Merck Exhibit 2179, Page 1
`Mylan Pharmaceuticals Inc. v. Merck Sharp & Dohme Corp.
`IPR2020-00040
`
`

`

`Merck Exhibit 2179, Page 2
`Mylan Pharmaceuticals Inc. v. Merck Sharp & Dohme Corp.
`IPR2020-00040
`
`

`

`2
`
`CRYSTALS. CRYSTAL GROWTH, AND NUCLEATION
`Allan S. Myerson and Rajiv Ginde
`
`2.1. CRYSTALS
`
`Crystals are solids in which the atoms are arranged in a periodic
`repeating pattern that extends in three dimensions. While all
`crystals are solids, not all solids are crystals. Materials that have
`short-range rather than long-range ordering, like glass, are non(cid:173)
`crystalline solids. A noncrystalline solid is often referred to as an
`amorphous solid. Many materials can form solids that are crystal(cid:173)
`line or amorphous, depending on the conditions of growth. In
`addition, some materials can form crystals of the same composi(cid:173)
`tion but with differing arrangements of the atoms forming differ(cid:173)
`ent three-dimensional structures. Other materials can have the
`same three-dimensional structure but appear different in shape
`when viewed under the microscope. To make sense of this, and
`to understand the nature of crystals and how they are identified
`requires some knowledge of crystals and their structure. The study
`of crystal structure is called crystallography and is described in a
`number of standard references (Bunn 1961; Cullity 1978). In this
`section, we will discuss the basics of crystals and their structure.
`
`2.1.1. LATTICES AND CRYSTAL SYSTEMS
`
`Crystals are solids in which the atoms are arranged in a three-
`dimensional repeating periodic structure. If we think of crystals in
`a purely geometric sense and forget about the actual atoms, we can
`use a concept known as a point lattice to represent the crystal. A
`point lattice is a set of points arranged so that each point has
`identical surroundings. In addition, we can characterize a point
`lattice in terms of three spatial dimensions: a, b, and c, and three
`angles: a, /3, and 7. An example of a point lattice is given in Figure
`2.1. Looking at Figure 2.1 we can see that the lattice is made up of
`repeating units that can be characterized by the three dimensions
`and three angles mentioned. We can arbitrarily choose any of these
`
`units, and by making use of the spatial dimensions and angles can
`reproduce the lattice indefinitely. The lengths and angles men(cid:173)
`tioned are known as lattice parameters and a single cell constructed
`employing these parameters is called the unit cell. A unit cell is
`shown in Figure 2.2.
`There are obviously a number of different lattice arrange(cid:173)
`ments and unit cells that can be constructed. It was shown, how(cid:173)
`ever, in 1848 by Bravais that there are only 14 possible point
`lattices that can be constructed. These point lattices can be divided
`into seven categories (crystal systems) that are shown in Table 2.1.
`Figure 2.3 shows all 14 of the Bravais lattices. Looking at the
`crystal systems we see that they are all characterized by these
`lattice parameters. For example, cubic systems all must have equal
`lengths (a = b = c) and angles equal to 90°. In addition, lattices
`can be classified as primitive or nonprimitive. A primitive lattice
`has only one lattice point per unit cell while a nonprimitive unit cell
`has more than one. If we look at the cubic system, a simple cubic
`unit cell is primitive. This is because each lattice point on a corner
`is shared by eight other cells so that 1/8 belongs to a single cell.
`Since there are eight corners, the simple cubic cell has one lattice
`point. Looking at a body centered cubic cell, the point on the
`interior is not shared with any other cell. A body centered cubic
`cell, therefore, has two lattice points. A face centered cubic cell has
`a lattice point on each face that is shared between two cells. Since
`there are six faces as well as the eight corners, a face centered cubic
`cell has four lattice points.
`Another property of each crystal system that distinguishes one
`system from another is called symmetry. There are four types of
`symmetry operations: reflection, rotation, inversion, and rotation-
`inversion. If a lattice has one of these types of symmetry, it means
`that after the required operation, the lattice is superimposed upon
`itself. This is easy to see in the cubic system. If we define an axis
`normal to any face of a cube and rotate the cube about that axis,
`the cube will superimpose upon itself after each 90° of rotation. If
`we divide the degrees of rotation into 360°, this tells us that a cube
`has three fourfold rotational symmetry axes (on axes normal to
`three pairs of parallel faces). Cubes also have threefold rotational
`symmetry using an axis along each body diagonal (each rotation is
`
`(100)
`
`[200]
`i
`
`.2/311
`
`Figure 2.1 A point lattice.
`
`Figure 2.2 A unit cell.
`
`33
`
`Merck Exhibit 2179, Page 3
`Mylan Pharmaceuticals Inc. v. Merck Sharp & Dohme Corp.
`IPR2020-00040
`
`

