Structures and Properties of Rubberlike Networks (Topics in Polymer Science)

STRUCTURES PROPERTIE S RUBBERLIKE AND OF NETWORK S TOPICS IN POLYMER SCIENCE II s.,it, I IId,,.,,01:. XubfH.rli...

Author: Burak Erman | James. E. Mark

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STRUCTURES

PROPERTIE S RUBBERLIKE

AND

OF

NETWORK S

TOPICS IN POLYMER SCIENCE

II

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STRUCTU RES AND PROPERTIES OF RUBBERLIKE NETWORKS

Burak Erman & James E. Mark

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Preface

In a sens<, thi s book isa sequ~ l t" ... ur 19881:><>01:. XubfH.rlike Elasticity. A Mol~c ular Pri_f. As such, ;1 has much in common "ilb it< predecessor. panicularly ils Mrong emphasis on molecular concepls and lheories. Similarly. only equilibrium properlies are co",,"'" in any dClai!. This i! n<)[ 10 q"~llion the im(>Orlan~ of the dynamic asp«ts "f vi<eoclastic phcn omcnO]l. nor c>"<'n the im(>Onan~ of ullimatciy developing. Ihoory that cov.:n; both rubber elast;c;ty and ,';sooclaslic;ly. Unfortunatel y, ]1O such cnoompassing Ihoory e_,i.", al pn"$Cnl, and il will probably be some I;me before one becomes ,,,'ailahlc;n a forn, thaI is u""fullo experimen. lalists. as w~1I as Iheorim . It i, also our opinion tltal a thorou gh understanding of equilibrium properlics must. of ne""",sily. pr<:<:<.>Je an understaooing of d}'namic phenomena.• 1 least in Ihe molecuLlr arca. This w~s Ih. approach taken in our earlier book. and in lhe now-cla,~i" 1975 book by Trcl""r (The Physic' of Ru/>bl'r Elarlidty). As waS specified in il' litle. R,,/>bI'rlik~ £1o~ treats mucb the ",me ,ubjtel maller, buI j, meanl to be a more oo"' l,rehen,i,,,, some".-hat more sophis· ticated. Ir~atm.nl. Ikcaus<: o f iI, more oompreh.",ive oh3.ao"" much n,ore delai) is gi,-cn but. for (he oonvcnicnce of tM reader. this has generally bt.'Cn placed imo Ihe appendixe". It will be useful if Ihc reader already has SOme famil· iarity ",-i th the funda'"'enl .... ~lIClt os Ihose di",,,g&.:d in the ;nt.oo"CIOI)' book. For Ihi' reason. some of Ihis mate rial i. presented in condensed form in the lirst chapter, and in Ine first welion. of severa l of the other chaple ... A number of Ihe books and rc"iew anicles c ilc'
l11is /xige illl<'lIIiollally left blallk

Contents

1 CI ... kll Theorln 0/ R~ bber EIast;city 7 2. 1 The Kuhn-Trel"a r Tbwry g 1.2 T~ I'ha "~
2()

1 1..... moIecular Effect" I. The eonsr.ralned·luna"'" Model 12 3. 1 TM ModclanditsA ..... nlplions 2) 3.2 ProhobiHly Distribulion of l-' ucHllllions ,n I~ IXfom>c:d Nelwork 1.3 ~ Elastic F"", Enc-rgy JO 1.1. 1 exneral AspocU JO 3.3.2 Th¢ EI~Sh" F m: EncrS)' 1>... lO Dmorl;"" of t:.R JO J.J .J The Elallic Free En
Effecu , II. Con" n.int. >.Ion, N ..... ork Chain' 4.1 T he Slip-Link Model 34 4.2 T he Conslrain Mndcl 4(1 4.4 Olher T n:aln>cnIS of En~angk:,nenl> 42 K.f<~

",2

11

27

viii CONTENTS 5 Relation.h ips bet_en Str..... nd S'r>in +4 5. 1 General ReLationships of Finite ELa"icity Theory .14 5.2 Stres.-Strain Rdation. for the Phanlom ~nd Affine Nd·Chain Modeb under Uniaxia l Stress 4l! 5.4.1 TI\(: Slip-Link Modd 4l! 5.4.2 The ConSiraint>d·Chain Mood 49 5.5 Comparison of Slress Sirain Relations wilh Experimental Data 49 References 52 6 Swelli ng 01 N .. wor~. 53 6.1 Free Energy or a Swollen Network

53

6.2 The Solvent Cl\(:micai Potent ial for an lsotropically S"'olkn NCI"wk 55 6.3 Thermodynamics of a Network Uniaxially StTetchcd in Sol"em 6.4 Elastic Activity of a Swollen Nelwork 60 6,5 MOTe Recent Treatments of Nelwork Sw.;lIing 62 6.6 Sorption and E "f"~C1ion of Diluents 63 6.6, I Line~r Diluent, 63 6.6,2 Branched Diluents 65 6.6,3 Cyclic Dil""nts 66 6.7 T rapping of C)'ct;';s witnin Network Struclures 66 6.7 ,1 Experimental Results 66 6.7 ,2 Theoretical Interpretations 67 6.7,3 Olympic Nelworks 69 Kef.renees 70 7 Crit ic.1 Phenomena and Ph..., Tran.itions in Gel. n 7. 1 Theory of Critical PhenOln."" and Phase Transitions 7.2 ThcrmOTC'ICr>ible Gels 79 References 83

57

74

8 Calcu lation •• nd Sim ulation. 87 8, I Spati"l Configurat ions of an Isolated Chin 87 8.2 Statistical A '''''''ges of Configurational Variables 92 8.3 Distributions for End.to·"nd Scpanltions for SptciflC T ypes of Chains 94 K4 Stress_Strain lso(henns Cakolatw from the Non·Gaussian J)i s( ributions 100 8.5 Molecular Dynamics Calculations 104 Reference. 104

Nctwor~

CONTENTS

i~

107 9.1 Tl>cory 108 9.2 T ypiclll Stress- Temperature Ddta 11 0 9. 3 IIlumati,"" Th.rn'oela"i~ R~~ul!~ 112 9.4 Relevan! Cal",imet';e SlUdic. of Elastic Deformation. 121 9.S Relevant Vi !Cosily-Tcmpo:1""~turc ResuilS On Dilute Polymer Solutions 122 9.6 Rotali 1.11

<) Th.r~IU(i(ity

10 Mod,,1EI . >tomen; 134 10.1 T he Dependence of Ihe 51""" Oil Network Structure 114 10.1.1 General Approach 134 10. t.:' BITect "f Jun<. ion hnd ionalit y OS 10.2 T he Issuc of Entanglemen!s I3h 10.3 ln terpret.l i"n of Ultim a Ie Properties 141 10.4 Some Other Unusua l Netw"rk. 143 10.4.1 Dangl ing_Ch"in Networh 143 10.4.2 NCN'orh Containing Reptating Ow in, 145 IOA3 Networks ~par<xJ ;n s<:>lulion or in a Stal.
163

12 N l)eI"n",uon a nd O"o" ,,,,ion 174

x CON TE NTS 12.L6

[""lropic--N~matic

Phase T ransitions in Deformed Polymer Nctworks 17S ]2,2 Strain·l nduced Crystall ization 178 12.2. 1 General Feat ures 178 12.2.2 Models for Strain- Induced Crystallization in Stretched Networks 179 12.2.3 Predictions of the Molecular Theories 180 12.2.4 The Effects of Str~in_I nduced Cryitalli,ahon on Mochan;.;al Properties 182 References 185 13 Network> Hoving Mu lli mod~1 Chain_ Length Dist ribution. 188 [J . I Ultimate Properties and Non_(iauuian EfTects 188 13.2 lIi mo,:!. 1 Network. 189 13.2.1 Materials a"d Synthetic Tochniqucs IIW 13.2.2 T esting or the Weakest- l ink T heory 191 13.2.3 Elongation Result. 193 13.2.4 Result. in Other Mechan ical Deformations 203 13.2.5 Results On Nu nmc
Sm . II _An~le Neutron S.ca'ltering 220 14.1 GellC'al Fcature, of SANS 220 14.2 Expe,imental Stod ies 223 14.3 l boory of SANS from Net ..... or\> 224 14,3.1 The Scattering Low 224 14,3.2 Scatteri ng from a Phantom Network Chain 225 14.3.3 Scattering from an Affine Network Chain 228 14.3.4 Sealleri~g from a Chain wh"", Individual Segments Deform Affinel)" 229 14,3.5 Scauering f,om a La~led I'"th in the Network 229 14.4 Typical Result. from Experimcnts and Comparison ,",'ith Theory 230 RefetCnces 233

15 8;.,.,1..,0""'''' lJ5 15. 1 Some Gene .... l O bscr>ations 235 15.2 CIlcmical Aspects of Protein Biocl"stomer1 237 15,2.1 O"cfall Amino Aeid Cuml"',ition 237 15, 2.2 Amino Acid Scque,,,,ing 2.19 152..1 Cr<:>ss_l.inking Chemi
CONTENTS 15.3 NOlwork Thormoc:llI.Slicily 241 15.l .1 ~""r;d Relovantt 24 1 15.3.2 Elul,n 242 1~.J.3 Resili n 241 Is.l .4 OIh" I'rolei" Ellulomers 247 1 ~ .4 Stm.a-Strain Rc havior 248 15.4.1 ~nc,..1 Resullll 248 15.4.2 Ellutin 249 1~.4.l R";I'n 2S2 15.4.4 Spider.Web Silk 2j2 15.4.5 OIMr Protein Eto"Dfncr1 !~2 15.S Dynamic·Meclwoniaol Properties 253 I S.S.1 V,tOOdastoc Responses i" Gene ....1 251 IS .U EtT«t. of Dehyd .... lIon 256 IS.6 Some OtMr D"",Luwmcric Gels 2j7 15.6. 1 Gc1IcnJ l'roperties 257 1506.2 Some S~tic Systeml 2~ Rcf=nces 262

t6 Muklpl\ue Sys<erM 265 16.1 Some Theoretical Approac:hes 267 16.2 In S,\u GCI'H:,-~tio" of Fillcr$ in F.lutomen 26'1 16.1. 1 Gene ....1 Comment' 269 16.1.2 P~""r"lion 270 16.2.3 Elec1(on Mie, os«>py 270 16.2.4 Scanorina Toch niques 27 1 16.2.5 NlIClcar Magnetic Rewna ~ 272 16.2.6 A8ina 275 16.2.1 0 0n.i1i.. 2H 16.1.8 Calorimetry 275 16.2.9 Thorm"iravirr"'lIic Analysi, 2 76 16.2.10 Mechanica l Propcllies and !;quilihriu m S ....ellin' 276 16.2.11 Compori",n. amons Vari"". Silica· Hased fillers 287 16.2.12 Oth" Polymers 288 16.2.1l OIMr u",mic- Typr "i lien 2~8 16.l Prcpo "'lion of 6 icOllI,n"",,1 Syslcml 2119 16.4 10 Situ Genc: .... 1ion of EJastomcn In Ccnn,ia 290 16.5 In Situ GenerlOt;on of CatalystS in Polymers 291 16.6 In Silu PoI)...,..ri>.aI'oru ofGbSf,Y PoIy"",rs 292 16.6.1 Isotropic SY$IC1nS 292 16.6.2 Anisotmpoc S)'lIcmJ 29S 16.7 Filler. R"''''''''!'n, 10 MalJlCl ic l'"icJ
Rcf=nces

299

~i

~ii

CONTENTS

Ap~ndi~""

A. Network Structural "" ... mete... Rcfen: n<:es 310

307

a. Definitions In , .... Are. oi Rubber Toechnolotl:y B.I Basic Defini, ions 311 Rcfcn:nces 314

c.

111

DelormationandStre.. 115 C.I Deforma.ion 315 CI.I G .... r... 1 Aspect; 315 CI.2 l;;otropic Sv..iling 317 C. I.) Simple Tcns,on 317 C.l.4 Biaxial Extension 317 C.1.5 Pur. Shear 318 C.2 Stress 318 Ref...,,= 320

D. Summ ....,. oIThermodyn.mlcs . nd Stul.. ,<. 1M,,< h.nKs Ref.,""n= 323

321

E. fluetu .. io", in Phantom Network. 324 E. I The Matn. l' and its Inverse 324 E.2 Expressions f"r Van"us Fluctuations 326 E.2.1 Tv.·o j unction. Join.d by a Singk Chain 326 E.2.2 Two Junctions Separated by Several Chains 327 E.2.3 Poin," on a Ntlwor' <'''ain 327 E.2.4 Points on Network Chain. that a rc Separa.ed by Several JunctiOnl 328 References 329 F. Di .. r ibutio<\sohhe Chli" End-to-End Vector 130 1'.1 Examples of DiStribution Function, 330 1'.1.1 Th.Fr~"'yJoinlCdCha;n 330 F.I.2 More Realistic Piclures of the Ne.work Chain 131 1'.2 Transfonnation of Distnbution Functio", under Deforma.ion References 333

333

G. Fortran Pr"Olram /or Monte Carlo Calculat ion. 115 G . I Program (Calculation of Persistence Length, and Mean·Squaw End·to· End DiSla,lCcs) 335 0.2 Form or.he [Ja,a Sc, 319 G .3 Output of the f'rogr;,m JJ9

CONTE NTS H. Some Historical Aspect.

341 ILl The Earliest History 341 11.2 Natural Rubber in th. Un.Cro.,;-Li~h'
Setected (;e""r~1 8ibli"' .... phy

"' ~hor

Indu

Subject Irtde~

355

16t

l51

~i ii

STRUCTURES AN D

PROPERTIES

Of RUBBERLIKE NETWORKS

1

Overview and Some Fundamental Information

This chapter i~ a bti ~f overview of the ' and Ihus 10 Ino". "s " phaoll"m.:' Although Ihis Sttms 10 be a very seWr. appro,imalion. many experimemai resulu arc nOI in .(arlling dilagretn'enl wilh theori<:. b;osed on this highly iMali,cd a$Sump!io". These (hwrie, ass-ociair Ihe l(>[a! Helmholtz free energy of a d~formM nelwork with the sum of the free energies of the individual chains-an important assump. tion adopted throughout the book. T hey t",at the ,ingle chain in iI' maximum simplicity, as a Gau,,;.n chain, which i. a type of "structu",kss" chain (where the ooly ehemi':al con'titulion spe.;itiw is the number of bond, in the nelwork chain) In Ihi. respe<:1. lhe dassicallheories foo:us on itka{ networks and, in fael, are also referrwlo a$ "k inetic" Ihrori" I:>oca use of th.ir res<:mblancc to ideal 11"$ t!w,o,i.,., Chain fte. ibilily and mobility arc lhe ..sential features of these models, according to which lhe network chain. Can ""pericore. 1I possiblt oonfonnatio n. or _pati.1 arrangemem. subj<>cl 10 the n,,,wNk '. connectivity, One of the prediction! of the elassical theori" is th31 the clastic modulus <1f the network i. inocpendcn t of strain, T hi. r•• u lt ~ from lhe assumplion Ihat on ly the entropy a l the chain Ic>'el conuibules 10 lhe Hdmhoh.., free bperimenl81 evidence. on 1M other hand, indieales Ihat Ihe modulus decre ..... significanlly ,,-it l1 incre"~.i ng lension or OOnl!'re",on, implica ling interehain in leraclion>, .""h a. enlanglemen," of some type or other, T his has ltd 10 the more modem theories of rubber ciasticily, $uch a. Ihe constrained-ju nction or the slip.lin k thc'Orics, which go bey<md the singlechain Itnglh scale "oJ inlroduce oddilional entropy to l~e Iklmhoil~ fr~.., energy al the .ub<;~"in Ie, d , T~i. appmach is supported by "",enl compu,er si mulalions on networks, using molecular d~n"mic>, which or" in ,u!'pon of cntropic con·

am""

''''''gy.

