Polymer Bonding CHEE102 Chapter II PDF

10/14/24 1 POLYMER BONDING (1) primary or chemical bonds Primary bonds are metallic, covalent, and ionic. (2) secondary or van der Waals bonds. London Dispersion, Hydrogen Bonding and Dipole dipole interaction. 2...

10/14/24 1 POLYMER BONDING (1) primary or chemical bonds Primary bonds are metallic, covalent, and ionic. (2) secondary or van der Waals bonds. London Dispersion, Hydrogen Bonding and Dipole dipole interaction. 2 1  lent bonds found in polymers have been estimated and are shown in Table 4.1. The disassociation energy (kJ/mole or k cal/mole) or cohesive energy density (J/cm3) is the energy required to move a molecule far enough away from another molecule so that the attractive force or energy between the two is negligible. The cohesive energy densities to break the bond between mer units of a number of linear polymers is shown in Table 4.2. In linear or thermoplastic polymers, it is only the secondary bonding forces that 10/14/24 hold the polymer together if entanglements are neglected. Therefore the energies in Table 4.2 give only an estimate of the breaking strength of a highly oriented samples of the various polymers listed. Table 4.1 Typical covalent bond lengths and energies found in polymers. (Data from Billymeyer (1984)) Bond Bond Length (Å) Dissociation Energy (kJ/mole) C C 1.54 347 C C 1.34 611 C H 1.10 414 C N 1.47 305 C N 1.15 891 C O 1.46 360 C O 1.21 749 C F 1.35 473 C Cl 1.77 339 N H 1.01 389 O H 0.96 464 O O 1.32 146 Of even more interest is a comparison of bond lengths and energies given in Table 4.3 for primary and secondary bonds which assists in understand- ing the differences between Science 102 Polymer Engineering linearand and crosslinked Viscoelasticity: polymers. An Introduction 3 Table 4.2 Cohesive energies of linear polymers. (Data from Billmeyer (1984)). Polymer Repeat Unit Cohesive Energy 102 Polymer Engineering Science and Viscoelasticity: An Introduction 3 Density (J/cm ) Polyethylene CH CH 2 259 2 (Data from Billmeyer (1984)). Table 4.2 Cohesive energies of linear polymers. Polyisobutylene Polymer CH2C(CH Repeat Unit 3)2 272 Cohesive Energy 3 Polyisoprene CH2C(CH3) 280 ) CHCH2 Density (J/cm Polyethylene CH2C(C CH CH 259 310 Polystyrene 2 26H5) Polyisobutylene PMMA CHCH 2C(CH C(CH 2 3)2 )COOCH 3 3 272 348 Polyisoprene CH2C(CH3) CHCH2 280 PVC CH2CHCL 381 Polystyrene CH2C(C6H5) 310 PET CH2CH2OCOC6H4COO 477 PMMA CH2C(CH3)COOCH3 348 PAN CH2CHCN 992 PVC CH2CHCL 381 PET CH2CH2OCOC6H4COO 477 Table 4.3 Comparsion of primary and secondary bond distances and energies. PAN CH2CHCN 992 (Data from Rosen (1993)) Bond Type Interatomic Distance (nm) Dissociation En- ergy Table 4.3 Comparsion of primary and secondary bond distances and(kcal/mole) energies. Primary covalent 0.1-0.2 (Data from Rosen (1993)) 50-200 Ionic Bond Type 0.2-0.3 Interatomic Distance (nm) 10-20 Dissociation En- Hydrogen 0.2-0.3 3-7 ergy (kcal/mole) Dipole Primary covalent 0.2-0.3 0.1-0.2 50-2001.5-3 van der IonicWaals 0.3-0.5 0.2-0.3 0.5-2 10-20 Hydrogen 0.2-0.3 3-7 ExceptDipole for the dispersion bond,0.2-0.3 all bonds are functions of1.5-3 temperature. As a van result, dervariations Waals in temperature for the same polymer 0.5-2 0.3-0.5 lead to different physical states as represented by Fig. 4.1. The relation of these states to 4 mechanical Except for the a result, ters. properties dispersion variations Notice will be discussed bond, all bonds are in temperature that both linear and for further in later functions the same polymer cross-linked sections andAschap- of temperature. polymers lead aretoindicated different and physical states temperature canas be represented by Fig. used to alter the 4.1. stateThe andrelation or the of these states chemistry of atopoly- mechanical properties will be discussed further in later sections and chap- mer. ters. Notice that both linear and cross-linked polymers are indicated and temperature can be used to alter the state and or the chemistry of a poly- mer. 2  10/14/24 4 Polymerization and Classification 103 High 4 Polymerization and Classification 103 Liquid Elastomer or Thermoset High Elastomer Liquid Flexible Thermoset or Polymer Flexible Temperature Polymer Temperature Structural Structural Polymer Polymer Crystalline Crystalline Solid Solid Fiber Fiber Polymer Polymer Low Low Low Intermolecular High Low Intermolecular Forces High Forces Fig. 4.1 The interrelation of states in a bulk polymer. (After Billmeyer (1984)). Fig. 4.1 The interrelation of states in a bulk polymer. (After Billmeyer (1984)). 