`

`34 CRYSTALS, CRYSTAL GROWTH, AND NUCLEATION
`
`/r7\
`
`SIMPLE
`CUBIC (P)
`
`BODY-CENTERED
`CUBIC (I)
`
`FACE-CENTERED
`CUBIC (F)
`
`a ^ic-
`
`izzr
`
`SIMPLE
`TETRAGONAL
`(P)
`
`BODY-CENTERED
`TETRAGONAL
`(I)
`
`A7\ w SIMPLE
`
`ORTHORHOMBIC
`(P)
`
`BODY-CENTERED
`ORTHORHOMBIC
`(i)
`
`WW
`
`BASE-CENTERED
`ORTHORHOMBIC
`(C)
`
`FACE-CENTERED
`ORTHORHOMBIC
`(F)
`
`c
`
`120°
`
`a
`
`±
`
`RHOMBOHEDRAL
`(Ft)
`
`HEXAGONAL
`(R)
`
`SIMPLE
`MONOCLINIC (P)
`
`BASE-CENTERED
`MONOCLINIC (P)
`
`TRICLINIC (P)
`
`Figure 2.3 The Bravais lattices (P, R = primitive cells, F = face centered, / = body centered, and C = base centered). (From B.D. CuUity,
`Elements of X-ray Diffraction, © 1978 by Addison-Wesley Publishing Company. Reprinted with permission of the publisher.)
`
`Merck Exhibit 2179, Page 4
`Mylan Pharmaceuticals Inc. v. Merck Sharp & Dohme Corp.
`IPR2020-00040
`
`

