Engineering Materials Vol II (microstructures_ processing_ design) 2nd ed. - M. Ashby_ D. Jones (1999) WW Part 10 potx

234 Engineering Materials 2 Fig. 22.6. A schematic drawing of a largely crystalline polymer like high-density polyethylene. At the top the polymer has melted and the chain-folded segments have unwound. metal crystal, and a unit cell can be defined (Fig. 22.5). Note that the cell is much smaller than the molecule itself. But even the most crystalline of polymers (e.g. high-density PE) is only 80% crystal. The structure probably looks something like Fig. 22.6: bundles, and chain-folded seg- ments, make it largely crystalline, but the crystalline parts are separated by regions of disorder – amorphous, or glassy regions. Often the crystalline platelets organise them- selves into spherulites: bundles of crystallites that, at first sight, seem to grow radially outward from a central point, giving crystals with spherical symmetry. The structure is really more complicated than that. The growing ends of a small bundle of crystallites (Fig. 22.7a) trap amorphous materials between them, wedging them apart. More crystallites nucleate on the bundle, and they, too, splay out as they grow. The splaying continues until the crystallites bend back on themselves and touch; then it can go no further (Fig. 22.7b). The spherulite then grows as a sphere until it impinges on others, to form a grain-like structure. Polythene is, in fact, like this, and polystyrene, nylon and many other linear polymers do the same thing. When a liquid crystallises to a solid, there is a sharp, sudden decrease of volume at the melting point (Fig. 22.8a). The random arrangement of the atoms or molecules in the liquid changes discontinuously to the ordered, neatly packed, arrangement of the crystal. Other properties change discontinuously at the melting point also: the vis- cosity, for example, changes sharply by an enormous factor (10 10 or more for a metal). Broadly speaking, polymers behave in the same way: a crystalline polymer has a fairly well-defined melting point at which the volume changes rapidly, though the sharp- ness found when metals crystallise is blurred by the range of molecular weights (and thus melting points) as shown in Fig. 22.8(b). For the same reason, other polymer properties (like the viscosity) change rapidly at the melting point, but the true discon- tinuity of properties found in simple crystals is lost. The structure of polymers 235 Fig. 22.7. The formation and structure of a spherulite. Fig. 22.8. (a) The volume change when a simple melt (like a liquid metal) crystallises defines the melting point, T m ; (b) the spread of molecular weights blurs the melting point when polymers crystallise; (c) when a polymer solidifies to a glass the melting point disappears completely, but a new temperature at which the free volume disappears (the glass temperature, T g ) can be defined and measured. 236 Engineering Materials 2 When, instead, the polymer solidifies to a glass (an amorphous solid) the blurring is much greater, as we shall now see. Amorphous polymers Cumbersome side-groups, atacticity, branching and cross-linking all hinder crystallisa- tion. In the melt, thermal energy causes the molecules to rearrange continuously. This wriggling of the molecules increases the volume of the polymer. The extra volume (over and above that needed by tightly packed, motionless molecules) is called the free- volume. It is the free-volume, aided by the thermal energy, that allows the molecules to move relative to each other, giving viscous flow. As the temperature is decreased, free-volume is lost. If the molecular shape or cross- linking prevent crystallisation, then the liquid structure is retained, and free-volume is not all lost immediately (Fig. 22.8c). As with the melt, flow can still occur, though naturally it is more difficult, so the viscosity increases. As the polymer is cooled fur- ther, more free volume is lost. There comes a point at which the volume, though sufficient to contain the molecules, is too small to allow them to move and rearrange. All the free volume is gone, and the curve of specific volume flattens out (Fig. 22.8c). This is the glass transition temperature, T g . Below this temperature the polymer is a glass. The glass transition temperature is as important for polymers as the melting point is for metals (data for T g are given in Table 21.5). Below T g , secondary