Crystal Structures
Crystal Structures Definitions Crystalline material – one in which atoms are situated in a repeating or periodic array over large atomic distances Amorphous or noncrystalline material – long-‐range atomic order is absent Polymorphous -‐ crystallization into two or more chemically identical but crystallographically distinct forms Crystal structure -‐ manner in which atoms, ions or molecules are spatially arranged Lattice – three-‐dimensional array of points coinciding with atom positions Coordination number – number of touching atoms Atomic Packing Factor (APF) = volume of atoms in unit cell/total unit cell volume Planar Packing Factor (PPF) = area of atoms per face/total face area Linear Packing Factor (LPF) = length of atoms along direction/total length of direction
Unit Cells
-‐smallest repeating entity in a crystal structure -‐usually parallelepipeds or prisms
Metallic Crystal Structure
Face-‐Centered Cubic Crystal Structure (FCC) -‐atoms located at each of the corners and centers of all cube faces -‐example: copper, aluminum, silver, gold -‐spheres (ion cores) touch across a face diagonal 𝑎 = 2𝑅 2 -‐total of four atoms in a given unit cell -‐coordination number: 12 -‐APF: 0.74
Body-‐Centered Cubic Crystal Structure (BCC) -‐atoms located at all eight corners and a single atoms in center -‐atoms touch along cube diagonals 4𝑅 𝑎 = 3 -‐examples: chromium, iron, tungsten -‐two atoms per unit cell -‐coordination number: 8 -‐APF: 0.68
Hexagonal Close-‐Packed Crystal Structure -‐top and bottom faces of unit cell consist of six atoms that form regular hexagons around a center atom -‐another plane provides three additional atoms between top and bottom planes -‐six atoms total in each unit cell (1/6 of the 12 top and bottom corner atoms, ½ of each of the top and bottom center atoms and the 3 interior atoms) -‐ideal c/a value is 1.633 -‐coordination number: 12 -‐APF: 074 (both same as FCC) -‐examples: cadmium, magnesium, titanium, zinc
Density Computations 𝑝= where: n = number of atoms A = atomic weight VC = volume of unit cell NA = Avogadro’s number
𝑛𝐴 𝑉! 𝑁!
Ceramic Crystal Structure -‐since composed of at least two elements, crystal structure is more complex -‐range from purely ionic to totally covalent
-‐for materials in which atomic bonding is predominantly ionic, the crystal structures may be thought of as being composed of electrically charged ions instead of atoms -‐cations: positively charged metallic ions (because cats are happy) -‐anions: negatively charged nonmetallic ions -‐two characteristics of ions influence crystal structure -‐> magnitude of electrical charge on each of the ions -‐> relative sizes of cations and anions -‐crystal must be electrically neutral (ie. cation positive charges must be balanced by an equal number of anion negative charges) -‐cations prefer to have as many nearest neighbor anions as possible -‐therefore, coordination number is related to the cation-‐anion radius ratio -‐for a specific coordination number, there is a critical (minimum) rC/rA ratio Note: relationships between CN and cation anion ratios are based on geometrical considerations and are approximations. Therefore there are some exceptions
AX-‐Type Crystal Structure -‐some common ceramic materials have equal numbers of cations and anions -‐these are referred to as AX compounds, where A denotes the cation and X the anion -‐there are several different crystal structures for AX compounds, each named after a common material that assumes the particular structure Rock Salt Structure (Sodium Chloride NaCl) -‐CN: 6 -‐one cation situated at cube center and ine at the center of each of the 12 cube edges -‐two interpenetrating FCC lattices, one composed of cations, other of anions -‐example: NaCl, MgO, MnS, LiF and FeO -‐FCC
Cesium Chloride Structure -‐coordination number is 8 for both ion types -‐anions are located at each of the corners of the cube, cube center is a single cation -‐this is not a BCC because ions of two different kinds are involved -‐simple cubic
Zinc Blende Structure -‐CN: 4 -‐zinc blende: ZnS
-‐all corner and face positions of cubic cell are occupied by sulfur atoms while zinc atoms fill interior tetrahedral positions -‐equivalent structure results if atom positions are reversed -‐each Zn atom is bonded to four S atoms and vice versa -‐FCC
Fluorite (CaF2) -‐AX2 type -‐ionic radii ratio is about 0.8, therefore CN is 8 -‐calcium ions are positioned at the centers of cubes with fluorine ions at corners -‐crystal structure is similar to CsCl except only half of the center cube positions are occupied by Ca ions -‐one unit cell consists of eight cubes -‐other examples: ZrO2, UO2, PuO2 and ThO2 -‐simple cubic
Perovskite (BaTiO3) -‐Ba ions are situated at all eight corners -‐single Ti is at the cube center -‐Oxygen ions located at the center of each of the six faces -‐FCC Density Computations for Ceramics 𝑛! ( 𝐴! + 𝐴! ) 𝑝 = 𝑉! 𝑁! where: n’ – number of formula units 𝐴! – sum of atomic weights of all cations in formula unit
Silicate Ceramics
