Fundamentals of Polymer Engineering Part 11 ppt

10 TheoryofRubberElasticity 10.1INTRODUCTION AsmentionedinChapter2,allpolymersarestiff,brittle,glassymaterialsbelow their glass transition temperature, T g . However, they soften and become pliable once above T g and, ultimately, flow at still higher temperatures. For crystalline polymers, the flow temperature is slightly above the crystalline melting point. In this chapter, we examine the mechanical behavior of solid polymers above T g , whereaspolymercrystallizationisconsideredinChapter11,andthedeformation andfailurepropertiesofglassypolymersarepresentedinChapter12.Thestress- versus-strain behavior of amorphous polymers above T g is similar to that of natural rubber at room temperature and very different from that of metals and crystalline solids. Although metals can be reversibly elongated by only a percent or so, rubber can be stretched to as much as 10 times its length without damage. Furthermore, the stress needed to achieve this deformation is relatively low. Thus, polymers above T g are soft elastic solids; this property is known as rubberlike elasticity. Other extraordinary properties of rubber have also been known for a long time. Gough’s experiments in the early 1800s revealed that, unlike metals, a strip of rubber heats up on sudden elongation and cools on sudden contraction [1]. Also, its modulus increases with increasing temperature. These properties are lost, however, if experiments are performed in cold water. Explaining these remarkable observations is useful not only for satisfying intellectual curiosity but also for the purpose of generating an understanding that is beneficial for tailoring 407 Copyright © 2003 Marcel Dekker, Inc. the properties of rubberlike materials (called elastomers) for specific applications. Recall that rubber (whether natural or synthetic) is used to manufacture tires, adhesives, and footwear, among other products. Note also that because polymer properties change so drastically around T g , the use temperature of most polymers is either significantly below T g (as in the case of plastics employed for structural applications) or significantly above T g (as in the case of elastomers). Chemically, rubber is cis-1,4-polyisoprene, a linear polymer, having a molecular weight of a few tens of thousands to almost four million, and a wide molecular-weight distribution. The material collected from the rubber tree is a latex containing 30–40% of submicron rubber particles suspended in an aqueous protein solution, and the rubber is separated by coagulation caused by the addition of acid. At room temperature, natural rubber is really an extremely viscous liquid because it has a T g of À70  C and a crystalline melting point of about À5  C. It is the presence of polymer chain entanglements that prevents flow over short time scales. In order to explain the observations made with natural rubber and other elastomers, it is necessary to understand the behavior of polymers at the microscopic level. This leads to a model that predicts the macroscopic behavior. It is surprising that in one of the earliest and most successful models, called the freely jointed chain [2,3], we can entirely disregard the chemical nature of the polymer and treat it as a long slender thread beset by Brownian motion forces. This simple picture of polymer molecules is developed and embellished in the sections that follow. Models can explain not only the basics of rubber elasticity but also the qualitative rheological behavior of polymers in dilute solution and as melts. The treatment herein is kept as simple as possible. More details are available in the literature [1–7]. 