bonds bind the molecules into an amorphous solid; above, they start to melt, allowing molecular motion. The glass temperature of PMMA is 100°C, so at room temperature it is a brittle solid. Above T g , a polymer becomes first leathery, then rubbery, capable of large elastic extensions without brittle fracture. The glass temperature for natural rubber is around −70°C, and it remains flexible even in the coldest winter; but if it is cooled to −196°C in liquid nitrogen, it becomes hard and brittle, like PMMA at room temperature. That is all we need to know about structure for the moment, though more informa- tion can be found in the books listed under Further reading. We now examine the origins of the strength of polymers in more detail, seeking the criteria which must be satisfied for good mechanical design. Further reading D. C. Bassett, Principles of Polymer Morphology, Cambridge University Press, 1981. F. W. Billmeyer, Textbook of Polymer Science, 3rd edition, Wiley Interscience, 1984. J. A. Brydson, Plastics Materials, 6th edition, Butterworth-Heinemann, 1996. J. M. C. Cowie, Polymers: Chemistry and Physics of Modern Materials, International Textbook Co., 1973. C. Hall, Polymer Materials, Macmillan, 1981. R. J. Young, Introduction to Polymers, Chapman and Hall, 1981. Problems 22.1 Describe, in a few words, with an example or sketch where appropriate, what is meant by each of the following: The structure of polymers 237 (a) a linear polymer; (b) an isotactic polymer; (c) a sindiotactic polymer; (d) an atactic polymer; (e) degree of polymerization; (f) tangling; (g) branching; (h) cross-linking; (i) an amorphous polymer; (j) a crystalline polymer; (k) a network polymer; (l) a thermoplastic; (m) a thermoset; (n) an elastomer, or rubber; (o) the glass transition temperature. 22.2 The density of a polyethylene crystal is 1.014 Mg m –3 at 20°C. The density of amorphous polyethylene at 20°C is 0.84 Mg m –3 . Estimate the percentage crystal- linity in: (a) a low-density polyethylene with a density of 0.92 Mg m –3 at 20°C; (b) a high-density polyethylene with a density of 0.97 Mg m –3 at 20°C. Answers: (a) 46%, (b) 75%. 238 Engineering Materials 2 Chapter 23 Mechanical behaviour of polymers Introduction All polymers have a spectrum of mechanical behaviour, from brittle-elastic at low temperatures, through plastic to viscoelastic or leathery, to rubbery and finally to viscous at high temperatures. Metals and ceramics, too, have a range of mechanical behaviour, but, because their melting points are high, the variation near room temperature is unimportant. With polymers it is different: between −20°C and +200°C a polymer can pass through all of the mechanical states listed above, and in doing so its modulus and strength can change by a factor of 10 3 or more. So while we could treat metals and ceramics as having a constant stiffness and strength for design near ambient temper- atures, we cannot do so for polymers. The mechanical state of a polymer depends on its molecular weight and on the temperature; or, more precisely, on how close the temperature is to its glass temper- ature T g . Each mechanical state covers a certain range of normalised temperature T/T g (Fig. 23.1). Some polymers, like PMMA, and many epoxies, are brittle at room tem- perature because their glass temperatures are high and room temperature is only 0.75 T g . Others, like the polyethylenes, are leathery; for these, room temperature is about 1.0 T g . Still others, like polyisoprene, are elastomers; for these, room temperature is well above T g (roughly 1.5 T g ). So it makes sense to plot polymer properties not against temperature T, but against T/T g since that is what really determines the mechanical Fig. 23.1. Schematic showing the way in which Young’s modulus E for a linear polymer changes with temperature for a fixed loading time. Mechanical behaviour of polymers 239 state. The modulus diagrams and strength diagrams described in this chapter are plotted in this way. It is important to distinguish between the stiffness and the strength of a polymer. The stiffness describes the resistance to elastic deformation, the strength describes the re- sistance to collapse by plastic yielding or by fracture. Depending on the application, one or the other may be design-limiting. And both, in polymers, have complicated origins, which we will now explain. Stiffness: the time- and temperature-dependent modulus Much