-‐materials composed primarily of silicon and oxygen -‐rather than characterizing the crystal structures of these materials in terms of unit cells, it is more convenient to use various arrangements of an SiO4 4-‐ tetrahedron -‐each atom of silicon is bonded to four oxygen atoms, which are situated at the corners of the tetrahedron with silicon atom positioned at the center -‐not considered ionic because there is a significant covalent character which is directional and relatively strong -‐various silicate structures arise from the different ways in which the tetrahedron units can be combined into one, two or three-‐dimensional arrangements Silica -‐most simple silicate material -‐structurally, it is a 3-‐D network generated when the corner oxygen atoms in each tetrahedron are shared by adjacent tetrahedra -‐thus the material is electrically neutral and all atoms have stable electronic strucures -‐ratio of Si to O atoms is 1:2 -‐three primary polymorphic crystalline forms: quartz, cristobalite and trydymite -‐atoms are not closely packed together, relatively low densities -‐high melting temperature of Si-‐O bond Silicates -‐one two or three of the corner oxygen atoms are shared by other tetrahedral
Simple Silicates -‐most structurally simple ones involve isolated tetrahedra -‐Si2O7 ion is formed when two tetrahedral share a common oxygen atom Layered Silicates -‐two dimensional sheet or layered structure can be produced by sharing of three oxygen ions in each of the tetrahedral -‐repeating unit formula may be represented by (Si2O5)2-‐ -‐net charge comes from unbonded oxygen atoms projecting out of plane -‐electroneutrality is ordinarily established by a second planar sheet structure having an excess of cations -‐found in clay and other minerals
Carbon
-‐exists in various polymorphic forms as well as amorphous state -‐does not fall in any of the metal, polymer or ceramic classifications -‐graphite is sometimes classified as ceramic though Diamond -‐metastable carbon polymorph at room temperature and atmospheric pressure -‐crystal structure is a variant of the zinc blende, in which carbon atoms occupy all positions (both Zn and S) -‐bonds are totally covalent, called the diamond cubic crystal structure
Graphite -‐crystal structure is more stable than diamond at ambient temperature and pressure -‐composed of layers of hexagonally arranged carbon atoms, within the layers each carbon atom is bonded to three coplanar neighbor atoms by strong covalent bonds -‐fourth bonding electron participates in a weak van der Waals type of bond
Fullerenes -‐polymorphic form of carbon -‐exists in discrete molecular form and consists of a hollow spherical cluster of sixty carbon atoms, single molecule is denoted by C60 -‐each molecule is composed of groups of carbon atoms that are bonded to each other to form both hexagon and pentagon geometrical configurations -‐pure crystalline solid, packed together in a face centered cubic array -‐electrically insulating but can be made highly conductive
Polymorphism and Allotropy
-‐some metals may have more than one crystal structure, phenomenon called polymorphism -‐when found in elemental solids, condition is called allotropy -‐prevailing crystal structure depends on both temperature and external pressure -‐one example is found in carbon: graphite is stable polymorph at ambient conditions, whereas diamond is formed at extremely high pressures -‐most often, physical properties are modified by a polymorphic transformation
Crystal Systems
-‐lattice parameters: edge lengths (a,b,c) and three interaxial angles(alpha, beta and gamma) -‐seven different possible combinations of a, b and c and alpha, beta and gamma each of which represents a distinct crystal system -‐seven crystal systems are cubic, tetragonal, hexagonal, orthorhombic, rhombohedral (trigonal), monoclinic and triclinic -‐both FCC and BCC structures belong to cubic crystal system -‐HCP is hexagonal
Hexagonal Indices
[u’ v’ w’] -‐> [u v t w]
1 𝑢 = (2𝑢! − 𝑣 ! ) 3 1 𝑣 = (2𝑣 ! − 𝑢! ) 3 𝑡 = −(𝑢 + 𝑣) 𝑤 = 𝑤′
Crystallographic planes -‐if the plane passes through the selected origin than either a parallel plane must be constructed or a new origin must be established in the corner of another unit cell
-‐crystallographic plane either intersects or parallels each of the three axes -‐reciprocals of intersects are taken -‐number a chanted to a set of integers using a common factor -‐indices are enclosed by parantheses -‐for cubic crystals: planes and directions having the same indices are perpendicular to one another -‐family of planes: contains all planes that are crystallographically equivalent (same atomic packing) -‐family is indicated by indices enclosed in braces i.e. {1 0 0} -‐for cubic systems: all planes having the same indices, irrespective of order and sign belong to the same family (example both (1 2 3) and (3 1 2) are part of the (1 2 3) family Hexagonal Crystals -‐equivalent planes have same indices -‐four index (hkil) scheme -‐I is determined by the sum of h and k through I = -‐(h+k) -‐h, k and l indices are identical for both indexing systems Ceramics -‐interstital sites exist in two different types -‐tetrahedral position: four atoms surround one type -‐octahedral position: six ion spheres -‐for each anion sphere, one octahedral and two tetrahedral positions will exist -‐Ceramic crystal structures depend on two factors: stacking of close packed anion layers and manner in which interstitial sites are filled with cations Single Crystal -‐periodic and repeated arrangement of atoms extends throughout the entirety of the specimen without interruption -‐all unit cells interlock the same way and have the same orientation -‐exist in nature but may also be produced artificially Polycrystalline Materials -‐most crystalline solids are composed of a collection of many small crystals or grains -‐small crystals or nuclei form at various positions -‐random crystallographic orientations indicated by the square grids -‐small grains grow y the successive addition from the surrounding liquid of atoms to the structure of each -‐extremities of adjacent grains impinge on one another as the solidification process approaches completion -‐crystallographic orientation varies from grain to grain -‐exists some atomic mismatch within the region where two grains meet (grain boundary)