4.2. Polymerization 5 4.2. Polymerization The polymerization process can be illustrated by the conversion of ethyl- ene into polyethylene which is one of the most widely produced polymers The inpolymerization process can the world. The unsaturated be illustrated ethylene molecule orby the conversion monomer is shown inof ethyl- ene Fig. into 4.2 below. (In general, polyethylene whichtheis term one unsaturated of the mostrefers to molecules widely produced with polymers POLYMERIZATION double or triple bonds while those with only single bonds are termed satu- in the world. The unsaturated ethylene molecule or monomer is shown in rated.) Fig. 4.2 below. (In general, the term unsaturated refers to molecules with double or triple bonds while those with only single bonds are termed satu- rated.) 104 Polymer Engineering Science and Viscoelasticity: An Introduction Fig. 4.2 The ethylene molecule. Under appropriate conditions of heat and pressure in the presence of a catalyst, the double bond between the two carbon atoms can be “opened” or broken and replaced by a single saturated bond with other similarly opened monomeric units Fig.on4.2 Theside either ethylene to formmolecule. a long replicated strand of mer units as illustrated in Fig. 4.3. Under appropriate Fig.conditions of heat 4.3 Repeating and pressure mer units in the presence of a of polyethylene. catalyst, the double bond between the two carbon atoms can be “opened” or an In broken and replaced by a single saturated bond with other similarly 6 actual polymer each individual chain may contain from several thou- opened sand monomeric to hundreds of units on either thousand side to repeating form mers oraunits. long replicated strand of mer units as illustrated in Fig. 4.3. The resulting solid polyethylene will contain a great many chains but each chain will vary in length. This leads to the need to have special meth- 3 ods to quantify the molecular weight of polymers and these will be dis- cussed in a subsequent section. In the case of polyethylene, the molecular bonds between carbon atoms along the length of the chain are all primary or covalent. However, the bonds between individual chains are secondary. For this reason, under a sufficient increase in temperature, the secondary  4 Polymerization and Classification 105 manner in which one long molecule can move through a seemingly con- tinuous mass of other chains (Aklonis and McKnight (1983)). The method of displaying the atoms as bar-mass4 linkages Polymerization and Classification in Figs. 4.3 and 4.4105is tradi- 10/14/24 tional. It is often used to visualize bonding arrangements and many types manner in which of computations one long with associated molecule can move molecular through and geometry a seemingly motion. con- tinuous mass of other chains (Aklonis and McKnight (1983)). The method of displaying the atoms as bar-mass linkages in Figs. 4.3 and 4.4 is tradi- tional. It is often used to visualize bonding arrangements and many types of computations associated with molecular geometry and motion. Fig. 4.4 Zigzag shape of polyethylene molecule. Fig. 4.4 Zigzag shape of polyethylene molecule. Fig. 4.5 Random nature due to rotation of carbon molecules. Fig. 4.5 Random nature due to rotation of carbon molecules. 7 106 Polymer Engineering Science and Viscoelasticity: An Introduction Fig. 4.6 Shape of a 1,000 link polyethylene chain (Treolar (1975), reprinted by permission of Oxford University Press). The mer units of a number of frequently used thermoplastic polymers are 8 given in Fig. 4.7. 4  10/14/24 4 Polymerization and Classification 107 Polymer Repeating (Mer) structure Polyethylene (PE) Polyvinyl Chloride (PVC) Polytetrafluoroethylene (PTFE) Polypropylene (PP) Polystyrene (PS) Polymethylmethacrylate (PMMA) Polycarbonate Fig. 4.7 Mer units of selected thermoplastic polymers. 9 108 Polymer Engineering Thermosetting or Science “cross-linked”and Viscoelasticity: polymers Ancatalytic are also formed under Introduction conditions of heat and pressure (often pressure is not needed). However, in this case covalent bonds do exist between individual chains. This “cross- linking” may vary considerably from polymer to polymer but generally leads to a solid material which cannot be melted. Examples of several chemical units that lead to cross-linked polymers are shown Fig. 4.8. Phenol-formaldehyde (bakelite) Polyurethane Bisphenol-A epoxy based polymer Fig. 4.8 Mer units of selected thermoset polymers. Pheno-formaldehyde or Bakelite was one of the first polymers introduced 10 in the US by Leo Bakeland in 1907. Polyurethane can be polymerized with other elements to give either elastomeric or rigid polymers. The epoxy precursor shown can be reacted with several other compounds to give well known epoxy resins. 5 4.3. Classification by Bonding Structure Between Chains and Morphology of Chains One simple classification scheme according to bonding structure is shown in Fig. 4.9. Here it is appropriate to emphasize the distinction between  Bisphenol-A epoxy based polymer Fig. 4.8 Mer units of selected thermoset polymers. Pheno-formaldehyde or Bakelite was one of the first polymers introduced in the US by Leo Bakeland in 1907. Polyurethane can be polymerized with other elements to give either elastomeric or rigid polymers. The epoxy precursor shown can be reacted with several other compounds to give well 10/14/24 known epoxy resins. 