Imperfections in Solids PDF

CHAPTER 4: IMPERFECTIONS IN SOLIDS ISSUES TO ADDRESS... What are the solidification mechanisms? What types of defects in solids? Can the number and type of defects be varied and controlled? How do defects affect material properties? Are defects undesirable?...

CHAPTER 4: IMPERFECTIONS IN SOLIDS ISSUES TO ADDRESS... What are the solidification mechanisms? What types of defects in solids? Can the number and type of defects be varied and controlled? How do defects affect material properties? Are defects undesirable? Chapter 4 - 1 Nothing is perfect. The crystalline structures that we have looked at all have imperfections. We will quantify these imperfections here. Chapter 4 - 2  Imperfections in Solids There is no such thing as a perfect crystal. What are these imperfections? Why are they important? Many of the important properties of materials are due to the presence of imperfections. Chapter 4 - 3 Crystalline Defects A crystalline defect is a lattice irregularity having one or more of its dimensions on the order of an atomic dimension. Chapter 4 - 4  Imperfections in Solids Solidification- result of casting of molten material( metals and alloys) the size and shape of the structure depends on the cooling rate – 2 steps Nuclei form Nuclei grow to form crystals – grain structure Start with a molten material – all liquid nuclei crystals growing grain structure liquid Crystals grow until they meet each other Chapter 4 - 5  Defect are created during the processing of materials Chapter 4 - 6  (1) POINT DEFECTS Vacancies: -vacant atomic sites in a structure. Vacancy distortion of planes Self-Interstitials: -"extra" atoms positioned between atomic sites. self- distortion interstitial of planes Chapter 4 - 3  Impurites: May be intertional or un intertional Ex, carbon added in small amount to iron make steel,which is stronger than pure iron Ex,Poron added to silicon to change its electrical properties Chapter 4 - 8  Impurities in Solids “Alloys”: impurity atoms are added intentionally to modify specific properties of the material. “Solvent” is the material with the higher concentration. “Solute” is the element present in the minor concentration, also called “impurity”. “Solid Solution” forms when, as the solute atoms are added to the host material, the crystal structure is maintained. It is compositionally homogeneous Chapter 4 -  The addition of impurities atoms to a impurities metals will result in the formation of a solid solution and /or a new second phase ,depending on the kind of impurities ,its concentration, and the temperature on alloy. Chapter 4 - 10  Solid solution maybe substitutional or interstitial depending in atomic radii in solute and host atoms Chapter 4 - 11  POINT DEFECTS IN ALLOYS Two outcomes if impurity (B) added to host (A): Solid solution of B in A (i.e., random dist. of point defects) OR Substitutional alloy Interstitial alloy (e.g., Cu in Ni) (e.g., C in Fe) Solid solution of B in A plus particles of a new phase (usually for a larger amount of B) Second phase particle --different composition --often different structure. Chapter 4 - 8 Impurities defects: Substitutional or interstitial - Substitutional : impurity atoms type A substitutional atoms type B in structure - Interstitial : atoms type A fill the voids or interstices among the atoms type B substit Chapter 4 - 13 interstitial  The Degree of Solubility(substitutional)depend on: The solvency of solutes in solvents depends on: – Atomic Size Factor (±15%) – Crystal Structure (same) – Electronegativity (low) – Valences (solvents have lower valency) Chapter 4 -  Copper and nickle have the same complet solubility in one another at all properity: – Atomic Size Factor (±15%) Atomic radii cu=.128nm, ni=.125 nm – Crystal Structure (same) Both Cu and ni have fcc crystal strucure – Electronegativity (low) Cu=1.9, ni =1.8 – Valences 1 and 2 for cu,and 2 for ni Chapter 4 - 15  Imperfections in Solids Dislocations are visible in electron micrographs Chapter 4 - 16  Types of Imperfections Point defects Vacancy atoms Interstitial atoms Impurities defects ( for solid solution) Line defects ( dislocation) Edge dislocation Screw dislocation mixed dislocation Chapter 4 - 17  TYPES OF IMPERFECTIONS 1. Point defects Vacancy atoms Interstitial atoms Substitutional atoms 2. Line defects Dislocations - Edge Dislocations - Screw Dislocation - mixed dislocation 3. Area defects Grain Boundaries 4. Volume defects Cracks, Pores, Inclusions Chapter 4 - 2  Line Defects Dislocations: are line defects, slip between crystal planes result when dislocations move, produce permanent (plastic) deformation. Schematic of Zinc (HCP): before deformation after tensile elongation slip steps Chapter 4 - 19  Imperfections in Solids Linear Defects (Dislocations) – Are one-dimensional defects around which atoms are misaligned Edge dislocation: – extra half-plane of atoms inserted in a crystal structure – b  to dislocation line Screw dislocation: – spiral planar ramp resulting from shear deformation – b  to dislocation line Burger’s vector, b: measure of lattice distortion Chapter 4 - 20  Imperfections in Solids Edge Dislocation Fig. 4.3, Callister 7e. Chapter 4 - 21  Motion of Edge Dislocation Dislocation motion requires the successive bumping of a half plane of atoms (from left to right here). Bonds across the slipping planes are broken and remade in succession. Atomic view of edge dislocation motion from left to right as a crystal is sheared. Chapter 4 - 22  Imperfections in Solids Screw Dislocation Screw Dislocation b Dislocation line Burgers vector b (b) (a) Chapter 4 - 23 Edge, Screw, and Mixed Dislocations Mixed Edge Screw Chapter 4 - 24  (2) LINEAR DEFECTS 2.a. Edge Dislocations: an extra portion if a plane of atoms, or half-plane, the edge of which terminates within the crystal. Chapter 4 - 2.b. Screw Dislocation: Shear stresses cause shifts between two parts of the crystal. Chapter 4 - 2.c. Mixed Dislocations: Edge and Screw dislocations combined. Chapter 4 -  Area (interfacial)Defects in Solids External surfaces one of most obvious boundaries, along which the crystal structure terminates. Surface atoms are not bonded to the maximum number of nearest neighbors, and are therefore in a higher energy state than the atoms at interior positions Grain boundaries occur where the crystallographic direction of the lattice abruptly changes. This usually occurs when two crystals begin growing separately and then meet. Chapter 4 - 28  (3) INTERFACIAL/AREA DEFECTS: GRAIN BOUNDARIES Grain boundaries: are boundaries between crystals. are produced by the solidification process, for example. have a change in crystal orientation across them. impede dislocation motion. Metal Ingot Schematic ~ 8cm grain boundaries heat flow Adapted from Fig. 4.10, Callister 6e. (Fig. Adapted from Fig. 4.7, Callister 6e. 4.10 is from Metals Handbook, Vol. 9, 9th edition, Metallography and Microstructures, Am. Society for Metals, Metals Park, OH, 1985.) Chapter 4 - 15  The grain boundary are classified according to the angel of misorientation between the grains to (low and high) angle grain