Anisotropy
-‐physical properties of single crystals depend on the crystallographic direction in which measurements are taken -‐example: elastic modulus, electrical conductivity and index of refraction have different values in the [100] and [111] directions -‐directionality of properties is called anisotropy -‐associated with variance of atomic or ionic spacing with crystallographic direction -‐isotropic: substances in which measured properties are independent of direction -‐extent and magnitude of anisotropic effects in crystalline materials are dependent on the symmetry of the crystal structure -‐degree of anisotropy increases with decreasing structural symmetry – triclinic structures are normally highly anisotropic -‐for many polycrystalline materials, crystallographic orientations of individual grains are total random -‐even though each grain may be anisotropic, specimen composed of the grain aggregate behaves isotropically -‐magnitude of measured property represents some average of the directional values -‐materials with a preferential crystallographic orientation are said to have “texture” -‐magnetic properties of some iron alloys used in transformer cores are anisotropic -‐grains magnetize in a type direction more easily than in any other crystallographic direction -‐energy losses in transformer cores are minimized by utilizing polycrystalline sheets of these alloys into which have been introduced a “magnetic texture”
X-‐Ray Diffraction
-‐diffraction occurs when a wave encounters a series of regularly spaced obstacles that are capable of scattering the wave and have spacings that are comparable in magnitude to the wavelength Bragg’s Law 𝑛𝜆 = 2𝑑!!" 𝑠𝑖𝑛𝜃 where: n – order of reflection d – interplanar spacing (magnitude of distance between two adjacent and parallel planes of atoms) For crystals with cubic symmetry: 𝑎 𝑑!!" = ℎ! + 𝑘 ! + 𝑙 ! Bragg’s law is a necessary but not sufficient condition for diffraction by real crystals -‐specifies when diffraction will occur for unit cells having atoms positioned only at cell corners -‐atoms situated at other sites act as extra scattering centers which can produce out of phase scattering
Diffractometer
-‐apparatus used to determine the angles at which diffraction occurs for powdered specimens -‐specimen S in the form of a flat plate is supported so rotations about the axis labeled O are possible -‐axis is perpendicular to plane of the page -‐monochromatic x-‐ray beam is generated at point T and intensities of diffracted beams are detected with a counter labeled C in the figure -‐specimen, x-‐ray source and counter are all copla the ease with which nar -‐counter is mounted on a movable carriage that may also be rotated about the O axis
Noncrystalline Solids
-‐noncrystalline solids lack a systematic and regular arrangement of atoms over relatively large atomic distances -‐sometimes such materials are also called amorphous -‐whether a crystalline or amorphous solid forms depends on the ease with which a random atomic structure in the liquid can transform to an ordered state during solidification -‐amorphous materials are characterized by atomic or molecular structures that are relatively complex and become ordered only with some difficulty -‐rapidly cooling through freezing temperature favours the formation of a noncrystalline solid since little time is allowed for the ordering process -‐metals normally form crystalline solids -‐inorganic gases are amorphous -‐polymers may be completely noncrystalline consisting of varying degrees of crystallinity -‐silicon dioxide in the noncrystalline state is called fused silica -‐common inorganic glasses are used for containers, windows etc. are silica glasses which have been added to other oxides
Polymer Structures
Polymer Molecules -‐macromolecules: molecules in polymers -‐repeat units: structural entities successively repeated along the chain -‐monomer: small molecule from which a polymer is synthesized
-‐when all repeating units along a chain are of the same type, resulting polymer is a homopolymer -‐chains may be composed of two or more different repeat units called copolymers -‐bifunctional: monomers with an active bond that may react to form two covalent bonds with other monomers forming a 2-‐D chain like molecular structure -‐functionality: number of bonds a given monomer can form Molecular Weight -‐number-‐average molecular weight is obtained by dividing the chains into a series of size ranges and determining the number fraction of chains within each size range 𝑀! =
𝑥! 𝑀!