4.3. Classification by Bonding Structure Between Chains and Morphology of Chains CLASSIFICATIONS BASED ON BONDING STRUCTURE One simpleBETWEEN CHAINS classification scheme AND MORPHOLOGY according is 4shown OF CHAINS to bonding structure Polymerization and Classification 109 in Fig. 4.9. Here it is appropriate to emphasize the distinction between thermoplastic and thermosetting polymers, Crosslinked or Thermosetting Polymers: Intrachain bonds are primary. Intrachain Linear or Thermoplastic Polymers: Interchainbonds bondsareare both secondary and covalent. Very primary (covalent). Interchain bonds heavily crosslinked are secondary polymers (hydrogen, are often induction, di- called network polymers. pole, etc.). 4 Polymerization and Classification 109 Crosslinked or Thermosetting Polymers: Intrachain bonds are primary. Interchain bonds are both secondary and covalent. Very heavily crosslinked polymers are often called network polymers. Fig. 4.9 A simple classification scheme for polymers. 11 It is noted that there are variations in each type and schematically these may be represented as given in Fig. 4.10. The branches in branched poly- 4 Polymerization and Classification 109 mers may vary from very short to very long. Long branches may be further Fig. 4.9 A simple classification scheme for polymers. Crosslinked or classified Thermosetting Polymers: Intrachain bondsasarecomb-like, random or star shaped as shown in Fig. 4.11. primary. Interchain bonds are both secondary and covalent. Very heavilyItcrosslinked is noted that are polymers there oftenare calledvariations in each type and schematically these network polymers. may be represented as given in Fig. 4.10. The branches in branched poly- mers may vary from very short to very long. Long branches may be further classified as comb-like, random or star shaped as shown in Fig. 4.11. Fig. 4.9 A simple classification scheme for polymers. It is noted that there are variations in each type and schematically these may be represented as given in Fig. 4.10. The branches in branched poly- mers may vary from very short to very long. Long branches may be further classified as comb-like, random or star shaped as shown in Fig. 4.11. Fig. 4.10 Variations in thermoplastic (top) and thermosetting polymers (bottom). Fig. 4.10 Variations in thermoplastic (top) and thermosetting polymers (bottom). 12 Fig. 4.10 Variations in thermoplastic (top) and thermosetting polymers (bottom). 6  10/14/24 110 Polymer Engineering Science and Viscoelasticity: An Introduction 110 Polymer Engineering Science and Viscoelasticity: An Introduction Fig. 4.11 Variations in branched polymers. Crystalline regions ofFig. 4.11 Variations a linear polymerininbranched Fig. 4.10polymers. are shown schemati- cally as small parallel segments within a chain. This ordering of the struc- tureCrystalline regions in will be discussed ofaalater linear polymer section in Fig. 4.10 However, on morphology. are shownitschemati- is im- cally as small parallel segments within a chain. This ordering of the struc- 13 portant to note here the relative amount of ordering (crystallinity) for ture will beand polyethylene discussed in aonlater the effect section density andon morphology. mechanical However, properties. it is This in-im- portant istogiven formation noteinhere the relative the Table 4.4. amount of ordering (crystallinity) for polyethylene and Characteristics and applications the effect on of density andlinear several mechanical properties. polymers This are given in in- 4.5. is given in the Table 4.4. formation Table Characteristics and applications of several linear polymers are given in Table Table 4.4 4.5. Effect of crystallinity on density and strength of polyethylene. (Data from Hertzberg (1989)) Table 4.4 Effect of crystallinity on density and strength of polyethylene. (Data 3 Density from (g/cmHertzberg ) % Crystallinity (1989)) Ultimate Tensile Strength MPa ksi Density (g/cm3) % Crystallinity Ultimate Tensile Strength 0.920 65 13.8 2.0 MPa ksi 0.935 75 17.8 2.5 0.920 0.950 85 65 13.8 27.6 4.02.0 0.935 0.960 87 75 17.8 31.0 4.52.5 0.950 0.965 95 85 27.6 37.9 5.54.0 0.960 87 31.0 4.5 0.965 95 37.9 5.5 14 7  10/14/24 4 Polymerization and Classification 111 Table 4.5 Characteristics of several linear polymers. Material Characteristics Applications Low Density Branched crystalline, inex- Films, moldings, squeeze Polyethylene pensive, good insulator bottles, cold water plumbing Polypropylene Crystalline, corrosion and fa- Fibers, pipe, wire covering tigue resistance Nylon 66 Crystalline, tough, resistance Gears, bearings, rollers, pul- to wear, high strength leys, fibers PTFE (Teflon) Crystalline, corrosion resis- Coatings, cookware, bear- tance, very low