10.2 PROBABILITY DISTRIBUTION FOR THE FREELY JOINTED CHAIN One of the simplest ways of representing an isolated polymer molecule is by means of a freely jointed chain having n links each of length l. Even though real polymers have fixed bond angles, such is not the case with the idealized chain. In addition, there is no correspondence between bond lengths and the dimensions of the chain. The freely jointed chain, therefore, is a purely hypothetical entity. Its behavior, however, is easy to understand. In particular, as will be shown in this section, it is possible to use simple statistical arguments to calculate the probability of finding one end of the chain at a specified distance from the other end when one end is held fixed but the other end is free to move at random. This probability distribution can be coupled with statistical thermodynamics to obtain the chain entropy as a function of the chain end-to-end distance. The 408 Chapter 10 Copyright © 2003 Marcel Dekker, Inc. expression for the entropy can, in turn, be used to derive the force needed to hold the chain ends a particular distance apart. This yields the force-versus-displace- ment relation for the model chain. If all of the molecules in a block of rubber act similarly to each other and each acts like a freely jointed chain, the stress–strain behavior of the rubber can be obtained by adding together contributions from each of the chains. Because real polymer molecules are not freely jointed chains, the final results cannot be expected to be quantitatively correct. The best that we can hope for is that the form of the equation is correct. This equation obviously involves the chain parameters n and l, which are unknown. If we are lucky, all of the unknown quantities will be grouped as one or two constants whose values can be determined by experiment. This, then, is our working hypothesis. To proceed along this path, let us conduct a thought experiment. Imagine holding one end of the chain fixed at the origin of a rectangular Cartesian coordinate system (as shown in Fig. 10.1) and observe the motion of the other end. You will find that the distance r between the two ends ranges all the way from zero to nl even though some values of the end-to-end distance occur more frequently than others. In addition, if we use spherical coordinates to describe the location of the free end, different values of y and f arise with equal frequency. As a consequence, the magnitude of the projection on any of the three axes x, y, and z of a link taken at random will be the same and equal to l= ffiffiffi 3 p . To determine the probability distribution function for the chain end-to-end distance, we first consider a freely jointed, one-dimensional chain having links of length l x ¼ l= ffiffiffi 3 p , which are all constrained to lie along the x axis. What is the probability that the end-to-end distance of this one-dimensional chain is ml x ? The FIGURE 10.1 The unconstrained freely jointed chain. Theory of Rubber Elasticity 409 Copyright © 2003 Marcel Dekker, Inc. answer to this question can be obtained by analyzing the random walk of a person who starts out from the origin and takes n steps along the x axis; n þ of these steps are in the positive x direction and n À are in the negative x direction, and there is no relation between one step and the next one. Clearly, m equals ðn þ À n À Þ. From elementary probability theory, the probability that an event will occur is the ratio of the number of possible ways in which that event can occur to the total number of events. As a consequence, the probability, p(m), of obtaining an end-to-end distance of ml x is the number of ways in which one can take n þ forward steps and n À backward steps out of n steps divided by the total number of ways of taking n steps. The numerator, then, is the same as the number of ways of putting n objects (of which n þ are of one kind and n À are of another kind) into a container having n compartments. This is n!=ðn þ !n À !