,

• STR UCTURE S AN D PR OPER TIES OF RUBBE RLI KE NETWORK S tribut;ons at sealts btlow the chain_length level. These topics, within the Gaussian chain fonnali,m, are diseussed in chapter! 3. 4, and 8. One important prediction of the molecutar th.ori .... especially the more modern on<:'$, is th. dependence of the modulus on 'itr.in .• nd thiS is discussed in chaplcr 5. TIl(: successful descriplion of the .tress-strain relationship that has ~n achie"ed i. one of the major achic."""nts of the mot<xular theories, wilh good agreo:ment betwttn theory a nd c,perimcn t being achievtd o,-cr a t"c"tyrold I'(Cgion of tensile and cumpressive strains. The type of defonnation most widely u«-'1.l to charneteri,e elastomers in til<: stfl:SS-5train ""periments just mentioned i•• implc clongation , It i. the ea,iest to impose, and seems to be the ca.iest to thin k about intuiti\'Cly. It is for t~ reasons that it is discussed so frequently throughoutthc bonk. I/owev.r. networks also deform by imbibing sol,.. nt. that is, by ."",lIing. to yield fragile but none' theles, solid mater;als called .. gel •. ·• As might be anticipated. for their characteri,.ation they require sume of the methodology used to st udy polymer SOlutions, The swelling prOttSS and the prope"i.. or Ihe "",ulting gels are discu>sed in chapters 6 and 7. M<:>r. srecilically. chap\cr 6 provido:s a detailed analysis of the chemical potential of $wollen 'I<;t ,,·orks. paving Ihe way to Ihe stnd)" of their large-scale volume changes in phase tr~nsit io"s. as presented in chapter 7. Some additional. less <:Qmrnon types of deforma tion. lueh as biaxial cdcnsi"". torsion. and shear. are described elsewhere, particularly in chapters 13 and 16, Departures from Gaussian beha,io. may ~ome significant in networks haying unusually short chains. such as those inteUlionally introduced into some of the ,,'ell-characteri7.ed. end-lin ked model ""tworks disc ... scd in chapl." 10 and 13, Alsu •• 'udi.. or rubbers at the segmental or the subehain k,-cl .-equirc .Irueln",l inf"rmation bey"nd that of the Gau"ia n chain , F"r these rea",n•. ch'p\er ~ Koe> beyond the Gaussian chain model to diseuss the properties of the isolated chain from the Yicwpoint of the 'Olational iso"'eric state (R1 S) fo"nali ,m. This apprn.ach ta kcs into account the usual structural features of interest in any molecule (i .... bond lengths, bond "n<s. and the locations and energies of bond rotational states). Specific application" of the R1 S seheme to problems of rubber elasticity are gi"en lale. in tbe chapter. along with \he expressions of the end-toend ."",tor distribution fun<:lion5 for sh"rlcr chains, Some of the m010cular d)'nami"" simulatiOn! mcnlioned earlier are discussed 31 Ihc cnd of chapter 8S<.'Vcral controvcrsial i,sue> of the 1?70s and the 1980i may, in principle be. resol,'ed by such simulation •. Thi. t""hniquc should therefore nol be undoresti_ maled. c,'en though Ihe size of the networks thai may be lrealed by molecular dynamic. i., at prescnt. very small. Some initial " lle,npt. to model 1110 l'(Cinforcement of elastomers by the incorporation of fillers are also described in chapter 8. Although Ih. toO" important relationship in rubberlike elasticity is thaI bctwttn Itres' and strain. the relalionships bet""",,n stn'S<. str~in. and tcmper~IU'" arc also of great intert$L They can be used. for example. to rcsol>e tlaSIOmeric quantitie, into cntropic ~nd energetic components, a nd to lest some of Ihe major postulate, or ,"" 'nolccular theory deseribed ill chapter 2. Thi . ,uhject or nctwork "thcm,oeb$ticity" il th~refof< co,-crcd scpam tcly. in chapter~,

OVE RVI EW A N D SO ME FUN DAM ENTA L INf OR MATION

5

Nea rly all of the remaining cha plers are more s~ialized . For example, chapler 10 is ~boullh e already meolioo«l model elaslomers oblained by carr,'ing oul nelw",k.fonn>llion r~aclions ,nUC'h more <'arcfully and specific>lUy Ihan in 1m: uSual tochnolog}'. The fu ndamental ad"."'age of lhese elastome r. i. Ihe fact thallhey have known structures, p."ticulatly network chains of known mok:<:ular ,,-eight. and mO~\l lar weight di.lribulion., and junctions of ~nown fUI>Clionalily. Chapler 10 describes the prepllralion of such "elworks, and Iheir use in clarifying a number of SlrUClu rt>- pTopeny relalionships . Uhimalc properties are touched upon, and.", then considem:i furlher in chapters 12 and 13. ChaplcT II f""u$CO on chain segmenlal orienlation acwmpanying ""fonn.lio n of an elaslomeric "etworx. An c""mplc of ilS fundame",al importance is in""Ili. galion of lhe mok:<:ular deformalion ansing from an imposed macroscopic slrain. the importance of which i$ sl"'<Sed in chaplcrs 2. 3. and 4. A more frequently applied c.ample of in importance i, ils ronnection to sl .. in·induced cryslalliza· lion, and the "'inforcement il provid"" in Ihe cas< of nelworks Ihal can undergo il The classical lheorieS of ",bbcr elaslicity are nf>! valid when Ihe cbains ma king up Ihc nelwork h••'c ",miflexible ",gmenl$. This is Ihe area of liquid-cryilalline nelwork,. and great advances have now been made;n Ihi. field. The trcaln1C111 of such networks requires the descriplion of the packing enlropy of semirigid chains in a deformed lanlet, Thi. is covered in chapler 12. as is slrain·induced cryslal. 1i7-l1tion of networks in general. Cbapler 13 describes a particular lyre of model n.lwork, th.1 is, one in which the diSl ribulion of network chains is intenlionally made mu lti modal . 6 imodal elaslomer! a", Ihe .implesl and most imporlanl example of Ihi' type, and have been much siudied beeause Ihey have unusually good mO\Ohanical properli"", s~iFJeally simuhaneo usly large values of the .1""" ~nd exlensibilily. Th is, in lurn, gives la rge energies for Tuplure. lhe standard measure of the 10ught>eSS of a malerial. T he u", of Ihese nClwOfks in bolh fund.menul and applied slooies is oovcm:i, usi ng Ihi, as an OIXa,ion 10 MiiCribc mochani.al properly '<:su ll-! in a varkt)' of olh.. mochanical deformaliuns: biaxial eXlen,ion. shear. lorsion. lear, and cyclic defonn"lions. The u'" of neulron scallering measuremenls in Ihe . ",a of elaslomeric maleri. a ls i. described in chapler 14. T he cover-age ronsists of Ihe general fealures of lhe lechnique, Ihcorclica l lrealmenlO, and comparisons belween lheory and experi· ment. " Biod. slomer<." or elaslomen; in which lhe nel"'ork chain, art biopolymeric and produced by nature, arc the suhject of chapter I~. Again, this lopic is inhcrCally inleresling. but il can .lso provide practical informalion; for example. how .ynthetic chemistS mighl mimic na' ure in . he wa)'$ il produttS elastomeric malo· riab. The fina l chapler discusses " ~ubjec1 on which Ihere has !leen relalively lillie molecular inlerpretation, namely Ihe reinforctmenl of eluSlomehc networks by fillers. Some theoretical approache, are describo..-d. hUI mosl or lhe informalion concern, c,peri,""n t ~1 r",ult, on da,lomcrs in which the rcinfon:ing pba", i, generaled in .i.u through a sui'abl)' choscn chemical rc~clion . The justificalion otTer<'d is Iha ' under 'hc.'I<: rondilion. it may be I"'s;iblc to oh,ain relali",ly .implc

IUUO!I!PP" .oj =U""'J"')O =nos (npSn ~ ""I Plno~, pUU '~ooq '"'I'"" mo U! p,p!",ud ~uo '~I)O uoo=, p"l"pdn II~ AUF! IU,,""" I! II ·p.>pnpu! "'I" '! AI!~!li"l" '''''Iqns uo jl"P!lJ~ .""!h~' puc s~ooq <"!1 ' ''''U''"0J''' IU,:)Uoi JO '(~d ... io!lq!q V 's:x:>!PU! 1»fqus pu~ '01(10" I"nm "'II ",. "'"~I '~II~U!;1 j""'! lnp ''''''In, JO UQu>wOU:>4d 041 01 P~$:lJpP" ~',,, :x:>U"~ J>W.(lod)o II" "' ""P""I(I ""11"1!IIl "nb l'<JU;>ojl 'I'"J II I 'iuu:x:>uliuo pu~ ,.:'!".~ J,m'IOO ""I"I!IUCnb JO dUluuli>q "'''' ,~! 01 P~ 08 .. u.>wdol"~'p ='lI. ''''''ld'''''lP OI(l JO uo!w,=dd. J'll""l ~ JOpC'" '1(1 ""'~ "'[" pu~ '""'C INI U! ~Jo." Ap"' JO /I\,!.<,','o lie ,p!hoJd lI!/I\ UO!I"WlOJU! '!41 1"1(1 0<101( OM · .(I!~!"UI' ' l 'IJoqqnJ puc ""wo,""p ~U!AIO"'U' I1U'.<' [C,OJo"!1( ':""O~!P H ~!pu>ddV ''(I!''IISCI" ' l !I'Jqqn, '''J I"'!ni>:u AI!I!'l!" U '1(1 SlIIj ~UI·)[\JlS "!"I(~ "'A!1' " 1('!4·" 01 ,U)Ir;) "41 ~U!guo:'iI h! ·pn",'" 'Sl:>:>ds:u JO J""wnu" "' InJ""" 'I pun " ppow "C)! ,!"W""! I""O"CIOJ 110 t»S'1ddu "! u,... ,S "I 'U!~I() ')"'/."IOOjo =uU1S IP PUo-<)1'PU' ~"U'" ·uu,w puc StllI1U)[ :x:>\"IS'Uo.l[qns ''11)0 lU)WI":Ul "1"'OO~ '141 Aq P",oJSJ,puu i! ;):luel'odw! '!"I(' P"" 'Al!'!,""p '~ !jl""qn' JO A'<»I(' J"I"""IOW AU" 011"lU,,,mpUn) ,J. -Ol'PU' U'"~~ '41 JO 'UO!lnq!'lS!P \noqu I! ~ "!pu:Kld" '" x,puodd~ y, POlUos:ud so "do, ~!41 JO u"!$ln"'l!p P"I!"'~P ~ '>'''0'4) A1"'''~IO uJOpow JO .)UJW,p IUIU"'''"pUn) OJB .UOlSUOW'p U'"~~ JO .UO,,""'''''~ """'S -AI!,,!"UP ~~'jJ>qqru 01 ""UC."I'" JeIO"'lJOO JO ' 1:>:>dIQ UO I! I!S~4d,," '4' ·p.11""!0!,U8 >q Plno,,, I V '0 ' !puolu ..",d I! !o:l!u ~I(""w IB,,!iIIIBII puc S'IW"U~poUlloql)O u"'u '1(1 U! Sidx.uo, JO "'Bwwn~ 'IS!"UOJ V 'UO!SU'1 ~Idw!. I"nln ~4' Se [p." S~ 'fI"J~~O) :uu JU'I(' 3Jnd pu~ 'U01<".", I~! ' "!OddV 'IL[l1!'U! JO lIu'PU"IU,'pUn Jeln""low ,u,M lSSO'll ~" ""4 1""''''00 ~o '"1(1 ..:>:>dso "41 ' ~ooq <141)0 ~"IIJ'[qO poI"IS "41 411-" lu."fSuoJ -IBJ,uJlI u, NOIOUI("'" JX[qnJ gUI~ 'lo~U! 1.J)41O 1"''''$ pUB '";).IB '!41 U! "d>Juo.:> )41)0 ow", ",,!J:>SOP 01 .. dUL>U" \I .. ptJ:KldB pu~ 'IU"UOOu11 I~'I('JUOU J'" s;>nb!U4:"" ~4.1 'wopue, A1411!4 puc P"IIOJIUOOUn "'. 1"4' 'UO,,8",'U",[n... Se 4'"W 's;>obIUI(""1 SU1~u'I-SWJJ IBnsn '~1 Aq poU!",qo u""I suI( ~JO/l\PU '1(1 u'4·" llu,'U,P AjJ~ln"lJ"" so "",""nJ1S ~"". JO UO!lB'!l)jJBJ" 4:J '1''''1,''0'411" II"" IC 1"1"'W)J>dX, 'SUO!,o:'iI!)SO" U111" AU"U""'., )0 '!S~ 'I(IIW)OJ lB~l :un,,,,,,,, lJON.IO" ""1 )0 uo"d,,:>s>p t"'l,epp" '.p",oJd V ' !pu:Kld. '''OS"lJ S!~1 '0.1 'P""I0.:!u"pmS J""P Ino~,c" 'I""'~JP IP," 'I "'''I:>OJ1I ~JOM1"" "'II 1"'11 ~wnSl~ ~'!,,!lS.l~ "''Iqn, JO .. U"""~,,, ll~ !SOW[" I"~I l~.J 01(' APOW'" 01 P"pnIJuI'1 'X.IPu)(ldo)o ~,,~ '1() JO lSJ~ ~U ',w'lds ~)ql J" 'wos n' poU!tlqO 'I)OJP 8u!:l.J0Ju1>j ~q' )1&.I1I"1I! 0, popnpu! S! IUO!I~WlOl'P JO ,(P!,U~ 'P'M 0 'uoriV 'A1""""1,' ~I"q 48!1( AI~u!puodm,<)(> r. pUB i\(1l!~M JOIO""[OW \(11'1( " •• ~ AII"JOuJll Q"l4," ,"WOI 'S"P U~ O)UI (UO!I'":U ~11lJ.d:>s • U! p"WJOJ) ""I" JO P"lq uoqJ"" t""IU"WOjllll~ AIP~ l,u'lq 01 lIundw'l1. Aq p""'B1qo <>&Qql 0) p;>W
S ~ 1I0MHN

nll1l 388m:l

~O

S31.11! 3dOlld ONI>' S311n l :lnlllS 9

2 Cla ssica l Th eo ri es of Rubber Ela sticity

Thi. chapler desc , ibes Ihm: mok'Cula, theories or rubber c1aSl;city. Scc1ion 2. 1 outl;n", lhe elementary theory of Kuhn 1-' and Tr<:loar"', which i. of particular imponancc si""" il presenls Ihe basic elements or rubberlike elasticity in a very ,run'parent way. Section 2.2 prescnt' th. phantom network model developed by Jame, and Gulh6 oJ. aM section ::'-3 pr=nls 'he affine ne,wor k modd de"eloped by Wall and Flory" 1O. Uis!Orical as"""ls of the theories have been given in an article by GUlh and Mark". and in a book l'Il'paml " •• memon.lto GUlh"". Finally. the major f~a {u re, of both th..."ics are bricny summarizcries: the pnantom nclwor~ and the affine net,,·o.k theories. Despite their diff.",ncn. th..., Iwo theories and the corresponding mol.,. cular modets ha"• ..".,xxI a. Nosump'i'ms and the predictions of the two models have led to .. riou. di .. grc-ements during their de"elopment. as may be .... n from the original papers ci too carlier. The main roim ()f disagree. ment was tne magni1Ude of the (ront facI", th" t appeared in Ihe eXltression for Ihe elastic free energy and the stress, For tcttafunct ional nelwork •• the Jame,·Gulh phantom nttwork lhwry prediNs one-half thc value of Ihe fro nl facto r Ohtained by the Wall . Flory affine network ,heo'y . A<"';II he di<eus""", in g"'~ te' de' ail, in later "haplers a, ,,·cll ,IS in this "nc. Ihe pktur" has hccom< cle",.r "nd it is now well CSlabli5hed th" the ph. llt onl nClwQr~ "'odd i, the appropri",,, one for , wolkn network>. for networks in tho un,wolkn ,tatc. the problem is Slill not

,

8 STRUCTU RE S AND PROPERTIES Of RUBBER LI KE NETWORKS

unambiguously senlN. and III.re is di"'greem.m among re .. arch." as 10 Ih. nalure of intcrmolocular correlations contributing to th.c clastic froc energy of the nel work al equilibrium. According to one group of =rcl\<,,,. II\<, action of entanglement. in the unswolkn network C'~n ultimalely suppress /luctuations of Ih. junctions. and Ihe network Il\<,n approaches Ihc Wa ll_Flory model , Ollie" maintai n Ihal entanglements further CQ"lrihule as chemical jU""lio" .. a nd Ihercfore inerease lhe elastic free .nergy and lhe modulus of tl\<, nel"'"or k alx",e 1hat of Ihe affine network . These qllCslions will be addressed in furlher del ail in chaplers ) and 4 in relation 10 differenl enlanglemenl theorie'S of rubber elasticity. In Ih. final secl;on of Ihi' chapter, Ihe basic feal ures of the Ikam -Et! ",·ard. lrealment l ' are qualitalively oUllined. and severa l other Ibrories related 10 lhe cla",;';.1 theories of elasticily are discussed. T h. Ihcory of Ik~m and Edwards unifies Ihe ph ysical picture I\<,hind the ' ames-Ont h and Wall_Flory theories .nJ may be regarded as constiluting the starling point of >ome of Ih. mosl modem Ihrories of rubberli ke elasticity.