Where Mi -‐ mean (middle) molecular weight of size range i -‐xi is the fraction of total number of chains within corresponding range Alternative way: degree of polymerization (DP) = average number of repeat units in chain 𝑀! 𝐷𝑃 = 𝑚 Where m is the repeat unit molecular weight -‐polymer properties are affected by the length of polymer chains -‐example: melting or softening temperature increases with increasing molecular weight -‐at room temp: polymers with short chains exist as liquids -‐polymers with weights of approximately 1000 g/mol are waxy solids -‐solid polymers have molecular weights ranging between 10, 000 to several million g/mol Molecular Shape -‐some of the mechanical and thermal characteristics of polymers are a function of the ability of chain segments to experience rotation in response to applied stresses or thermal vibrations -‐rotational flexibility is dependent on repeat unit structure and chemistry -‐example: region of a chain segment with double bond is rotationally rigid Linear Polymers -‐repeat units are joined together end to end in single chains -‐long chains are flexible (may be thought of as a mass of spaghetti) -‐extensive van der Waals and hydrogen bonding -‐some common polymers are polyethylene, PVC, polystyrene, nylon and fluorocarbons
Branched Polymers -‐may be synthesized where side-‐branch chains are connected to the main ones -‐branches may result from side reactions that occur during the synthesis of the polymer -‐chain packing efficiency is reduced with the formation of side branches which results in a lowering of the polymer density -‐polymers that form linear structures may also be branched Crosslinked Polymers -‐adjacent linear chains are joined to one another at various positions by covalent bonds -‐process is achieved during synthesis or a nonreversible chemical reaction -‐crosslinking is accomplished by additive atoms or molecules that are covalently bonded to the chains -‐in rubbers this is called vulcanization Network Polymers -‐multifunctional monomers forming three or more active covalent bonds make 3-‐D networks -‐distinctive mechanical and thermal properties -‐examples: epoxies, polyurethanes, phenol-‐formaldehyde Note: polymers are usually not only one distinctive structural type
Molecular Configurations -‐for polymers having more than one side atom or group bonded to the main chain: regularity and symmetry of side group arrangement can significantly influence the properties
-‐in most polymers, the head to tail configuration predominates, often a polar repulsion occurs between R groups for the head-‐to-‐head configuration -‐isomerism is found in polymer molecules where different configurations are possible for same composition Stereoisomerism -‐denotes situation in which atoms are linked together in the same order but differ in spatial arrangement -‐specific polymer has many configurations; predominant form depends on method of synthesis
a) isotactic b) syndiotactic c) atactic Thermoplastic Polymers -‐behaviour in response to rising temperature -‐thermoplastics soften when heated, harden when cooled -‐molecular level: as temperature increases, secondary bonding forces are diminished so that the relative movement of adjacent chains is facilitated when stress is applied -‐irreversible degradation occurs when a molten thermoplastic is raised to too high of a temperature -‐most linear polymers and those having branched structures with flexible chains are thermoplastic -‐these materials are normally fabricated by simultaneous application of heat and pressure -‐examples of common thermoplastic polymers include polyethylene, PVC and polystyrene Thermosetting Polymers -‐network polymers -‐become permanently hard during formation, do not soften upon heating -‐covalent crosslinks between adjacent molecular chains
-‐during heat treatment, bonds anchor chains together to resist the vibrational and rotational chain motions at high temperatures -‐10 – 50% of chain repeat units are crosslinked -‐heating to excessive temperatures causes severance of crosslink bonds and polymer degradation -‐generally harder and stronger than thermoplastics -‐better dimensional stability Copolymers 𝑚 =
𝑓! 𝑚!