Þ. Because any given step can either be a forward step or a backward step, each step can be taken in two ways. Corresponding to each way of taking a step, the next step can again be taken in two ways. Thus, the total number of ways of taking n steps is 2 n , which gives us pðmÞ¼ n! 2 n n þ !n À ! ð10:2:1Þ We can rewrite n þ and n À as follows: n þ ¼ 1 2 ðn þ mÞð10:2:2Þ n À ¼ 1 2 ðn À mÞð10:2:3Þ For large n we can use Stirling’s formula: n! ¼ ffiffiffiffiffiffi 2p p n ð2nþ1Þ=2 e n ð10:2:4Þ In Eq. (10.2.1), introduce Eqs. (10.2.2) and (10.2.3) in the result and simplify to obtain the following: pðmÞ¼ ffiffiffiffiffiffi 2 np r 1 þ m n  ðnþmþ1Þ=2 1 À m n  ðnÀmþ1Þ=2  À1 ð10:2:5Þ Taking the natural logarithm of both sides of Eq. (10.2.5) and recognizing that ln 1 þ m n  ffi m n À m 2 2n 2 ð10:2:6Þ provided that m=n is small, ln pðmÞ¼ 1 2 ln 2 np  À m 2 2n þ m 2 2n 2 ð10:2:7Þ 410 Chapter 10 Copyright © 2003 Marcel Dekker, Inc. NeglectingtheverylastterminEq.(10.2.7), pðmÞ¼ 2 np  1=2 e Àm 2 =2n ð10:2:8Þ whichisknownasaGaussianorNormaldistribution.Notethatforallofthese relationstobevalid,nhastobelargeandm=nhastobesmall. Equation(10.2.8)representsadiscreteprobabilitydistributionandisthe probabilitythatxliesbetweenml x andðmþ2Þl x .Thisisbecauseifn þ increases by1,n À hastodecreaseby1andmincreasesby2.Simultaneously,thedistance betweenthechainendsgoesupby2l x .Toobtainthecontinuousprobability distributionpðxÞdx,whichistheprobabilitythattheend-to-enddistanceranges fromxtoxþdx,wemerelymultiplypðmÞbydx=ð2l x Þ.Furthermore,becausem equalsx=l x , pðxÞdx¼ð2npl 2 x Þ À1=2 e Àx 2 =2nl 2 x dxð10:2:9Þ Inordertoextendtheone-dimensionalresultsembodiedinEq.(10.2.9)to thethree-dimensionalcaseofpracticalinterest,weusethelawofjointprob- ability.Accordingtothislaw,theprobabilityofanumberofeventshappening simultaneouslyistheproductoftheprobabilitiesofeachoftheeventsoccurring individually.Thus,theprobability,pðrÞdr,thattheunconstrainedendofthe freelyjointedchainliesinarectangularparallelepipeddefinedby x;y;z;xþdx;yþdy,andzþdz(seeFig.10.1)istheproduct pðxÞ dx pðyÞ dy pðzÞ dz, where pðyÞ dy and pðzÞ dz are defined in a manner analogous to pðxÞ dx. Therefore, pðrÞ dr ¼ð2npÞ À3=2 ðl 2 x l 2 y l 2 z Þ À1=2 exp À 1 2n x 2 l 2 x þ y 2 l 2 y þ z 2 l 2 z !"# dx dy dz ð10:2:10Þ Denoting the sum ðx 2 þ y 2 þ z 2 Þ as r 2 and recalling that l 2 x ¼ l 2 y ¼ l 2 z ¼ l 2 =3, pðrÞ dr ¼ 3 2npl 2  3=2 e À3r 2 =2nl 2 dx dy dz ð10:2:11Þ To obtain the probability that the free end of the chain lies not in the parallelepiped shown in Figure 10.1 but anywhere in a spherical shell of radius r and thickness dr, we appeal to the law of addition of probabilities. According to this law, the probability that any one of several events may occur is simply the sum of the probabilities of each of the events. Thus, the probability that the chain end may lie anywhere within the spherical shell is the sum of the probabilities of finding the chain end in each of the parallelepipeds constituting the spherical shell. Using Eq. (10.2.11) to carry out this summation, we see that the result is Theory of Rubber Elasticity 411 Copyright © 2003 Marcel Dekker, Inc. again Eq. (10.2.11), but with the right-hand side modified by replacing dx dy dz with 4pr 2 dr , the volume of the spherical shell. Finally, then, we have pðrÞ dr ¼ 3 2npl 2  3=2 e À3r 2 =2nl 2 4pr 2 dr ð10:2:12Þ which represents the probability that the free end of the chain is located at a distance r from the origin and contained in a spherical shell of thickness dr. This is shown graphically in Figure 10.2. Note that the presence of r 2 in Eq. (10.2.12) causes pðrÞ to be zero at the origin, whereas the negative exponential drives pðrÞ to zero at large values of r. As seen in Figure 10.2, pðrÞ is maximum at an intermediate value of r 2 . Also, because the sum of all the probabilities must equal unity, Ð 1 0 pðrÞ dr ¼ 1. At this point, it is useful to make the transition from the behavior of a single chain to that of a large collection of identical chains. It is logical to expect that the end-to-end distances traced out by a single chain as a function of time would be the same as the various end-to-end distances assumed by the collection of chains at a single time instant. Thus, time averages for the isolated chain ought to equal ensemble averages for the collection of chains. Using Eq. (10.2.12), then, the average values of the chain’s end-to-end distance and square of the chain’s end-to- end distance are as follows: hri¼ ð 1 0 rpðrÞ dr ¼ 2l 2n 3p  1=2 ð10:2:13Þ hr 2 i¼ ð 1 0 r 2 pðrÞ dr ¼ nl 2 ð10:2:14Þ FIGURE 10.2 Distribution function pðrÞ given by Eq. (10.2.12). (Reprinted from Treloar, L. R. G.: The Physics of Rubber Elasticity, 3rd ed., Clarendon, Oxford, U.K., 1975, by permission of Oxford University Press.) 