`

`TABLE 2.1 Crystal Systems and Bravais Lattices
`
`System
`
`Cubic
`
`Tetragonal
`
`Orthorhombic
`
`Rhombohedral^
`
`Hexagonal
`
`Monoclinic
`
`Triclinic
`
`Axial Lengths and Angles
`
`Three equal axes at right angles
`a = b = c,a = l3 = ^ = 90°
`
`Three axes at right angles, two equal
`a = b^c, a = (3 = j = 90°
`Three unequal axes at right angles
`a^by^c, a = (3 = y = 90°
`
`Three equal axes, equally inclined
`a = b = c, a = p = ^^ 90°
`Two equal coplanar axes at 120°, third axis at
`right angles
`= (3 = 90°, 7 - 120°
`a = b^c,a
`Three unequal axes, one pair not at right angles
`a y ^ b / c, Q; = 7 = 90°//?
`Three unequal axes, unequally inclined and none
`at right angles
`a^ b^ c,a^
`
`(5 ^-i ^90°
`
`2.1. CRYSTALS 35
`
`Bravais Lattice
`
`Simple
`Body-centered
`Face-centered
`Sinnple
`Body-centered
`Simple
`Body-centered
`Base-centered
`Face-centered
`Simple
`
`Simple
`
`Simple
`Base-centered
`Simple
`
`(Data adapted from Cullity 1978.)
`^Also called trigonal.
`
`TABLE 2.2 Symmetry Elements
`
`System
`
`Cubic
`Tetragonal
`Rhombohedral
`
`Hexagonal
`Monoclinic
`Triclinic
`
`Minimum Symmetry Elements
`
`Four threefold rotation axes
`One fourfold rotation (or rotation-inversion) axis
`Three perpendicular twofold rotation
`(or rotation-inversion) axes
`One sixfold rotation (or rotation-inversion) axis
`One twofold rotation (or rotation-inversion) axis
`None
`
`(Data adapted from Cullity 1978.)
`
`120°) and twofold rotational symmetry using the axis formed by
`joining the centers of opposite edges. Each lattice system can be
`defined in terms of the minimum symmetry elements that must be
`present. Table 2.2 lists the minimum symmetry elements that must
`be present in a given crystal system. A system can have more than
`the minimum but not less. A more complete discussion of lattices
`and symmetry can be found in Culhty (1978).
`
`M12O]
`
`Figure 2.4 Indices of directions. (From B.D. Culhty, Elements of
`X-ray Diffraction, © 1978 by Addison-Wesley Pubhshing Company.
`Reprinted with permission of the publisher.)
`
`2.1.2. MILLER INDICES AND LATTICE PLANES
`
`If we take any point on a lattice and consider it the origin, we may
`define vectors from the origin in terms of three coordinates. If, for
`example, we started with a cubic cell and defined a vector going
`from the origin and intersecting point 1,1,1, the line would go in
`the positive direction along the body diagonal of the cube and
`would also intersect the points 2,2,2 and all multiples. This direc(cid:173)
`tion is represented in shorthand by [1,1,1] where the numbers are
`called the indices of the direction. Negative numbers are indicated
`by putting a bar over a number so that [1,1,1] means the first index
`is negative. We can represent a family of direction by using the
`symbol <1,1,1>. This represents all the directions using both
`positive and negative indices in all combinations. In this case, it
`represents all the body diagonals of a cube. By convention, all
`indices are reduced to the smallest set of integers possible either by
`division, or by clearing fractions. An illustration of various indices
`and the directions they represent is shown in Figure 2.4.
`The representation of planes in a lattice makes use of a con(cid:173)
`vention known as Miller indices. In this convention, each plane is
`represented by three parameters {hkl), which are defined as the
`reciprocals of the intercepts the plane makes with three crystal
`axes. If a plane is parallel to a given axis, its Miller index is zero.
`Negative indices are written with bars over them. Miller indices
`refer not only to one plane but a whole set of planes parallel to the
`plane specified. If we wish to specify all planes that are equivalent,
`we put the indices in braces. For example, {100} represents all the
`cube faces. Examples of Miller indices in the cubic system are
`shown in Figure 2.5.
`Real crystals are often described in terms of the Miller indices
`of the faces (planes) present. Examples of some common
`crystals with their faces given in terms of Miller indices are shown
`in Figure 2.6.
`
`2.1.3. CRYSTAL STRUCTURE AND BONDING
`
`In the previous section we have developed a geometric system that
`can be used to represent the structure of actual crystals. In the
`simplest actual crystal, the atoms coincide with the points of one of
`the Bravais lattices. Examples include chromium, molybdenum,
`
`Merck Exhibit 2179, Page 5
`Mylan Pharmaceuticals Inc. v. Merck Sharp & Dohme Corp.
`IPR2020-00040
`
`

`

`3 6 CRYSTALS, CRYSTAL GROWTH, AND NUCLEATION
`
`^100-
`
`h•^20o-^
`
`(TOO)
`
`(200)
`
`v
`
`(110)
`
`/
`
`A
`
`(iio)
`
`(111)
`
`(102)
`
`Figure 2.5 Miller indices of planes in the cubic system. The distance d is the interplanar spacing.
`(From B.D. CuUity, Elements of X-ray Diffraction, © 1978 by Addison-Wesley Publishing Company.
`Reprinted with permission of the publisher.)
`
`111
`
`001
`
`503
`
`TsVff
`
`.011
`111
`
`210-
`
`110
`
`V
`
`111
`
`100
`
`001
`
`-+110
`210
`
`" V,
`111
`
`100'
`
`^120
`
`100
`
`121.
`
`112
`
`211
`
`Oio'A^^OOl^
`
`Sucrose (monoclinic)
`
`Copper sulphate (triclinic)
`
`Calcite (trigonal)
`
`Oil
`
`001
`
`111 Oil
`131
`
`102
`
`120'
`
`11040
`
`120 010
`
`110 110
`
`^^M.1
`
`101 102
`
`i.;i011
`
`021
`
`Ammonium sulphate
`(orthorhombic)
`
`Sodium thiosulphate
`(monoclinic)
`
`Sodium chloride
`(regular)
`
`Figure 2.6 Crystal forms of some common materials with their Miller indices shown. (Reproduced with
`permission from Mullin 1972.)
`
`Merck Exhibit 2179, Page 6
`Mylan Pharmaceuticals Inc. v. Merck Sharp & Dohme Corp.
`IPR2020-00040
`
`