boundary Chapter 4 - 30  Polycrystalline Materials Grain Boundaries regions between crystals transition from lattice of one region to that of the other slightly disordered low density in grain boundaries Chapter 4 - 31  Grain boundaries have higher energy than the grains themselves. The magnitude of the grain boundary energy is proportional to the angel of misorientation, being larger for high angel boundary Chapter 4 - 32  Thus the grain boundaries are more chemically reactive than the grains themselves. Impurity atoms often preferentially segregate along these boundaries because of their high energy The total energy is lower in course grained materials , Than the fine grained one, since there is less boundary area in the former. Chapter 4 - 33  COMPOSITION Definition: Amount of impurity (B) and host (A) in the system. Two descriptions: Weight % Atom % Conversion between wt % and at% in an A-B alloy: C'BAB CB = x 100 C'AAA + C'BAB Basis for conversion: Chapter 4 - 10  ALLOYING A SURFACE Low energy electron microscope view of a (111) surface of Cu. Sn islands move along the surface and "alloy" the Cu with Sn atoms, to make "bronze". The islands continually move into "unalloyed" regions and leave tiny Reprinted with permission from: A.K. Schmid, bronze particles in N.C. Bartelt, and R.Q. Hwang, "Alloying at Surfaces by the Migration of Reactive Two- their wake. Dimensional Islands", Science, Vol. 290, No. 5496, pp. 1561-64 (2000). Field of view is 1.5 Eventually, the islands mm and the temperature is 290K. disappear. Chapter 4 - 9  BOND BREAKING AND REMAKING Dislocation motion requires the successive bumping of a half plane of atoms (from left to right here). Bonds across the slipping planes are broken and remade in succession. Atomic view of edge dislocation motion from left to right as a crystal is sheared. (Courtesy P.M. Anderson) Chapter 4 - 13 DISLOCATIONS & CRYSTAL STRUCTURE Structure: close-packed view onto two planes & directions close-packed planes. are preferred. Comparison among crystal structures: FCC: many close-packed planes/directions; HCP: only one plane, 3 directions; BCC: none Results of tensile Mg (HCP) testing. tensile direction Al (FCC) Chapter 4 - 14  (3) INTERFACIAL/AREA DEFECTS: GRAIN BOUNDARIES Grain boundaries: are boundaries between crystals. are produced by the solidification process, for example. have a change in crystal orientation across them. impede dislocation motion. Metal Ingot Schematic ~ 8cm grain boundaries heat flow Adapted from Fig. 4.10, Callister 6e. (Fig. Adapted from Fig. 4.7, Callister 6e. 4.10 is from Metals Handbook, Vol. 9, 9th edition, Metallography and Microstructures, Am. Society for Metals, Metals Park, OH, 1985.) Chapter 4 - 15 Twin Boundaries Chapter 4 -  Solidification Grains can be - equiaxed (roughly same size in all directions) - columnar (elongated grains) ~ 8 cm heat flow Shell of Columnar in equiaxed grains area with less due to rapid undercooling cooling (greater T) near wall Grain Refiner - added to make smaller, more uniform, equiaxed grains. Chapter 4 - 40  Planar (interfacial)Defects in Solids External surfaces one of most obvious boundaries, along which the crystal structure terminates. Surface atoms are not bonded to the maximum number of nearest neighbors, and are therefore in a higher energy state than the atoms at interior positions Grain boundaries occur where the crystallographic direction of the lattice abruptly changes. This usually occurs when two crystals begin growing separately and then meet. Chapter 4 - 41  Planar Defects in Solids twin boundary (plane) – Essentially a reflection of atom positions across the twin plane. there is a specific mirror lattice symmetry; that is, atoms on one side of the boundary are located in mirror-image positions of the atoms on the other side. The region of material between these boundaries is appropriately termed a twin Chapter 4 - 42 Stacking faults: occur in a number of crystal structures, but the common example is in close-packed structures. Face centered cubic (FCC) structures differ from hexagonal close packed (HCP) structures only in stacking order For FCC metals an error in ABCABC packing sequence Ex: ABCABABC Anti phase boundaries: occur in ordered alloys: in this case, the crystallographic direction remains the same, each side of the boundary has an opposite phase: For example if the ordering is usually ABABABAB, an anti phase boundary takes the form of ABABBABA Chapter 4 - 43  problem For each of the following stacking sequences found in FCC metals, cite the type of planar defect that exists: a) …..A B C A B C B A C B A b) ……A B C A B C B C A B C Chapter 4 - 44 Solution: a) The interfacial defect that exists for this stacking sequence is a twin boundary, which occurs at the indicated position The stacking sequence on one side of this position is mirrored on the other side b) The interfacial defect that exists within this FCC stacking sequence is a stacking fault, which occurs between the two lines. Within this region, the stacking sequence is HCP.Chapter 4 - 45  Bulk(volume) defects These include pores, cracks, foreign inclusions, and other phases. They are normally introduced during processing and fabrication steps Voids are small regions where there are no atoms, and can be thought of as clusters of vacancies. Impurities can cluster together to form small regions of a different phase. These are often called precipitates. Chapter 4 - 46  Microscopic Examination Crystallites (grains) and grain boundaries. Vary considerably in size. Can be quite large – ex: Large single crystal of diamond – ex: Aluminum light garbage can see the individual grains Crystallites (grains) can be quite small (mm or less) – necessary to observe with a microscope. Chapter 4 - 47  Microscopic Examination Several important applications of micro-structural examinations are as follows: To ensure that the association between the properties and structure ( and defects ) are properly understood To predict the properties of materials once these relationship have been established To design the alloys with new property combinations To determine whether or not a material has been correctly heat treated To ascertain the mode of mechanical fracture Chapter 4 - 48  Optical Microscopy Useful up to 2000X magnification. Polishing removes surface features (e.g., scratches) Etching changes reflectance, depending on crystal orientation. the chemical reactivity of the grains of some crystallographic planes single-phase materials depends on crystallographic Micrograph of brass (a Cu-Zn alloy) orientation 0.75mm Chapter 4 - 49  Optical Microscopy Grain boundaries... are imperfections, are more susceptible to etching, may be revealed as polished surface dark lines, change in crystal surface groove orientation across grain boundary (a) boundary. Fe-Cr alloy (b) Chapter 4 - 50  Optical Microscopy Polarized light – metallographic scopes often use polarized light to increase contrast – Also used for transparent samples such as polymers Chapter 4 - 51  Microscopy Optical resolution ca. 10-7 m = 0.1 mm = 100 nm For higher resolution need higher frequency – X-Rays? Difficult to focus. – Electrons wavelengths ca. 3 pm (0.003 nm) – (Magnification - 1,000,000X) Atomic resolution possible Electron beam focused by magnetic lenses. Chapter 4 - 52  Scanning Tunneling Microscopy (STM) Atoms can be arranged and imaged! Carbon monoxide Iron atoms arranged molecules arranged on a copper (111) on a platinum (111) surface. These Kanji surface. characters represent the word “atom”. Chapter 4 - 53  Summary Point, Line, and Area defects exist in solids. The number and type of defects can be varied and controlled (e.g., T controls vacancy conc.) Defects affect material properties (e.g., grain boundaries control crystal slip). Defects may be desirable or undesirable (e.g., dislocations may be good or bad, depending on whether plastic deformation is desirable or not.) Chapter 4 - 54 Chapter 4 Polymer Structures (a) Schematic representation of the arrangement of molecular chains for a crystalline region of polyethylene. Black and gray balls represent, respectively, carbon and hydrogen atoms. (a) (b) Schematic diagram of a polymer chain-folded crystallite—a plate-shaped crystalline region in which the molecular chains (b) (red lines/curves) fold back and forth on themselves; these folds occur at the crystallite faces. (c) Structure of a spherulite found in some semicrystalline polymers (schematic). Chain-folded crystallites radiate outward from a common center. Separating and connecting these crystallites are regions of amorphous material, wherein the molecular chains (red curves) assume misaligned and disordered configurations. (c) (d) Transmission electron micrograph showing the spherulite structure. Chain-folded lamellar crystallites (white lines) approximately 10 nm (e) A polyethylene produce bag thick extend in radial directions from containing some fruit. the center. 15,000⫻. (d) [Photograph of Figure (d) supplied by P. J. Phillips. First published in R. Bartnikas and R. M. Eichhorn, Engineering Dielectrics, Vol. IIA, Electrical Properties of Solid Insulating Materials: Molecular Structure and Electrical Behavior, 1983. Copyright ASTM, 1916 Race Street, Philadelphia, PA 19103. Reprinted with permission. Figure (e) from Glow Images.] (e) 102 WHY STUDY Polymer Structures? A relatively large number of chemical and structural 2. Degree of crosslinking—on the stiffness of rubber-like characteristics affect the properties and behaviors of poly- materials (Section 8.19). meric materials. Some of these influences are as follows: 3. Polymer chemistry—on melting and glass-transition 1. Degree of crystallinity of semicrystalline polymers— temperatures (Section 11.17). on density, stiffness, strength, and ductility (Sections 4.11 and 8.18). Learning Objectives After studying this chapter you should be able to do the following: 1. Describe a typical polymer molecule in terms of (b) the three types of stereoisomers, its chain structure and, in addition, how the (c) the two kinds of geometrical isomers, and molecule may be generated from repeat units. (d) the four types of copolymers. 2. Draw repeat units for polyethylene, poly(vinyl 5. Cite the differences in behavior and molecular chloride), polytetrafluoroethylene, polypropyl- structure for thermoplastic and thermosetting ene, and polystyrene. polymers. 3. Calculate number-average and weight-average 6. Briefly describe the crystalline state in polymeric molecular weights and degree of polymerization materials. for a specified polymer. 7. Briefly describe/diagram the spherulitic structure 4. Name and briefly describe: for a semicrystalline polymer. (a) the four general types of polymer molecular structures, 4.1 INTRODUCTION Naturally occurring polymers—those derived from plants and animals—have been used for many centuries; these materials include wood, rubber, cotton, wool, leather, and silk. Other natural polymers, such as proteins, enzymes, starches, and cellulose, are important in biological and physiological processes in plants and animals. Modern scientific research tools have made possible the determination of the molecular struc- tures of this group of materials and the development of numerous polymers that are synthesized from small organic molecules. Many of our useful plastics, rubbers, and fiber materials are synthetic polymers. In fact, since the conclusion of World War II, the field of materials has been virtually revolutionized by the advent of synthetic poly- mers.The synthetics can be produced inexpensively, and their properties may be managed to the degree that many are superior to their natural counterparts. In some applica- tions metal and wood parts have been replaced by plastics, which have satisfactory properties and can be produced at a lower cost. As with metals and ceramics, the properties of polymers are intricately related to the structural elements of the material. This chapter explores molecular and crystal structures of polymers; Chapter 8 discusses the relationships between structure and some of the mechanical properties. 4.2 HYDROCARBON MOLECULES Because most polymers are organic in origin, we briefly review some of the basic concepts relating to the structure of their molecules. First, many organic materials are hydrocarbons; that is, they are composed of hydrogen and carbon. Furthermore, the intramolecular bonds 103 104 Chapter 4 / Polymer Structures are covalent. Each carbon atom has four electrons that may participate in covalent bonding, whereas every hydrogen atom has only one bonding electron. A single covalent bond exists when each of the two bonding atoms contributes one electron, as represented schematically in Figure 2.10 for a molecule of methane (CH4). Double and triple bonds between two car- bon atoms involve the sharing of two and three pairs of electrons, respectively. For example, in ethylene, which has the chemical formula C2H4, the two carbon atoms are doubly bonded together, and each is also singly bonded to two hydrogen atoms, as represented by the struc- tural formula H H C C H H where — and “ denote single and double covalent bonds, respectively. An example of a triple bond is found in acetylene, C2H2: H C C H unsaturated Molecules that have double and triple covalent bonds are termed unsaturated. That is, each carbon atom is not bonded to the maximum (four) other atoms; there- fore, it is possible for another atom or group of atoms to become attached to the orig- saturated inal molecule. Furthermore, for a saturated hydrocarbon, all bonds are single ones, and no new atoms may be joined without the removal of others that are already bonded. Some of the simple hydrocarbons belong to the paraffin family; the chainlike paraffin molecules include methane (CH4), ethane (C2H6), propane (C3H8), and butane (C4H10). Compositions and molecular structures for paraffin molecules are contained in Table 4.1. The covalent bonds in each molecule are strong, but only weak hydrogen and van der Waals bonds exist between molecules, and thus these hydrocarbons have relatively low melting and boiling points. However, boiling temperatures rise with increasing molecular weight (Table 4.1). Table 4.1 Name