2.1 The Kuhn-Treloar Theory The theory of Kuhn. as improved subsequenlly by T reloar. is based on Ih. following fundamenlal a" umption5

v fr .. ly.join led Gau"i.n ch.in,. whl:r<....:II • network chain i. < ..... "'1>1< of ne."'ork chai ... in ti>< "","",. f""ned nelwork are lhe same a•• h.,.. for an "nsemble of chains in ,he bulk. "n_ cross_linked ". Ie, The mean·sq""" eo.d·lo-cnd dimen""", of ,he lall.,.. in 1",n. are ~ .. I." ,hose of "'" ,; ngk <""-in in ,hi: unperturbed ........ p",n in eq. (F.9) of appendi' F, Thi, '"U'''pl;''''' i, , ul'f'O'led by neutron scanorinll .. perimen,,'''' by ""'"rari.."" of chain dimen,ion. in ,he bulk un..,ross-linktin1.. bu, don TIO\ impo>e any «mm. ints 01 poin ts along lhe chain ""n,ou" In thi, respelogy i. u,ually rese< I. The nO'''''ork ""n,;", of

_.,.,-sq...

seq"""'"

.,'id."",""'.

Jomeo.Quth ItIrory,

4. Thc lotal cia";' •• "'BY of lhe network i, the 'um of the elastic cnerVos of ,hi: ir>di,'id ... 1cru.in •. Due to the .",umpti"" ttul the ohains are {=Iy join,ed•• 11 spotial arra ngements arc or lhe same .... rgy. lhe netwnr ~ < rel .. i~ n ;l,ol .. a 6£ _ TtlS b«:<»neo ;l.... ~ - T6S. 11>< '",.tm.nt m.y be 1I'no.. lizcd 10 nnn-f,tcly-rinted ch.in .. h",,'C,'Cr, t 'oU"",i"'8 131« diSCU""'"' by ... " Ike,.,,,",o'" "nd Floo"y' • 'he fro<: energy or Ihe ol... ic fftC onergy,

CtASSICAt THEORIES OF RUBBE R EtASTICITY

9

The elaslic f"", .n.rgy of an isolaled deformed Gaussian chain wilh ilS Iwo ends fixed 31 r is givcn by eq. (F.14). Summing Ih,s equalion over all chains oflhe nccwork. lhe change .:lA,! in Inc 10lal c!aslic energy at conStanl ICmper.l1urc (relati'"e to Ihal of lhe undeformed
~ 2(;{ ~:
"' 2~Vk7"( (r). {?)

(2. I)

-I)

In going from Ihe first 10 lhe """"nd C<.jualily in «I. (2.1). lhe rdalion,hip (,.l ) = L. ,.lIft i. used 10 repre""nl lhe average of Ihe s.qua ...:d .nd_lo.cnd >"octorS. Wriling Ihe cnd·lo.cnd >"eeior in lerms of lhe Carlesian C<)mp
+ V) + (?)

(2.2)

Dividing both sid .. of lhis equal ion by (,.') • . using Ih. isolroPY of Ihe nclwork chain. in Ihe undeformed ~ale. and using Ihe .»umplion (see seo:tion 11.2) Ihal Ihe chain ends are di'placed in prop
(2.3) Here. ),•. .l.y> and )" are Ihe conlp
(24) 11 sho uld be: nOlod Iha1 imefmolccular inleraclion' are zero in Ihis modd: Ih al is. {he .y.lem is e&"'nl;.lIy like an We.l g~s_ The expres,ion for lhe fo.-.;efaCling on a prismal;e .. mple under unia, i~1 len$'O" along the x directi,," i. oblained from Ihe lhermodynamic e'pression ("". (D.S)!:

f _ (iJ~A") ,J/.

IY

_..., (:' ~ A,, ) iJ~

(2 .~) TV

",-here ), = .I., = L/ Lo (with Land Lo denoting Ih~ le nglh, "f lhe .. mple along (he direel;on of slrelch ill Ihe ddurmoo and ""dcfofll1ed 'lates. respeclively). Following the a>sumpl;on Ihm the volume of (he .. mplc re mains coo,lanl during deformation. Ihe y and ! comp
(26) Eq ual;"n! 0 .4) and t 2.6) aft· Ihc mo_'1 i,nporlam a,~ic,·cn,cnt.< or lhe ele",e" · lary Ihc"Ory of ruhber dasli;:il y_ I, sh"uld hi, cml'h".j"~l {ha, ahllough {he modd

10 STRUCTURES A ND PROPERTIES OF RUBBERLI KE NETWORKS

m!, On "uump!ion, ,uch a, the ,imple additivity of !he fr« ene rgies of !h. individual chains and !he lack of in!cnnole<::ular interactions, il< predictions approximate e. perimen!al data within reasonable bound •. a. will be: outlined in more de!ail in laler chaplers.

2,2 Th e Phantom Networl< Theory of James and Guth 2,2.1 General Aspec!s The James-Gulh Iheo!), is based on !hc following . .. ump!ion.: I. Tb< ""twolk duin, ore G. wr,ian 2. Son.. of lhe iUnc1ion. al the .una<:<: of Ih. "",work. a", fi.<ed and doform amn.ly ,",'ilh lhe"~ ",..i", J. 1ll< chain. a .. , u\ljoc<1 only 10 con""aints th., ori .. ~ i ""' ly from ,he conrl an.cl>eod at i" end •. Thi, eh."",,.rillic of 0 p .... nlom 1IC"',ork booto al . 11 defo"",,,tion.

'wo

The second assumption. according'" ""hich Ihe m"cro,",opic conSlr.i nlS are inlrodua:d by fixing micro:;copic variables. h'" been criticized by Edward. a nd f'l<.'<.'d j , . Fr=l .... and Floryl. Placing the fixed june,ion; a' Ih. , utfa"" has no significa r>ee and Ihey could be inside the body instead. Fixing a 'Ubscl of microscopic varia bles. howt:""r. is n O! strict ly a rigur<.>u~ appr(>;}ch in sta!iM;';.1 mechaniC1j]j. Ne,,,,,he!.,.,.. the theo!)' based on these " "umplions leads to significanl imp",,,,men!> in the underst,nd ing of Ihc properties of I>ct",-orks. such a. microscopic " "", ua!ion, "nd neUlron sca llering behavior. A, a result of lhe third assumption. Ih. configur;nional panition function Z~, of 'he nctw"rk may be wriuen as ,Ix prOOUCI of lhe configurational part;!ion fU<>Ctions of il. indi"idu.1 chains, A$ described d"'wherc~'. each of Ihc taller pan ilion functions is fully determined by Ihc cnd_Io.end "tetor ' " for Ihe chain connecting junctions j .ndj. Thus.

z '" =

I1 Z,. = r.n W{r~) ~

(2.7)

,"

wht:rc i" is ,ht: partit ion function for Ihe Chain wilh end_Io-.nd veclor , 'I' and Z is Ibe parlil;On funclion corresponding '0 the frC'C chain . and IV is defined in <XI. (2.8). F urther descripli<)n is given in appendix O. The second eq ualily in <XI . (2 -1) folio .... , from the faet that Ihe dislribu lion funelion is C<j uallo Ibe ratio of the partition fun<,ions i" and Z . T h. produCI" in "'I . (2.7) include all pa irs of Ju",,'io", eonneelc'
t"

X-"'" o>f ,ub>«l """"y "ilh-

,,,,". om,,, '-
( '_ _ 1'1<,.,1<> by

~lorj

in I%< .rod ~- . . .'iokl,' ej".ial",J ..

,h.o, limo::_

,,~ .,

r""" """,.""", ioo f,~ ""","",, ""'_ ,,., ,.,...1., ""'" ··,.to.on".. "'-.~-,~ I " ij ' " OJ'!'<" «'_

,I .. n

CLASSICAL THEORIES OF RUBBER ELASTICI TY

11

tOI'$ in the undefonned network is idenlicalto thai of. free chain. This distribution may be thought of by conceptually f"""ing the network in,;tantanffiu,ly and counting the number of chains with end-to-end voctorS of specified IlliIgnitude and direction. In rrality. the in
m '" c>p(.-,.~ )

(U)

W(,,) = GJ where

(2.9)

Substituting cq. (2.~) into eq. (2 .7). and Mnoting the position. of junction. i and j by It, and It; . eq. (2.7) may be written

ZN - e n 1<)

Hi

e,p(-~LL 1;IR, ,

(2.10)

Itlf )

I

i~ the square of the magnitude of the chain ,'octor rii ' and J oonnected by a cha in. and ,.,0 othcrwi ... Lowercase ,)"mix>ls denote chain vectors and upflCrca .. letters denOle positions of points in the network. The specification of ,.; th~i introduces the iIr<>eturc and topology of the network into the statistic~1 thffiry. an aspect thaI is mi<sing in tile Kuhn-Trcioar (hcoT)' Equation (2. )0) e>pouses Ihe parI it ion function in tennS of the positions of junction point' ralher than in lenns of the chain voctors. The position of Ihe network may be fi~ed in Spa«: by 10<31ing one of the junetio". at the origin of a la ooratory·fixcd COIlfdinate s)'stem and measuring all olher Ri values relati"e to thi , ~ ystem. The po,ition vectors R/ (with i ranging from I to the 10lal number of junctiollS. I') m~y he ~rrangru in column form. repl"C$Cntcd as {lt l· I"!quation (2. 10) may tben be written

Her<:.

lit, -

,.~ _ J / (2 (1;).) if I and

.r<,

z,. ~ eex p(- { R J , r{ It ))

(2,11)

where Ihe . uperscript T !knotes the tran'pOse. The c!ement. 1~' of the s}'mmetr;c matrix r ar<: obtained hy expanding the quadratic form &i.'en by cq. (2. 11) as

,... z_,.;. r '" ( -rt '" ~:>;

,

i"j

'" L, . ,.;

(2,1 2)

In expanding~ . (l.11 J. product> such a. Mi lt, arc to be tre3,ed a ....alar products. The matri .. r complete I)' describes the conn,-cth'ity of thc network. If all ~'ha;ns of the network are idenlical and arc ,'naraderi'cd b}' the $limo ,.Iue of then all non?ero element. of r arc C.lucos to Ihc Kirchoff adjacency

<
12 STRUCTUR ES AN D PROPERT IES OF RU BB ERLI KE NE TWO RKS

malri. ~dopted l>y Eichinger" in the study of perfect phantom networks by graph theory. A small "'t 1"'1 of the I' nelwork juoctioni ar. assumed to be n:qui~ by the s«ond assumption ~tated pre,·iou,ly. T hese may be thought 10 be junction< al th. surface of the nelwork. describing lhe size and Iht shape of Ihe ""t work and directly subject 10 macroscopic manipulations. The rem..ining sci !,.) of junctions are assumed 10 Ix free 10 fluclu", • . The t ...o "'ts of junctions may nOW ht used to partition the quadratic fQrm ofeq. (2. 11) as

fi.e
{ M)' I" { R) _

{R.- ) (( R, )

)'(r..r.

r .....

r,

)() {II, } { II~ }

(2.1l)

or equivalently as (2. 14) where {R. ) and {M, ) arc the column ,'eetors for the position ,'oclors of the f,xed and fr.,. junctions. rtSpcClivcly. Th. q""nlity r. i. a squar.: ma!fix eompris.ing row; and rolumns of r for fixed j unctions . AnalQgQusly. r , is a square matri x charactcn,.jng f= junel;Qn•. aoo I".. ~ I":;' ;.the ",,:tangular malri. whose ro"'.. rorf't"S]">
(2. (5) G.~ r. - I".....;

'r ..

(2.16)

{A M, ) _ {R,) -{ it, )

(2.17)

- r; 'r _ { R~ }

(2.18)

{il , ) _

The .. t Qf va,iables defined by "'-I . (2. 17) indicates the Huctuat;ons Qf tnc free junclions from Ihei, equilibrium ]">
(2.19)

Equalio" (2. 19) thus lead, to the partition functiQ" where term! relaling to fi xed and f= vari"hics are ..,p.rated . The two product, of the CX]">
CLASSICAL THEORIES OF RUBBER ELASTICITY LNo _

LN,

..

C' C~P(~{ K. ) TGo{ R. })

C· e~p( ~· I6. M, ) Tr , {6. M, )

(2.21)

The product of the ft""tualions of two junctiQns i and j. 3vcragc:d network, may be ohtainc:d from e'
O\'~r

thc

(2.22)

8 102 _ __ I).,.

..,

13

(2 .23)

,

,II .u R, ) "" d 6. RI . d 6.K" ... d "' K~,

(2 .24)

2, _ ]cxpl-1.u R, j" ', II.'!. K, ildII.'!. R, ) _ (d:;-r ,) 'fl

(2.2S)

Equation (2.13) is an identity obtained by noting that the denominator of eq. (2.22) is Z , . and the Gaussian prO[1
+ I.'!..~

Here, r,,· is the mean chain """to,. and I.'!.' ij is its fluctuation. The of chain """toTS ' il an: thcr<:foT<: nonalT,"e.

(2.27) trun iform3lio n~

2.2.2 The Ela$lil' F"", E""'gy In ord"r to rctate the .tati,tica l prop<"rties of th~ network to mc
t4

STRUCT URES AND PROPERTIES OF RUBBERLI KE NETWOR KS

juneli(>n•. Inlegralion o,,,,r ftuctuations of the I' , frc<; junctions in eq, (2 . 19) kads 10 Ihe confIguration ~rlition funeti"n "f the network. as determine<.! by the I"',iti"n, "f Ihe sel kl "f fixed j unel;"n, (, .. in number). l1Ic result is

(228) The ela'lie free energy tiA" "f the nel"-o,k is "blaine<.! by substiluling cq . (2.2S) inl" <:<J. (0 .8) of appendi, O. multing in

The sel "f "<'CIOI'$ 'pCCif~ by {R.. ! represent. Ihe I"'.ilion, or Ihe f, .• ed junNion,. which traRSionn aflindy wilh maer<>-'iCopie strain . Ocnoti!lj! the po,ition of Ihe fi xed ju nclion i in Ihe und.formcd sW Ic by R~.. and using the relation R_J = ~ R~J for Ihe pmition of til< junclion in Ihe dcforme<.! statc (,,'hkh holds fo, all fixed junctions I). eq (2 .29) may be wrillen as (2.30)

The factOf in the square br.ICkct. in eq. (2,30) depends only Qn network SIr...:tu,..,. Its evaluation in ctO>Od form secm, to be diffICult. but one may use m;cron.t. worh"'" tngether ",;th computer Monte Carlo simulation. ro, il!; evalualion. In the following ~ragraph'. the '-al"" of Ihi. faelor for a unimooal phantom net· work will be obtai<>td by considering the statistics of forming. network with " chains and I' junctions, The instantaneous conflgur~tion of. network chain OCt"l."tn junctions i and j is shown in figure 2.1. The quantilY ' Ii is Ihe inSla ntaneou, cnd.tn-<:nd ''eCtor. and i ij rerresen .. til< tim<; aV1;r.>g
f

, .••

':::';;;'' ' ',..... T fil:... l . t 'rl>< instantaneou, '"""('M"tatiO" "r. ;_ TI>o I,.ri,bles ,"","'II.tt defined in the ,.AI,

-"

,-

..

r ,""w",\

ch';n bet"'""" junc1;',,,,, , ""d

CLASS ICAL THEORIES OF RUBBER ElASTICITY

15

(2.]1 ) " 'here O-' v = O- Rj - 0- 1{, rep,..,..,"t. the instantaneous fluctuation in the chain vector ' (i hom it. average value i ii' T he instantaneous distribution of the mea" vectors in the network and Au.;{uati<)ns O- r~ ate ~r>rescnloo hy l (rQ) and ", (0-, ,). respectivC!y. T hey a'e related to W(' q) b)' lhe con"olution C' I't=lon

'jf

W(•• ) = l('~) • "' (~ri/l !!"