412 Chapter 10 Copyright © 2003 Marcel Dekker, Inc. wheretheangularbracketsdenoteensembleaverages.Becausethefullyextended lengthofthechain(alsocalledthecontourlength)isnl,Eq.(10.2.14) demonstratesthatthemeansquareend-to-enddistanceisveryconsiderably lessthanthesquareofthechainlength.Therefore,thefreelyjointedchain behaveslikearandomcoilandthisexplainstheenormousextensibilityofrubber molecules. Havingobtainedtheaveragevalueofthesquareofthechainend-to-end distanceandthedistributionofend-to-endvaluesaboutthismean,itisworth pausingandagainaskingifthereisanyrelationbetweentheseresultsandresults forrealpolymermolecules.Inotherwords,howcloselydofreelyjointedchains approximateactualmacromolecules?Iftheansweris‘‘notveryclosely,’’thenhow dowemodifythefreelyjointedchainresultstomakethemapplytopolymers? Thefirstresponseisthatmostpolymermoleculesdo,indeed,resemblelong flexiblestrings.Thisisbecauselinear(unbranched)polymerswithalargedegree ofpolymerizationhaveaspectratiosthatmaybeashighas10 4 .Theyarethus fairlyelongatedmolecules.Furthermore,despitetherestrictiontofixedbond anglesandbondlengths,thepossibilityofrotationaboutchemicalbondsmeans thatthereislittlecorrelationbetweenthepositionofonebondandanotherone thatisfiveorsixbondlengthsremoved.However,twoconsequencesofthese restrictionsarethatthecontourlengthbecomeslessthantheproductofthebond lengthandthenumberofbondsandthatthemeansquareend-to-enddistance becomeslargerthanthatpreviouslycalculated. Ifbondanglesarerestrictedtoafixedvaluey,thefollowingcanbeshown [4]: hr 2 i¼nl 2 ð1ÀcosyÞ ð1þcosyÞ ð10:2:15Þ If,inaddition,thereishinderedrotationaboutthebackbonedueto,say,steric effects,thenwehave hr 2 i¼nl 2 ð1ÀcosyÞð1þcoshfiÞ ð1þcosyÞð1ÀcoshfiÞ ð10:2:16Þ wherehf 2 iistheaveragevalueofthetorsionangle.Small-angleneutron scatteringdatahavesupportedthispredictedproportionalitybetweenhr 2 iandnl 2 . Becausehr 2 iincreaseswitheachadditionalrestrictionbutremainspropor- tionaltohr 2 iforafreelyjointedchain,wecanconsiderapolymermoleculea freelyjointedchainhavingn 0 links,wheren 0 islessthanthenumberofbonds,but thelengthofeachlinkl 0 isgreaterthanthebondlength,sothathr 2 iisagainn 0 l 02 andthecontourlengthisn 0 l 0 . Example10.1:PolyethylenehastheplanarzigzagstructureshowninFigure 10.3.Ifthebondlengthislandthevalenceangleyis109:5  ,whatarethe Theory of Rubber Elasticity 413 Copyright © 2003 Marcel Dekker, Inc. contourlengthRandthemeansquareend-to-enddistance?Letthechainhaven bondsandlettherebefreerotationaboutthebonds. Solution:FromFigure10.3,itisclearthattheprojectedlengthofeachlinkis lsinðy=2Þ.Usingthegivenvalueofyandnotingthattherearenlinks,thefully extendedchainlengthisgivenby R¼nlsinð54:75  Þ¼ ffiffiffiffi 2 3 r nl Themeansquareend-to-enddistanceisobtainedfromEq.(10.2.15)asfollows: hr 2 i¼2nl 2 Whenthemeansquareend-to-enddistanceofapolymerisgivenbyEq. (10.2.16),thepolymerissaidtobeinits‘‘unperturbed’’state.Whatcausesthe polymertobe‘‘perturbed’’isthefactthatinthederivationofEq.(10.2.16),we haveallowedforthepossibilityofwidelyseparatedatomsthatmakeupdifferent portionsofthesamepolymermoleculetooccupythesamespace.Inreality,those arrangementsthatresultinoverlapofatomsareexcluded.Thisisknownasthe excluded-volumeeffect,anditresultsindimensionsofrealpolymermolecules becominglargerthantheunperturbedvalue.Itiscustomarytoquantifythis phenomenonbydefiningacoilexpansionfactorthatistheratiooftherootmean squareend-to-enddistanceoftherealchaintothecorrespondingquantityforthe unperturbedchain.Inaverygoodsolvent,thereisafurtherincreaseinsize,as determinedbyintrinsicviscositymeasurements,andthecoil-expansionfactorcan becomeaslargeas2.Inapoorsolvent,ontheotherhand,themoleculeshrinks, andifthesolventqualityispoorenough,thecoilexpansionfactorcanbecome unity.Insuchacase,thesolventiscalledathetasolvent,andwehavethetheta conditionencounteredearlierinChapter9.Itis,therefore,seenthatthetheta conditioncanbereachedeitherbychangingtemperaturewithoutchangingthe solventorbychangingthesolventunderisothermalconditions. Inclosingthissection,were-emphasizethatthesizeofapolymermolecule measuredusingthelight-scatteringtechniquediscussedinChapter8isthemean FIGURE10.3Theplanarzigzagstructureofpolyethylene. 