Composition Structure Boiling Point (⬚C) Compositions and Molecular Structures H for Some Paraffin Methane CH4 H C H ⫺164 Compounds: CnH2n⫹2 H H H Ethane C2H6 H C C H ⫺88.6 H H H H H Propane C3H8 H C C C H ⫺42.1 H H H Butane C4H10 ⭈ ⫺0.5 Pentane C5H12 ⭈ 36.1 Hexane C6H14 ⭈ 69.0  4.3 Polymer Molecules 105 Hydrocarbon compounds with the same composition may have different atomic isomerism arrangements, a phenomenon termed isomerism. For example, there are two isomers for butane; normal butane has the structure H H H H H C C C C H H H H H whereas a molecule of isobutane is represented as follows: H H C H H H H C C C H H H H Some of the physical properties of hydrocarbons depend on the isomeric state; for example, the boiling temperatures for normal butane and isobutane are ⫺0.5⬚C and ⫺12.3⬚C (31.1⬚F and 9.9⬚F), respectively. There are numerous other organic groups, many of which are involved in polymer structures. Several of the more common groups are presented in Table 4.2, where R and R⬘ represent organic groups such as CH3, C2H5, and C6H5 (methyl, ethyl, and phenyl). Concept Check 4.1 Differentiate between polymorphism (see Chapter 3) and iso- merism. [The answer may be found at www.wiley.com/college/callister (Student Companion Site).] 4.3 POLYMER MOLECULES The molecules in polymers are gigantic in comparison to the hydrocarbon molecules al- macromolecule ready discussed; because of their size they are often referred to as macromolecules. Within each molecule, the atoms are bound together by covalent interatomic bonds. For carbon-chain polymers, the backbone of each chain is a string of carbon atoms. Many times each carbon atom singly bonds to two adjacent carbons atoms on either side, rep- resented schematically in two dimensions as follows: C C C C C C C Each of the two remaining valence electrons for every carbon atom may be involved in side bonding with atoms or radicals that are positioned adjacent to the chain. Of course, both chain and side double bonds are also possible. 106 Chapter 4 / Polymer Structures Table 4.2 Characteristic Representative Some Common Family Unit Compound Hydrocarbon Groups H Alcohols R OH H C OH Methyl alcohol H H H Ethers R O R⬘ H C O C H Dimethyl ether H H OH H OH Acids R C H C C Acetic acid O H O R H Aldehydes C O C O Formaldehyde H H R OH Aromatic hydrocarbonsa Phenol H C H a C C The simplified structure denotes a phenyl group, C C H C H H repeat unit These long molecules are composed of structural entities called repeat units, which monomer are successively repeated along the chain.1 The term monomer refers to the small molecule from which a polymer is synthesized. Hence, monomer and repeat unit mean different things, but sometimes the term monomer or monomer unit is used instead of the more proper term repeat unit. 4.4 THE CHEMISTRY OF POLYMER MOLECULES Consider again the hydrocarbon ethylene (C2H4), which is a gas at ambient temperature and pressure and has the following molecular structure: H H C C H H 1 A repeat unit is also sometimes called a mer. Mer originates from the Greek word meros, polymer which means “part”; the term polymer was coined to mean “many mers.”  4.4 The Chemistry of Polymer Molecules 107 If the ethylene gas is reacted under appropriate conditions, it will transform to poly- ethylene (PE), which is a solid polymeric material. This process begins when an active center is formed by the reaction between an initiator or catalyst species (R⭈) and the ethylene monomer, as follows: H H H H R· ⫹ C C R C C· (4.1) H H H H The polymer chain then forms by the sequential addition of monomer units to this actively growing chain molecule. The active site, or unpaired electron (denoted by # ), is transferred to each successive end monomer as it is linked to the chain. This may be represented schematically as follows: H H H H H H H H R C C· ⫹ C C R C C C C· (4.2) H H H H H H H H The final result, after the addition of many ethylene monomer units, is the polyethylene molecule.2 A portion of one such molecule and the polyethylene repeat unit are shown in Figure 4.1a. This polyethylene chain structure can also be represented as VMSE Repeat Unit Structures H H (C C )n H H Figure 4.1 For polyethylene, H H H H H H H H (a) a schematic representation of C C C C C C C C repeat unit and chain structures, H H H H H H H H and (b) a perspective of the molecule, indicating the zigzag Repeat unit backbone structure. (a) C H (b) 2 A more detailed discussion of polymerization reactions, including both addition and condensation mechanisms, is given in Section 14.11. 108 Chapter 4 / Polymer Structures or alternatively as —( CH2 — CH2 — )n Here the repeat units are enclosed in parentheses, and the subscript n indicates the num- ber of times it repeats.3 The representation in Figure 4.1a is not strictly correct, in that the angle between the singly bonded carbon atoms is not 180⬚ as shown, but rather is close to 109⬚. A more accurate three-dimensional model is one in which the carbon atoms form a zigzag pattern (Figure 4.1b), the C—C bond length being 0.154 nm. In this discussion, depiction of poly- mer molecules is frequently simplified using the linear chain model shown in Figure 4.1a. Of course polymer structures having other chemistries are possible. For example, the tetrafluoroethylene monomer, CF2 “ CF2, can polymerize to form polytetrafluo- roethylene (PTFE) as follows: VMSE F F F F Repeat Unit Structures n C C (C C )n (4.3) F F F F Polytetrafluoroethylene (trade name Teflon) belongs to a family of polymers called the fluorocarbons. The vinyl chloride monomer (CH2 “ CHCl) is a slight variant of that for ethylene, in which one of the four H atoms is replaced with a Cl atom. Its polymerization is represented as VMSE Repeat Unit Structures H H H H n C C (C C )n (4.4) H Cl H Cl and leads to poly(vinyl chloride) (PVC), another common polymer. Some polymers may be represented using the following generalized form: H H (C C )n H R where the R depicts either an atom [i.e., H or Cl, for polyethylene or poly(vinyl chloride), respectively] or an organic group such as CH3, C2H5, and C6H5 (methyl, ethyl, and phenyl). For example, when R represents a CH3 group, the polymer is polypropylene (PP). VMSE Poly(vinyl chloride) and polypropylene chain structures are also represented in Figure 4.2. Repeat Unit Structures Table 4.3 lists repeat units for some of the more common polymers; as may be noted, some of them—for example, nylon, polyester, and polycarbonate—are relatively complex. Repeat units for a large number of relatively common polymers are given in Appendix D. When all of the repeating units along a chain are of the same type, the resulting homopolymer polymer is called a homopolymer. Chains may be composed of two or more different repeat units, in what are termed copolymers (see Section 4.10). copolymer The monomers discussed thus far have an active bond that may react to form two covalent bonds with other monomers forming a two-dimensional chainlike molecular 3 Chain ends/end groups (i.e., the R’s in Equation 4.2) are not normally represented in chain structures.  