1x(ri')"'(o.r~)dru

(2.32)

H e~. lhe a$,~risk denotes til<: ronvolution e, pre<.,,-"
I,

; c_,p( _ 01>(.1 , .)1) (.)'" X)'""~p( - l~ ) X(i •.) .. (-;

.jI (01' ul "'

where

.-

x

"

2«(01,.) ) X

r __

(2..13)

(2.34 )

. 2«••)')

Replac;ng ( 0-. ;) in "'I . (2 .33) by the square of the magnitude of r. - i'~ and performing th~ in!~gn"ion "'~d$ to til<: following rdation,hip bf,twttn the p"ca' ITIC!CfS 'Y. X. and ~!;

J/IIl+

I/ X -

Ih

(2. 35 )

which is equiv.knt to (2 36) Thus. the magnitudes of the inst" "t"neous "0.1 "Verdge chain .""tor~ and their AUCluat;ons all; not independent. hul arC ~Iated through C. the mean chain d i m ~ "sions and ftuotuat;ons a re rdatoo to nNwork ~h"i n d;monsio'" by the relat;ons:

(2.H)

16 STRU CTURES AND PROPERTIES Of RUBBERLIKE NETWORKS

. '" {U8) Eq~.tions (2.37) and (2,38) follow from the work of Eichinger", Gr~essley". and flory"- A detailoroach of Edwards" and ,.."" pUl'lucd in the latcr th«>ri« of f.dwards and his collabomtors. Ahemali~l)..o.."may<. a .twith a giam "C)'die tree, as is .hown in figure A.I of 'Pl"'ndix A, Formation and deformation ofa network in this man""r i. shown in figure 2.2. As ,hOI''''' in this figure, one may obtain the deformed phantom network following either tile palh 2. J', and 6. or the path 4. 5, and 6. In tlte latter ""se, lhe clastic f"", energy ll.lI " of the deformed phantom network rdati,.. to tho undcform«l network i,

{2.39)

Tile clastic r"", energy for each of the stagC!l will I>t: described separately in this section.

. -...--. --2[;l; ==..... ._-" .... -_ .... ot_.. _..... __. --• N oh _

1n

... . . - _ - lot__

""'~poIn

5

_otC)
,.,..,

_,,"'"

______ __ _

,

otC)
~~=-::t

...... In ,

n il"'" 2.2 FOfm .. ion ..... deformation or . n< I"'th 2. 3'.• nd 6. or the I"'th 4. 5, and 6, The ""'W(>tk;, obt.inoo by following th. patb 2.00 l' or 4.nd S

.ffi..,

CLASSICAL THEORIES OF RUBBE R elASTICI TY

17

1.1.2.1 Caku/mioo of ll.A l Tw(> reactive groups must be within a volume ele_ menl 6V rar rcaelion 10 be possible. Of primary inleml i, Ibe probabilily Ihal any given one of Ihe v label' ha, anolher label 0"1 of lhe v ~ I remaining al'l(S Ibal is siluate
<.

( 2.40)

1.2.2.3 C,,/cu/ali<)l1 of .ction (del.l.) II'( A. ,)d. , and ",ay be laken 'IS Gaussian for sufficient ly lang chains. The probability of hlI"ing all of tho chains of the 1= in the stale
v,

v,.

0'

n

l(dcllllI'( A. ,)d.,)'"'

(2.41 )

n

Howe",r. there arc v!j v,! wa)'> of combinmg Ihc chains into lb. ' I""'ified groups. T he probability gi"en by eq. (2,41) must Ihererore be mulliplkd by this number to yield the probabilily f1, of having the slaled dimibulion of chains in thc deformed slalc, fit -

v'

n[(detl ) 1I'(1. ,)"1,j'" I",!

'" II )(v/ ",)(dCl l j W(l . ,j
(2 42)

The second line folluws from use of Stirling'> approximaiion of the facl orial. thai is. In v! _ " In v - " The r.tio of !!, in lhe deform." Ihc eonf'gurallon parlilion of lbe ,,"deformed on'" Iha t i<. runcti'",

ze

18 STR UCTU RES AND PROPERTIES Of RU BB ERLI KE NETWOR KS

Z.(l) /~ -

n

( detl )W(l· , )I W (, I))"

.. exp{ ~

L "rl [(-'!xf + -,;1, + -';=;)~ (X; + I, + z!») } (Y I v" )'

= exp{ - V)'l(-';

-

I )(~ ).

+ (-'; -

1)0),

+ (-'; -

I )(: 1).1l( 1'1v")" (2.43 )

Here. Ihe 1,.,1 line fOllow> from the relationship of average chain dimension, in the deformed slate 1o the undofwmed ones. and (dell) ha s been ,..,placed by 1'/ v". The eb stic free energy liA. of deforming the tree folio",', then. from cq. (2.43). as liA • .. (1/ 2)vk T( / , _ 3) _ "k1'1n( VI 0')

{2.44)

where the first strain invariant I , is definrd as

,

, ,

1, = -'; +-', + -',

(2.4 S)

2.1,2.4 C"lrul"lio~ of liA, Step 5 requires the joining of the { pairs of deformed chain •. The change in free energy may be wrillcn in analogy ""ith th~t in Slep 2. where nOw the •..,fCTCOCC volume is the deform,'d vol ume. Thus. OI\C o bta in,. from eq . (HO), (2,46)

2.1.2.5 C,,/ruIMiOl1 of liA. In step 6. the Huctuation. li R o f the" - { labeled poin1S are rc! .. rd to their equilibrium dist ribution. Thi s corre, pond s to t!.c rever_ $III of the p;.rt of the free energy change that took place during the affine defor, mation in step 4. Thu •. cq, (2.4 3) must be wrillen in inwn,'d form for the nuNuation< "r the V " (lal:>cled points:

4 _0 1%'(._0(1) =

11 ['I' (lir,)/(detl)"' (l lir,)j""

(2.4 7)

where the d~nom in 'lOr on the right-hand side indicates that Ihe initial $Iate "r the junctions i$ the affi nc\y Jef"rrned one . f>crf"rrning th. sub,ti tu. i"n, the di>tribution ,.. (li ••). the clast"' f",.., energy is obwined as

r",

li A. _ -(112)(" ~ ~)k T ( I, _ 3) + ("

_ ~)kT In ( VI VO )

(2.48 )

f',"

IJf ,he TOlat t:loslk l-:"agy flJ' Ihe Plwtl/Qm NrrwQ,k Substitutinll ""S. (2.40). (2 .44). (2 .46). arod (2.48) imo eq, (2.39) Icud, to the clastic free energy liA " ... of the phantom ncm'or~'

12.1.6 ('''/cll /m ien

liA",. _ ( 1/ 2)( kT(J ,

~

3)

(2.49)

One sees that the v" l~me-depend"nt term [n(V / V.) is not present in the fina l ",prc"ion for the clastic free energ)' "f 'he ph"ntom nct"""~. Thi, term ",'a. a poin t of CQntro,""n;y during the dc"cI"pment "r the amnc ,md phantom th")ric. si""c i, appeared in the rormcr but nOl the lalter (sec . e,g .. Wall and Flory"'.

ClASSICAl THEORIES OF RUBBER elAST ICITY

19

Jam", and GUlh "). T he c""",,)alion of Ihis lenn takes place in addi ng Ihe change in clastic free energy in lhe 6th slcl' 10 Ihe changes in ilCJIS 4 and S. The eia!lic free e""rgy given hy <:<.J . (2.49) i. valid for imrCtc'" (0 rdlher than Iwo (such as the lWO Lame conSlants in cla"i""llinear e"'<1icily theory) re,u lt' from Ihe a"umplion of incompressihility Ihat is buill into the Iheory in canceling the In( V/ Vo) lenn. For ordinary rubberlike material •• lhe changes in volume"", itldeed second order compared ""ith changes in linear dimeo'ions ~nd may safely be negloclCd. as ha$\>ocn pOinted Oul b)I Treloar (ref. 5. p_ 67 and (8)-

2.3 Tho Affino Network Theory of Wall and Flory In lhe amne network model. il is assumc"
t::.A"..,- .. 6 A),

~

6A, + 6 A, - 6 A,

(2 ,50)

Subsl ituling fro m "'I'. (2.4(1). (2.44), and (2.46). ic;,dl 10 the expressio "

t::.A" ..,- = (1/2)<4;T( I, -, J) - (2"NlkT ln( V/ JA')

(2 ,51 )

Equalion (2.51) is Ihc fn.'C energy e'pre"ion corresponding 10 Ihe Wall ....lory theory , The presence o f the "olume tenn .hows Ihat the m3ler;al i, nol assumed 10 bc in<."(lmpress.ibk_ Neglocling 'he ,-olume lerm and Subslituting for ~ converlS eq , (2,51) inlo

,

A A"..,- .. 2(~ _ 2) {kT(I, - 3)

(2 ,52)

Comparison of Ihi' e~pr.ssio" with e<j. (2.49) for the claSlie free cn"rgy of a phantom network ,1' 0 " -' thai the latter is .I,,·u)'. smaller th«" the el~.tie free energy of lhe affine nct " 'ork_ Fo,' " lelrafuneli onai nelwork (which is the ea>¢ trea ted mo,t extcnsively in the litermure). the elast>c free cnergy of the pha ntom network equal. on .... half of the eJastic frtt ~nerl!.\-' of lhe affine nelwor._ Equal ion (2 .52) is valid for imr> ,}'Slem, in ~e,,,,ral The amorphous rubbcr ;s taken "s " "'li d whose lopology Of con ,leuivity is conserwd ,lunng fab rication: Mmdy . nc,work forma ,i"" . Tl,~ theory r""ogni,,,,, difft're" ' topologies, ~a!"ul,, ' e' 'heir

:10 STRUCTURES A ND PROPERTIES OF RU BBE RLI KE NETWO RK S prohabilities. and formulates the conflllunllion,,1 partilion function in Ih< unde_ formed and Ihe dcformoo states_ The theory is more generallha" Ihe James-G ulh and lhe Wall-Flary Ihcories because il indudes Ih. inl ...... elion' betw.. n chains ori,ing from the dcform~tion of tl>< 'opologically conneclcd ,t ruc!U~ . These interaelio" ' arise from the discre,c entangkmenl< belween ehaim and form the b;uis of en tanglemen t Iheories. " 'hieh are treatoo in chapter, 3 and 4. In tho absence of discfCte entanglements. F.dwa . ds· theory red~ 10 either the ph.nton, or the a ffine network models depending on the .pecilk ...."mption made rega.ding the density of the polyme. confinl'
Re fe rences (I) Kuhn. W_ J_ Polym_ .'><:i 1946. 1.180_

m K uhn. W. KelloW /:. 19)6. 76.258

(1) K uhn. W. 11118""" CIu!m .. In<. t:d. tj,g. 1938.51. 640. (4) T .. I"",. l _ It . G_ 1'".",_ f;,,"'''',:><>c. I~ . 42. 17. (S) T .. lo ••• I ~ 11., G, T/Nr rh,..i<-J <>I R..M.-. EI",,;o;I),. Jrd cd. C1amtd"" Press: Q.ford. 1975 , (~) Jamn. H , M ,: Guth . E, Ind. C~'m . 1941. 614_ (7) Jamn. II M_: Gut •• E. Ind_ Eng_ CIu!m, 194,.34. 1365_ (8) jo,n... H :\t .: Gut •• E. J_ CMm_ PAy._ 1<J.I3. 10. 45S. (9) Jamt.... H , M.: Gulh. Ii. J. "pp/, Ph)" , 1944. IS. 2<J.I. (10) Jom'-'$. H 1.1 J_ CII<m_1'iI),'_ 19~J. 15. 6SI. (11) Jam... II, 1\1 ,: Gu'h.l!, J. C....". PAy,. 1'1-47. 15. 669. (12) lorn ... II 1.1 .: Gu'h. 10, J_ ""')'m, Sci. 1949. 4. ISl_ (Il) Jam",. 11 :\t .: Guth. E. J_ CMm_ Phy'. 1953.21. tOW_ (14) Wa ll. F_ T 1 CMm_ P~,.. 1')42.10.132_ (15) W~II . F. T J, CArm. p~Y'_ 1942. 10. 48S_ ( t~) W.II. P_ T_I . ClI<m . PhY" 1')43. II. $27.

f:.w,

n.

(17) FlQry. 1'_ J_ C"",,_ /I" ... 1944.lS. 51. (18) f lQry. p, J, lmi. £Itt, C/Itm, 1946. )8. 41 7. (19) Flory. p , J, J. CIN-.... Ph,•. 1950. 18. 108. (:10) W.U. 1'. T.: I ·lor~. I', J, J_ CIN .... Ph, • . 1951. \9. 143S.

(21) Gutn. Ii.: Mark. II . F. I . PoI},,,,_ .'><:;., 1'01)"". P~,., fJ, 1991.;!9. 627. (22) M" ~ . J. E.: ["',on. 8 .. (;d •. EI",,_.k Pol)",," N~t.'",h !'tont;", 11.1\: EngkwDOd Cliff,. NJ . 1992 (23) Kloczkow>ki. A, In P~y,ieQ/ I'rol'"I k. of Pol!,,",," If~ . J_ E. Mark. (;d.: A",.rican In"itute or Phy,\cs p""" Woodbury. NY . 19%; 1', (11. (24) Gut •. ~_ J . PoII"' __'><:1.: Po;( C 197(1. 31. 20;7 (21) Flory. P. J. Pm,'_ R.,;,_ St><_ 1.<JNJon. II 1976. lSI. lSI. (2~) l 'lor~ . I' l _1";"";pM' of Pol)'''''' CIN,n;",,_ Comcll l)ni\'Orsit)' p"",: Ithaca_ NY 19~1

CLASSICAL THEORIES OF RUBBER ELASTICITV

21

(21) Deam. R , T .; &J .... rd,. S, F. I'~il. Tr(PIS. R",.. S(K. WUJkt:IIk. 1974. 7. 863. (29) Chri,l R, G. : H.,. ..... C. A . J. J. P"I),,,, . Sci. 197(1. AI, 8. ISO). (30) Allen.G,; Ki ' ~ h"m. M. J,; PaJg<>. J.: Price. C. T",,,.. f,,,a4<>y SQC. 1971. 67. 12111. (31) Hindley. J. A.. H.n. C . C. Mos«. 11. . Yu. II . M"""""Q/«w/<J 19711. 11. 836, (32) Ilel,"u"g. 1>1,: I'irol. C; Her•• J. MlX'romoJr<:ul" 1984. 17. 663. (l3) Yu . H.: KilO"O. T .. Kim. Co Y .• Ami •. E. I .; o.."lI- T,; landry. M. R, ; We...., •. I . A.; Il,n. C. Co : Lodge. T. J,; Glink •• C. J, I . A ~.;" ~'ku'_TS aNI lI.ubiwr EI""i<'fly. I lal .nd J. E. Mork. l'.d •. PI"""", " res;; N<w York. 1986: p. 407. (34) VolkenSl.in. M C""fiKltTa,w-1 SID,/J,;" of P<>I),_, ('I,,,;,,.. Int.""ic:ncc: Ne ... York. 1963. 0$) W"'ard •. S. I'; I'rood. K J . Ph),<. 1~ 7(I . 0. 739. 7SO, (36) F....d. K. J. Cltn~ . fir)'• . 197 1. $~. $S88. (37) I!ich ;nll"r. R, Ii , M~(M",,*,.I.. 19n, $. 4%. (lS) 1'ca",,".D. S. Ma",m",i<cuk, 1977. IG. <\9Ii, (l9) K I()Czk,,"~ ki. A, : Ma rk . I . E, : Elma n. B. M(KI"".Q/"cul" 1989. 22. 142), (40) Gno",.Iey. W , W , M.,,:.<>mOkc.l... 1975. 8,186. (41) G ..... Ioy. w , W, M ",")mok<,.I<J 191~ .~ . 86S. (42) Ed"''lIrds. s, F, In 41h Inlel",,,ivna/ Coni'"""" "" AmtXpJwu.r .\laud,,!>, R. W. l)(ougl •• an<.! Ii, 1:11;.. Ed •. W ;Ie~. N.", Yo,k , 1910. (4J) Edward •. S, F I" PoI;'n",. N"",,,,b. S""dltTal and M{/. Sci. 1946, I . 2)$, (47) Slo""nn.o. A, J. Ad>, Pol)"", Sri. 1982. 44. 1), (48) Eichinger. B. ", AM, R.<-•• I'h •. 1983. 34> )$9 (49) F- I. E. A"". R,',. Ph.,•. CIrt .... )989. 4(1. )S I,

Co""".

