Engineering Materials vol 2 Part 11 ppsx

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.. So good elastomers, likepolyisopr

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 glassymodulus as a function of the fraction of covalent bonding

3

1

This function is plotted in Fig 23.3 The glassy modulus of random, linear

poly-mers ( f = 1) 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 polymersare 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 lowtemperatures this is correct But the tangled packing of polymer molecules leavessome “loose sites” in the structure: side groups or chain segments, with a little helpfrom 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 theycannot be ignored But their effect is small compared with that of the visco-elastic, orglass transition, which we come to next

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 thechains can slip relative to each other like bits of greasy string, and the modulus fallssteeply (Fig 23.1) It is helpful to think of each polymer chain as contained within atube made up by the surrounding nest of molecules (Fig 23.4) When the polymer isloaded, 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 packingdensity, allowing more regions to slide, and giving a lower apparent modulus But thereare still non-sliding (i.e elastic) parts On unloading, these elastic regions pull thepolymer 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-statecreep 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 withtemperature as

ε˙ ss = ε = exp (− / )

0 0

This result says that a simple shift along the time axis by log (a T) will bring the

response at T into coincidence with that at T (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

Polymers are a little more complicated The drop in modulus (like the increase increep rate) is caused by the increased ease with which molecules can slip past eachother In metals, which have a crystal structure, this reflects the increasing number ofvacancies and the increased rate at which atoms jump into them In polymers, whichare amorphous, it reflects the increase in free volume which gives an increase in therate of reptation Then the shift factor is given, not by eqn (23.11) but by

predict the effect of temperature on polymer behaviour If T0 is taken to be the glass

temperature, then C1 and C2 are roughly constant for all amorphous polymers (and

inorganic glasses too); their values are C1= 17.5 and C2= 52 K

Rubbery behaviour and elastomers

As the temperature is raised above T g, one might expect that flow in the polymershould become easier and easier, until it becomes a rather sticky liquid Linear poly-mers with fairly short chains ( DP < 103) do just this But polymers with longer chains( DP > 104) pass through a rubbery state.

The origin of rubber elasticity is more difficult to picture than that of a crystal orglass The long molecules, intertwined like a jar of exceptionally long worms, form

entanglements – points where molecules, because of their length and flexibility, become

knotted together (Fig 23.6) On loading, the molecules reptate (slide) except at ment points The entanglements give the material a shape-memory: load it, and thesegments between entanglements straighten out; remove the load and the wriggling of

entangle-the molecules (being above T g) draws them back to their original configuration, and

Fig 23.6. A schematic of a linear-amorphous polymer, showing entanglement points (marked “E”) which act like chemical cross-links.

thus shape Stress tends to order the molecules of the material; removal of stress allows

it to disorder again The rubbery modulus is small, about one-thousandth of the glassy modulus, T g, 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 entanglementsare 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, and this makes the polymer above T g into a true elastomer, capable of elastic

extensions of 300% or more (the same as the draw ratio of the polymer in the plasticstate – see the next section) which are recovered completely on unloading Over-frequent cross-links destroy the rubbery behaviour If every unit on the polymerchain has one (or more) cross-links to other chains, then the covalent bonds form athree-dimensional network, and melting of the secondary bonds does not leave longmolecular spans which can straighten out under stress So good elastomers, likepolyisoprene (natural rubber) are linear polymers with just a few cross-links, well

above their glass temperatures (room temperature is 1.4 T g for polyisoprene) If they

are cooled below T g, the modulus rises steeply and the rubber becomes hard andbrittle, with properties like those of PMMA at room temperature

Viscous flow

At yet higher temperatures (>1.4Tg) the secondary bonds melt completely and even theentanglement points slip This is the regime in which thermoplastics are moulded:linear polymers become viscous liquids The viscosity is always defined (and usuallymeasured) in shear: if a shear stress σs produces a rate of shear ˙γ then the viscosity(Chapter 19) is

Its units are poise (P) or 10−1 Pa s

Polymers, like inorganic glasses, are formed at a viscosity in the range 104 to 106poise, when they can be blown or moulded (When a metal melts, its viscosity dropsdiscontinuously to a value near 10−3 poise – about the same as that of water; that iswhy metals are formed by casting, not by the more convenient methods of blowing ormoulding.) The viscosity depends on temperature, of course; and at very high tem-peratures the dependence is well described by an Arrhenius law, like inorganic glasses

(Chapter 19) But in the temperature range 1.3–1.5 Tg, where most thermoplastics are

formed, the flow has the same time–temperature equivalence as that of the viscoelasticregime (eqn 23.12) and is called “rubbery flow” to distinguish it from the higher-

temperature Arrhenius flow Then, if the viscosity at one temperature T0 is η0, the viscosity at a higher temperature T1 is

Fig 23.7. A modulus diagram for PMMA It shows the glassy regime, the glass–rubber transition, the rubbery regime and the regime of viscous flow The diagram is typical of linear-amorphous polymers.