4.4 The Chemistry of Polymer Molecules 109 Figure 4.2 Repeat unit and chain structures F F F F F F F F for (a) polytetrafluoroethylene, (b) poly(vinyl C C C C C C C C chloride), and (c) polypropylene. F F F F F F F F Repeat unit (a) H H H H H H H H C C C C C C C C H Cl H Cl H Cl H Cl Repeat unit (b) H H H H H H H H C C C C C C C C H CH3 H CH3 H CH3 H CH3 Repeat unit (c) bifunctional structure, as indicated earlier for ethylene. Such a monomer is termed bifunctional. In general, the functionality is the number of bonds that a given monomer can form. For ex- functionality ample, monomers such as phenol–formaldehyde (Table 4.3) are trifunctional: they have trifunctional three active bonds, from which a three-dimensional molecular network structure results. Table 4.3 Repeat Units for 10 of the More Common Polymeric Materials Polymer Repeat Unit H H Polyethylene (PE) C C VMSE Repeat Unit Structures H H H H Poly(vinyl chloride) (PVC) C C H Cl F F Polytetrafluoroethylene (PTFE) C C F F H H Polypropylene (PP) C C H CH3 (continued) 110 Chapter 4 / Polymer Structures Table 4.3 (continued) Polymer Repeat Unit H H Polystyrene (PS) C C H H CH3 C C Poly(methyl methacrylate) (PMMA) H C O O CH3 OH CH2 CH2 Phenol-formaldehyde (Bakelite) CH2 H O H O Poly(hexamethylene adipamide) N C N C C C (nylon 6,6) H H 6 H H 4 O a O H H Poly(ethylene terephthalate) C C O C C O (PET, a polyester) H H a CH3 O Polycarbonate (PC) O C O C CH3 H H C C a The symbol in the backbone chain denotes an aromatic ring as C C C C H H Concept Check 4.2 On the basis of the structures presented in the previous section, sketch the repeat unit structure for poly(vinyl fluoride). [The answer may be found at www.wiley.com/college/callister (Student Companion Site).]  4.5 Molecular Weight 111 0.3 0.3 Figure 4.3 Hypothetical polymer molecule size distributions on the basis of (a) number and 0.2 0.2 Number fraction (b) weight fractions of Weight fraction molecules. 0.1 0.1 0 0 0 5 10 15 20 25 30 35 40 0 5 10 15 20 25 30 35 40 3 3 Molecular weight (10 g/mol) Molecular weight (10 g/mol) (a) (b) 4.5 MOLECULAR WEIGHT Extremely large molecular weights4 are observed in polymers with very long chains. During the polymerization process not all polymer chains will grow to the same length; this results in a distribution of chain lengths or molecular weights. Ordinarily, an average molecular weight is specified, which may be determined by the measurement of various physical properties such as viscosity and osmotic pressure. There are several ways of defining average molecular weight. The number-average molecular weight Mn is obtained by dividing the chains into a series of size ranges and then determining the number fraction of chains within each size range (Figure 4.3a). The number-average molecular weight is expressed as Number-average Mn ⫽ 兺 xi Mi (4.5a) molecular weight where Mi represents the mean (middle) molecular weight of size range i, and xi is the fraction of the total number of chains within the corresponding size range. A weight-average molecular weight Mw is based on the weight fraction of molecules within the various size ranges (Figure 4.3b). It is calculated according to Weight-average molecular weight Mw ⫽ 兺 wi Mi (4.5b) where, again, Mi is the mean molecular weight within a size range, whereas wi denotes the weight fraction of molecules within the same size interval. Computations for both number-average and weight-average molecular weights are carried out in Example Problem 4.1. A typical molecular weight distribution along with these molecular weight averages is shown in Figure 4.4. 4 Molecular mass, molar mass, and relative molecular mass are sometimes used and are really more appropriate terms than molecular weight in the context of the present discussion—in fact, we are dealing with masses and not weights. However, molecular weight is most commonly found in the polymer literature and thus will be used throughout this book. 112 Chapter 4 / Polymer Structures Figure 4.4 Distribution of molecular weights for a typical polymer. Number-average, Mn Weight-average, Mw Amount of polymer Molecular weight degree of An alternative way of expressing average chain size of a polymer is as the degree polymerization of polymerization, DP, which represents the average number of repeat units in a chain. DP is related to the number-average molecular weight Mn by the equation Degree of polymerization— dependence on Mn number-average and DP ⫽ (4.6) m repeat unit molecular weights where m is the repeat unit molecular weight. EXAMPLE PROBLEM 4.1 Computations of Average Molecular Weights and Degree of Polymerization Assume that the molecular weight distributions shown in Figure 4.3 are for poly(vinyl chlo- ride). For this material, compute (a) the number-average molecular weight, (b) the degree of polymerization, and (c) the weight-average molecular weight. Solution (a) The data necessary for this computation, as taken from Figure 4.3a, are presented in Table 4.4a. According to Equation 4.5a, summation of all the xiMi products (from the right-hand column) yields the number-average molecular weight, which in this case is 21,150 g/mol. Table 4.4a Data Used for Number-Average Molecular Weight Computations in Example Problem 4.1 Molecular Weight Mean Mi Range (g/mol) (g/mol) xi xiMi 5,000–10,000 7,500 0.05 375 10,000–15,000 12,500 0.16 2000 15,000–20,000 17,500 0.22 3850 20,000–25,000 22,500 0.27 6075 25,000–30,000 27,500 0.20 5500 30,000–35,000 32,500 0.08 2600 35,000–40,000 37,500 0.02 750 Mn ⫽ 21,150  4.6 Molecular Shape 113 (b) To determine the degree of polymerization (Equation 4.6), it is first necessary to compute the repeat unit molecular weight. For PVC, each repeat unit consists of two carbon atoms, three hydrogen atoms, and a single chlorine atom (Table 4.3). Furthermore, the atomic weights of C, H, and Cl are, respectively, 12.01, 1.01, and 35.45 g/mol. Thus, for PVC m ⫽ 2112.01 g/mol2 ⫹ 311.01 g/mol2 ⫹ 35.45 g/mol ⫽ 62.50 g/mol and Mn 21,150 g/mol DP ⫽ ⫽ ⫽ 338 m 62.50 g/mol (c) Table 4.4b shows the data for the weight-average molecular weight, as taken from Fig- ure 4.3b. The wiMi products for the size intervals are tabulated in the right-hand column. The sum of these products (Equation 4.5b) yields a value of 23,200 g/mol for Mw. Table 4.4b Data Used for Weight-Average Molecular Weight Computations in Example Problem 4.1 Molecular Weight Mean Mi Range (g/mol) (g/mol) wi wi Mi 5,000–10,000 7,500 0.02 150 10,000–15,000 12,500 0.10 1250 15,000–20,000 17,500 0.18 3150 20,000–25,000 22,500 0.29 6525 25,000–30,000 27,500 0.26 7150 30,000–35,000 32,500 0.13 4225 35,000–40,000 37,500 0.02 750 Mw ⫽ 23,200 Many polymer properties are affected by the length of the polymer chains. For example, the melting or softening temperature increases with increasing molecular weight (for M up to about 100,000 g/mol). At room temperature, polymers with very short chains (having molecular weights on the order of 100 g/mol) will generally exist as liquids. Those with molecular weights of approximately 1000 g/mol are waxy solids (such as paraffin wax) and soft resins. Solid polymers (sometimes termed high poly- mers), which are of prime interest here, commonly have molecular weights ranging between 10,000 and several million g/mol. Thus, the same polymer material can have quite different properties if it is produced with a different molecular weight. Other properties that depend on molecular weight include elastic modulus and strength (see Chapter 8). 