friction, non- ings, gaskets, insulation sticking. tape, non-stick linings PVC Amorphous, inexpensive, Film, water pipes, insulation good processability PMMA Amorphous, high transpar- Signs, windows, decorative ency products 4.4. Molecular Configurations 15 The terms configuration and conformations are often used to describe the arrangement of atoms in a polymer and sometimes it seems as if they can be used interchangeably. However, herein the description for each given by Billmeyer (1962) will be used. Configurations describe those arrange- MOLECULAR ments of atoms CONFIGURATIONS that cannot be altered except by breaking or reforming chemical bonds. Conformations are arrangements of atoms that can be al- 112 by tered Polymer Engineering rotating groups ofScience atomsand Viscoelasticity: about a single bond. An Introduction Each will be dis- Isomers - Polymers that have the same composition but with different atomic arrangements. cussed in the subsections below. figuration Two Types of Isomers predominates as polar repulsion occurs between R groups in Stereoisomers head-to-head configurations. Geometrical 4.4.1. isomers Isomers Polymers that have the same composition but with different atomic ar- rangements are called isomers. Two basic types are: stereoisomers and geometrical isomers. Isomers occur because polymers may have more than one type of side atom or side group bonded to the main chain (e.g. PVC, (a) Fig. see Basic4.7) mersuch unit that (b) a mer unit Head would to head Fig.Head appear as in (c) config. 4.12(a) to tailinconfig. which R represents an atom or side group other than hydrogen. Polymers with Fig. 4.12 Sequences for isomers. only one extra side group are called “vinyl” polymers. A “head-to-head” arrangement of mers occurs when the R groups are adjacent to each other, For a polymer chain with a given sequence of mer groups, stereoisomers or and a “head-to-tail” arrangement occurs when the R groups bond to alter- geometrical isomers can then be distinguished. The three types of stereoi- nate carbon atoms in the chain as shown in Fig. 4.12. The head-to-tail con- 16 somers (isotactic, syndiotactic and atactic) for a head-to-tail sequence are shown in Fig. 4.13 and the two types of geometric isomers (trans and cis) for a mer unit containing a double bond are shown in Fig. 4.15. In steroi- somerism, the atoms are linked together in the same order (e.g., head-to- tail) but their spatial arrangement is different. The isotactic configuration 8 (a) is when the R groups are all on the same side of the chain. The syndio- tactic configuration (b) is when the R group is on alternate sides of the chain and the atactic configuration (c) is when the R group alternates from one side to the other in a random pattern. Examples of stereoisomers for polypropylene are given in Fig. 4.14.  tail) but their spatial arrangement is different. The isotactic configuration (a) is when the R groups are all on the same side of the chain. The syndio- tactic configuration (b) is when the R group is on alternate sides of the chain and the atactic configuration (c) is when the R group alternates from one side to the other in a random pattern. Examples of stereoisomers for polypropylene are given in Fig. 4.14. 10/14/24 Conversion from one type to another is only possible by breaking a car- bon to carbon bond, rotating and reattaching. This constraint can be seen best by use of molecular models or three dimensional chain representations (eg, Fig. 4.20). A specific polymer may contain more than one type of steroisomer but one may predominate depending only on the synthesis procedure used. (a) Isotactic (b) Syndiotactic (c) Atactic Fig. 4.13 Stereoisomers 17 4 Polymerization and Classification 113 Fig. 4.14 Atactic (a), Isotactic (b) and Syndiotactic (c) polypropylene. 18 An example of geometrical isomerism is given by the isoprene mer and is shown in Fig. 4.15. The cis-isoprene is when the structure is such that the CH2 groups are on the same side of the carbon to carbon double bond and the trans-isoprene is when the CH2 groups are on the opposite side of the carbon to carbon double bond. Conversion between the two configura- 9 tions is not possible by a simple rotation as the double bond is rotationally rigid.  An example of geometrical isomerism is given by the isoprene mer and is shown in Fig. 4.15. The cis-isoprene is when the structure is such that the CH2 groups are on the same side of the carbon to carbon double bond and the trans-isoprene is when the CH2 groups are on the opposite side 10/14/24 of the carbon to carbon double bond. Conversion between the two configura- tions is not possible by a simple rotation as the double bond is rotationally rigid. (a) Cis-isoprene (b) Trans-isoprene Fig. 4.15 Geometrical isomers. With these various molecular characteristics, it is now possible to have a more precise classification scheme of polymers as is illustrated in Fig. 4.16. 