C/o''''.

3

Intermolecular Effects I. The Constrained-Junction Model

The d.>Sicallheo,;c$ of rubber elasticity presented in chapler 2 are ba$<XI On a hypo!h ••;",,1 elwin which may pass freely through its neighbors as well as through it..lf. In " real chain. ho""o'",. 1he volume of a segment i. excluded 10 other segments bdons,ing c~th.r t" 1he $amt chain or 10 olhcrs in [he ""''''eM-known effecl arising from defonnation.dependenl contrihutions from en\anglem~n1S. The constrainescrvalion$. thi, mOOel

THE CONSTRAINED·JUN CTION MODE L 23

,

,,, ,, ,

.',

',

~Unswolen

•• •

•

, ,,

,,

'"i-""'; ,,

/

, , ,,,

,

--

,

00

'0

~l ..... 3.1 TI>o off."" of o""'galion 0< oomp.-eo.ion and ."",lIin, on 11>0 modulu. of . ,,,:lwork . 'The absci ... rep.-nlS lile in" ...... «",io" r.. io as ,ugge>led by lhe scmi· .mpili""l Mooney. Ri,'lin ropr<S0 C. are co,","an". 'The I"'in" «p ....", ""pc,imen,.1 nd compon.on' be'""",n ,he 'wo eunte1 show ,he of , .."Uini. The upturn in the: modul ... , hip o".,,!.ion. oould be do<
d.,•.•

&""',.

lak ... the f= ene rgy of Ill<; deformed ""Iwork a. lhe Sum of two cOnlribut;on •. one from Ihe phanlom nelwor. and II .. olher fr<>m lhe nstrained.junction modd was firsl propo>ef the Ihrory was ind<~nd.ntly givon by Flory'. and Ihi, was subsequenlly modified by Ennan and Flory'-'. The model has been further describet! in pal"''' by Flory~', ., well as in review P"~rs by Eichingcr9. Edward. and Vilgi.'''. Heinrich el and Erman and Mark ".

"I.".

3.1 The M odel and iTS AssumpTions

T h. eonstl'llined-junction model i•. in ~ sense. intcnnedialC 10 lhe Jamt'$-Gu1h phan10m m<>del anll lhe l-10ry_Wall affine ne1work model di.<eusse
"mile

24

STRUCTURES AND PROPERTtES OF RUBBERLIkE NETWOR KS

(J. t)

",here

t is the avcralf. junction functionality. Thus. for a tetrafunctional

networ~.

« AN) ) _ (3/8)(,) )• . Thederivatinn ofthi. reblion i. given in appendix E. The corresponding Hueluations ( A'}') in chain dim.n sion. (from appendix E) are (3.2) In the con.trained·junction moo.l. th. entanglement of ntt"·or~ chains with Iheir ntighbors i, assumed to diminish til<; .itt of the Huetual;on domains of the junctions. An increa.., in the constraining .ff«1 of the surroundings resullS in a decrease in the magnitude of the Huetualions of the junctions. ultin>ately leading to Ihe affme model. for which (( AR)') _ O. Co",~rsely . a decre.se in co nw"ints ffOn> the environments of junctions. such a. that obtained in stretching or swelling. result s in an incrcase in the magnitude of tilt Huctuation •. ultimately leading 10 the phanlom mooe!. It should be <mph.sited Ihat consideral;on of th. juoot;ons a. the centers being affected by the constraints. rather than points " long the chain<. i. only a maU .. of convenience since the Huet""t ion, of tl>o 1"·0 are proponional toone other. as ac~nowledged in "'I. (3.1). A mOre reoxn t trut ment . by Erman and Monneri."· ... of conmaints "pplied direclly onlO the ehaiQs ralhot Ihan to the junctions has indttd . hown thai the results do nOI change appreciably. Thi, moo.l, referred to in the lileralure as the ··cnnslrained-chai n moorr· ' s treated in some delail in chapter 4. Figure 3.2 shOWl! a nelwork junction. The rad ius of the domain in which it may Hoctu"te in the phantom nelwor k i~ ;ndicated hy the da,hed cirele. The crosscs show the o ther j unelions of thc network that share the spaec a'·ailable to Ihe ""nlno l junction: these other junclion, may Ix: topologically close 10 or ",m" le from the cenlral juncl ion under con,ide,ation . According 10 the constrained .june. tion mood. the iC'"¢rity of the entangl.ment s is measured by the degree of inter· penetration . This interpcncI .... lion is tlIC",urt"d b)· the ",·erage number C· '·lory number··)'·" N p of junctions within the nlcan T"~diu. of the . ph.re occupied by the n.lwork chain. namely. (3.3) where yO is the volume in the sla tc of reference (in which the network was formed). Since (r' )~ varies inversely with the number of nclwork juncli ons. N p must be in'·erscly proponional to the square root of Ihe degree of eross-l inkit.g. Th is is sho"'·n by the proportionality part of cq. (3.3). In lypical da.lOmerie nelworks. NF is in the range or 2$-100"' ''. The sign ificance or entanglement.< becomes immediately .... ident wlltn one considers the $pitCe available to a given junction to be shared by 25- tOO \>""funclional junction. connected to differenl points in Ihe network. The Flory number is diS(;us$C<.l further in chapler 10. T",·o types of forces ac l on a junction in a real nelwork when it insta ntaneously nllCtual~", awa)· from its mean position . ( I) The re.toring action of the phatl!om 'ICtwor~ pulls lhe ju\\Ction toward ils mcan po';lion . This effect result, from the connectivity of the ne twork. S;n"" lIuetu"t;on, "f junction, arc Gaussian in a

fHE CONSTfl .... INED·JU NCTION MO DEL

,

x

, ,

... ~ ••.......••..•.

.""

,

,

X

,

, ,

X

25

,

X···.... \X

, , , '\ , , ,

./

.~.:......~..•........•~

,

,

F"... 3.2 A cen,,..1 'c'rofu,,",i.on.1 i"",,' ioo ,uf'"unOlogically ooghboring iunc,ions and ...,.., ... 1 'I""ially neighboring iu""'ion1. Th< cro!>C< ,how ,be otltCli.on. "f ,ho .'work ,ha' , ~are 'he 'I"'<" . ..... ilablt '0 ,ho <>:"'ral iu"",;"., The flLK1uaiion. of 'he <>:"1,,,1 iu"""ion are <X>Il>lraiO>Od by 'he p.....""" of I,," '1"'t iolly .eigh· borin, 0 .... ond 'beir pendan' choi n" phantom nelwor k. 'he re<1oring for<'<: obeys 'he Iincor-.prinS law, (2) Anothe r spring-lik. force resnltins from IhI: ronm.ining effect of Ih. other junc,ion. (and chain.) pr=nt in the doma in of the choS<':n illnction, This clastic force is dir":loo (o,,-..d a ""nlU. called lho "ecnlcr of conSlr"ints :- fo r lhe chosen ju nClio n, The location of (his center i'l f. . cd dunng Ihe forma l,on of lhe nelwork One rna)' imagine lhal in the absence of nelwork CO"''''''ttVily. lhe junction would nOCI~"le around lhe «nter of constrai nt s, In fogure 3,3. variou, vectorS ",-"SCribing ,he phanlom network center and the ,enter of ""mtrain!> are shown relative 10 a laboratory-fixed coordin.te ,)'&\e01 Ox,.:. One of Ihc junctions of the network ma y

"' ~il... ).J

S<:< ,e" for

v"~O". ' -""'0'-' ,le":r;t>O"~ H ph.n!"", ' '''!''·M~ . ~"""rip,;ot, of!ho "'''''0'' ,h",",'" ,

a,ld ,II<
26 STRUCTURES ANO PROPERTIES OF RUBBERLIKE NETWORKS

be assumed to be

~~ed

at til<: origin, thereby fi>cal<$ the cen ler of fluctuation. of lhe junction oblainc'" under lhe joint action of th. phantom network and constrai nts. The ''''''tors II R, and 6s, are the instantaneous displacements of the junction from lhe phantom network center and Ihe constraint ""oter. respectively. and 6R, is the instantaneous fluctuation from Ihe joint ccnter of action of the n<:t ,,·o rk and oo","'dints. Thus. the instan. taneous position of the jUllCtion j is R, + II R" where (3.4)

The refcrence state of the network is prescribed by the temperature. volume. and 'hape prevailing during its formation . The volume V" may be adjusted as required by the !~mpernture cod1ku,nt of (r' ), if the expcrimomlil a rc c<>nducttd at a different temperature. The "ssumplion, of the oonstrained·junction model are a, followl. I. 1be distribution of chains;n the unMformrd state is unaffectffi byentan. glement. and the em.,·linking pr"""... T he instantaneous distribution of chain "oclOn in the network Ih.refore ,"quates to the Gau.ssian distribution of fn" chains:

W(r) -

(-;t~ exP{-1r' )

(35)

where 1 = 3/( 2(r')0). The flueluations of junctions in the pha nlom network are Gau"ian and independent of macroscopic dcfonn" tion. as follow. from the Iheory of Jam., and Guth . Thu~ the distribution of Auctuations 6. 1l, "f the ith jUllCtion of Ihe phantom "etwork moMI are expresstd by

(')'"cxp [-P(II R,) ~ I

R(II R,) ,. ;

(3.6)

where J' _ 3/ [2{{lI ll i ) ' )). and", _
-..1,

(3.7)

THE CONSTRAINED-JUNCTION MODE L 27

where II, is

, ,

II, " ,...,(U )

(3.8)

with '10 denoting a ocalar parameter to be '~ . where 6s? i, Ihe fluctua lion of thejuH'."tion from the COn· \lTaint center in the undcformed state. Th "", the probabil it y is wtitten as

6.,

6., .

S(.:ll,) .. .. - l/'(del (J ,)'/1 .'pl-( D.I,) T11",(64< )1 The parameter

(J ,

( 3.9 )

is
"" ,-'

0 , _ 001""

(J. IO)

where "0 is a scalar ch.racteri;jng the ",,'.. rity of the ~nt "nglement conslraints in the referen", state. Equation (3. 10) is a !1aIerncm of the ba~ic pOStulate of the theory that the cITecl of oo nstraints diminishc"S with incrcasi1lK utenlion or swel· ling. 4. The network i, of uniform structure, S. The (nla ngiement con.trainl about <,'ery jUllction i. the same.

"r

The formulation the conSlta inaJ.jllnclio n model may I>.: si mplific<J by ,;cpar· .ting Inc v'-'Ctori.1 qua ntities into their x. p. and z eomponents .• nd giving the c.pression~ for Ihe ... component. unly. T his is pOSsible inasmuch as the distribulions of chain dimcm;ons and their fluetu.tion, for the model are assumed to be Gaul'ian,. and lhe ,""ri<>", di"ribution fu nctions may be factot'e',. Additionally. 'he defomlation gratlient tensor ~ will be replaced by ~ or >.,. referring to the x component alone.

IJ, .

3.2 Probability Distribution of Fluctuations in the Deformed Network A junction in. real ""twork i. "n,kr the joint action of network conneclivi ly and entanglo:mcnlio. The probability 1'16.,,) of having the junction at ~X when its "ueluation from th~ constraint "-enter is t>.x ,s "bta;ne
J

P(.t>. X ) • R(6 X ) " S(6X - x)1 R{6 X )

~

S{6X - -,,),MX

(3.1 1)

where cq, (3.4) i. usOO for replac i,, ~ t>.s ,n the ,Iislrihution (""nction S . S,aled in Ihi. mann.; •. 1'(6,1' ) is a ",,,,Jilion,,) prnhabihl}' The product In tl>c numerator of

28 STRUCTURES AND PROPERTIES OF RUBBERLIKE NETWOR KS

eq. (3.11) i. Ihe joinl prob.bility of h".ing the junction 31 tiX. ~nd C"q . (l.II) i. wrillen On lhe basis of the Sa}·"" theorem of probability. SUbSlilUting e
(3. 12) where 6X _ 6.X - tiX 6.X _

(3.13)

I"./(p+ ".ll·.

The argument of lhc funclion P on th. left-hand side of eq, (3.12) is . ~p",sscd in term, of 6X ,i"", Ihis is the ,'ariable appearing on Ihe righl.hand side of the same ",pression_ Thu,. P represcnls Ihe pmbabilily diStribulion of Auclual;on. of the junelio" from the joint ""Oler of aClion of lhe nelwork and cntan!;lemenl 5_'111.. second I>'1rt of eq. (3.13) locales thc posilion of the joint centcr rdati"" to the average loxalion of ll>c j unetion in Ihe phanlom nClwork. Ina, much as th. conSlraint action is 'phcricall}' S)'mmelric in the undeformed stale. ". _ "0' and t"" v«.Or d X b«.-omes paralld 10 t_ S<>lving til<: ""'ond part of C
Ha ving define
I

!'(6K) x 6(ti.I')
(3.15)

The subscript astcrisk in the distrib~lion of.:l. X differentiates it from the a priori distribution /'(tiX) of vectors .olX obl.ined for the phantom n.lwork . In th. ref.rence Slate. the di,tribu.ion R. (6.X) 10 P(tiX ) and he"", is Gau"ian ;n thi' stale , From «t. (3.15). il fol1o"-, Ihat €I{.olX) i. al,o Gaussia n in Ihe referente sta tc. Finally. fI (x ) is Gaussian at alll.\,el, ofstr'din a<xording 10 lhe assumption .t"ted in eq, (3.7). 1·lenee. e
red....,.

(3, 16)

(3,17)

wbieh ca" be r.arranged to 1'/ (1., = I

+ (0' / 1')1(", / 1') (1"/"

_ I)

111 (1 + 0'/1')1

(l. 18)

THE CONSTRAINED-JU NCT IO N MOOEL

29

Compliance wit h th. requireme nt thai Ihe aCl ual distribulion R.(a.Y) should mlUet to R(aX) for the phantom model in the $tat. of ..fercnoc resullS in

OjJ ' _ p ' + o-.'