4.6 MOLECULAR SHAPE Previously, polymer molecules have been shown as linear chains, neglecting the zigzag arrangement of the backbone atoms (Figure 4.1b). Single-chain bonds are capable of rotating and bending in three dimensions. Consider the chain atoms in Figure 4.5a; a 114 Chapter 4 / Polymer Structures 109° (a) (b) (c) Figure 4.5 Schematic representations of how polymer chain shape is influenced by the positioning of backbone carbon atoms (gray circles). For (a), the rightmost atom may lie anywhere on the dashed circle and still subtend a 109⬚ angle with the bond between the other two atoms. Straight and twisted chain segments are generated when the backbone atoms are situated as in (b) and (c), respectively. (From Askeland. Science and Engineering of Materials, 3E. © 1994 Cengage Learning, a part of Cengage Learning, Inc. Reproduced by permission. www.cengage.com/permissions) third carbon atom may lie at any point on the cone of revolution and still subtend about a 109⬚ angle with the bond between the other two atoms. A straight chain segment results when successive chain atoms are positioned as in Figure 4.5b. On the other hand, chain bending and twisting are possible when there is a rotation of the chain atoms into other positions, as illustrated in Figure 4.5c.5 Thus, a single chain molecule composed of many chain atoms might assume a shape similar to that represented schematically in Figure 4.6, having a multitude of bends, twists, and kinks.6 Also indicated in this figure is the end-to-end distance of the polymer chain r; this distance is much smaller than the total chain length. Polymers consist of large numbers of molecular chains, each of which may bend, coil, and kink in the manner of Figure 4.6. This leads to extensive intertwining and entanglement of neighboring chain molecules, a situation similar to what is seen in a Figure 4.6 Schematic representation of a single polymer chain molecule that has numerous random kinks and coils produced by chain bond rotations. (From L. R. G. Treloar, The Physics of Rubber Elasticity, 2nd edition, Oxford University Press, Oxford, 1958, p. 47.) r 5 For some polymers, rotation of carbon backbone atoms within the cone may be hindered by bulky side group elements on neighboring chain atoms. 6 The term conformation is often used in reference to the physical outline of a molecule, or molecular shape, that can be altered only by rotation of chain atoms about single bonds.  4.7 Molecular Structure 115 heavily tangled fishing line. These random coils and molecular entanglements are responsible for a number of important characteristics of polymers, to include the large elastic extensions displayed by the rubber materials. 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. For example, the region of a chain segment that has a double bond (C “ C) is rotationally rigid. Also, introduction of a bulky or large side group of atoms restricts rotational movement. For example, polystyrene molecules, which have a phenyl side group (Table 4.3), are more resistant to rotational motion than are polyethylene chains. 4.7 MOLECULAR STRUCTURE The physical characteristics of a polymer depend not only on its molecular weight and shape, but also on differences in the structure of the molecular chains. Modern polymer synthesis techniques permit considerable control over various structural possibilities. This section discusses several molecular structures including linear, branched, crosslinked, and network, in addition to various isomeric configurations. Linear Polymers linear polymer Linear polymers are those in which the repeat units are joined together end to end in single chains. These long chains are flexible and may be thought of as a mass of “spaghetti,” as represented schematically in Figure 4.7a, where each circle represents a (a) (b) (c) (d) Figure 4.7 Schematic representations of (a) linear, (b) branched, (c) crosslinked, and (d) network (three-dimensional) molecular structures. Circles designate individual repeat units. 116 Chapter 4 / Polymer Structures repeat unit. For linear polymers, there may be extensive van der Waals and hydrogen bonding between the chains. Some of the common polymers that form with linear struc- tures are polyethylene, poly(vinyl chloride), polystyrene, poly(methyl methacrylate), nylon, and the fluorocarbons. Branched Polymers Polymers may be synthesized in which side-branch chains are connected to the main branched polymer ones, as indicated schematically in Figure 4.7b; these are fittingly called branched polymers. The branches, considered to be part of the main-chain molecule, may result from side reactions that occur during the synthesis of the polymer. The 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. For example, high- density polyethylene (HDPE) is primarily a linear polymer, whereas low-density poly- ethylene (LDPE) contains short-chain branches. Crosslinked Polymers crosslinked polymer In crosslinked polymers, adjacent linear chains are joined one to another at various positions by covalent bonds, as represented in Figure 4.7c. The process of crosslinking is achieved either during synthesis or by a nonreversible chemical reaction. Often, this crosslinking is accomplished by additive atoms or molecules that are covalently bonded to the chains. Many of the rubber elastic materials are crosslinked; in rubbers, this is called vulcanization, a process described in Section 8.19. Network Polymers Multifunctional monomers forming three or more active covalent bonds make three- network polymer dimensional networks (Figure 4.7d) and are termed network polymers. Actually, a polymer that is highly crosslinked may also be classified as a network polymer. These materials have distinctive mechanical and thermal properties; the epoxies, polyurethanes, and phenol- formaldehyde belong to this group. Polymers are not usually of only one distinctive structural type. For example, a pre- dominantly linear polymer may have limited branching and crosslinking. 