`

`and vanadium, which have a body centered cubic crystal structure,
`and copper and nickel, which are face centered cubic.
`In a more complex arrangement, more than one atom of the
`same type can be associated with each lattice point. A structure
`that a number of metals have, which is an example of this, is the
`hexagonal close-packed structure. It is called close packed because,
`if the molecules are assumed to be spherical, this arrangement is
`one of only two possible ways spheres can be packed together to
`yield the greatest density yet still be in a periodic structure. The
`packing arrangement in crystals is another part of the information
`that helps to understand crystal structure. More information on
`packing in crystals can be found in Ruoff (1973) and CulHty
`(1978).
`Many inorganic molecules form ionic crystals. An example of
`an ionic structure common to a number of molecules is that of
`sodium chloride, shown in Figure 2.7. Ionic crystals are made up
`of the individual ionized atoms that make up the species in their
`stoichiometric proportion. They are held in place by electronic
`forces. The sodium chloride structure is face centered cubic and
`the unit cell contains four sodium ions and four chloride ions.
`Because the unit cells contain two types of atoms some additional
`constraints on the structure exist. For example, a symmetry oper(cid:173)
`ation on the crystal must superimpose atoms of the same type.
`Most organic species form molecular crystals in which discrete
`molecules are arranged in fixed positions relative to the lattice
`points. This of course means that the individual atoms making
`up the molecules are each arranged at fixed positions relative to
`each other, the lattice point, and the other molecules. The forces
`between molecules in molecular crystals are generally weak when
`compared with the forces within a molecule. The structure of
`molecular crystals is affected by both the intermolecular forces
`and the intramolecular forces since the shapes of the individual
`molecules will affect the way the molecules pack together. In
`addition, the properties of the individual molecule, such as the
`polarity, will affect the intermolecular forces. The forces between
`the molecules in molecular crystals include electrostatic inter(cid:173)
`actions between dipoles, dispersion forces, and hydrogen bond
`
`O Na-^
`ci-
`
`(010]
`
`NaCI
`
`Figure 2.7 The structure of sodium chloride (NaCl).
`
`2.1. CRYSTALS 37
`
`interactions. More information about the structure and energetics
`of molecular crystals can be found in the work of Kitaigorodski
`(1973) and in Wright (1987).
`An important tool used to identify crystals and to determine
`crystal structure is that of x-ray diffraction. Crystals have atoms
`spaced in a regular three-dimensional pattern. X-rays are electro(cid:173)
`magnetic waves with wavelengths of similar size as the distance
`between the atoms in a crystal. When a monochromatic beam of
`x-rays is directed at a crystal in certain directions, the scattering
`of the beam will be strong and the amplitudes of the scattering will
`add creating a pattern of Unes on photographic film. The relation(cid:173)
`ship between the wavelength of the x-rays and the spacing between
`atoms in a crystal is known as Bragg's law, which is given below
`
`\ = 2d sin 6
`
`(2.1)
`
`where A is the wavelength of the incident x-rays, d is the inter-
`planar spacing in the crystal, and 0 the angle of the incident x-rays
`on the crystal.
`Bragg's law shows us that, if x-rays of a known wavelength are
`used and the incident angle of the radiation is measured, determin(cid:173)
`ation of the interplanar spacing of a crystal is possible. This is the
`foundation of x-ray diffraction methods that are used to analyze or
`determine the structure of crystals. Several different experimental
`methods making use of x-ray diffraction and Bragg's law have
`been developed and are used depending on the type of sample that
`is available and the information desired.
`The most powerful method that can be used to determine
`unknown crystal structures is the rotating crystal technique. In
`this method a single crystal of good quaUty (of at least 0.1 mm in
`the smallest dimension) is mounted with one of its axes normal to a
`monochromatic beam of x-rays and rotated about in a particular
`direction. The crystal is surrounded by cylindrical film with the
`axis of the film being the same as the axis of rotation of the crystal.
`By repeating this process of rotation in a number of directions, the
`rotating crystal method can be used to determine an unknown
`crystal structure.
`It is unhkely that you will ever need to use the rotating crystal
`method to determine an unknown structure since most materials
`you are likely to crystallize have structures that have been deter(cid:173)
`mined. This will not be true for a newly developed compound, and
`is rarely true for proteins and other biological macromolecules.
`An x-ray method more commonly used is called the powder
`method because, instead of using a single crystal, a very fine
`powder of the crystal is used. This is convenient since you do not
`have to grow a single crystal of the size and quality needed for
`single crystal methods. The powder method reUes on the fact that
`the array of tiny crystals randomly arranged will present all pos(cid:173)
`sible lattice planes present for reflection of the incident mono(cid:173)
`chromatic x-ray beam. The powder pattern of a particular
`substance acts as a signature for that substance so that a powder
`diffraction pattern can be used for identification, chemical analysis
`(presence of impurities), and determining if a material is crystalline
`or amorphous. This is made easier by the fact that a reference
`known as the powder diffraction file (Joint Committee on Powder
`Diffraction Standards 1990) is available with the powder x-ray
`patterns of more than 30,000 materials and is arranged in a way
`that makes searches based on a measured pattern quite possible.
`Description of powder diffraction methods and analysis can be
`found in Bunn (1961) and Cullity (1978). Recently (Engel et al.,
`1999) computer methods have been developed, which can solve
`crystal structures from high quaHty powder patterns. This is of
`great value when it is not possible to grow a crystal of sufficient
`size for single crystal structure determination.
`
`Merck Exhibit 2179, Page 7
`Mylan Pharmaceuticals Inc. v. Merck Sharp & Dohme Corp.
`IPR2020-00040
`
`