engineering design – particularly with polymers – is based on stiffness: the designer aims to keep the elastic deflections below some critical limit. Then the mater- ial property which is most important is Young’s modulus, E. Metals and ceramics have Young’s moduli which, near room temperature, can be thought of as constant. Those of polymers cannot. When a polymer is loaded, it deflects by an amount which increases with the loading time t and with the temperature T. The deflection is elastic – on unloading, the strain disappears again (though that, too, may take time). So it is usual to speak of the time- and temperature-dependent modulus, E(t, T) (from now on simply called E). It is defined, just like any other Young’s modulus, as the stress σ divided by the elastic strain ε E tT (, ) .= σ ε (23.1) The difference is that the strain now depends on time and temperature. The modulus E of a polymer can change enormously – by as much as a factor of 1000 – when the temperature is changed. We will focus first on the behaviour of linear-amorphous polymers, examining the reasons for the enormous range of modu- lus, and digressing occasionally to explain how cross-linking, or crystallisation, change things. Linear-amorphous polymers (like PMMA or PS) show five regimes of deformation in each of which the modulus has certain characteristics, illustrated by Fig. 23.1. They are: (a) the glassy regime, with a large modulus, around 3 GPa; (b) the glass-transition regime, in which the modulus drops steeply from 3 GPa to around 3 MPa; (c) the rubbery regime, with a low modulus, around 3 MPa; (d) the viscous regime, when the polymer starts to flow; (e) the regime of decomposition in which chemical breakdown starts. We now examine each regime in a little more detail. The glassy regime and the secondary relaxations The glass temperature, T g , you will remember, is the temperature at which the second- ary bonds start to melt. Well below T g the polymer molecules pack tightly together, either in an amorphous tangle, or in poorly organised crystallites with amorphous 240 Engineering Materials 2 Fig. 23.2. A schematic of a linear-amorphous polymer, showing the strong covalent bonds (full lines) and the weak secondary bonds (dotted lines). When the polymer is loaded below T g , it is the secondary bonds which stretch. material in between. Load stretches the bonds, giving elastic deformation which is recovered on unloading. But there are two sorts of bonds: the taut, muscular, covalent bonds that form the backbone of the chains; and the flabby, soft, secondary bonds between them. Figure 23.2 illustrates this: the covalent chain is shown as a solid line and the side groups or radicals as full circles; they bond to each other by secondary bonds shown as dotted lines (this scheme is helpful later in understanding elastic deformation). The modulus of the polymer is an average of the stiffnesses of its bonds. But it obviously is not an arithmetic mean: even if the stiff bonds were completely rigid, the polymer would deform because the weak bonds would stretch. Instead, we calculate the modulus by summing the deformation in each type of bond using the methods of composite theory (Chapter 25). A stress σ produces a strain which is the weighted sum of the strains in each sort of bond ε σσ σ ( ) ( ) .=+− = + −      f E f E f E f E 1212 1 1 (23.2) Here f is the fraction of stiff, covalent bonds (modulus E 1 ) and 1 − f is the fraction of weak, secondary bonds (modulus E 2 ). The polymer modulus is E f E f E ( ) .== + −      − σ ε 12 1 1 (23.3) If the polymer is completely cross-linked ( f = 1) then the modulus (E 1 ) is known: it is that of diamond, 10 3 GPa. If it has no covalent bonds at all, then the modulus (E 2 ) is that of a simple hydrocarbon like paraffin wax, and that, too, is known: it is 1 GPa. Mechanical behaviour of polymers 241 Fig. 23.3. The way in which the modulus of polymers changes with the fraction of covalent bonds in the loading direction. Cross-linking increases this fraction a little; drawing increases it much more. Substituting this information into the last equation gives an equation for the glassy modulus as a function of the fraction of covalent bonding E f f ( ) .