4.8 MOLECULAR CONFIGURATIONS For polymers having more than one side atom or group of atoms bonded to the main chain, the regularity and symmetry of the side group arrangement can significantly influence the properties. Consider the repeat unit H H C C H R in which R represents an atom or side group other than hydrogen (e.g., Cl, CH3). One arrangement is possible when the R side groups of successive repeat units are bound to alternate carbon atoms as follows:  4.8 Molecular Configurations 117 H H H H C C C C H R H R This is designated as a head-to-tail configuration.7 Its complement, the head-to-head configuration, occurs when R groups are bound to adjacent chain atoms: H H H H C C C C H R R H In most polymers, the head-to-tail configuration predominates; often a polar repulsion occurs between R groups for the head-to-head configuration. Isomerism (Section 4.2) is also found in polymer molecules, wherein different atomic configurations are possible for the same composition. Two isomeric subclasses— stereoisomerism and geometrical isomerism—are topics of discussion in the succeeding sections. Stereoisomerism stereoisomerism Stereoisomerism denotes the situation in which atoms are linked together in the same order (head to tail) but differ in their spatial arrangement. For one stereoisomer, all of the R groups are situated on the same side of the chain as follows: R H R H R H R H R H VMSE Stereo and Geometrical C C C C C C C C C Isomers H H H H H H H H isotactic This is called an isotactic configuration. This diagram shows the zigzag pattern of the configuration carbon chain atoms. Furthermore, representation of the structural geometry in three dimensions is important, as indicated by the wedge-shaped bonds; solid wedges repre- sent bonds that project out of the plane of the page, and dashed ones represent bonds that project into the page.8 7 The term configuration is used in reference to arrangements of units along the axis of the chain, or atom positions that are not alterable except by the breaking and then re-forming of primary bonds. 8 The isotactic configuration is sometimes represented using the following linear (i.e., nonzigzag) and two-dimen- sional schematic: H H H H H H H H H C C C C C C C C C R H R H R H R H R 118 Chapter 4 / Polymer Structures syndiotactic In a syndiotactic configuration, the R groups alternate sides of the chain9: configuration R H H R R H H R R H C C C C C C C C C VMSE Stereo and Geometrical H H H H H H H H Isomers and for random positioning R H R H H R R H H R C C C C C VMSE C C C C Stereo and Geometrical Isomers H H H H H H H H atactic configuration the term atactic configuration is used.10 Conversion from one stereoisomer to another (e.g., isotactic to syndiotactic) is not possible by a simple rotation about single-chain bonds. These bonds must first be severed; then, after the appropriate rotation, they are re-formed into the new confi- guration. In reality, a specific polymer does not exhibit just one of these configurations; the predominant form depends on the method of synthesis. Geometrical Isomerism Other important chain configurations, or geometrical isomers, are possible within repeat units having a double bond between chain carbon atoms. Bonded to each of the carbon atoms participating in the double bond is a side group, which may be situated on one side of the chain or its opposite. Consider the isoprene repeat unit having the structure CH3 H VMSE C C Stereo and Geometrical Isomers CH2 CH2 9 The linear and two-dimensional schematic for the syndiotactic configuration is represented as H H R H H H R H H C C C C C C C C C R H H H R H H H R 10 For the atactic configuration the linear and two-dimensional schematic is H H H H R H H H R C C C C C C C C C R H R H H H R H H  4.8 Molecular Configurations 119 in which the CH3 group and the H atom are positioned on the same side of the double cis (structure) bond. This is termed a cis structure, and the resulting polymer, cis-polyisoprene, is natural rubber. For the alternative isomer CH3 CH2 VMSE C C Stereo and Geometrical CH2 H Isomers trans (structure) the trans structure, the CH3 and H reside on opposite sides of the double bond.11 Trans- polyisoprene, sometimes called gutta percha, has properties that are distinctly different from those of natural rubber as a result of this configurational alteration. Conversion of trans to cis, or vice versa, is not possible by a simple chain bond rotation because the chain double bond is extremely rigid. To summarize the preceding sections: Polymer molecules may be characterized in terms of their size, shape, and structure. Molecular size is specified in terms of molecular weight (or degree of polymerization). Molecular shape relates to the degree of chain twisting, coiling, and bending. Molecular structure depends on the manner in which structural units are joined together. Linear, branched, crosslinked, and network structures are all possible, in addition to several isomeric configurations (isotactic, syndiotactic, atactic, cis, and trans). These molecular characteristics are pre- sented in the taxonomic chart shown in Figure 4.8. Note that some of the structural elements are not mutually exclusive, and it may be necessary to specify molecular structure in terms of more than one. For example, a linear polymer may also be isotactic. Concept Check 4.3 What is the difference between configuration and conformation in relation to polymer chains? [The answer may be found at www.wiley.com/college/callister (Student Companion Site).] 11 For cis-isoprene the linear chain representation is as follows: H CH3 H H C C C C H H whereas the linear schematic for the trans structure is H CH3 H C C C C H H H 120 Chapter 4 / Polymer Structures Figure 4.8 Molecular Classification scheme characteristics for the characteristics of polymer molecules. Chemistry Size Shape Structure (repeat unit (molecular weight) (chain twisting, composition) entanglement, etc.) Linear Branched Crosslinked Network Isomeric states Stereoisomers Geometrical isomers Isotactic Syndiotactic Atactic cis trans 4.9 THERMOPLASTIC AND THERMOSETTING POLYMERS The response of a polymer to mechanical forces at elevated temperatures is related to its dominant molecular structure. In fact, one classification scheme for these materials thermoplastic is according to behavior with rising temperature. Thermoplastics (or thermoplastic polymer polymers) and thermosets (or thermosetting polymers) are the two subdivisions. Thermoplastics soften when heated (and eventually liquefy) and harden when thermosetting polymer cooled—processes that are totally reversible and may be repeated. On a molecular level, as the temperature is raised, secondary bonding forces are diminished (by increased molecular motion) so that the relative movement of adjacent chains is facilitated when a stress is applied. Irreversible degradation results when a molten thermoplastic polymer is raised to too high a temperature. In addition, thermoplastics are relatively soft. Most linear polymers and those having some branched structures with flexible chains are thermoplastic. These materials are normally fabricated by the simultaneous application of heat and pressure (see Section 14.13). Examples of common thermoplastic polymers include polyethylene, polystyrene, poly(ethylene terephthalate), and poly(vinyl chloride). Thermosetting polymers are network polymers. They become permanently hard during their formation and do not soften upon heating. Network polymers have covalent crosslinks between adjacent molecular chains. During heat treatments, these bonds anchor the chains together to resist the vibrational and rotational chain motions at high temperatures. Thus, the materials do not soften when heated. Crosslinking is usually extensive, in that 10% to 50% of the chain repeat units are crosslinked. Only heating to excessive temperatures will cause severance of these crosslink bonds and polymer degradation. Thermoset polymers are generally harder and stronger than thermoplastics and have better dimensional stability. Most of the  4.10 Copolymers 121 crosslinked and network polymers, which include vulcanized rubbers, epoxies, and phenolics and some polyester resins, are thermosetting. Concept Check 4.4 Some polymers (such as the polyesters) may be either thermoplastic or thermosetting. Suggest one reason for this. [The answer may be found at www.wiley.com/college/callister (Student Companion Site).] 