`

`38 CRYSTALS, CRYSTAL GROWTH, AND NUCLEATION
`
`Another method of x-ray diffraction is the Laue method,
`which employs a single crystal oriented at a fixed angle to the
`x-ray beam. The beam, however, contains the entire spectrum
`produced by the x-ray tube. In this method, therefore, the angle
`6 in Bragg's law is fixed but a variety of wavelengths are impinging
`on the crystal. Each set of planes that will satisfy Bragg's law with
`a particular wavelength will diffract and form a pattern known as
`a Laue pattern. The Laue method is used as a way to assess crystal
`orientation and to determine crystal quality. Other x-ray techni(cid:173)
`ques are also available, making use of x-ray spectrometers and
`variations on the methods mentioned previously. A number of
`references can supply more details on any of the x-ray methods
`(Cullity 1978; Bunn 1961; Bertin 1975). It is important to remem(cid:173)
`ber that x-ray diffraction is the only unambiguous way to deter(cid:173)
`mine if a material is truly crystalline. In addition, as we will see in
`the next section, x-ray diffraction is often necessary to determine
`whether a material is cocrystallized or has crystallized into more
`than one crystal structure.
`
`2.1.4. POLYMORPHISM
`
`As we have seen in the previous section, crystalline materials can
`be characterized in terms of their crystal structures. A given
`chemical species, however, can have more than one possible crystal
`structure. The phenomena of a chemical species having more than
`one possible crystal form is known as polymorphism. The term
`allotropism is used to describe elements that can form more than
`one crystal form. Materials crystallize into different crystal forms
`as a function of the conditions of growth (temperature, pressure,
`impurity content, growth rate, etc.).
`When a material can crystallize into a different polymorph,
`the chemical nature of the species remains identical, however, the
`physical properties of the material can be different. For example,
`properties such as density, heat capacity, melting point, thermal
`conductivity, and optical activity can vary from one polymorph
`to another. Table 2.3 Hsts common materials that exhibit poly(cid:173)
`morphism. Looking at Table 2.3 we can see that density varies
`significantly for the same materials when the crystal structure has
`changed. In addition, the change in the crystal structure often
`means a change in the external shape of the crystal, which is often
`an important parameter in industrial crystallization that has to be
`controlled. Many substances crystallize into structure in which the
`solvent is present as part of the crystal lattice. These crystals are
`known as solvates (or hydrates when the solvent is water). A
`substance can have multiple solvates with different crystal
`structures as well as a solvent free crystal form with a unique
`crystal structure. The solvates are often referred to as pseudopoly-
`morphs. They are not true polymorphs because of the addition of
`the solvent molecule(s) to the crystal lattice. Conformational
`polymorphism refers
`to
`the situation where
`the molecular
`conformation of the molecules of a given substance are different
`in each polymorph.
`Materials that exhibit polymorphism present an interesting
`problem. First, it is necessary to control conditions to obtain the
`desired polymorph. Second, once the desired polymorph is
`obtained, it is necessary to prevent the transformation of the