where: f – mole fraction m – molecular weight of repeat unit j
Polymer Crystallinity -‐may exist in polymeric materials -‐atomic arrangements will be more complex for polymers -‐packing of molecular chains to produce an ordered atomic array -‐molecular substances with small molecules are normally either totally crystalline or totally amorphous -‐degree of crystallinity by weight can be determined according to:
-‐depends on rate of cooling during solidification as well as chain configuration -‐during crystallization upon cooling through the melting temperature, chains assume an ordered configuration -‐for linear polymers, crystallization is easily accomplished because there are few restrictions to prevent chain alignment -‐branched polymers are never highly crystalline -‐network and crosslinked polymers are almost completely amorphous because crosslinks prevent polymer chains from rearranging and aligning -‐atactic polymers are difficult to crystallize -‐isotactic and syndiotactic crystallize much more easily because of regularity of geometry -‐copolymers: the more irregular and random, the greater the tendency for development of noncrystallinity -‐alternating/block copolymers have some likelihood of crystallization -‐crystalline polymers are usually stronger and more resistant to dissolution and softening by heat
Unit Cells
-‐smallest repeating entity in a crystal structure -‐usually parallelepipeds or prisms
Metallic Crystal Structure
Face-‐Centered Cubic Crystal Structure (FCC) -‐atoms located at each of the corners and centers of all cube faces -‐example: copper, aluminum, silver, gold -‐spheres (ion cores) touch across a face diagonal 𝑎 = 2𝑅 2 -‐total of four atoms in a given unit cell -‐coordination number: 12 -‐APF: 0.74
Body-‐Centered Cubic Crystal Structure (BCC) -‐atoms located at all eight corners and a single atoms in center -‐atoms touch along cube diagonals 4𝑅 𝑎 = 3 -‐examples: chromium, iron, tungsten -‐two atoms per unit cell -‐coordination number: 8 -‐APF: 0.68
Hexagonal Close-‐Packed Crystal Structure -‐top and bottom faces of unit cell consist of six atoms that form regular hexagons around a center atom -‐another plane provides three additional atoms between top and bottom planes -‐six atoms total in each unit cell (1/6 of the 12 top and bottom corner atoms, ½ of each of the top and bottom center atoms and the 3 interior atoms) -‐ideal c/a value is 1.633 -‐coordination number: 12 -‐APF: 074 (both same as FCC) -‐examples: cadmium, magnesium, titanium, zinc
Density Computations 𝑝= where: n = number of atoms A = atomic weight VC = volume of unit cell NA = Avogadro’s number
𝑛𝐴 𝑉! 𝑁!
Ceramic Crystal Structure -‐since composed of at least two elements, crystal structure is more complex -‐range from purely ionic to totally covalent
-‐for materials in which atomic bonding is predominantly ionic, the crystal structures may be thought of as being composed of electrically charged ions instead of atoms -‐cations: positively charged metallic ions (because cats are happy) -‐anions: negatively charged nonmetallic ions -‐two characteristics of ions influence crystal structure -‐> magnitude of electrical charge on each of the ions -‐> relative sizes of cations and anions -‐crystal must be electrically neutral (ie. cation positive charges must be balanced by an equal number of anion negative charges) -‐cations prefer to have as many nearest neighbor anions as possible -‐therefore, coordination number is related to the cation-‐anion radius ratio -‐for a specific coordination number, there is a critical (minimum) rC/rA ratio Note: relationships between CN and cation anion ratios are based on geometrical considerations and are approximations. Therefore there are some exceptions
AX-‐Type Crystal Structure -‐some common ceramic materials have equal numbers of cations and anions -‐these are referred to as AX compounds, where A denotes the cation and X the anion -‐there are several different crystal structures for AX compounds, each named after a common material that assumes the particular structure Rock Salt Structure (Sodium Chloride NaCl) -‐CN: 6 -‐one cation situated at cube center and ine at the center of each of the 12 cube edges -‐two interpenetrating FCC lattices, one composed of cations, other of anions -‐example: NaCl, MgO, MnS, LiF and FeO -‐FCC
Cesium Chloride Structure -‐coordination number is 8 for both ion types -‐anions are located at each of the corners of the cube, cube center is a single cation -‐this is not a BCC because ions of two different kinds are involved -‐simple cubic
Zinc Blende Structure -‐CN: 4 -‐zinc blende: ZnS
-‐all corner and face positions of cubic cell are occupied by sulfur atoms while zinc atoms fill interior tetrahedral positions -‐equivalent structure results if atom positions are reversed -‐each Zn atom is bonded to four S atoms and vice versa -‐FCC
Fluorite (CaF2) -‐AX2 type -‐ionic radii ratio is about 0.8, therefore CN is 8 -‐calcium ions are positioned at the centers of cubes with fluorine ions at corners -‐crystal structure is similar to CsCl except only half of the center cube positions are occupied by Ca ions -‐one unit cell consists of eight cubes -‐other examples: ZrO2, UO2, PuO2 and ThO2 -‐simple cubic
Perovskite (BaTiO3) -‐Ba ions are situated at all eight corners -‐single Ti is at the cube center -‐Oxygen ions located at the center of each of the six faces -‐FCC Density Computations for Ceramics 𝑛! ( 𝐴! + 𝐴! ) 𝑝 = 𝑉! 𝑁! where: n’ – number of formula units 𝐴! – sum of atomic weights of all cations in formula unit
