Chapter 1
Some Facets of Molecular Disorder in Crystalline and Amorphous Pharmaceuticals

Marc Descamps and Jean-François Willart

Most drugs, agrochemicals, and so on, are formulated in the solid state, which may be either crystalline or amorphous (i.e., glassy). It is well known that glassy compounds are very disordered solids. However, molecular crystals can also have varying degrees of disorder. Even perfect crystals are always disordered because of the thermal agitation of atoms and molecules. The intentional use of disordered solids and amorphous materials can be of great interest in pharmaceutical formulations because they may have favorable biopharmaceutical properties, for example, enhanced solubility and dissolution capabilities [1-3]. The drawback of this approach is that, often, the disordered solids can be metastable or unstable, physically or (and) chemically [4-6]. Glassy materials are in a nonequilibrium state and evolve upon aging. Formation of disordered solids may also be accidental, during the processing of crystalline materials. That can dramatically undermine the expected stability of the drug [7].

The differences in behavior between the different types of solids are fundamentally associated with the peculiarities of the molecular disorder. In the pharmaceutical literature related to amorphous solids, most interest has been focused on the molecular mobility and the way it may impact stability [8]. If mobility obviously plays an important role, many other aspects of the disorder also can determine the stability level of solids [9]. In this chapter we examine some of the facets of disorder that make it possible to differentiate between crystalline and amorphous states and also between various amorphous states of the same compound. That requires considering the structural, dynamic, thermodynamic, and even kinetic aspects of disorder.

1.1 The Crystal/Amorph Alternative

1.1.1 Crystal/Amorph Alternative: Terminology and Solidity Concept

Condensed matter can normally come in two forms: liquid and solid. The distinction between liquid and solid states lies in how they respond to the application of a shear stress.

A liquid deforms continuously when it is subjected to a shear stress: it is a fluid form of condensed matter.

A solid, on the contrary, can support a shear stress without flowing. If the shear is not too strong, it will deform elastically. This means that when the external forces are removed, an elastically deformed solid returns to its initial state: it is a rigid form of matter.

Solids can be either crystalline or amorphous: they differ in their structure and in the way they are solid.

Perfect crystals are those in which a "motif" - formed by a limited number of atoms or group of atoms - is repeated periodically in a three-dimensional array (lattice). This is reflected in the specific external shape that the crystals can adopt. The equilibrium state of a material at low temperature is expected to be crystalline.

The amorphous state of a material does not possess the long-range translational order (periodicity) of a crystal. It has no specific external shape. A liquid is always amorphous, but amorphous materials can be either solid or liquid. Noncrystalline solids formed as the result of the deep undercooling of a liquid are conventionally called glasses.

Crystalline and amorphous solids are not solids in the same way:

• Crystalline solids are "real solids" in that the elastic reversibility does not depend on the length of time that the shear stress acts. The elastic behavior is characterized by the shear modulus G, which is the constant of proportionality between the strain and the applied stress.
• Noncrystalline (amorphous) solidity needs considering the viscoelastic property of a real liquid. Generally, a real liquid is capable of responding at first in an elastic way (instantaneous shear modulus G8). But, in the words of Maxwell [10], the elastic behavior is fugitive. After a given time t (t = relaxation time, which is a function of the temperature T), the behavior is that of a liquid. The behavior is then characterized by the viscosity ?(T). According to Maxwell's model [11], the value of t is linked to that of ? and G8 by

  1.1

For liquid water at room temperature, t ? 10-3 Pa·s/109 Pa = 10-12 s. This very low value of t gives rise to a high effective fluidity. The designation of an amorphous compound as a liquid or solid depends on the value of t relative to the time of observation t0. If t0 < t, the material behaves as a solid. If t becomes on the order of a few hours or days, a viscoelastic liquid can be considered operationally as a solid: it is a glass. G8 depends on the material and temperature, but its order of magnitude is always some gigapascals (GPa) or tens of GPa. When a liquid can be undercooled (i.e., cooled without crystallizing), the value of G8 is observed to vary with temperature. However, its temperature dependence is insignificant compared to the large temperature dependence of ?. As a consequence, t and ? are roughly proportional. Values of 103-104 s for t correspond to viscosity values of about 1012 Pa·s (= 1013 poises). When the viscosity of an undercooled liquid - which fails to crystallize - reaches such values, we start observing a solid behavior. This marks the entry into the glassy domain. The glass transition temperature Tg has often been defined as that at which the viscosity of an undercooled liquid reaches a value of 1013 poise. We will later give another definition of Tg based on calorimetric observations.

1.1.2 Crystal/Amorph Alternative: Structural Order and Disorder

In this subsection we focus on the structural aspects of order and disorder. We are concerned about the relative positions and orientations of the molecules, ignoring the possible dynamic aspects of the disorder (molecular mobility). In a simple manner, an amorphous solid is sometimes defined as a "disordered solid"; however, crystals, even perfect, are always disordered at some level, and sometimes very disordered. On the other hand, the structure of many amorphous solids is, in fact, non-random at certain length scales. We will briefly discuss the various forms of disorders that will help us to specify the boundary between crystalline and amorphous disorders.

X-ray (and neutron) diffraction by a sample are probably the best techniques to provide direct information about the structural organization of condensed matter. The effects of the different types of disorders on X-ray diffractograms will be presented to help identifying them.

1.1.2.1 Perfect Crystals

A perfect crystal is the periodic repetition in three dimensions (in principle to an infinite extent) of unit cells containing an atomic or molecular motif composed of a few atoms only. The unit cell is built on three noncoplanar vectors a1, a2, and a3 (Figure 1.1). The structural situation at some point in space is exactly reproduced at every other point obtained by adding a lattice translation vector:

Figure 1.1 Two-dimensional representation of the periodic property of a molecular crystal.

In brief, a .

The translational invariance defines the long-range order (LRO) of a crystalline state.

X-Ray diffraction by perfect crystals (for details, see [[12]-14])

The geometry of an X-ray diffraction experiment is shown in Figure 1.2.

Figure 1.2 Usual setup for X-ray diffraction experiments. ? is the wavelength. Q is the scattering vector. IQI = |(s-s0)/?| = 2sin(?)/? |s|=|s0|=1.

The general expression of the intensity diffracted by the atoms of a sample - whatever its structure, crystalline or not - is given by

where Q is the scattering vector, which is a vector in the reciprocal space as defined in Figure 1.2. fn is the atomic scattering factor of the nth atom of the sample situated at rn from the origin. Summations are taken over the full sample. This expression simply shows that X-ray diffraction provides an image, in the reciprocal space, of the structure of the sample.

In the case of a perfect crystal, the translational repetition of the motif allows simplifying the expression of I(Q) in the form of a product:

F(Q) is the structure factor of the unit cell. It reflects the distribution of positions of the molecules (the motif) within the unit cell relative to the lattice points.

where the summation is taken on the atoms j of the unit cell only.

I(Q) is called the interference function. It reflects the geometry of the lattice.

where rm is the vector specifying the origin of the mth unit cell. Summations are taken on the full sample.

• Because of the translational...