19 114 Polymer Engineering Science and Viscoelasticity: An Introduction Fig. 4.16 Classification of polymers by molecular characteristics. 20 4.4.2. Copolymers The polymers described previously are generally referred to as ho- mopolymers because the mer units along the backbone chains are identical. However, it is possible to form copolymers such that the mer units along 10 the backbone chain may vary. Depending on the process of polymeriza- tion, various sequences of mers may occur along the backbone chain in random, alternating, block or graft arrangement as shown in Fig. 4.17.  10/14/24 4 Polymerization and Classification 115 COPOLYMERS 21 Fig. 4.17 Types of copolymers: (a) random, (b) alternating, (c) block, (d) graft. 128 Polymer Engineering Science and Viscoelasticity: An Introduction 4.4.3. 128 Molecular Conformations Polymer Engineering Science and Viscoelasticity: An Introduction In an earlier section, it was suggested that the shape of a polymer molecule could change because of a rotation about the bond between carbon atoms. An example of a possible rotation is given in Fig. 4.18 where two possible positions, staggered and eclipsed, of hydrogen atoms attached to two adja- cent carbon atoms are shown for the ethane molecule. t1 t2 t3 t4 Fig. 4.32 Stages in the growth of a spherulite. t1 t2 t3 t4 Fig. 4.32 Stages in the growth of a spherulite. Fig. 4.33 Examples of spherulites: (a) Branched spherulite in polypropylene Fig. 4.18from AFM (Zhouand Staggered et al.eclipsed (2005), reprinted by permission from conformations Elsevier.) of ethane. (b) Field of growing spherulites in polystyrene (Reprinted by permis- sion from Beers et al. Copyright 2003, American Chemical Society) 22 Birefringence occurs because polarized light passing through a crystal is broken Fig. 4.33 up into two Examples of components spherulites:propagating (a) Branchedalong spherulite a plane perpendicular to in polypropylene the principal from AFM axis(Zhou of the et crystal. Each component al. (2005), reprinted bytravels at a different permission ve- from Elsevier.) locity(b) and therefore Field one is retarded of growing relative spherulites to the other. On in polystyrene emergingby (Reprinted andpermis- passing sionthrough a second from Beers et al.polarizer, Copyright interference betweenChemical 2003, American the two waves Society) gives rise to the dark “fringes” and hence the maltese cross. Actually the maltese cross is indicative of the directions of principal stress which are Birefringence occurs perpendicular because to each other inpolarized the plane oflight passingThethrough the section. nature ofa bire- crystal is 11 brokenfringence up intointwo components crystals propagating has been known for well along over a acentury plane(as perpendicular discussed to the principal axis of theand in the Introduction) crystal. Eachforcomponent is the basis travels the well known at a different photoelastic stress ve- locity analysis method. one and therefore In this procedure, relative is retarded initially amorphous polymers to the other. become and On emerging passingoptically through anisotropic a second due polarizer, to the application of external interference forces. That between the is, twothewaves gives rise to the dark “fringes” and hence the maltese cross. Actually the maltese cross is indicative of the directions of principal stress which are perpendicular to each other in the plane of the section. The nature of bire- fringence in crystals has been known for well over a century (as discussed  sphericalthatnature the polymer reacts to light is apparent andasitif is it were a crystal. to be notedAsthat a result, thetheindividual stress fi- inside the material can be visualized and analyzed using the birefringence brils/lamellae grow radially. The individual fibrils have a folded effect. The isochromatic fringes (lines of equal shear stress) in a sample of chain structurepolycarbonate and the chain traverses containing a crack both crystalline are shown regions and amorphous in Fig. 4.35. regions as Cross-linked Fig. 4.31 of illustrated orinthermosetting the folded polymers chain used are typically model. for photoelas- tic stress analysis. Thus, it is clear that a certain amount of crystallinity can be induced by stresses in network polymers but the degree of crystallinity is necessarily very small. 