Silicate Ceramics
-‐materials composed primarily of silicon and oxygen -‐rather than characterizing the crystal structures of these materials in terms of unit cells, it is more convenient to use various arrangements of an SiO4 4-‐ tetrahedron -‐each atom of silicon is bonded to four oxygen atoms, which are situated at the corners of the tetrahedron with silicon atom positioned at the center -‐not considered ionic because there is a significant covalent character which is directional and relatively strong -‐various silicate structures arise from the different ways in which the tetrahedron units can be combined into one, two or three-‐dimensional arrangements Silica -‐most simple silicate material -‐structurally, it is a 3-‐D network generated when the corner oxygen atoms in each tetrahedron are shared by adjacent tetrahedra -‐thus the material is electrically neutral and all atoms have stable electronic strucures -‐ratio of Si to O atoms is 1:2 -‐three primary polymorphic crystalline forms: quartz, cristobalite and trydymite -‐atoms are not closely packed together, relatively low densities -‐high melting temperature of Si-‐O bond Silicates -‐one two or three of the corner oxygen atoms are shared by other tetrahedral
Simple Silicates -‐most structurally simple ones involve isolated tetrahedra -‐Si2O7 ion is formed when two tetrahedral share a common oxygen atom Layered Silicates -‐two dimensional sheet or layered structure can be produced by sharing of three oxygen ions in each of the tetrahedral -‐repeating unit formula may be represented by (Si2O5)2-‐ -‐net charge comes from unbonded oxygen atoms projecting out of plane -‐electroneutrality is ordinarily established by a second planar sheet structure having an excess of cations -‐found in clay and other minerals
Carbon
-‐exists in various polymorphic forms as well as amorphous state -‐does not fall in any of the metal, polymer or ceramic classifications -‐graphite is sometimes classified as ceramic though Diamond -‐metastable carbon polymorph at room temperature and atmospheric pressure -‐crystal structure is a variant of the zinc blende, in which carbon atoms occupy all positions (both Zn and S) -‐bonds are totally covalent, called the diamond cubic crystal structure
Graphite -‐crystal structure is more stable than diamond at ambient temperature and pressure -‐composed of layers of hexagonally arranged carbon atoms, within the layers each carbon atom is bonded to three coplanar neighbor atoms by strong covalent bonds -‐fourth bonding electron participates in a weak van der Waals type of bond
Fullerenes -‐polymorphic form of carbon -‐exists in discrete molecular form and consists of a hollow spherical cluster of sixty carbon atoms, single molecule is denoted by C60 -‐each molecule is composed of groups of carbon atoms that are bonded to each other to form both hexagon and pentagon geometrical configurations -‐pure crystalline solid, packed together in a face centered cubic array -‐electrically insulating but can be made highly conductive
Polymorphism and Allotropy
-‐some metals may have more than one crystal structure, phenomenon called polymorphism -‐when found in elemental solids, condition is called allotropy -‐prevailing crystal structure depends on both temperature and external pressure -‐one example is found in carbon: graphite is stable polymorph at ambient conditions, whereas diamond is formed at extremely high pressures -‐most often, physical properties are modified by a polymorphic transformation
Crystal Systems
-‐lattice parameters: edge lengths (a,b,c) and three interaxial angles(alpha, beta and gamma) -‐seven different possible combinations of a, b and c and alpha, beta and gamma each of which represents a distinct crystal system -‐seven crystal systems are cubic, tetragonal, hexagonal, orthorhombic, rhombohedral (trigonal), monoclinic and triclinic -‐both FCC and BCC structures belong to cubic crystal system -‐HCP is hexagonal
Hexagonal Indices
[u’ v’ w’] -‐> [u v t w]
1 𝑢 = (2𝑢! − 𝑣 ! ) 3 1 𝑣 = (2𝑣 ! − 𝑢! ) 3 𝑡 = −(𝑢 + 𝑣) 𝑤 = 𝑤′
Crystallographic planes -‐if the plane passes through the selected origin than either a parallel plane must be constructed or a new origin must be established in the corner of another unit cell
-‐crystallographic plane either intersects or parallels each of the three axes -‐reciprocals of intersects are taken -‐number a chanted to a set of integers using a common factor -‐indices are enclosed by parantheses -‐for cubic crystals: planes and directions having the same indices are perpendicular to one another -‐family of planes: contains all planes that are crystallographically equivalent (same atomic packing) -‐family is indicated by indices enclosed in braces i.e. {1 0 0} -‐for cubic systems: all planes having the same indices, irrespective of order and sign belong to the same family (example both (1 2 3) and (3 1 2) are part of the (1 2 3) family Hexagonal Crystals -‐equivalent planes have same indices -‐four index (hkil) scheme -‐I is determined by the sum of h and k through I = -‐(h+k) -‐h, k and l indices are identical for both indexing systems Ceramics -‐interstital sites exist in two different types -‐tetrahedral position: four atoms surround one type -‐octahedral position: six ion spheres -‐for each anion sphere, one octahedral and two tetrahedral positions will exist -‐Ceramic crystal structures depend on two factors: stacking