(3, 19)

Repla""me nt of /Ill" in eq , (3 , 18) by /11),1 ~ (~/'b )( 1 + (>/ 0-0)

(3 ,2(1)

lead. 10 (3,21)

where ,he dcfin;, ion (3 ,22)

has b«n adoptc-d. In the case of a phantom networ k. no constraint. operate on the jU1l<:tiooJ; IhU$. «(a.)'). i. infi"ildy large and ~ bocomes ZCro. 10 Ihe mher lim;t. constituted by t"" affine model. the nuctuali"n. of j unction s art f.ottn and «(Cu)')o equates In ,orn, leading tn an infinitely large~ . The parameler 1<, whid, rcpre""nts a mca.ure nf ent anglemenls of chain. with their ,urro un di ng' ill the .-.:al nelwnrk , is as,umod' 10 be propOrtional 10 the Flory number. namely. tile number of junetion. in the "olume ,,"upied by a gi"en j unet;on I""" eq, (3.3)]. Thus. (3.23)

where f i. the consl,nt ofpropOnionali ty, and I' i. the number of junct ions in Ihe volume 0' of the .tate of r.ferenc<:. For I
Yo here N A is A vogadw', number. d i. th' density. and M i. the mnlceula r weight of a chain who .. mean·square end·w-eoo dist~nce .. {,l)t ' Equation (3.24) rela tes the constraint parameter" to mo"","la r IMramele ... Since. for" lelnofun<:tiona! n<:\Yo·ork. Ihe cycle rank is rdu ted to tbe moi<'cul., weight M , of network chains

.,'

(3.2S) nne""", from eq, (3.24) that lhe " par.. m.ter increase, with the square rool of neh"'ork chain length . The distribution S, (l'l..r) of ,'""tor component' ~ ... in the deformed network i. giv<:n by Ihe ."n""lution of P(6X ) witk tlte dimibution of ,'ceIO" joi ning poi nt A 10 pO;nl R (""" figure ),3). De noting the .r COnlponenl~ of these ,·""IO. s by a~, one L1). ollm;n •. u,ing eq.

n

, , """"

---

(3.26)

30 STRUC TURES AND PR OPERTIES OF RU BBERLI KE NETWOR KS

The dimibution of tl.u i.the ... me as H {x). where x is related to tl.u by eq. (3.26) , Thus. S, (tl.x) _

I'(~X )

. 1I (x )

(3.27)

whe ... ~X .. i + tl.u. froon figure ) ,3. Equations (3.12) and (3.7) furnish expr(1sion. for I' and II, respecti,.. ly, Replacing in i eq. (3.27) by tl.u of eq. (3.26), cornp~ling the square in tl.u. a nd integrating over d ~u. leads to (l. 28)

{1 ,29}

3.3 The Elastic Ftee Energy ) ,3.1 Genera l Aspects The elastic f"", energy tl.A" of the network is t~ kon as the suon of the dastic free encrgy tl.A,. of the pha ntom net wo rk and tl. A, of the ron'ln. ints:

!loA" _ !loA,. ...

~A ,

(3 3(1)

The exprc.. ion for the dastic free energy of the phantom networ k model is gi''Cn by c'q, (2.49). Contribution, to the elastic free energy from oonmaint, resu lt from 1",0 effoct •. One i. from the ,pring·like action of the oonstraints o n the Huctuations 6.R of j unctions froon their ti me-avc"'ged position' in lhe network. The $OCOnd oontribution 10 the f"'" energy resulu from distortion of the domains of oonstraint . A""ording to the model, the domains of cons!",i nl are r.prtsente
EI.s~ie

Fr« Energy Du. to [)i,torlion of 6 R

The cakula lion of ~hc probability of I'j junctions having displ.«ments in ~"" range 6. :tj and AX, + 6(AXj } f"lIo",., lhe loa "" a rgum~nlS giwn in chap~CT 2 for the phan~om network. Thus. til" numher of configurations n ... x oon!istcnt .... i,h lhe distrihution V'j) of the I'j junctions is g;vcn by (3.31) He .... "'J i. Ih. a prion probabili ty <:>f <.>(>Cur",,,,,,, o f a d ispla,;"mcnt A Xj in Ihe r.nge 6(A Xj } and i. given by R(AX,} ~(AXJ) ' The: relative incidence of junctions in the Slated ra nge in tht deformed ottwork is " iii, ,, R. (AXi} ~( AXj)

(3,)2)

Using the c.' p",,,ions for R{A XJ} and R.{!lo XJ} from cqs, (.'.6) a nd (3.16), and ", king lhe loga rilhm of cq . (3.2 8). lead. In

THE CO NSTAAtNEO-J UNCTtO N MOOE l

In

n.-x '" (p / 2) In(p/p.. ) ~ (p ~ 1'•• ) L

p/(tlXJl'

'"' (p / 2}!1n(p/ p.. ) ~ {I'll' .. - 1)1

31

(l.ll)

The second line of eq. (3.33) is obtained by "sing tM u[,r=ion for tM ave"'g. fluctuation.: (3. 34) Using eq. (3.2l) for 1'/1' •• and the relation tlA /lx ~ ~kTlnn /lx. the t,[,res,ion for the elastic froc eJ\(Crg}' due 10 tluct uations of junctions from their mean 10Ct.ained as tlA M .. ~ j.ikT

L, 1°, ~ In (B, + III

(3.35)

(3.36)

3.3.3 The EIa,tic Free Energy 0 "" \0 Distortion of tl. The contributions tu the elastic free energy due to distortion of the constraint domains i, , imilarly obtained as

tlA /I, =

, ."

'2 ,.1<1 L. , ID,

- In(l), + I )J

(3.37)

where

D, " (u",lu •• ) - I .. ("~/ :":" .• ) - I _ ,,;..;( A; - I )(A;

+ "r ' =

:..; " . , 8,

(3 .38)

3.3. 4 The To,al ( Iaslic rm; Encrgy Combining "'Is. (3.H) and (3.37) leads to the total clastic free energy due to con.tnlin($. Combining the eX[,rcs$ion oblained in this man"", with eq. (2.49). re[,resenting the elastic free energy of the phantom ""twork mndd. leads to 'he 10Iai elastic Free energy for the constrained·junction model' tlA" -

4(kT L, {pI - I)"," (",/ ( )IB,

t D, - In (B, +

I) - In(D,

+ 1)1} (3.l9)

In the limit ",hert the Slrtngth of the conSl,aints ,·anish ... " _ O. 8, and D, equate to "'TO, and the clastic free encrgy given bv 00.. (3.39) reduces to that of lh. phantom netwo'~ modd. When" .... 00.

,

H, .. , '\', _ l ,

IJ,

0

(3.41))

32

STRUCT URES AND PR O PE RTIE S O F RU BBER li KE NETWORK S

and eq. (3.39) gives

I

,,.,

Ll.Arl-2 vkT(~'+~' -'- ~J-3)-

( .• 2,,/<». 1<1( In

V/ ~ ·)

(3 .41 )

which is ,he d..s,i. fr"" energy of ,he affine "",work The daslic rrtt energy or,he conSlrai ncd·junction mooel gi,,,,n hy «i (3 .4 1) i, "sed in chapler 5 for deri ving Ihe rda';onship betwttn 't",,1S and strain. The effect of the '" p"r~m.t.r on 'he m""hani.lI l properties of the nClwork is .1", ill"'Ir>, ted in
References (I) ROOClI. G ., A I]", ... , G. J . C~,,,, . /'10)'" 1915. 63. 49\10. (2) Flory. 1'. J. J. Clotnt. Phy •. \~11. 66. S120. (3) Erman, II : Flory. P. J. J. Clt"",. PhJ~. 1978 , 6&, 5)6).

(4) (S) (6) (7) (8) (9) (IO) ( 11) (t2)

(13) (14)

(15)

Hory, P. J.: EnII,n. II. M",,-""'{.. 1982. ".800 EnJI.n, II. : Flory. P. J. Mam",,,,",,,,{.. 1982. IS. 806. Flory. P. J. l'oJ,_, \97~, 20, 1317. Flory, P. J. IMI. 1'0/),"" J. I98S. 11. '16. Hory. P. J. Pol}"",. J. 1~85, 17. 1. Eiching<),,1'. J. Proc. 1(0, . S;x:. '"",don. A 1976, lSI. J~l.

4

Intermolecular Effects II. Constraints al ong Network Chains

In the ",n"ruined_junction ",ode) presenl~d ,,' chapler 3. intermolecular <: !Upprcss the fluctuations of junctions. Acoording 10 this model. the dastic rn.~ energy ofa network ,'aric-s between (he f,c<: energies oftM phantom and the ~n;nc n,,(worb, In a second &roup of mood •. 10 be introduced here. there is a con.training act inn of entanglements "long Ihe chains that may fUf lher contribute 10 Ihe ciastic free energy . ~, if they wen: additional (alhe" 'ernp<Jmry) jm>Cliun •. Con ... quenll y. Ihe upper ""und of the elastic f'ee !;lle'S)' of such networks m;oy . xC«d thai of an affine ncl",·urk. SillCC the enlangj<:mcms along the ch.in wntour arc •• plicitly taken inlo account in Ihe modds. thO!}' no rcfc""ecom. peml .... nlly tra pped by lhe cross_linking and act as addilional cross_links. The .. trapped entanglements. unlike Ihe chemical cro,;s-links, ha,·. some freedom, and the t",·orh.in. forming the emangicmenl m.y .Iide rci,ui"e 10 one ol her. The 1",'0 chai", may Iherefore be rcgardc-d as bcillS allached 10 each olher by mean, of a ftClitious "slip-link: ' as is illu,tmled !lChemalically in figure 4.1. The enlangled 'ysltm of chains represenling the real , il""tiolt " ,ho""n in part (a). "nd the reprc'SCnla, ion of Iwo entangled chain, in Ihi, system joined logether by a sl iplink is shown in pari (b). The 'lip-lin k may mow along the chain, by a distance Q. which is invOnio n,,1 10 the sc,..,rily of the en'anglNn.,,'" A modd based on Ihi' picture of 'lip-link' wa. fi,..1 proposed b}' Ora ... ley'. and. more rig<mJu, I,ealnl~nl of the , lip-lin k model was giw n by Ball el a i, ' and .ubse_ q""nlly oimplified by Edw,rd. and Vilgis", SOrroach,,'_ Ihe diflusc:d-<:Qllstrailll.< model. is lhen dc""rii>cd brictly ;n >Celion 4.} Finally. "",'lion 4.4 d~""ribc , I'arious "Iher I",alments of entanglements.

3( STRUCTURES AND PROPERTIES OF RUBBER LI KE NETWORK S

,., ~i~... "-I (0) An onl.ngk e" .. ngle~ chain, joit1«l by ••l i~link. The ,lip-lin k may move alo"S lhe chai.. by a d ; 'la.-.co~. whkh i, i.'...... ly propotlion.ll0 Ihe "" .. rily of lbe en"'"gIome"" .

Two ex,dlenl ",vicw aMkles on this IOpic. in general. arc by Heinrich el al.·. and by Edwards and Vilgisio. 4.1 The Slip-link Model The probability I'[II (,}[ of the end·l
He .... . ¥ is a no'm~li"'1ion oon"anl. It i'1he posilion vee10r ill space.• is the are lengt h alonJ? 1he chain oonlour (varying b<:t""ttn ~ero and the chain conlour lenglh 1-). [1I(,lI '" ll ll (.l / Ds. and I is Ihe Kuhn lenglh (Ihe lenglh of a bond in an e(juivalcm freely jo;nted chain), Equa tion (4, I) gi'-es the dislribulion fU!>Clion for a Gau .. ian chain, A sl ip-lin k. as shown in figu r. 4.1(b). may be visu;±liwj as a slidi ng CrOSS' lin k Ihal join. a point of Ihe ilh ne1work ch. in 10 • poinl of Ihe jlh nelwork chain . This condilion is cxpm$«l as (4.2)

where II, Mnotes Ih. f>O'il;On """lor of It.. slip-lillk on Ihe 'Ih .hain. The quant ily f>O'i tion along It.. ,)h ~hain con lou r, wilh the sUPCl"I(;ript J indicaling tha1lhi. poin\ i,common 10 1he slip.tin k pos;li'>n on lbejlh ~hain, The

1, loca te, the slip- link CO"Slr-d im

n

--"

~llI.bD

-

II,(..{ )]

(4. 3)

appears in 1he network pOrlit;on funclion if Ihe slip_link s are frottn. as i. Ihe caS<: for Ihe I"'rrnancnl junclion. or .ms.·links. Il ere. 6(.~ ) represents lhe Dirac runclion . which is if x I- O. and (Xl if x ~ (I, If a link i, not permanent bm slides by

'"<0

CONSTRAINTS ALONG NETWOR K: CHAINS

35

a di,tance ±a along th. contours of Ihe 11'10 chains which il joins. then the constraint appearing in the product of eq . (4.3) becomes

-b L Ld~Jt[N,{,,, + ~,) df:,

R/';:

~~j)l

(4.4)

Tbese conSlrainli are bui ll inlo lhe lheory. by Ocam and Edwards" for crosslinks and by Bal! el al ' for slip-linb. by mean, of lhe replica fonnalism of Slalislical mechanics, As 'Iatoo by the aUlhors. 10 build Ihese constrainlS inlO Ihe Iheory is a difficult task and Ihe replica melhod has serious algebraic compltx_ ily. Subsequenl lre~lm.nl ofth. problem by Ed ....ards and Vilgi,· b)'passed these complexities by introducing Ill<: sl ip-link picture into lhe Iheory using Ih. Floryby eq (2.8) is wrinen Wall affine model. The Gaussian distribution funclion for a f=ly joimoo chain as

gi''''"

( ')'"

2JL ("')

.~p -

pe R. L ) ., hlL

(4.5)

Equalion (4.S) is oblained by Ihe Subslilution or Ihe relalion fo r a freely jointed chain:

(H ) into eq . (2.8). Hel\". L = ,,/ denOles Ibe contour Itnglh oflhe chain. The Gaus.si.n e'plession given by <"
,

P( R) ..

n

f'(X ;)

(4.7)

"

where

(4.8) For brevity. Ihe contour lenglh L is nol wril1en in Ihe argnment of P in eq. (4.1). The sl ip-link P'''''''''' is modd<:
(21 2I{1.)Xi+t)Lf1 )

u p P{X,) .. fd! p(!)

[,U H)]

(4.9 )

Here. £ i. the ,lipp"lIe Itnglh and P( ~) ;$ the pcobabi\ily of Ihe arc length of lhc .Uppage. As.suming Pic) 10 be uniform along lhe slippage knglh, Ihal is_

,

P(.} ~ ,

"

(4 . 10)

36 fq

STRUC TURE S AND PROPERTIES OF RUBBERLIKE NETWORKS

(4.9) may be writtcn

[ ", I n/

,

P(X,) ", _. 2a~xPl - 2/(L + ~) - 2 In (L 1<:)

)

(4_ 11 )

Thi s integral should be normaliz~d for every ,-a luc of f, This is n<>l easy w~n tile complete expression giyon by "'I. (4.11) i. used. For further progress. the imcgr<md i~ .xpa nd<:
["' i ' ,'-, ,[ (') (' )'1 - e,p - - L.A, X,. 1 _.2a 21f. i_ l

-

I.

+ -

I.

(4.1 2)

-l[(fH(f)'J) where,Y is the nOmlalinl1ion consl,,,1. In order to perform the inlcgr~tion o'-er t . lh. expon~nt in « 1_ (4_1 2) is further . xpand<:
5

- '2

, ts ) ( ) {;>.;x, J 211.

~"

(4_13)

J5

Hcn:, ;I denotes the """ond momenl of t defined by (4. 14) The free energy A(A) of the network at the state of dcform~I;On A is evaluated accordi ng 10 lhe expre,"ion

J

A(A) ~ - kT Jl RP(R, L) In P(~R . L)

(4.15)

The . xpres,;"" f<>r the Helmhoh froc energy givcn by "'I. (4. 15) results from the replica approach. which has been . ummarized by Vilgi. ". The argumentS are based on the ditTe",,,,,, betwcco ao un-cross-linked polymer melt and a cron-li nked one . The former is treated as an ·· annea!<:
CONSTRAINTS ALONG NETWOR K CHAINS :P

A,(Al/kT " !LAi , - 8f.;" " ~ i. '

(( LA} ' )')~ 2 ('L~:)' -lo L, ~;+ 15 (~i

,. ,

,. ,

(4.16) H."" lhe subscTipl ~ denotes COII$lrainll. This cxpttSsio" ohtain<>
a.