=+ −       − 10 1 1 3 1 GPa (23.4) This function is plotted in Fig. 23.3. The glassy modulus of random, linear poly- mers ( f = 1 2 ) is always around 3 GPa. Heavily cross-linked polymers have a higher modulus because f is larger – as high as 0.75 – giving E = 8 GPa. Drawn polymers are different: they are anisotropic, having the chains lined up along the draw direc- tion. Then the fraction of covalent bonds in the loading direction is increased dramatic- ally. In extreme drawing of fibres like nylon or Kevlar this fraction reaches 98%, and the modulus rises to 100 GPa, about the same as that of aluminium. This orientation strengthening is a potent way of increasing the modulus of polymers. The stiffness normal to the drawing direction, of course, decreases because f falls towards zero in that direction. You might expect that the glassy modulus (which, like that of metals and ceramics, is just due to bond-stretching) should not depend much on temperature. At very low temperatures this is correct. But the tangled packing of polymer molecules leaves some “loose sites” in the structure: side groups or chain segments, with a little help from thermal energy, readjust their positions to give a little extra strain. These second- ary relaxations (Fig. 23.1) can lower the modulus by a factor of 2 or more, so they cannot be ignored. But their effect is small compared with that of the visco-elastic, or glass transition, which we come to next. 242 Engineering Materials 2 Fig. 23.4. Each molecule in a linear polymer can be thought of as being contained in a tube made up by its surroundings. When the polymer is loaded at or above T g , each molecule can move (reptate) in its tube, giving strain. The glass, or visco-elastic transition As the temperature is raised, the secondary bonds start to melt. Then segments of the chains can slip relative to each other like bits of greasy string, and the modulus falls steeply (Fig. 23.1). It is helpful to think of each polymer chain as contained within a tube made up by the surrounding nest of molecules (Fig. 23.4). When the polymer is loaded, bits of the molecules slide slightly in the tubes in a snake-like way (called “reptation”) giving extra strain and dissipating energy. As the temperature rises past T g , the polymer expands and the extra free volume (Chapter 22) lowers the packing density, allowing more regions to slide, and giving a lower apparent modulus. But there are still non-sliding (i.e. elastic) parts. On unloading, these elastic regions pull the polymer back to its original shape, though they must do so against the reverse viscous sliding of the molecules, and that takes time. The result is that the polymer has leathery properties, as do low-density polyethylene and plasticised PVC at room temperature. Within this regime it is found that the modulus E at one temperature can be related to that at another by a change in the time scale only, that is, there is an equivalence between time and temperature. This means that the curve describing the modulus at one temperature can be superimposed on that for another by a constant horizontal dis- placement log (a T ) along the log (t) axis, as shown in Fig. 23.5. A well-known example of this time–temperature equivalence is the steady-state creep of a crystalline metal or ceramic, where it follows immediately from the kinetics of thermal activation (Chapter 6). At a constant stress σ the creep rate varies with temperature as ˙ exp ( ) ε ε ss /== − t AQRT (23.5) Mechanical behaviour of polymers 243 giving ε (t, T) = tA exp (–Q/RT). (23.6) From eqn. (23.1) the apparent modulus E is given by E tT tA QRT B t QRT (, ) exp ( ) exp ( ).== = σ ε σ // (23.7) If we want to match the modulus at temperature T 1 to that at temperature T 0 (see Fig. 23.5) then we need exp ( ) exp ( )QRT t QRT t // 1 1 0 0 = (23.8) or t t QRT QRT Q RT T 1 0 1 010 11 exp ( ) exp ( ) exp .==−       / / (23.9) Thus ln , t t Q RT T 0 110 11      =− −       (23.10) and log (a T ) = log (t 0 /t 1 ) = log t 0 − log t 1 = − −       . . Q RT T23 11 10 (23.11) This result says that a simple shift along the time axis by log (a T ) will bring the response at T 1 into coincidence with that at T 0 (see Fig. 23.5). Fig. 23.5. Schematic of the time–temperature equivalence for the modulus. Every point on the curve for temperature T 1 lies at the same distance, log ( a T ), to the left of that for temperature T 0 . [...]