of close packed anion layers and manner in which interstitial sites are filled with cations Single Crystal -‐periodic and repeated arrangement of atoms extends throughout the entirety of the specimen without interruption -‐all unit cells interlock the same way and have the same orientation -‐exist in nature but may also be produced artificially Polycrystalline Materials -‐most crystalline solids are composed of a collection of many small crystals or grains -‐small crystals or nuclei form at various positions -‐random crystallographic orientations indicated by the square grids -‐small grains grow y the successive addition from the surrounding liquid of atoms to the structure of each -‐extremities of adjacent grains impinge on one another as the solidification process approaches completion -‐crystallographic orientation varies from grain to grain -‐exists some atomic mismatch within the region where two grains meet (grain boundary)
Anisotropy
-‐physical properties of single crystals depend on the crystallographic direction in which measurements are taken -‐example: elastic modulus, electrical conductivity and index of refraction have different values in the [100] and [111] directions -‐directionality of properties is called anisotropy -‐associated with variance of atomic or ionic spacing with crystallographic direction -‐isotropic: substances in which measured properties are independent of direction -‐extent and magnitude of anisotropic effects in crystalline materials are dependent on the symmetry of the crystal structure -‐degree of anisotropy increases with decreasing structural symmetry – triclinic structures are normally highly anisotropic -‐for many polycrystalline materials, crystallographic orientations of individual grains are total random -‐even though each grain may be anisotropic, specimen composed of the grain aggregate behaves isotropically -‐magnitude of measured property represents some average of the directional values -‐materials with a preferential crystallographic orientation are said to have “texture” -‐magnetic properties of some iron alloys used in transformer cores are anisotropic -‐grains magnetize in a type direction more easily than in any other crystallographic direction -‐energy losses in transformer cores are minimized by utilizing polycrystalline sheets of these alloys into which have been introduced a “magnetic texture”
X-‐Ray Diffraction
-‐diffraction occurs when a wave encounters a series of regularly spaced obstacles that are capable of scattering the wave and have spacings that are comparable in magnitude to the wavelength Bragg’s Law 𝑛𝜆 = 2𝑑!!" 𝑠𝑖𝑛𝜃 where: n – order of reflection d – interplanar spacing (magnitude of distance between two adjacent and parallel planes of atoms) For crystals with cubic symmetry: 𝑎 𝑑!!" = ℎ! + 𝑘 ! + 𝑙 ! Bragg’s law is a necessary but not sufficient condition for diffraction by real crystals -‐specifies when diffraction will occur for unit cells having atoms positioned only at cell corners -‐atoms situated at other sites act as extra scattering centers which can produce out of phase scattering
Diffractometer
-‐apparatus used to determine the angles at which diffraction occurs for powdered specimens -‐specimen S in the form of a flat plate is supported so rotations about the axis labeled O are possible -‐axis is perpendicular to plane of the page -‐monochromatic x-‐ray beam is generated at point T and intensities of diffracted beams are detected with a counter labeled C in the figure -‐specimen, x-‐ray source and counter are all copla the ease with which nar -‐counter is mounted on a movable carriage that may also be rotated about the O axis
Noncrystalline Solids
-‐noncrystalline solids lack a systematic and regular arrangement of atoms over relatively large atomic distances -‐sometimes such materials are also called amorphous -‐whether a crystalline or amorphous solid forms depends on the ease with which a random atomic structure in the liquid can transform to an ordered state during solidification -‐amorphous materials are characterized by atomic or molecular structures that are relatively complex and become ordered only with some difficulty -‐rapidly cooling through freezing temperature favours the formation of a noncrystalline solid since little time is allowed for the ordering process -‐metals normally form crystalline solids -‐inorganic gases are amorphous -‐polymers may be completely noncrystalline consisting of varying degrees of crystallinity -‐silicon dioxide in the noncrystalline state is called fused silica -‐common inorganic glasses are used for containers, windows etc. are silica glasses which have been added to other oxides
Polymer Structures
Polymer Molecules -‐macromolecules: molecules in polymers -‐repeat units: structural entities successively repeated along the chain -‐monomer: small molecule from which a polymer is synthesized
-‐when all repeating units along a chain are of the same type, resulting polymer is a homopolymer -‐chains may be composed of two or more different repeat units called copolymers -‐bifunctional: monomers with an active bond that may react to form two covalent bonds with other monomers forming a 2-‐D chain like molecular structure -‐functionality: number of bonds a given monomer can form Molecular Weight -‐number-‐average molecular weight is obtained by dividing the chains into a series of size ranges and determining the number fraction of chains within each size range 𝑀! =
𝑥! 𝑀!