414Chapter 10 Copyright © 2003 Marcel Dekker, Inc. squareradiusofgyrationhs 2 i.Forafreelyjointedchainthisquantity,definedas thesquaredistanceofachainelementfromthecenterofgravity,isgivenby hs 2 i¼ 1 6 hr 2 ið10:2:17Þ Theradiusofgyrationisespeciallyusefulincharacterizingbranchedmolecules havingmultipleendswheretheconceptofasingleend-to-enddistanceisnot particularlymeaningful. 10.3ELASTICFORCEBETWEENCHAINENDS IfwereturntotheunconstrainedchaindepictedinFigure10.1andmeasurethe time-dependent force needed to hold one of the chain ends at the origin of the coordinate system, we find that the force varies in both magnitude and direction, but its time average is zero due to symmetry. If, however, the other chain end is also held fixed so that a specified value of the end-to-end distance is imposed on the chain, the force between the chain ends will no longer average out to zero. Due to axial symmetry, though, the line of action of the force will coincide with r, the line joining the two ends. For simplicity of analysis, let this line be the x axis. In order to determine the magnitude of the force between the chain ends, let us still keep one end at the origin but apply an equal and opposite (external) force f on the other end so that the distance between the two ends increases from x to x þ dx. The work done on the chain in this process is dW ¼Àfdx ð10:3:1Þ where the sign convention employed is that work done by the system and heat added to the system are positive. If chain stretching is done in a reversible manner, a combination of the first and second laws of thermodynamics yields dW ¼ TdSÀ dU ð10:3:2Þ where S is entropy and U is internal energy. Equating the right-hand sides of Eqs. (10.3.1) and (10.3.2) and dividing throughout by dx gives f ¼ÀT dS dx þ dU dx ð10:3:3Þ From statistical thermodynamics, the entropy of a system is related to the probability distribution through the following equation: S ¼ k ln pðxÞð10:3:4Þ Theory of Rubber Elasticity 415 Copyright © 2003 Marcel Dekker, Inc. wherekisBoltzmann’sconstant.Inthepresentcase,pðxÞisgivenbyEq.(10.2.9) sothat f¼ÀT d dx À k 2 lnð2npl 2 x ÞÀ kx 2 2nl 2 x  þ dU dx ð10:3:5Þ andcarryingoutthedifferentiation, f¼ kTx nl 2 x þ dU dx ¼ 3kTx nl 2 þ dU dx ð10:3:6Þ Theinternalenergyterminthisequationisrelatedtochangesintheinternal potentialenergyarisingfromthemakingandbreakingofvanderWaalsbonds. Becauserubberselongateveryeasily,wefindthatthesecondtermontheright- handsideofEq.(10.3.6)issmallcomparedtothefirstterm.Consequently, f¼ 3kT nl 2 xð10:3:7Þ whichisalinearrelationshipbetweentheforceandthedistancebetweenchain endsandissimilartothebehaviorofalinearspring.Theconstantof proportionality,3kT=nl 2 ,isthemodulusofthematerialanditsvalueincreases astemperatureincreases.Thisexplainswhyastretchedrubberbandcontractson heatingwhenitisabovethepolymerglasstransitiontemperature. ThepositiveforcefinEq.(10.3.7)isexternallyappliedandisbalancedby aninward-actinginternalforce,which,intheabsenceoftheexternalforce,tends tomaketheend-to-enddistancegotozero.This,however,doesnothappenin practicebecausethespringforceisnottheonlyoneactingonthechain;the equilibriumend-to-enddistanceisgivenbyabalanceofalltheforcesactingon thepolymermolecule.Thisaspectofthebehaviorofisolatedpolymermolecules willbecoveredingreaterdetailinthediscussionofconstitutiveequationsfor dilutepolymersolutionsinChapter14. Ifwewerenotawareoftheassumptionsthathavegoneintothederivation ofEq.