10/14/24 A schematic visualization of a spherulite is given in Fig. 4.36. Here the spherical nature is apparent and it is to be noted that the individual fi- brils/lamellae grow radially. The individual fibrils have a folded chain structure and the chain traverses both crystalline regions and amorphous regions as illustrated in Fig. 4.31 of the folded chain model. Fig. 4.34 Examples of spherulites: (a) Spherulites in polyethylene (Armistead et al. (Reprinted by permission from Armistead et al. Copyright 2003, American Chemical Society). (b) Ringed spherulites of poly(hydroxybutyrate), (Hobbs et al. (2000)) Reprinted by permission of John Wiley and Sons, Inc. Fig. 4.34 Examples of spherulites: (a) Spherulites in polyethylene (Armistead et al. (Reprinted by permission from Armistead et al. Copyright 2003, American Chemical Society). (b) Ringed spherulites of poly(hydroxybutyrate), (Hobbs et al. (2000)) Reprinted by permission of John Wiley and Sons, Inc. 132 Polymer Engineering Science and Viscoelasticity: An Introduction containing 6 protons and 6 neutrons, with an atomic mass of exactly 12.000 amu. The atomic mass shown in the periodic table for Carbon is slightly higher (12.011) as it accounts for small amounts of the isotope 13C. Fig. 4.35 Birefringence photograph of polycarbonate showing isochromatic fringes surrounding a crack. Since one does Fig. not 4.35typically Birefringence work photographwith single atoms of polycarbonate or molecules, quan- showing isochromatic 23 tities of chemical substances are given in moles. A mole of an element is fringes surrounding a crack. 23 defined as 6.02214 x 10 (Avogadro’s number) atoms; a mole of a given type of molecule is 6.02214 x 1023 molecules. Avogadro’s number is de- fined to provide a simple conversion to grams: 6.022 x 1023 atoms (or molecules) have the mass in grams of the atomic mass of a single atom (molecule). MOLECULAR For example, WEIGHT 1 mole (6.022 x 1023 atoms) of 12C has a mass of exactly 12.0g. The conversion is therefore 23 6.02214 x 10 amu = 1 gram -24 or 1 amu = 1.66054 x 10 g As an example, consider a mole of water molecules (H2 O) which contains 6.022 x 1023 atoms of oxygen and 2 x (6.022 x 1023) atoms of hydrogen. atomic mass unit (amu) is defined on the basis of Carbon-12, the most common isotope of The atomic masses Carbon of oxygen and hydrogen are 15.9994 amu and 1.0079 amu respectively. Therefore a mole of water has a mass of 2x(1.0079g) + 15.9994g = 18.015 grams. This example also emphasizes that moles are the necessary units to use for chemical reactions as the proper number of atoms must 24 be tracked: e.g. one mole of oxygen and two moles of hydro- gen can be combined to form 1 mole of water; 1 gram of oxygen and 2 grams of hydrogen are not in the proper ratio to form a gram of water ow- ing to the differing masses of the elements. 12 Historically, the terms gram-atom and gram-molecule were originally coined to refer to the mass in grams of Avogadro’s number of atoms or molecules, respectively; with the introduction of the term mole, these early terms are less used but can still be found in the literature. The term “mo- a molar mass of Mass of 1 mole of PE chains: 280,000 grams (or grams/mole) neglecting chain end effects. Note that the molecular mass of a chain end (or at a branch point) is not the same as the molecular mass of a mer unit but the difference is neglected because the effect is small in terms of the 10/14/24 total molecular mass of a chain. While “relative molecular mass” is the official and more correct termi- nology for polymers (as used in McCrum, 1997), in the following the term molecular weight will be most often used as is common in many polymer texts. Mass of 1 PE chain: 104 (2 x 12 + 4 x1) = 280,000 amu A useful term to describe the extent of polymerization in polymers is the “degree of polymerization” Mass of 1 mole (DP) which of PE chains: isgrams 280,000 defined as the number of mer (or grams/mole) units per chain or, M n= = DP (4.6) Mr where M is the molar mass (weight) of a chain and Mr is the molar mass (weight) of a mer where M is the molar mass (weight) of a chain and M is the molar mass or repeat (weight) of a merunit. (Number or repeat unit. r average and weight average de- grees of polymerization are also used as will be evident directly.) The degree of polymerization or the length of a polymer chain is an in- dicator of the nature and mechanical characteristics of a polymer com- posed of similar length chains. The following table illustrates the relation- ship between chain length and the character of a polymer at 25 ºC and a pressure 25 of one atmosphere. 134 Polymer Engineering Science and Viscoelasticity: An Introduction Table 4.6 Degree of polymerization – phase relationship (data from Clegg and Collyer (1993), p. 11) Number of –CH2-CH2 Molar Mass Softening Character at Repeat units per chain kg mol-1 Temperature 25°C and 1 (Degree of Polymerization °C at. 1 28 -169 Gas 6 170 -12 Liquid 35 1000 37 Grease 140 4000 93 Wax 430 12000 104 Resin 1350 38000 112 Hard Resin It is now clear how to calculate the molecular weight of a single chain or of a mole of polymer chains of identical lengths. Unfortunately, however, the lengths of chains in a polymer vary greatly and depend to a large de- gree on the circumstances and the manner in which the polymerization re- action proceeds. That is, a wide distribution of chain lengths (DP’s or chain molecular weights) exist in a typical polymer as shown in Fig. 4.38. The distribution is seldom symmetrical and the breath of distribution var- 26 ies with the type of reaction. For example, the distribution is often quite broad for polyethylene while the distribution for polystyrene may be quite narrow (Fried, 1995). Because of the distributed nature of the lengths of chains in a polymer it is necessary to define the molecular weight using an averaging process. 