`material to another polymorph. Materials that form polymorphs
`often will transform from one form to another. This is known as
`a polymorphic transition. Often a simple change of temperature
`will cause a material to change form. In many cases, a particular
`polymorph is metastable, meaning that after crystaUizing the mater(cid:173)
`ial will eventually transform into a more stable state. This trans(cid:173)
`formation can be relatively rapid in some systems while in others
`it can be infinitely slow. At room temperature, a diamond is a
`metastable form of carbon.
`
`TABLE 2.3 Polymorphic Forms of Some Common Sub(cid:173)
`stances
`
`Element or
`Compound
`
`Chemical
`Composition
`
`Cesium chloride
`
`CsCI
`
`Calcium carbonate
`
`CaCOa
`
`Carbon
`
`Iron
`
`Mercuric iodide
`
`Phosphorus
`
`Fe
`
`Hgl2
`
`Silica
`
`Si02
`
`Sulfur
`
`Tin
`
`Sn
`
`Zinc sulfide
`
`ZnS
`
`Known Polymorphic Forms^
`
`Cubic (CsCI type) (s), d = 3.64
`Cubic (NaCI type) (m), d = 3.64
`Calclte (s), rhombohedral,
`uniaxial, d = 2.71
`Aragonite (m), orthorhombic,
`biaxial, d = 2.94
`Diamond (m), cubic, very
`hard, d = 3.5, covalent
`tetrahedral binding,
`poor conductor
`Graphite (s), hexagonal,
`soft, d = 2.2 layer structure,
`good conductor
`7 iron (m), f.c.c.
`a iron (s) and 6 iron (m), b.c.c.
`Red (s), tetragonal
`Yellow (m), orthorhombic
`White phosphorus (m),
`c/= 1.8, melts 44°C
`Violet phosphorus (s), d = 2.35;
`melts around 600 °C
`Quartz (s) {a and l3 forms),
`d = 2.655
`Tridymite (m) {a and /? forms),
`d = 2.27
`Cristobalite (m) {a and p forms),
`d = 2.30
`a, orthorhombic (s), d = 2.05,
`melts 113 °C
`j3, monoclinic (m), d = 1.93,
`melts 120°C
`White tin, tetragonal, d = 7.286,
`stable above 180 °C
`Gray tin, cubic (diamond type),
`d = 5.80, metastable
`above 180 °C
`Wurtzite (m), hexagonal
`Sphalerite (s),
`cubic (diamond type)
`
`(Data adapted from Verma and Krishna 1966.)
`'^d = density(g/km^); m = metastable and s = stable.
`
`These transitions from one polymorph to another usually
`occur most rapidly when the crystals are suspended in solution,
`however, some materials will undergo transformation when in a
`dry powder form. It is possible, with some effort, to obtain a phase
`diagram that shows where a particular polymorph is stable and
`where it is unstable. When a material has multiple polymorphs and
`one of the polymorphs is the stable form at all temperatures the
`system is known as monotropic. If different polymorphs are stable
`at different temperatures the system is known as enantiotropic. In
`enantiotropic systems, the polymorph with the lowest solubility is
`the stable form. If the solubility is plotted versus temperature in an
`enantiotropic system, the transition temperature will be where the
`solubihty curves cross. This transition temperature is independent
`of solvent used. It is also possible through experiment to determine
`which conditions of growth favors the formation of a particular
`polymorph. In many, if not most industrial crystallizations involv(cid:173)
`ing polymorphs, it is necessary to obtain the same polymorph in all
`cases. If the wrong polymorph is produced, the properties of the
`material would change, making the product unacceptable. A good
`example of this is in the production of precipitated calcium
`carbonate that is used for coating and filler in paper. Calcium
`
`Merck Exhibit 2179, Page 8
`Mylan Pharmaceuticals Inc. v. Merck Sharp & Dohme Corp.
`IPR2020-00040
`
`