(10.3.7),wemightconcludethattheforcebetweenthechainends increaseslinearlyandwithoutboundasxincreases.Actually,Eq.(10.3.7)is validonlyforvaluesofxthataresmallcomparedtothecontourlengthofthe chain.Forlargerextensionsexceedingone-thirdthecontourlength,fincreases nonlinearlywithx,andweknowthatforvaluesofxapproachingnl,chemical bondsbegintobestretched.Itcanbeshownthattheright-handsideofEq. (10.3.7)ismerelythefirstterminaseriesexpansionforf[1]: f¼ kT l 3x nl þ 9 5 x nl  3 þ 297 175 x nl  5 þÁÁÁ  ¼ kT l L À1 x nl  ð10:3:8Þ 416Chapter10 Copyright © 2003 Marcel Dekker, Inc. [...]... of Particulate Fillers in Elastomer Reinforcement: A Review, Polymer, 20, 691–704, 1979 Kraus, G., Reinforcement of Elastomers by Carbon Black, Rubber Chem Technol., 51, 297–321, 1978 Polmanteer, K E., and C W Lentz, Reinforcement Studies—Effect of Silica Structure on Properties and Crosslink Density, Rubber Chem Technol., 48, 795–809, 1975 Bragaw, C G., The Theory of Rubber Toughening of Brittle Polymers,... called the modulus G of the rubber The left-hand side of Eq (10.4.7) is recognized to be the stress based on the undeformed area Example 10.3: When rubber is brought into contact with a good solvent, it swells in an isotropic manner Consider a cube of rubber, initially of unit volume, containing N polymer chains If in the swollen state the polymer volume fraction is f2 and the length of each edge is l,... University Press.) increase in free energy due to stretching of polymer chains If we use the Flory– Huggins expression, Eq (9.3.20), for the former free-energy change and the result of Example 10.3 for the latter free-energy change, then the total change in free energy on mixing unit volume of polymer containing n2 moles of chain segments with n1 moles of solvent is DGM ¼ RT ðn1 ln f1 þ n2 ln f2 þ w1 n1 f2... which f* is the polymer volume fraction at equilibrium Also, 1=m has been 2 neglected in comparison with unity A measurement of the equilibrium amount of swelling, together with a knowledge of the polymer solvent interaction parameter, then allows us to compute the chain density n2 Indeed, Eq (10.8.5) has proved to be a popular alternative to Eq (10.4.7) for the determination of the number of chain segments... are explored elsewhere [5] The utility of the theory, however, does not end with explaining the behavior of cross-linked rubber A knowledge of the fundamentals of rubber elasticity allows us to synthesize other elastomers and to modify and optimize their properties Indeed, the total production of synthetic rubbers such as styrene–butadiene rubber today exceeds that of natural rubber, and synthetic routes... and abrasion resistance of the elastomer A second issue is the impact modification of polymers during the formation of microcomposites or macrocomposites by the addition of a rubbery phase An example of this is high-impact polystyrene (HIPS); adding rubber to glassy polymers can raise their impact strength by an order of magnitude [23] On a macrolevel, we use polyvinyl butyral as an interlayer in laminated... cube of rubber of edge l0 Under the influence of this force, Copyright © 2003 Marcel Dekker, Inc Theory of Rubber Elasticity 419 the cube transforms into a rectangular prism having dimensions l1 ; l2 , and l3, as shown in Figure 10.5 If we define l1 as l1 =l0 , l2 as l2 =l0 , and l3 as l3 =l0 , then the affine deformation assumption implies that the coordinates of the end-to-end vector of a typical polymer. .. a strip of the butyl rubber of Example 10.5 If, at 25 C, the strip is extended to twice its original length and used to hurl a 10-g projectile, what will be the maximum possible speed of the projectile? Let the volume of the rubber band be 1 cm3 What is the Young’s modulus of the rubber sample used in Figure 10 .11? How does it compare with the corresponding value for steel? Does a block of rubber... in the best-fit modulus, but the cross-link density estimated from the extent of crosslinking is usually lower than the experimentally determined