When you have to estimate how a change of temperature changes the viscosity of apolymer (in calculating forces for injection moulding, for instance), this is the equation

to use

Cross-linked polymers do not melt But if they are made hot enough, they, likelinear polymers, decompose

Decomposition

If a polymer gets too hot, the thermal energy exceeds the cohesive energy of some part

of the molecular chain, causing depolymerisation or degradation Some (like PMMA)decompose into monomer units; others (PE, for instance) randomly degrade into manyproducts It is commercially important that no decomposition takes place during high-temperature moulding, so a maximum safe working temperature is specified for each

polymer; typically, it is about 1.5 Tg.

Modulus diagrams for polymers

The above information is conveniently 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 diagram) The

modulus E is plotted, on a log scale, on the vertical axis: it runs from 0.01 MPa to

10,000 MPa The temperature, normalised by the glass temperature Tg, is plotted early on the horizontal axis: it runs from 0 (absolute zero) to 1.6 Tg (below which the

lin-polymer decomposes)

The diagram is divided into five fields, corresponding to the five regimes described

earlier In the glassy field the modulus is large – typically 3 GPa – but it drops a bit as thesecondary transitions cause local relaxations In the glassy or viscoelastic–transitionregime, the modulus drops steeply, flattening out again in the rubbery regime Finally,true melting or decomposition causes a further drop in modulus

Time, as well as temperature, affects the modulus This is shown by the contours of

loading time, ranging from very short (10−6 s) to very long (108 s) The diagram showshow, even in the glassy regime, the modulus at long loading times can be a factor of 2

or more less than that for short times; and in the glass transition region the factorincreases to 100 or more The diagrams give a compact summary of the small-strainbehaviour of polymers, and are helpful in seeing how a given polymer will behave in

a given application

Cross-linking raises and extends the rubbery plateau, increasing the rubber-modulus,and suppressing melting Figure 23.8 shows how, for a single loading time, the con-tours of the modulus diagram are pushed up as the cross-link density is increased.Crystallisation increases the modulus too (the crystal is stiffer than the amorphouspolymer because the molecules are more densely packed) but it does not suppressmelting, so crystalline linear-polymers (like high-density PE) can be formed by heatingand moulding them, just like linear-amorphous polymers; cross-linked polymerscannot

Fig 23.8. The influence of cross-linking on a contour of the modulus diagram for polyisoprene.

Strength: cold drawing and crazing

Engineering design with polymers starts with stiffness But strength is also important,sometimes overridingly so A plastic chair need not be very stiff – it may be morecomfortable if it is a bit flexible – but it must not collapse plastically, or fail in a brittlemanner, when sat upon There are numerous examples of the use of polymers (lug-gage, casings of appliances, interior components for automobiles) where strength, notstiffness, is the major consideration

The “strength” of a solid is the stress at which something starts to happen whichgives a permanent shape change: plastic flow, or the propagation of a brittle crack, forexample At least five strength-limiting processes are known in polymers Roughly inorder of increasing temperature, they are:

(a) brittle fracture, like that in ordinary glass;

(b) cold drawing, the drawing-out of the molecules in the solid state, giving a largeshape change;

(c) shear banding, giving slip bands rather like those in a metal crystal;

(d) crazing, a kind of microcracking, associated with local cold-drawing;

(e) viscous flow, when the secondary bonds in the polymer have melted

We now examine each of these in a little more detail

Brittle fracture

Below about 0.75 T g, polymers are brittle (Fig 23.9) Unless special care is taken to

avoid it, a polymer sample has small surface cracks (depth c) left by machining or

abrasion, or caused by environmental attack Then a tensile stress σ will cause brittlefailure if

Fig 23.9. Brittle fracture: the largest crack propagates when the fast-fracture criterion is satisfied.

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.