Where Mi -‐ mean (middle) molecular weight of size range i -‐xi is the fraction of total number of chains within corresponding range Alternative way: degree of polymerization (DP) = average number of repeat units in chain 𝑀! 𝐷𝑃 = 𝑚 Where m is the repeat unit molecular weight -‐polymer properties are affected by the length of polymer chains -‐example: melting or softening temperature increases with increasing molecular weight -‐at room temp: polymers with short chains exist as liquids -‐polymers with weights of approximately 1000 g/mol are waxy solids -‐solid polymers have molecular weights ranging between 10, 000 to several million g/mol Molecular Shape -‐some of the mechanical and thermal characteristics of polymers are a function of the ability of chain segments to experience rotation in response to applied stresses or thermal vibrations -‐rotational flexibility is dependent on repeat unit structure and chemistry -‐example: region of a chain segment with double bond is rotationally rigid Linear Polymers -‐repeat units are joined together end to end in single chains -‐long chains are flexible (may be thought of as a mass of spaghetti) -‐extensive van der Waals and hydrogen bonding -‐some common polymers are polyethylene, PVC, polystyrene, nylon and fluorocarbons
Branched Polymers -‐may be synthesized where side-‐branch chains are connected to the main ones -‐branches may result from side reactions that occur during the synthesis of the polymer -‐chain packing efficiency is reduced with the formation of side branches which results in a lowering of the polymer density -‐polymers that form linear structures may also be branched Crosslinked Polymers -‐adjacent linear chains are joined to one another at various positions by covalent bonds -‐process is achieved during synthesis or a nonreversible chemical reaction -‐crosslinking is accomplished by additive atoms or molecules that are covalently bonded to the chains -‐in rubbers this is called vulcanization Network Polymers -‐multifunctional monomers forming three or more active covalent bonds make 3-‐D networks -‐distinctive mechanical and thermal properties -‐examples: epoxies, polyurethanes, phenol-‐formaldehyde Note: polymers are usually not only one distinctive structural type
Molecular Configurations -‐for polymers having more than one side atom or group bonded to the main chain: regularity and symmetry of side group arrangement can significantly influence the properties
-‐in most polymers, the head to tail configuration predominates, often a polar repulsion occurs between R groups for the head-‐to-‐head configuration -‐isomerism is found in polymer molecules where different configurations are possible for same composition Stereoisomerism -‐denotes situation in which atoms are linked together in the same order but differ in spatial arrangement -‐specific polymer has many configurations; predominant form depends on method of synthesis
a) isotactic b) syndiotactic c) atactic Thermoplastic Polymers -‐behaviour in response to rising temperature -‐thermoplastics soften when heated, harden when cooled -‐molecular level: as temperature increases, secondary bonding forces are diminished so that the relative movement of adjacent chains is facilitated when stress is applied -‐irreversible degradation occurs when a molten thermoplastic is raised to too high of a temperature -‐most linear polymers and those having branched structures with flexible chains are thermoplastic -‐these materials are normally fabricated by simultaneous application of heat and pressure -‐examples of common thermoplastic polymers include polyethylene, PVC and polystyrene Thermosetting Polymers -‐network polymers -‐become permanently hard during formation, do not soften upon heating -‐covalent crosslinks between adjacent molecular chains
-‐during heat treatment, bonds anchor chains together to resist the vibrational and rotational chain motions at high temperatures -‐10 – 50% of chain repeat units are crosslinked -‐heating to excessive temperatures causes severance of crosslink bonds and polymer degradation -‐generally harder and stronger than thermoplastics -‐better dimensional stability Copolymers 𝑚 =
𝑓! 𝑚!
where: f – mole fraction m – molecular weight of repeat unit j
Polymer Crystallinity -‐may exist in polymeric materials -‐atomic arrangements will be more complex for polymers -‐packing of molecular chains to produce an ordered atomic array -‐molecular substances with small molecules are normally either totally crystalline or totally amorphous -‐degree of crystallinity by weight can be determined according to:
-‐depends on rate of cooling during solidification as well as chain configuration -‐during crystallization upon cooling through the melting temperature, chains assume an ordered configuration -‐for linear polymers, crystallization is easily accomplished because there are few restrictions to prevent chain alignment -‐branched polymers are never highly crystalline -‐network and crosslinked polymers are almost completely amorphous because crosslinks prevent polymer chains from rearranging and aligning -‐atactic polymers are difficult to crystallize -‐isotactic and syndiotactic crystallize much more easily because of regularity of geometry -‐copolymers: the more irregular and random, the greater the tendency for development of noncrystallinity -‐alternating/block copolymers have some likelihood of crystallization -‐crystalline polymers are usually stronger and more resistant to dissolution and softening by heat