best-fit value A part of the discrepancy is thought to be due to the presence of physical entanglements that act as cross-links over the time scale of the experiment A consequence of cross-linking is that the resulting gigantic molecule does not dissolve in... equilibrium extent of swelling is determined by an interplay between the reduction in free energy due to polymer solvent mixing and an Copyright © 2003 Marcel Dekker, Inc 430 Chapter 10 F IGURE 10 .11 Simple extension Comparison of experimental curve with theoretical form (Reprinted from Treloar, L R G., The Physics of Rubber Elasticity, 3rd ed., Clarendon Oxford, UK, 1975, by permission of Oxford University . Inc. NeglectingtheverylastterminEq.(10.2.7), pðmÞ¼ 2 np  1=2 e Àm 2 =2n ð10:2:8Þ whichisknownasaGaussianorNormaldistribution.Notethatforallofthese relationstobevalid,nhastobelargeandm=nhastobesmall. Equation(10.2.8)representsadiscreteprobabilitydistributionandisthe probabilitythatxliesbetweenml x andðmþ2Þl x .Thisisbecauseifn þ increases by1,n À hastodecreaseby1andmincreasesby2.Simultaneously,thedistance betweenthechainendsgoesupby2l x .Toobtainthecontinuousprobability distributionpðxÞdx,whichistheprobabilitythattheend-to-enddistanceranges fromxtoxþdx,wemerelymultiplypðmÞbydx=ð2l x Þ.Furthermore,becausem equalsx=l x , pðxÞdx¼ð2npl 2 x Þ À1=2 e Àx 2 =2nl 2 x dxð10:2:9Þ Inordertoextendtheone-dimensionalresultsembodiedinEq.(10.2.9)to thethree-dimensionalcaseofpracticalinterest,weusethelawofjointprob- ability.Accordingtothislaw,theprobabilityofanumberofeventshappening simultaneouslyistheproductoftheprobabilitiesofeachoftheeventsoccurring individually.Thus,theprobability,pðrÞdr,thattheunconstrainedendofthe freelyjointedchainliesinarectangularparallelepipeddefinedby x;y;z;xþdx;yþdy,andzþdz(seeFig.10.1)istheproduct pðxÞ. Inc. wherekisBoltzmann’sconstant.Inthepresentcase,pðxÞisgivenbyEq.(10.2.9) sothat f¼ÀT d dx À k 2 lnð2npl 2 x ÞÀ kx 2 2nl 2 x  þ dU dx ð10:3:5Þ andcarryingoutthedifferentiation, f¼ kTx nl 2 x þ dU dx ¼ 3kTx nl 2 þ dU dx ð10:3:6Þ Theinternalenergyterminthisequationisrelatedtochangesintheinternal potentialenergyarisingfromthemakingandbreakingofvanderWaalsbonds. Becauserubberselongateveryeasily,wefindthatthesecondtermontheright- handsideofEq.(10.3.6)issmallcomparedtothefirstterm.Consequently, f¼ 3kT nl 2 xð10:3:7Þ whichisalinearrelationshipbetweentheforceandthedistancebetweenchain endsandissimilartothebehaviorofalinearspring.Theconstantof proportionality,3kT=nl 2 ,isthemodulusofthematerialanditsvalueincreases astemperatureincreases.Thisexplainswhyastretchedrubberbandcontractson heatingwhenitisabovethepolymerglasstransitiontemperature. ThepositiveforcefinEq.(10.3.7)isexternallyappliedandisbalancedby aninward-actinginternalforce,which,intheabsenceoftheexternalforce,tends tomaketheend-to-enddistancegotozero.This,however,doesnothappenin practicebecausethespringforceisnottheonlyoneactingonthechain;the equilibriumend-to-enddistanceisgivenbyabalanceofalltheforcesactingon thepolymermolecule.Thisaspectofthebehaviorofisolatedpolymermolecules willbecoveredingreaterdetailinthediscussionofconstitutiveequationsfor dilutepolymersolutionsinChapter14. Ifwewerenotawareoftheassumptionsthathavegoneintothederivation ofEq.(10.3.7),wemightconcludethattheforcebetweenthechainends increaseslinearlyandwithoutboundasxincreases.Actually,Eq.(10.3.7)is validonlyforvaluesofxthataresmallcomparedtothecontourlengthofthe chain.Forlargerextensionsexceedingone-thirdthecontourlength,fincreases nonlinearlywithx,andweknowthatforvaluesofxapproachingnl,chemical bondsbegintobestretched.Itcanbeshownthattheright-handsideofEq. (10.3.7)ismerelythefirstterminaseriesexpansionforf[1]: f¼ kT l 3x nl þ 9 5 x nl  3 þ 297 175 x nl  5 þÁÁÁ  ¼ kT l L À1 x nl  ð10:3:8Þ 416Chapter10 Copyright. T g , whereaspolymercrystallizationisconsideredinChapter11,andthedeformation andfailurepropertiesofglassypolymersarepresentedinChapter12.Thestress- versus-strain behavior of amorphous polymers above T g is similar to that of natural rubber