pi}\right)^{1/3} h^{1/6} D\_{\text{surf}}\left(\beta' \frac{\Gamma^\*}{\varkappa\_s}\right)^{2/3} (V\_m \Gamma N\_A)^{5/6} \tag{20}
$$

$$
\pi\_2 = \frac{\pi}{3} V\_m h \left(\frac{\gamma\_{sl}}{k}\right)^2 \tag{21}
$$

$$D\_{surf} = A\_{surf} \exp\left(\frac{-E\_{a,surf}}{RT}\right) \tag{22}$$

where *τ*<sup>1</sup> and *τ*<sup>2</sup> are lumped model parameters, *h* is the step height, *β*<sup>0</sup> is a correction factor *<sup>β</sup>*<sup>0</sup> ð Þ <sup>≤</sup><sup>1</sup> , <sup>Γ</sup> is the molecular adsorption coverage of the solute and <sup>Γ</sup><sup>∗</sup> is that property at equilibrium, *xs* is the mean displacement of the adsorbed units over the surface, *Vm* is the molecular volume of the solute, *Dsurf* is the surface diffusion coefficient, *Asurf* is a pre-exponential factor, *Ea*,*surf* is the activation energy for surface diffusion and the rest of the parameters have the same meanings as before [6].

Despite being a very comprehensive model, there is a specific inconsistency between predicted data and experimental observation that should be pointed out, and that is that once growth from a surface nucleation site has spread all the way to the edge of the crystal face, forming a new crystalline layer, any further growth would require adsorption to a smooth surface. Since this kind of adsorption is energetically less favored than binding to dislocations, it would require high levels of supersaturation to overcome the energy barrier. However, experimental data have shown that crystal growth can occur at much lower degrees of supersaturation [8]. To address this inconsistency, a screw dislocation model can be used, which assumes that screw dislocations on a crystal surface serve as self-perpetuating growth sites.

#### **4.4 Screw dislocation model**

The Burton–Cabrera–Frank (BCF) model [29] is a screw dislocation model, that assumes that crystal growth stems from screw dislocations on a crystal surface and

that growth from these ledges leads to more ledges in a self-perpetuating fashion, thereby attempting to explain how crystal growth can be obtained from supersaturations lower than that needed for the mechanism described by the B+S model. The BCF model can be expressed mathematically as Eqs. (23)–(25):

$$G\_{BCF} = \frac{\tau\_3 T}{\tau\_4} \left( \ln \left( \sigma \right) \right)^2 \tanh \left( \frac{\tau\_4}{T \ln \left( \sigma \right)} \right) \tag{23}$$

$$\pi\_3 = \frac{\Gamma^\* D\_{surf} V\_m}{\varkappa\_s^2} \tag{24}$$

$$\pi\_4 = \frac{19V\_m\gamma\_{sl}}{2k\chi\_s} \tag{25}$$

where *τ*<sup>3</sup> and *τ*<sup>4</sup> are lumped parameters and all the other parameters have the same meanings as before [6].

Now that we have presented the B+S and BCF models, we can appreciate the simplicity of the power-law model. In the introduction, we mentioned that Quilló and coworkers [6] used the power law, B+S, and BCF models, separately and in different combinations, to model the crystal growth kinetics of a proprietary API and thereby attempted to ascertain the underlying growth mechanism(s). During the combination tests, they also tested for dominance of one model over the other, sudden mechanism crossover, and simultaneous mechanisms. They found that the growth process could be best described if both the B+S and BCF models were simultaneously active, in an additive manner, with a smooth crossover from initial surface nucleation dominated growth to screw dislocation (spiral growth) dominated growth.