100 MPa But if deeper cracks or stress concentrations are cut into the polymer, thestress needed to make them propagate is, of course, lower When designing with

polymers you must remember that below 0.75 T g they are low-toughness materials,and that anything that concentrates stress (like cracks, notches, or sharp changes ofsection) is dangerous

Cold drawing

At temperatures 50°C or so below T g, thermoplastics become plastic (hence the name).The stress–strain curve typical of polyethylene or nylon, for example, is shown inFig 23.10 It shows three regions

At low strains the polymer is linear elastic, which the modulus we have just cussed At a strain of about 0.1 the polymer yields and then draws The chains unfold (if

dis-chain-folded) or draw out of the amorphous tangle (if glassy), and straighten andalign The process starts at a point of weakness or of stress concentration, and asegment of the gauge length draws down, like a neck in a metal specimen, until the

draw ratio (l/l0) is sufficient to cause alignment of the molecules (like pulling cottonwool) The draw ratio for alignment is between 2 and 4 (nominal strains of 100 to300%) The neck propagates along the sample until it is all drawn (Fig 23.10)

The drawn material is stronger in the draw direction than before; that is why theneck spreads instead of simply causing failure When drawing is complete, the stress–strain curve rises steeply to final fracture This draw-strengthening is widely used toproduce high-strength fibres and film (Chapter 24) An example is nylon made by meltspinning: the molten polymer is squeezed through a fine nozzle and then pulled (drawratio ≈ 4), aligning the molecules along the fibre axis; if it is then cooled to roomtemperature, the reorientated molecules are frozen into position The drawn fibre has

a modulus and strength some 8 times larger than that of the bulk, unoriented, polymer.Crazing

Many polymers, among them PE, PP and nylon, draw at room temperature Others

with a higher T g, such as PS, do not – although they draw well at higher temperatures

If PS is loaded in tension at room temperature it crazes Small crack-shaped regions

within the polymer draw down, but being constrained by the surrounding undeformedsolid, the drawn material ends up as ligaments which link the craze surfaces (Fig 23.11).The crazes are easily visible as white streaks or as general whitening when cheapinjection-moulded articles are bent (plastic pen tops, appliance casings, plastic caps).The crazes are a precursor to fracture Before drawing becomes general, a crack forms

at the centre of a craze and propagates – often with a crazed zone at its tip – to givefinal fracture (Fig 23.11)

Shear banding

When crazing limits the ductility in tension, large plastic strains may still be possible

in compression shear banding (Fig 23.12) Within each band a finite shear has taken

place As the number of bands increases, the total overall strain accumulates

Fig 23.11. Crazing in a linear polymer: molecules are drawn out as in Fig 23.10, but on a much smaller scale, giving strong strands which bridge the microcracks.

Fig 23.12. Shear banding, an alternative form of polymer plasticity which appears in compression.Viscous flow

Well above T g polymers flow in the viscous manner we have described already Whenthis happens the strength falls steeply

Strength diagrams for polymers

Most of this information can be summarised as a strength diagram for a polymer.

Figure 23.13 is an example, again for PMMA Strength is less well understood than

Fig 23.13. A strength diagram for PMMA The diagram is broadly typical of linear polymers.

stiffness but the diagram is broadly typical of other linear polymers The diagram ishelpful 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 nounced The right-hand scale gives the strength in MPa; the left-hand scale gives thestrength normalised by Young’s modulus at 0 K The horizonal scale is the temper-

pro-ature, unnormalised across the top and normalised by T g along the bottom (Thenormalisations make the diagrams more general: similar polymers should have similarnormalised diagrams.)

The diagram is divided, like the modulus diagram, into fields corresponding to the

five strength-limiting processes described earlier At low temperatures there is a brittlefield; here the strength is calculated by linear-elastic fracture mechanics Below this liesthe crazing field: the stresses are too low to make a single crack propagate unstably,but they can still cause the slow growth of microcracks, limited and stabilised by thestrands of drawn material which span them At higher temperatures true plasticitybegins: cold drawing and, in compression, shear banding And at high temperaturelies the field of viscous flow

The strength of a polymer depends on the strain rate as well as the temperature The

diagram shows contours of constant strain rate, ranging from very slow (10−6 s−1) to veryfast (1 s−1) The diagram shows how the strength varies with temperature and strainrate, and helps identify the dominant strength-limiting mechanism This is importantbecause the ductility depends on mechanism: in the cold-drawing regime it is large,but in the brittle fracture regime it is zero

Strength is a much more complicated property than stiffness Strength diagramssummarise nicely the behaviour of laboratory samples tested in simple tension Butthey (or equivalent compilations of data) must be used with circumspection In anengineering application the stress-state may be multiaxial, not simple tension; and theenvironment (even simple sunlight) may attack and embrittle the polymer, reducingits strength These, and other, aspects of design with polymers, are discussed in thebooks listed under Further reading

Further reading

J A Brydson, Plastics Materials, 6th edition, Butterworth-Heinemann, 1996.

International Saechtling, Plastics Handbook, Hanser, 1983.

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-densitypolyethylene at the glass transition temperature

Answer: 270 days.