... because the molecules are more densely packed) but it does not suppress melting, so crystalline linear-polymers (like high-density PE) can be formed by heating and moulding them, just like linear-amorphous polymers; cross-linked polymers cannot 248 Engineering Materials 2 Strength: cold drawing and crazing Engineering design with polymers starts with stiffness But strength is also important, sometimes... for alignment is between 2 and 4 (nominal strains of 100 to 300%) The neck propagates along the sample until it is all drawn (Fig 23 .10) Fig 23 .10 Cold-drawing of a linear polymer: the molecules are drawn out and aligned giving, after a draw ratio of about 4, a material which is much stronger in the draw direction than it was before 250 Engineering Materials 2 The drawn material is stronger in the draw... stress σs produces a rate of shear γ˙ then the viscosity (Chapter 19) is η= σs 10 ˙ (23.13) Its units are poise (P) or 10 1 Pa s Polymers, like inorganic glasses, are formed at a viscosity in the range 104 to 106 poise, when they can be blown or moulded (When a metal melts, its viscosity drops discontinuously to a value near 10 3 poise – about the same as that of water; that is why metals are formed by... polymers by creating —O— cross-links between polymer chains; it is a sort of unwanted vulcanisation The cross-links raise Tg, and make the polymer brittle; it is a particular problem with rubbers, which lose their elasticity Ozone (O3) is especially damaging because it supplies oxygen in an unusually active form Sunlight (particularly ultraviolet again) promotes oxidation, partly because it — creates... still hot, the cycle time is as short as 10 seconds for small components, 10 minutes for large thick-walled mouldings Pressures are lower than for injection mouldings, so the capital cost of the equipment is much less Fig 24.4 (a) Vacuum forming is good for making simple shapes out of sheet (b) Blow moulding is used to make plastic containers 260 Engineering Materials 2 Fig 24.5 Compression moulding:... is small, about one-thousandth of the glassy modulus, Tg , but it is there nonetheless, and gives the plateau in the modulus shown in Fig 23.1 Much more pronounced rubbery behaviour is obtained if the chance entanglements are replaced by deliberate cross-links The number of cross-links must be small – about 1 in every few hundred monomer units But, being strong, the covalent cross-links do not melt,... typical of linear polymers 252 Engineering Materials 2 stiffness but the diagram is broadly typical of other linear polymers The diagram is helpful in giving a broad, approximate, picture of polymer strength The vertical axis is the strength of the polymer: the stress at which inelastic behaviour becomes pronounced The right-hand scale gives the strength in MPa; the left-hand scale gives the strength... circumspection In an engineering application the stress-state may be multiaxial, not simple tension; and the environment (even simple sunlight) may attack and embrittle the polymer, reducing its strength These, and other, aspects of design with polymers, are discussed in the books listed under Further reading Further reading J A Brydson, Plastics Materials, 6th edition, Butterworth-Heinemann, 1996 International... P C Powell and A J Ingen Honsz, Engineering with Polymers, 2nd edition, Chapman and Hall, 1998 D W Van Krevlin, Properties of Polymers, Elsevier, 1976 I M Ward, Mechanical Properties of Solid Polymers, 2nd edition, Wiley, 1984 R J Young, Introduction to Polymers, Chapman and Hall, 1981 Problems 23.1 Estimate the loading time needed to give a modulus of 0.2 GPa in low-density polyethylene at the glass... block copolymer (Fig 24.1b) 256 Engineering Materials 2 Fig 24.1 (a) A copolymer of vinyl chloride and vinyl acetate; the “alloy” packs less regularly, has a lower Tg , and is less brittle than simple polyvinylchloride (PVC) (b) A block copolymer: the two different molecules in the alloy are clustered into blocks along the chain Fig 24.2 A two-phase polymer alloy, made by co-polymerising styrene and butadiene . the molecules increases the volume of the polymer. The extra volume (over and above that needed by tightly packed, motionless molecules) is called the free- volume. It is the free-volume, aided. bottom. (The normalisations make the diagrams more general: similar polymers should have similar normalised diagrams.) The diagram is divided, like the modulus diagram, into fields corresponding to the five. summarised in the modulus diagram for a poly- mer. Figure 23.7 shows an example: it is a modulus diagram for PMMA, and is typical of linear-amorphous polymers (PS, for example, has a very similar