#### **5. Crystallization from the amorphous state**

One of the ways to improve the solubility of a drug is to prepare an amorphous solid from it. The molecules inside these amorphous solids lack the long-range ordered packing of their crystalline counterparts, giving the amorphous solids higher thermodynamic activity and generally enhanced solubility. Methods to enhance the solubility of drugs have become a hot topic over recent years, and will likely remain a topic of interest, especially considering that approximately 36% of the 698 drugs currently available as immediate-release oral preparations and 60% of the 28912 new APIs under development are classified as biopharmaceutics classification system (BCS) class II or IV drugs [30, 31]. The BCS attempts to classify a drug into one of four categories, depending on its aqueous solubility and membrane permeability [32]. BCS class II and IV drugs have poor aqueous solubility but good membrane permeability, and poor aqueous solubility and poor membrane permeability, respectively. In other words, the market share of poorly soluble drugs is likely to increase in the future.

Currently, there are a limited number of amorphous preparations on the market [33]. A specific cause for concern is that the high thermodynamic activity of an amorphous solid, responsible for its enhanced solubility, also makes it thermodynamically unstable and likely to convert back to a more stable, but less soluble, crystalline form. A mechanistic understanding of this conversion and the resulting glass-to-crystal (GC) growth is an important topic in pharmaceutical sciences.

GC growth rates cannot be readily explained by the thermodynamic driving force responsible for crystallization from solution, as described previously [34]. Initially, it was believed that crystallization from the amorphous state was dependent on the storage conditions, such as the storage temperature relative to the glass transition temperature *Tg* , and the bulk properties of the amorphous solid, like α-relaxation processes [34–37]. However, it is now generally accepted that GC growth can take place at temperatures well below *Tg* and follows two mechanisms, namely fast surface crystal growth and slower growth in the bulk (interior) [31].

Both the surface and bulk crystallization processes are believed to be controlled by the self-diffusion of the molecules in the amorphous solid. Using indomethacin as model API, both Wu and Yu [38] and Swallen and Ediger [39] found that the crystal growth rates in the interior of their amorphous solids were proportional to the selfdiffusion coefficients for temperatures close to *Tg*. It is believed that surface crystallization is so much faster than bulk crystallization, because of the increased molecular mobility at the surface. This was corroborated by Zhu and coworkers [40] who measured the surface smoothing of indomethacin glasses and found the self-diffusion on the surface to be at least one million times faster than in the interior.

With the mechanism of surface crystallization understood, immobilization of the molecules at the surface makes for an appealing target to delay the crystallization process. Yu and coworkers [41] coated indomethacin glasses with gold (10 nm) and two polyelectrolytes. They found that even a single layer of polyelectrolyte was enough to inhibit the growth of exiting crystals and that the molecular mobility of molecules at the surface of an amorphous solid can be sufficiently suppressed by a coating only a few nanometers thick. Their results suggest that nanocoating is a promising technique to stabilize amorphous solids.

#### **6. Conclusions**

The crystallization of small-molecule APIs plays an important role in the pharmaceutical industry. Because the properties of these crystals, that is, crystal habit, size distribution, and polymorphic form, influence many downstream processes, ranging from quality control testing to formulation; it is desirable to be able to consistently produce crystals with specific properties. To achieve this kind of control over the crystallization process, a thorough understanding of the underlying mechanisms is required. In this chapter, we looked at the thermodynamic driving force behind crystallization, and how it can be rearranged into a supersaturation driving force. Methods to estimate the degree of supersaturation were discussed. The mechanisms behind crystal nucleation and growth, and the mathematical models describing these mechanisms, were discussed and examples of how these mechanisms can be used as intervention points to control the properties of the resulting crystals were given. We ended this chapter with a look at amorphous solids, which have a natural tendency to crystallize back to a more stable, but less soluble form, and saw that immobilization of the molecules at the surface of these solids with even a single layer of the polymer was enough to stabilize the amorphous solid. With an increasing movement in the pharmaceutical industry toward streamlining manufacturing processes through control, the techniques discussed in this chapter might see even more general use in the future.

*Crystallization: Its Mechanisms and Pharmaceutical Applications DOI: http://dx.doi.org/10.5772/intechopen.105056*