A,{.I) "" _,I N,U ·t ,~ ,

f .11(1+"~{) ~ ~) + In (1 + ~~;» ) 1(1

(4.17)

(4.18) A~ sho""n in chapler 2. the daSli~ frcc C"trgy of the phanlom raD k of ( is

n ~lwork

",'it h a qde

(4.19) T he 101.1 clastic tia"ic frcc energy is given as lhe . um of «I', (4.1 7) and (4.19): ~A '" ~A ...

+ A<

+',>S)] )

.. ! {ut i (>J _ I) + N, [ ~i -

i_,l

2

~

'--- + In (I ( 1 + ~~1l (1+.,)

(4.20)

Comparison of "'I. (4.2() Wilh "'I. (3.39) .how! thai the d(formalion depen. dences of the tlH' c>. pr=ion s are qillllilali~cly ,"cry similar. If one idemifies 'he 'I parameler of lhe slip-link model wi,h ,,- ', "'I . (4 20) may he wriuen' )

t.

~A ~ ~ (kT {(.I.; - I ) + ';' 18; + 0 , -

In( 8, + I),

+ J) -

In

.l.1J}

(4.21 )

wher. H, and D i are given by ""l_ I )(A}+ .(') I

(4.22)

AI$<>, eq . (4.2 1) r«luctS '0 lho: [>I'anlom nelwork result ,,·h.n N, becomes :.crO. Il o"",,,..,r. 6A of «I. (4.21) m ~ y become indefini tely large as the number of . Iif)links incrtasc<. ""hcreas th. clastic free energy obl"ined by the con"rain<> "
38 STRUCTURES AND PROPER TIES OF RUBBERLI KE NETWOR KS

4.2 The Constra ined-Chain Model In the con"ra inoo-junctioll model of rubber elasticity described in the pre_ious thaplcr. intermolecular crfecls are K\.$umcd !O act c.clusively at the junctions. Thi, a!iSumplion .implifics the way .hat (onstrai nt. are introduced in,n the for_ mu lation. How."",. the magnitudes of junction O""IUalion$ obtained from the neutron spin-«ho le of this. However. this type of generalization immediately leads to S(vtral questions. The constrained·junction modd ... umC$ a oo n",.int center . Can one introduce analogous constraint cenlers for points along the chain contour? If this can be done. how many of I~ e<>nstraint centers .hould thero be? An upper bound for the number of e<>nstrainl centers for a chain conlour may be taken to be M,/M,. where M, is Ihe molc<:ula r "'eight between Iwo entanglenlent points"·", Unfortunately. this point has remained conlro,,,,.ial e"er . ince lhe introduction of the constr4int Or entanglement concept into Ih. arca of rubber elasticity _Edwards and Vitgii h."" ta k~h the most conWtlCtivc step in this direclion by e<>llsidering a 'ingle 'lip_link th~t may slide along the chain e<>nlour. On the other hand. the eonstrained..,h.in model introduced by Efltl.n and Monnorie'~ proceeds along a different line. according 10 which the constraints operate on the ma .. center of each chain, Thi ' section outlinos the Efltlan· Monnerie theof)'_ [n figure 4.2. the instantaneous !'Osition of a rlCtwor~ chain is sbown and the vari.ble. used in the oonstraincd-chain modd arc identiflCd_ The lint A ll denolc, the instantaneous configuration of a network chain, and ii. and Ii arc the time· averaged 1)O>'itions ofth. junctions A and II, Points G and D locate the instanta. neo". ma.. center of the chain "nd of the center of constraints operating on Ihe

.1j; " ~

4.1 Th<: ;o>lant.nw.. I'<"i,,,,o or. ",, 'work o"'in .nd ,..., .... ria""" u.... in

con",·.io>ed..,hain m.xIcl, So:<: tiIC ",,' rOt ,iIC deti"it ion. of tile ,,,,;able> ,hQ"'' _

\~

CONSTRAIN TS ALO NG NETWOR K CHAI NS 39

chain . respectively. T he lime_ave. aged posilion of Ih< mass cenler in Ihe phan!om network is at poiol O. and point C is the timc-~'·.ragw posilion of the mass cenler under Ih. joint aelion of the phantom network :. nd the const.ainlS . T he qu.antitit$ a ... and b.s.;; are th. instantaneous """Iuation ,"<>CIon of the rna .. cenler from 0 and D. respectivdy. and 6Rc is tl>e /luctuation VOXto. from the joint center C. The =to<$ b. R.. and t" l<)rotc lhe posi tions of Ih. joint cenle"]" C and Ihe oonstrain! center D .•espe<:lively, from O . The nolation used in lhe derivalion of Ihe elaslic free energy fo r Ihe oonslrained-chain model follows Ihm of the constrained·junction model deseribed in th. p,""vious chapter. The distributions of the variables b.",. "0. !Is,;, a RG • and 6... . ,"" .ssumed to be Gaussian . In termS of xoomponents. they are given by

dl

R(b.XG) '" (1', )/ ~)lll .'pl- I',( AX 1f("G) = (7]'/ " jl l l uP[- 7],(.i " j'l S( a .
(<,./.-j'

,

2",p[_ u, (Ax.,)ll (9,/~ ) III • • p( - O, (b. X-Gj 'I

p(H~.J "" [(P,

(4.2)

+ ",)f"I'1l expl- {p. + o, )(6Xdl

1', - 1/ {2{(AX,, )' )) ' I> = 1/ {2{( x,,)l ))

(4.24)

" , " J/ {2{(il.
8, = 1f{2{( axd )) The subscript .\ inlpli"" that these variables an: generally funClions of deforma· tion. f ormulal ion of the elast ic free energy follows essentially the same steps tahn in ,he derivation of Ihe constrainw.junclion mod.I ". Tht rarumder " G' re/le<;ting the ",verily of constraints. is give" by lhe relationship (4.25) which i•• imilar to thai of Ihe constrained-junelion model. [)efmed in this manner. "" becomes zero when the conslrainl~ do not operate along the Chain COnlour and (( Ax,,)l ). is infinitely la rge. When Ihe conmainlS are infinitely strong. «(AxG)l)o equates 10 »:ro a nd "" then bc<:om<$ infin itely k"g<:. The ralio <7,17], or the con'trained·junetion model" is oblained fur Ihe con· slrained-ehain model as (4.26 )

where (4.27)

~o

STRUCTURES AND PROPERTIES OF RUBBERLI KE NETWOR KS

Th~ paramOlcr in

«I. (4.26) is gi"~n by Erman and Mon""rie' as

(428) wl>e functions 8~ and D, characteri,jng tl>< ela'tic free e""rgy are given by

_1 8< =1!.!. -

,,'

(4, 29)

(430) Function, for the other t",·o direction, are similarly defined. Following tl>< argu· rnenlS in dcri,-ing Ihe elastic f..., energy of the network in the ooostrained-junclion nl
,

+ (v/(){B, + /), -

In(8,

+ I) -

In ( /), + IJI)

(4 31) It should be noted that the contribution of constraints to LiA" i, proportional to 'he numller v of chains. whereas in Ihe constrained·junetion mood it is proportional 10 the number of junetion"

4.3 Th e Diffused·Constraints Model This thoory improVC$ the con'trained-ch~in modd by distributing the con,lraints continuously along lhe chain OOlllOlIr. rather than a l the chain mass cente,.>". In it> applicalion 10 ,lrc __ train isotherm. in elongation"'. it has the advantage of having only a single constraint parameter. and the \'alllC< it e.
t" n. """..1< Ikfi", """ ror ,"" .. 1""'''''''«' ... >«l on • m"",6«J r""" of ,.. «",,'r,,0<\1 , •• ,n.""""". " . 1...

~ "'"

;n

ITr,-",,~,- ~ .

I I I

I

--'

-

!

, ,,,,,",,,

..,

-----_ .. ---

I

........... ,.,

I _

_

,

...... Ion i . '

FiR. " 4..J Some "'e'<~ '''''wing dillor<..,.. in lbe

,o. con",..jn«l-J"""lIon '''''''ry, ""n"raincd-cl\>in

"'''Y lilt onnSlroin l$ arc .pplied in

,Itro'}', ~"'!
ory. ""pectivciy_ 1\ pos.>ihle fu,,"" r<~ ... mcnl in p".ilioning. b.>OTI.l " .... rimental ""ideo",,> i. ill~'l""ct1 in ,he bo Ll om graph

.2 S TflUCTUFIES AND PROPERTIES OF Ruee EfH IK E NETWORKS

suggesling how 10 position llle conmainlS along a chain in refining such model,. :;ine<: '" should ha>'C a p",nounccd .O'<:<:t on the magnitudes or the n""luation, of lhe jUl\Clions. A possible further n:fincrnent in positioning based 011 additional •.'perirnenlal ovide""" is illustrated in thc lowest graph of figu~ 4.3. 4.4 Other Treatmen ts of Enta nglemetlls

The "lip-link mod.1 i. <:$Semially an o.' ionsion of th. more gencraltubc modd of entangled polymer system,. This tube modcl'o is ~ oo,wenie", t<>ol. <sfJt"'ially for predicting lhe viscoela!lic beh.vior of polymer mclts~I.l·. and it has been c~tcnded to the st ud y of networh by >c"eral authors"'"'''''' Heinrich Cl a1" for example. r<:pr<:scntcd the tube as a harmonic pipe and deri .....-d the correspond· ing slres,---'Slrain ",Iat;ons for a rubbcr. and G ...l.... ley and !)os, in'" ".tended the Doi·F..dward" lheory to study thc viscoelastic bchavior o f netwo,ks. Ga)'lord" and Marucci " used alternative forms of the tube model to dcri>'C the Str<:l>S- stra in beh",'io, ror "",,,·orks. References (I) Langley. N. R. MIN"""""""I., 1'.1611. I. 34>1. (2 1 I.:.mgky. N R.: Polrno"ker. K E. 1. l'oI}"t", Sci.. PoI,'m. I '~,.• . f:d. 1974. t2, 1(2). 0) Graessley. W. W. Ad". Pol>,'"' Sri 1974. 16. 1 (4) Gra""ley. W. W. Ad•. 1'1)/),"" Sci. tY82. 47. ~7 . I~) 11"11. 11.. C .; Dci. M .• r:dv.'ord,. S. I' .: W.tn01. M. PuI)'",,, 1981. 22. IOltl I~) M .....d... S. F.: Vilgi •. T. II . PoI)'_r 19~6. 27. 48 3. 17) Erma". II .. M on..... ic. L. M"",,,,,,ol,,,ult. 1989. 22. 3342. (8) Erm.n. II.: Monnen.. L .\I ",,""""<'WI~. 19\12, 1'.4456. (9) lleinrich. G.: S'ra8be. E.; fl.I",;" G . lid<. I'''',·m. Se;, 19S8. 8S. 33. (10) &\ ....,d •. S. F.: Vit" •. T.II. R,p. Pr'>t. Phy•. t988. St. 24). (11) o.:a01. R,. T.: fO<>dcl\Ce. (1 4) 1Odv.'.rd •• s. I'. J. Phy" C 1%9. 2. I. (15) Il<>t:. ( I..,.u}oo) 1%7.92.9. (16) Il<>t:. 1970.49.43. (11) F.rry. J. ]). l'iKocl",'ir Prop''''' '' of Pol),,,,,,,,, 3, d ro Wi ley: New York. 1980. (22) Fette". L J .. Lohse. I). I.; Colby. R. fl . In Ph,..it:aI I'rop'''~' of PoI!'m", 11_ . J. E. Mark. Ed, IImenc:." , In>1it"1< of Phyok$ Pre": Woodbury. NY, 1996: 1'. 33S (ll) l'lory. l'. I, : Emlo". II, MQc"'mokro~" t982. IS. .wo, (24) Kloc>ko",·.ki. II.; M. ,k. J. E.: Eno.n. 11 , .If <>Ok. J, I:. M" r~. (d, Amcrk •• Institute or Ph y,,,,, lI'oodb",y, ;""y , 19\16: p. 341.

1'"""

CON STRAINTS ALO NG NETWOR K CHA INS (11) (28) (29) (30) (31) (J2)

Vilgil. T. A, p,,'1, Coil Pm"",. &1. 1987, 75, 4 vi lgis, T. A,; II.i~,i< h. O . ,f"l~". Mah "moi. Chem. 1992, 202[lfJ3. 243 Wagn... M. II.; Scha.ffor. J. J, fI~"'. 1'!'93. )1. 64). O ...... )ey. W. W.: l><»>io. t. M. 101",,,,,,,,,",,,,1,, 1979. 12. 123. Oa )'i<>,d. 11. , J. Poi" .... 8"11. 198),9.1 8 1. Ma rucci, G. M"",~~I" 1981. 14. 434

43

5 Relationsh ips between Stress and Strain

In thc first sectinn of Ihis chapler. the rciali,,"shiJ)S bet,..",," the Helmh<>lv f"", energy. thc ,\""" tensor. and th. deformalion tenso,'.1 an: &i"en for uni .. ;al

stress. These rdations folio,,' from Ihe general discussion <>f meSS and .Irain give" in apP"",lix C. and thc notation and approach d~ly follow {he classic treatmenl of Flory' , T he detailed forms of th. Sl~train relation, in simpl. ICnsion (or comp",ssionj arc given in Ihe renUlining sections of Ih. chapter for Ih. (1 ) phantom network. (2) afflno

~twor k .

(3) cuns\ rain«l-junClion modd, and (4)

slip-link model. Results of Iheo,y are then compar"" wilh expcrimen!. The drccn of swelling On the Sl",ss - sl r~in relatium are also included in th. di s<us.sion. 11 is to be nOloo thai thc stress-'Irain relations in lhis chaplcr arc oblain<Xl by lrealing the swollen netv.-orks as closed systems, The conditions for sucb s},stems are fulfilled if solvenl docs nO! mo.'C ;n and "ul of lhe nel"'ork during deform"I;On A nelwork ,wo!len wilh a nonvolalile oolvenl and m. of Ih • .ol"onl und.r incroased inlernal pressure', and is therefore not a closed system, For ",miopcn 5)'stems, ,uch a, those under oomp
5.1 General Relationships of Finite Elasticitv Theory The
(5.1 )

whtre 8A' ( 1". V ) i, a function oflemperalur. rand volume V, and denotes lhe «mlribUlion from inlermol""ular fo""" such a. tho"" in simple liquid. _ The quanlity tlA,.;( 7".1. ) is the elaslic frcc energy .ri,;ng from the el.sticily of Ihe network chain,_ The componcn. !lA ,.;( r ,l ) in eq (5.1) Wa< dcri,·,:J in Ch"Plcl'l

.

(n)

\0 (-""I've '

__ _ =1 , _ ', , ;W) (" (")" ". l

0 --_ "

I've'

:', - " X>U""~ll! 1' ~'11 S" :>Jo''':)Jd ~!1~4d.o\ul¥ '" ,,"'!'UI:)..I llO!I:>:I,III' IX ~41 hOI" ~ "".I)S ~ql ~u!l'P >:ud "'''4d,o\Ul~ :>;lU:XJ;X!YO 11'''' pu~ IX SUO,,=!P 1""1"1 "-''II 'lIU!I'OOI I~! x ","n u, ""'OJ "'11 JO UO! ,:>:J.J'P "4"" U,~"' S! 'x Jl '(u"!'U.ld """, 10) UO!'U'll"!X,,!un JO "InI' ~41 UO 'JOld"4~ '!'11 JO "('UIUtu:U ~41 U! UO'SS1\:.;!P " 41 AJ'ldwlS 0) JOp.!o U I

"'I '

'x

'"""J ""

(on)

,', [(..BL) '" _(..BL) .,,[ ;i .. "_ ""VIl ' "I've ' 1:

I,

,~t»S=dx •.: , 'UOIl"1"" JOOI"",L "'II 0) '1'"'1 SHU "\lOI,,,,,,!P 11ld,"uud "M, SU(>IU 'I JQ :>;>u"'~J.I!P '41 l!U! ~ "l Aq (n:) 'Ix> wO'J .Ii 'ICU1wlP ACw ouo -'P!ob" Oldwo, u, p;>U'Ulqo ""''II '" SU01P""\UI '"I .n""lou ... ,),,! WO'} "'''sRld :)ql 01 uO!lnq!JIUOJ "'II ~ (;le! YV (}) - '" ,Ii :>.JJq."

i,

'I""

0 = '1 (-=)..i...+ "'live :''1: .

