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 – longrange 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 – threedimensional 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 FaceCentered 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
BodyCentered 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 ClosePacked 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 nA VCN A where: n = number of atoms A = atomic weight VC = volume of unit cell NA = Avogadro’s number p=
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 cationanion 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
AXType 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 AA ∑ AC + ∑ ¿ ¿ n' ¿ p=¿ where: n’ – number of formula units ∑ AC – 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 threedimensional arrangements Silica most simple silicate material structurally, it is a 3D 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 SiO 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= (2 u' −v ' ) 3 1 v = (2 v ' −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”
XRay 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=2 d hkl 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 hkl= 2 2 2 √ 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 xray beam is generated at point T and intensities of diffracted beams are detected with a counter labeled C in the figure specimen, xray 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 2D chain like molecular structure functionality: number of bonds a given monomer can form Molecular Weight numberaverage 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 n= ∑ x i M i 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 DP=
Mn 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 sidebranch 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 3D networks distinctive mechanical and thermal properties examples: epoxies, polyurethanes, phenolformaldehyde 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 headtohead 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) syndiotacticc) 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 j m j 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 FaceCentered 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
BodyCentered 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 ClosePacked 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 nA VCN A where: n = number of atoms A = atomic weight VC = volume of unit cell NA = Avogadro’s number p=
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 cationanion 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
AXType 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 AA ∑ AC + ∑ ¿ ¿ n' ¿ p=¿ where: n’ – number of formula units ∑ AC – 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 threedimensional arrangements Silica most simple silicate material structurally, it is a 3D 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 SiO 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= (2 u' −v ' ) 3 1 v = (2 v ' −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”
XRay 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=2 d hkl 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 hkl= 2 2 2 √ 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 xray beam is generated at point T and intensities of diffracted beams are detected with a counter labeled C in the figure specimen, xray 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 2D chain like molecular structure functionality: number of bonds a given monomer can form Molecular Weight numberaverage 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 n= ∑ x i M i 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 DP=
Mn 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 sidebranch 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 3D networks distinctive mechanical and thermal properties examples: epoxies, polyurethanes, phenolformaldehyde 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 headtohead 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) syndiotacticc) 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 j m j 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