13 The most common averaging processes used are the number average, the weight average and the z-average. Only the number and weight average methods will be described here. Both discrete and continuous distributions are possible. For  action proceeds. That is, a wide distribution of chain lengths (DP’s or chain molecular weights) exist in a typical polymer as shown in Fig. 4.38. The distribution is seldom symmetrical and the breath of distribution var- ies with the type of reaction. For example, the distribution is often quite broad for polyethylene while the distribution for polystyrene may be quite narrow (Fried, 1995). Because of the distributed nature of the lengths of chains in a polymer it 10/14/24 is necessary to define the molecular weight using an averaging process. The most common averaging processes used are the number average, the weight average and the z-average. Only the number and weight average methods will be described here. Both discrete and continuous distributions are possible. For Fig. 4.38 Typical molecular weight distribution of the number of chains in a polymer. 27 4 Polymerization and Classification 135 example the continuous distribution in Fig. 4.38(b) is obtained by drawing a smooth curve through the discrete4distribution shown Polymerization and in Fig.1354.38(a). Classification For a discrete distribution the number average molecular weight is de- example the continuous distribution in Fig. 4.38(b) is obtained by drawing fined as,a smooth curve through the discrete distribution shown in Fig. 4.38(a). For a discrete distribution kthe number average k molecular weight is de- fined as, !k NM i i NM k ! i i i=1 i=1 Mn = ! kN i M i =NM ! i i (4.7) N Mn = i=1 k!NN = i i=1 N (4.7) ! i=1 i i=1 where Nwhere i is Nthe number of chains within an interval, Mi is the median i is the number of chains within an interval, Mi is the median (middle)(middle) molecular weight molecular weightin in an interval, an interval, k is k theistotal thenumber total ofnumber intervalsof intervals and N and N is the total number of chains. If a continuous curvetheisdis- is the total number of chains. If a continuous curve is fit to fit to the dis- crete data such that N is given as a function of M, i.e., N=N(M), the sum- crete data suchcanthat mations be N is given replaced by anas a function integration of M, to obtain N=N(M), i.e., and (Kumar Gupta, the sum- mations can be replaced by an integration to obtain (Kumar and Gupta, (1998)), (1998)), M M ! N(M) dM ! N(M) M0 M dM 0 Mn = = (4.8) ! N(M) ! dN dM !NN(M) dM The product ofM = 0 of chains in = then number 0 an interval, Ni, and the molecular (4.8) 28 ! dN weight of an interval, Mi, equals the total weight Nan interval. Exchang- of ing Ni in Eq. 4.10 by Ni Mi defines the weight average molecular weight and can be written as, The product of the number of chains in an interval, Ni, and the molecular k k weight of an interval, Mi, equals ! N i Mthe 2 total weight of an interval. Exchang- i ! N i M 2i ing Ni in Eq. 4.10 by NMi M=i i=1 w defines=the i=1 weight average molecular weight (4.9) 14 k and can be written as, M NM ! i i k i=1 k ! where M is the total molar mass Mthe N i of i 2 N iSome sample. ! M 2i have likened the number average and weight average molecular weights to the first and sec- ond moments of masses = i=1 M w (or = i=1 mechanics courses. Such areas) in elementary k (4.9) an analogy is appropriate if the number of chains,M Ni, is replaced by a lever ! arm d with units of length. One N textMincorrectly relates the weight average  fined as, i=1 k k where Ni is the number of chains within an interval, Mi is the median (middle) molecular weight in an ! N iM i ! N iM i interval,i=1k is the total number of intervals M n = i=1k and N is the total number of chains. If= a continuous N (4.7) curve is fit to the dis- crete data such that N is given as ! a ifunction of M, i.e., N=N(M), the sum- N mations can be replaced by ani=1integration to obtain (Kumar and Gupta, where Ni is the number of chains within an interval, Mi is the median 10/14/24 (1998)), (middle) molecular weight in an interval, k is the total number of intervals M of chains. If a M and N is the total number continuous curve is fit to the dis- ! crete data such that N is N(M) given asdM a functionN(M) dMN=N(M), the sum- of M, i.e., ! mations can be replaced M n = 0 by an integration = 0 to obtain (Kumar and Gupta, (4.8) (1998)), dN M ! N M The product of the number!of N(M) dM in !an chains N(M) dM N , and the molecular interval, i 0 0 = = weight of an interval, Mn i, equals the total weight of an interval. (4.8) M Exchang- ! dN N ing Ni in Eq. 4.10 by Ni Mi defines the weight average molecular weight and can beproduct The writtenofas, the number of chains in an interval, Ni, and the molecular weight of an interval, Mi, equals k the total weight of an interval. Exchang- k ing Ni in Eq. 4.10 by Ni Mi defines2the weight average molecular weight and can be written as, N i M! i N i M 2 i ! M w = i=1 k k = i=1 k (4.9) ! !N M N i M 2i i i ! NM 2 iM i Mw = i=1 = i=1 (4.9) i=1k M where M is the total molar mass !N M i i of the sample. Some have likened the i=1 number average and weight average molecular weights to the first and sec- where M is the total molar mass of the sample. Some have likened the ond moments of masses number average (or areas) and weight average inmolecular elementary mechanics weights to the firstcourses. and sec- Such an analogy is appropriate ond moments of massesif(or theareas) number of chains, in elementary Ni, is replaced mechanics by a lever courses. Such 29 d136 arm with ian unitsisEngineering analogy Polymer of length. appropriate One if theand Science text numberincorrectly of chains, Viscoelasticity: relates AnN the weight i, is replaced Introduction average by a lever arm dweight molecular i with units to aofradius length.ofOne text incorrectly relates the weight average 136 Polymer molecularEngineering Science gyration. and Viscoelasticity: An Introduction Consider theweight to awhere, example radius of gyration. Consider the example i where, Mi Ni Interval No. g/mole of chains No. of chains i in interval Mi in interval Ni 1 No. Interval 5,000 g/mole of chains 2 No. of chains 2 in15,000 interval in 4interval 13 30,000 5,000 5 2 24 50,000 15,000 1 4 3 30,000 5 The number average and the weight average molecular weight from Eqs. 4 be, 4.10 and 4.11 will 50,000 1 k The number average and the weight Naverage molecular weight from Eqs. 4.10 and 4.11 will be, ! iM i i=1 Mn = k (4.10a) k !!NN M i=1 i i i i=1 Mn = k (4.10a) 2(5,000) + 4(15,000) + 5(30,000) +1(50,000) Mn = 12 Ni ! = 22,500 g/mole (4.10b) i=1 Some experimental approaches separate the chains in a polymer into dis- 30 2(5,000) + 4(15,000) + 5(30,000) +1(50,000) xi, is defined Mcrete n = number or weight fractions. The number fraction, = 22,500 g/moleas(4.10b) the 12 in an interval to the total number of chains in ratio of the number of chains the sample, Some experimental approaches separate N the chains in a polymer into dis- crete number or weight fractions. = inumber fraction, xi, is defined x i The (4.11) as the 15 N ratio of the number of chains in an interval to the total number of chains in theand the weight fraction, wi, is the ratio of the total weight of the chains in sample, an interval to the total weight of the sample, N x i = M ii (4.11) wi = N (4.12) M and the weight fraction, wi, is the ratio of the total weight of the chains in  in interval in interval 1 5,000 2 2 15,000 4 3 30,000 5 4 50,000 1 10/14/24 The number average and the weight average molecular weight from Eqs. 4.10 and 4.11 will be, k !N M i i i=1 Mn = k (4.10a) !N i i=1 2(5,000) + 4(15,000) + 5(30,000) +1(50,000) Mn = = 22,500 g/mole (4.10b) 12 Some experimental approaches separate the chains in a polymer into dis- crete number or weight fractions. The number fraction, xi, is defined as the ratio of the number of chains in an interval to the total number of chains in the sample, 140 Polymer Engineering Science and Viscoelasticity: An Introduction N the extent of chain entanglement and the amount ofi melt elasticity. For x = i to measure the molecular these and many other reasons, it is necessary (4.11) N weight and molecular weight distribution. Indeed, as mentioned in the in- troduction, the lack of accurate methods to measure high molecular and the weight fraction, wi, is the weights impeded the initial development ratio of the total weight and understanding of polymers. of the chains in 31 an interval tothethe Many of totalusedweight methods ofmolecular to measure the sample, weight are listed in the Table 4.8. The usual range of weights that can be found by each method is M also given. Note that the end group analysis and colligative i property meth- ods give number average molecular weight w =i while the light scattering (4.12) M method gives the weight average molecular weight. The other methods give only a relative measure of the molecular weight and one of the former An example methods must illustrating this be used to provide approach a calibration is shown of the method in the and therefore hypothetical distri- Methods it isfor thetoMeasurement possible of Molecular Weight obtain either quantity. bution given below, Table 4.8 Average molecular weight measurement. End Group Analysis M n