4.10 COPOLYMERS Polymer chemists and scientists are continually searching for new materials that can be easily and economically synthesized and fabricated with improved properties or better property combinations than are offered by the homopolymers previously discussed. One group of these materials are the copolymers. Consider a copolymer that is composed of two repeat units as represented by 䊉 and 䊉 in Figure 4.9. Depending on the polymerization process and the relative fractions of these repeat unit types, different sequencing arrangements along the polymer chains are possible. For one, as depicted in Figure 4.9a, the two different units are randomly Figure 4.9 Schematic representations of (a) random, (b) alternating, (c) block, and (d) graft copolymers. The two (a) different repeat unit types are designated by blue and red circles. (b) (c) (d) 122 Chapter 4 / Polymer Structures Table 4.5 Chemical Repeat Units That Are Employed in Copolymer Rubbers Repeat Unit Repeat Unit Repeat Unit Repeat Unit Name Structure Name Structure H H H CH3 H H Acrylonitrile C C Isoprene C C C C VMSE H C N H H Repeat Units-Rubbers H CH3 H H Isobutylene C C Styrene C C H CH3 H CH3 Dimethylsiloxane Si O H H H H CH3 Butadiene C C C C H H H Cl H H Chloroprene C C C C H H random copolymer dispersed along the chain in what is termed a random copolymer. For an alternating copolymer, as the name suggests, the two repeat units alternate chain positions, as illus- alternating copolymer trated in Figure 4.9b. A block copolymer is one in which identical repeat units are clus- tered in blocks along the chain (Figure 4.9c). Finally, homopolymer side branches of one block copolymer type may be grafted to homopolymer main chains that are composed of a different graft copolymer repeat unit; such a material is termed a graft copolymer (Figure 4.9d). When calculating the degree of polymerization for a copolymer, the value m in Equation 4.6 is replaced with the average value m determined from Average repeat unit molecular weight for m⫽ 兺 fj mj (4.7) a copolymer In this expression, fj and mj are, respectively, the mole fraction and molecular weight of repeat unit j in the polymer chain. Synthetic rubbers, discussed in Section 13.13, are often copolymers; chemical repeat units that are employed in some of these rubbers are shown in Table 4.5. Styrene–butadiene rubber (SBR) is a common random copolymer from which automobile tires are made. Nitrile rubber (NBR) is another random copolymer com- posed of acrylonitrile and butadiene. It is also highly elastic and, in addition, resistant to swelling in organic solvents; gasoline hoses are made of NBR. Impact-modified poly- styrene is a block copolymer that consists of alternating blocks of styrene and butadiene. The rubbery isoprene blocks act to slow cracks propagating through the material. 4.11 POLYMER CRYSTALLINITY The crystalline state may exist in polymeric materials. However, because it involves molecules instead of just atoms or ions, as with metals and ceramics, the atomic arrange- polymer crystallinity ments will be more complex for polymers. We think of polymer crystallinity as the packing  4.11 Polymer Crystallinity 123 Figure 4.10 Arrangement of molecular chains in a unit cell for polyethylene. (Adapted from C. W. Bunn, Chemical Crystallography, Oxford University Press, Oxford, 1945, p. 233.) 0.255 nm 0.741 nm 0.494 nm C H of molecular chains to produce an ordered atomic array. Crystal structures may be spec- ified in terms of unit cells, which are often quite complex. For example, Figure 4.10 shows the unit cell for polyethylene and its relationship to the molecular chain structure; this unit cell has orthorhombic geometry (Table 3.6). Of course, the chain molecules also extend beyond the unit cell shown in the figure. Molecular substances having small molecules (e.g., water and methane) are nor- mally either totally crystalline (as solids) or totally amorphous (as liquids). As a con- sequence of their size and often complexity, polymer molecules are often only partially crystalline (or semicrystalline), having crystalline regions dispersed within the remain- ing amorphous material. Any chain disorder or misalignment will result in an amor- phous region, a condition that is fairly common, because twisting, kinking, and coiling of the chains prevent the strict ordering of every segment of every chain. Other struc- tural effects are also influential in determining the extent of crystallinity, as discussed shortly. The degree of crystallinity may range from completely amorphous to almost entirely (up to about 95%) crystalline; in contrast, metal specimens are almost always entirely crystalline, whereas many ceramics are either totally crystalline or totally non- crystalline. Semicrystalline polymers are, in a sense, analogous to two-phase metal alloys, discussed in subsequent chapters. The density of a crystalline polymer will be greater than an amorphous one of the Percent crystallinity same material and molecular weight because the chains are more closely packed (semicrystalline together for the crystalline structure. The degree of crystallinity by weight may be deter- polymer)— dependence on mined from accurate density measurements, according to specimen density, rc 1rs ⫺ ra 2 and densities of totally crystalline and % crystallinity ⫽ ⫻ 100 rs 1rc ⫺ ra 2 (4.8) totally amorphous materials 124 Chapter 4 / Polymer Structures where ␳s is the density of a specimen for which the percent crystallinity is to be deter- mined, ␳a is the density of the totally amorphous polymer, and ␳c is the density of the perfectly crystalline polymer. The values of ␳a and ␳c must be measured by other experi- mental means. The degree of crystallinity of a polymer depends on the rate of cooling during solidi- fication as well as on the chain configuration. During crystallization upon cooling through the melting temperature, the chains, which are highly random and entangled in the viscous liquid, must assume an ordered configuration. For this to occur, sufficient time must be allowed for the chains to move and align themselves. The molecular chemistry as well as chain configuration also influence the ability of a polymer to crystallize. Crystallization is not favored in polymers that are composed of chemically complex repeat units (e.g., polyisoprene). On the other hand, crystallization is not easily prevented in chemically simple polymers such as polyethylene and poly- tetrafluoroethylene, even for very rapid cooling rates. For linear polymers, crystallization is easily accomplished because