`

`2.1. CRYSTALS 39
`
`amorphous solid phase followed by a transformation to the least
`stable crystalline phase and finally transformation to the stable
`phase. While this is observed in a wide range of systems, it is most
`likely to be observed in the crystallization of organic, especially
`higher molecular weight materials. More information on poly(cid:173)
`morphism can be found in Gilh (1992) and Myerson (1999).
`There are some materials that can crystallize into two different
`crystal forms that are mirror images of each other. This is given the
`special name of enantiomorphism. In general, but not always,
`materials that form enantiomorphs also display a property called
`optical activity. This means that the crystal will rotate the plane of
`polarized light that passes through it. Optically active enantio(cid:173)
`morphs are given the designation D (for dextro) and L (for levo),
`which indicate the direction they rotate polarized light (to the right
`or left). When an optically active material crystallizes as a mixture
`of the D and L forms, it is known as a racemic mixture. The
`separation of racemic mixtures into the pure D or L form is an
`important practical industrial problem that is often attacked by
`using differences in properties of the two forms. There are four
`main techniques for the separation of racemic mixtures (Leusen
`1993). Two of the methods involve the use of a chemical that
`interacts specifically with one racemate, either through the rate
`of chemical reaction (kinetic resolution) or by absorption (chro(cid:173)
`matographic separation). The two other methods involve crystal(cid:173)
`lization. In direct crystallization, one racemate is selectively
`crystallized from a solution of the racemic mixture. This can be
`accomplished by seeding with the desired enantiomer or by using
`chiral solvents or additives that will aid the crystallization of one
`enantiomer (or inhibit the crystallization of the other enantiomer).
`Another crystallization method involves the use of an agent that
`will form a complex with each of the enantiomers with the com(cid:173)
`plexes having different solubilities so that one complex can be
`selectively crystallized. This method is known as resolution by
`diasteromeric salts. Examples of optically active enantiomorphs
`are tartaric acid and sodium chlorate, shown in Figures 2.8 and 2.9.
`
`Figure 2.8 (a) D and (b) L tartaric acid. (Reproduced with
`permission from Mullin 1972.)
`
`carbonate can crystallize into three forms: calcite, aragonite, and
`vaterite. For reasons of external crystal shape, calcite is the form
`needed. When aragonite is produced accidentally, which is a
`common occurrence, it is not usable and must be discarded and
`reprocessed.
`In crystallization processes involving a material that displays
`polymorphism, it is quite common for an unstable polymorph to
`appear first and then transform into a stable form. This observa(cid:173)
`tion is summarized by Ostwald's step rule, sometimes referred to as
`the "Law of Successive Reactions," which says that in any process,
`the state which is initially obtained is not the stablest state but the
`least stable state that is closest in terms of free energy change, to
`the original state. What this means, therefore, is that a crystal-
`Hzation process, the initial soHd phase, can be the least stable
`polymorph that will then transform into successively more stable
`forms until the stable form, at the conditions of the system, is
`reached. With some systems this can mean the formation of an
`
`Figure 2.9 L and D sodium chlorate crystal habit (top). Orientation of CIO3 groups
`on 111 faces (bottom). (Reproduced with permission from Bunn 1961.)
`
`Merck Exhibit 2179, Page 9
`Mylan Pharmaceuticals Inc. v. Merck Sharp & Dohme Corp.
`IPR2020-00040
`
`

`

`40 CRYSTALS, CRYSTAL GROWTH, AND NUCLEATION
`
`Figure 2.10 Imperfection in crystals: (a) perfect crystal; (b) substitutional impurity; (c) interstitial impurity;
`(d) Schottky defects; and (e) Frenkel defect. (From R.A. Laudise, The Growth of Single Crystals, © 1970, pp. 12-13.
`Reprinted by permission of Prentice-Hall, Englewood Cliffs, New Jersey.)
`
`2.1.5.
`
`ISOMORPHISM AND SOLID SOLUTIONS
`
`It is quite common for a number of different species to have
`identical atomic structures. This means that the atoms are located
`in the same relative positions in the lattice. We have seen this
`previously with the sodium chloride structure. A number of other
`species have this structure. Obviously, species that have the same
`structure have atoms present in similar stoichiometric proportion.
`Crystals that have the same structure are called isostructural.
`If crystals of different species are isostructural and have the
`same type of bonding, they also will have very similar unit-cell
`dimensions and will macroscopically appear almost identical. This
`is known as isomorphism. Examples of isomorphic materials
`include ammonium and potassium sulfate and KH2PO4 and
`NH4H2PO4. In each of these materials, the potassium and ammo(cid:173)
`nium ions can easily substitute for each other in the lattice since
`they are of almost the same size. This illustrates one of the proper(cid:173)
`ties of isomorphous materials, that is they tend to form soUd
`solutions, or mixed crystals. Crystallization from a solution of
`two isomorphous materials, therefore, can result in a solid with
`varying composition of each species with unit-cell dimensions
`intermediate between the two components. The purification of
`isomorphous substances can, therefore, be difficult.
`SoHd solutions do not all result from the substitution of iso(cid:173)
`morphous materials in the lattice sites, other types of solid solu(cid:173)
`tions are possible and are described in Vainshtein (1981).
`
`2.1.6.
`
`IMPERFECTIONS IN CRYSTALS
`
`In our discussions of the internal structure of crystals, we have
`shown that each atom (or molecule) has a precise location in a
`repeating structure. If this structure is disrupted in some way
`the crystal is said to have imperfections. There are a number of
`different kinds of imperfections that can occur. If a foreign atom
`(or molecule in a molecular crystal) is present in the crystal lattice,
`this is known as a chemical imperfection. The foreign atom can be
`present at a lattice site having substituted for an atom in the
`structure as we saw in our brief discussion of isomorphism and
`soHd solutions. This is called a substitutional impurity. The foreign
`atom can also be present in the crystal by fitting between the atoms
`in the lattice. This is called a interstitial impurity. Both of these
`types of impurities can cause the atoms in the crystal to be slightly
`displaced since the impurity atoms do not really fit in the perfect
`lattice structure. The displacement of the atoms causes a strain in
`the crystal.
`Another type of imperfection is due to vacancies in the crystal.
`A vacancy is simply a lattice site in which there is no atom. A
`
`vacancy caused by the migration of an atom to an interstitial
`region is called a Frenkel defect, while one in which a vacancy is
`just an empty lattice site missing an atom is called a Schottky
`defect. These types of imperfections are very important in semi(cid:173)
`conductors and microelectronics. Figure 2.10 illustrates the vari(cid:173)
`ous types of vacancies and chemical imperfections.
`The imperfections in crystals discussed so far are called point
`defects because they involve a single unit of the crystal structure,
`that is an atom or molecule. Another type of imperfection is
`known as a line defect or dislocation. There are two types of
`dislocations known as edge dislocations and screw dislocations.
`An edge dislocation is illustrated in Figure 2.11, which is a cross
`section of a crystal lattice. Looking at the figure you can see half of
`a vertical row (the bottom half) in the middle of the lattice is missing.
`This row of atoms is missing in each plane of the lattice parallel to
`the page. The dislocation is marked at point A. If