~'l

I

(n )

s, IlnS'l1 OI[J. -J x!pu:>ddu Of p.:>.'!'~P s! J!nuo »IJ "I'~P '41 0 1 P:>1"pJ :>J. lOSUOI '=1< '41 J" SlllOUOOw"J lW!"""d "'ll .II!." '4.1

'APA!p;>
'IU=":xl~"

All"

"!I'"]" 041 !u!Jnp puo · ~Jo."I"U)O UO!I"IUJOJ !ulJnp JOW.'lud JO "011"".1) :>Otn]""' "'ll "'n <. pU~ ><. >'l!I!lu.nb :I\U. ·own lo., I~uy "41 IU P.""S •.'W 4Il1u,] p.POI "'!pun "'II 01 UO!W
(n)

'0

(~) " 'n(ll(oAIA ) " JI(

",

01 <"'J! ldw,i ([ ',) 'il:> 'WJoJ IU""<>dUtn~ Uj ·1<.>5UOI UO!""I lOP P"E ' ~JO"'IOU "Il JO UO!I"W'Oj o4111U!JOP ,wniOA '41 S! 011 'OJ,,,

" rI,(oA I II ) '" r

(n:)

:,wnIOA IU~I'UO:> I" UO!1l0,,!P" ptlv. UOII"PP :>Jnd "jO pnpo,d '41 n J»<S",du "'I AOW I( "iJl!""4J ownloA wO'J ,,"!lJOI< 01 J,PJO ul '('IUI' j»w.JoJopun '41 tI! 1"41 <>l 01"1~ p:>uuoJOP ~41 U! 'x iroo!" w, ,,d "J""P!' '41 JU 41i1U"I "41)0 011'" '41 An"'yt:l:x!S)
<, " o

In )

o " o "

I)

I '" r

"

~ls.(S

"1'. {"-!"- '''-o

'Ir,U!I"OOO " U! UO!I"UllOj"P , "oouollowoq JO ' IU'UOOWO, IW1,uud "41 "'J p:>u~op 'JO'OU01 1U:)!P"'~ (UO!I"w.J0Pl' .10) IUOW"""ld'lP ",,"I sl u~;).Jd:)J I( IU;>W "nllJ~ '41 '([' .)"!>:I Uj ·' ~JO."IOU '!'~W(l1S1lj~J" sprow IU;).J '.U 'P JOJ t ~lInOJ41 t

St

NI""~lS

ON.".

SS3~lS

N33Ml38

SdIHSNOI 1""'3~

46 STRUCTUReS AND PROPERTIES OF RUIIIIERLI KE NETWORKS

ExperirrK.'1ltal data for w~-ss-slrain experiment' are flf eq. (5.8) is obtained hy ... ing Ihe rcl~tion AJ/ A = _ill". which follows fmm eq . (5.4). In the limit of small 0, the rcduec<J foltt IFI of eq. (5.8) c~n be related 10 the elastic mexlulu, /:.' or the .hea r modulus G. The dastie mexlulu, can be obtained from Hoo ke's law, U '" f.t. ",he", Ihe cnginoxring .trest " i. con'·cnlion.lly ~fined as the foltt per unit area measured at the beginning of the e'perimen!, and . is the infinitesimal strain (defined as the change in length per unit unde· formed length). The inilial stale of lhe sample is defined as the swollen but mochanically undeformed st.te. A. Slated earlier, elastomers are ~ppro. imately iDCOmpres
1/1

~"'~ k , '" Ovj'/l

V ' ) '"

(5.9)

G - [1").;"

5.2 Stress- Strain Relations for the Phantom and Affin e Network Models under Uniaxial Stress The clastic free energies for the phantom .nd amne network models an' given by eq •. (2.49) and (2.51). respeeti'·ely. Dilferemialing the .. expression. with respeet to.l.} and sOb~litutin8 into «I. (5.7) kads to the Stress for the phantom and affine network model. in
(5. 10)

Affine

IF) "

"T, ;.<'" ~of

1 (ot>

Phamom

{kT '"

2) v~

.".

(5.11)

Am ....

It folloW! from "'I . (5. 11 ) that the reduced force, or equi",... lcmly the clastic modulus. is independent of ddormation and dcgrtt of .",-el1ing fu r both the phantom and the .ffine ne\""ork model •. SubstilOling eq. (5.11) into eq. (5.9)

RElATIONSHIPS BETWE EN STRESS AND STRAIN

n

. hows that the shear modulus of the phantom and the affi ne nctwOfls dccreases with increasing swelling. 5.3 Stress-Strain Relations for the Constrained.Junction Model under UniaKial Stress The lotal elutic frcc encrgy of Ihc eon,(rained-jun
8:,1" ~ {kT [I+ ~ K(';)]

( 5.12)

=

where

,

'

K (;I., ) ~ [B,8,(8 , + I ) '" D,D, (D,

+ 1)[

(5.13 )

wilh

(5. 14)

OD, . D.. ' ", _ "- " ("8, + 11, )

'.

Subslituting eqs. ( 5. 12) and (5 . 13) into «I. ( 5.7) I•• ds w the following expression for tile true suess;

", ) "'(0 ,- - 0 -,+1' <,"'( "« [0 K (;I., ) (i )"

T=-VO

"

( 5. 15)

Tho roduc«l foro: is obtained by u,ing eq . (5 . I5): (5.1 6)

When 1M constr aint parameter ~ beco""", '''TO, the runClion in tile sq uare brack... of «1. (5.1 6) becomes uro and Ihe red""cd fo"", equate> ' 0 ,h . , of 'he phantom networ' . In the other extreme. when" approachc< inflnity, the term in the braces "'Iualcs to I +J'/{ and the reduced fon:e becomes equa l to that of the .. mnc nell,'ork The contribut ion VO l, of 'he ~"{)II$lmintS rebli,'c 10 lb., of 'he phanlom "",wo, k. 11"1.... may ho; $tcn rmm 1he ratio

46 STRUCTURES AND PROPE RT IES DF RUBBERLI KE NETWOR KS

,

• -"

:: 05

•

•

,.

0.'

lI a .,..", '5.1 Th. d
"'I. (S. J1)

on (t ood "" V! V. fQr

~.)~ = Z [<> K(>'iJ - " - lK(>.l))(a _

(I

- I) ,

(5.1 7)

In "'I. (5.17). the efTect of ."'<'Iling i. gi'-cn in t~e argu,,,,,nt. >.1 and >.j of "ph The dependence of the ralio given in cq. (5. 17) On (I ~nd On Vl v" is ,Ito",·" in figure 5.1 r<)r" .. 10.' In lhi' figure. lh. ordinate val ues of uoilY and '''0 COrrt'pond 10 the affine and phantom network model •. rc'PI"'lh-cly. The curv~, mo'·e lOward the phanlom network ,·al.es ",;th increasing values of V/ v". in ~g.rccmen1 with lhe postulate of the C<)n.,r..ined_junclion tht!ory that ,welling de"Cre."", the e!Teets of C<)n'lraint. on jUnc1ion /luclO"tion' . Similarly. s,n;,~~ing tbe network doxrtaS($ the CQ".trainl. alo"g that dir«liQn and the rcd","'Cd fut1X approacht!s ,hal of lhe phantom network model (as ubserved from the ,,- ' _ 0 intc"."cpt~ of the ""tv ..).

5. 4 Stress- Strain Relations for the Slip-Link and Const ra inedChain Models under Unia~ial Stress ~.4.1

The Slip·lin k Model

The total cla.t;'; free energy of lhe . lip-link modd is given as the ' um of the phantom clastic free energy and the free .""fgy due to the .Iip.linh. !J.A"., !J.lI... + !J.lI, . Following I~. same prOttdurc fur oblaining the K(>.I } function outlined in the previou, se<;li'>n. onc obtains

K(>.' ) '" '

[(8, + D,)( Ii, ~ b ,) - >. 'I 8 , _/), + 1

'

(5.18 )

RelATIO NSHIPS BETWEEN STRESS AND STRA IN .9

wilh

( S.19)

The Sire... Ihe reduced fr,..:". and Ihe rJlio tFl/If·I"" all follow from nj<. (5. IS)... (S.11) by rcpladng I' wilh N,. 5.4.2 The Constrained_C hain Model

T he 101.1 el"lic free energy for Ihe oonslrained-ch. in model i. defined in chapkr 4 . Di[fercnliating II>c energy "ilb respect 10 \ ; yield. the K(Afj function as

K(A') .. 8 ,B, , t ~ B,

+ .!!.,IJ,

(S.W)

I + D,

where

.aB,( , . am,u, [, h(A,l

I

B, il aA; " S, (A, - I )

" 6

[~ - h(A,ll"')

+ h(A, ) [AI + "'(A,l]

D,"!i'i1' = fl,

(5 .21)

( 5.22)

5.5 Comparison of Stress- Strain Relati ons with Ex perimental Data T he l-onSlrained _juneti<m modd ha. bcen very .uCC<;S$f ul in pTCdie.ing the 5"es.sirain ",Ialions of c laSlOme ... The slir>-lin k modd has oot been leMed e",ensi.",ly against c~perimental ,tala but one ,1I"uid e-<,rimcntal dala . Some comparisons "f Ihe pTCdicli"ns "f Ih. conslraineecau .. ..,f "",in·induced crysla ll i,ation. and Ibe ""uli. al small deformalion, may be: susceplible 10 large o"or>. Gi,,,,, Ihcse complicalion, . lh. lhrorelical curve gi .... a n:marhbly good aocounl of Ihe c, [><,rimelllal result<. o."r an imp"""ivci )" wide r"ngc of dongalion and c" mpre" ion. Some results cmph;"i,ing lhe d Tcet< of . "",Hing on Sln:,,-. lraln isotherms in ~l "ngation an: shown in figure 5.3". All of Ihe cur".. ,hown are for Ihc SCI o f

"""It.

'0

••

••• •., •• 0

j.

•

•

' .6

• •

.,

.,

,

1.0 0 ~';gur~ 5.l

•

•

,

.,

"

Ros.l .. of Ri,-lin '00 Saunders" for nalu .. ' ,ub", vukan ,=1 .,itb •• Ifu, .

• bown by ,he d.," poinl<. in COIn!",,",," ... i,h II>< oon"raincd'j""",,,wn ,heo,et",,1 calculaled ",i,h ~ _ 12 . nd! cn"",," to lfl - I.OS kscm- ' at ),, ' .. 0.'

,i,..,

,,

•,

0.25

•

!

• •

• •

0.20

~:~~"'."A"!'

. . . ' ••••

0

<

. '.00•

•••0.11 r-

Z ~

" N.,ural RuM"" Swo llen with n-Oecane

""rYe

0.2 4

0.15

,.,

" r"""" .,.

,.

.

"

'8

.,

•1J:u~ S.l RO<Joced f""Cli"" of " 'r(}, "a,u,a' rubber , ...olkn t" , .. 'yill$ dcg><es ,.-j,b n_,I<""'I><', The poi"!> "1',,,..nl ",,,,,rimenla' ""u l" of Allen t1 aI.' f<>r $P<_

,"i,

<.""""

oin>3r>mctcr ch"",-" from amne 'ran,f" rma,ion, of """'" • prn>,; pt"b;obilil y f.''''''on. on 'he '''''''.y ''. Rep"n,." ...·;'h !""on; ,,;,," f.om ~,,""n. It, .n~ 1' 10')" p. J. (1'1112). ,1/",,,,,,,,,1<"'010>, 15 . Co!,),';g)" 1997 A,nori
50.",,,,,),,

w.".

RELATIO NSH I PS BETWEE N ST RE SS A ND STRAI N 51 par~ mc\c,S indic~tcd , ",uh th~ t~~"'Y p",viding the chang~s in 'he i;"thcnns ",;th in:0). E,cept at the lo",est "alue of . , . Ihe dq>endcncc of Sire" on >tr~in. and the effcct of ',","lIing 0'" this dependence. arc secn t" be reproduced by Ihe lhcory, Fi~ure 5.4 c"mparc. mulls from the con>I ..~i"ed·junctiOll Ihco,y with ".peri"",ntul resuhs· r"r muhia_.ial ,Ial", uf ~lr"in', The Ih"My is Sl:(:n 10 d"sely appruxima lc Ihe rxpc,imental rdaliunshlp' amung Ihe sp...><:ifi<.-d cla.lic f'e<: enc,gi<"S and Ihe principal extensiun ralius, Use of the Mooney_Fl i"lin relationship' wou ld gi,'e Ihe huriwnlal dashed lone ,huwn al Ihe lop uf the figure_ I'hanlo,n nel,,'urk re,uI1' wou ld al"" give" hor;zolll"II;",," Ihal ",,,ul,1 make li>c", eq ually "ns:uisf:ICto,y fo, c, p l ~inin g Ihc •• perin",nla' obsc,,'aliuns, S""", comparisons i,,"olving Ihc "onSI,,~i"cd-<:h"in Ihc"ry" arc giwn in figure 5,5"_ In thi, figure ,l hc mluccd rore.: fu, a "",urdl rubber nctwOfk;s .hown as ~ fun,;,ion of Ihe invers
""II

..

, •

~

...

-•" .. Z

~

•

"• . _

~2

•

•

•;; '.0 "•

• •

,

•

•

•

•

•

,

.ll'l'" s.~ Com""ri..,n of ""-""rime",,,1 and ,tIw""i<-. 1 r"'lults fo, n",ural rubb<, ,n pure shear .n~ pure sh<.. 'u pel'[>OSe!ell.ion. wllere I, and I, ar< pri''''i",,1 <,'<"ow ""ul" e.kul.led fron, li>c CO">l,"'ncd -jUDCti"" ,heory "i,h K ~ J '1.1 ..... of Illc IIth<wk ,"""Iural po""I\",ler "I.; 1'/2 .....,re .,bitrarily el\m.cn 'u ma'ch one <>I'<,ill'> o>f <.1<111"",,", r", pure ,hea, "nd ,hear ;"1"', p"••:d On """pk """"';0". ""i><",;,ely, rhe huri,,,,,,',,t d,,,h,,I TcDC .. for 100 MooOn_

, h,p',

52

STRUCT URES ANO PROPER TIES OF RUBBERLIKE NETWORI(S

.~

on

,-

"

on

;;;

.n

...

a

~

b

.,

o,le

0,'. 0 ,"

o.

0.'

••

"

.

••

~

."

••

-. ,

•l~ • • r S.S 11>< reduced force for a n" ural rubbor ne, ... or. '''' ''., a function of ,lie im'er!< e.l<:nsion ""i<>" '. Tftc cirde. ,how data pOin" obuincd fo' dilTo",nl pylil!hl 1991 American a..",lcal Socio:1Y.

Refe ren ces (I) Mumagh.n, F, I). f'i,,/"

"ow York,

1>
1:'1~J1k

&>ii,l. John

W ilo~

&. Son.

I~SI.

c.:

(2) Truo«ldl, T oupin. R .• &.I •. T~ (;hwkal Fkld ~k.', Springor.Vetl.g: Be' lin . 1%0: Vol. III/ I. P) Fklry. !', J. 1'""", Faraday So<. 1961 . 51, 829. (4) 1'",k>a•• L. R. G . 1'''' Phpi.-s of l{uMn HIQ.• ,wil}" 3,d cd, Clarendon P..... Odor••I"",';on 10 .. r, !4). (1 4) " onlaine. F.; Noel. Monneli<, L.; Erman. 8. 1989. 22. JJS2 (>r< ",f, 13)

n"",.

,I..,

c.:

Mac,_",,",

6

Swelling of Networks

In the pn."ttding chapler. the $I.....,,-,ed &)'s\em (such lhat ",lvent molecules did not enler or leave Ihe 'I<,"'ork during dofomlalion). In this chapto" a more gcnerulthcnnodyoamic an"lysis will be given for the ca« wh.re the ne1'owrk- rolwnl sy,tenl will be rcgank<J as scm;open. In such systems. the solvent may emer Or leave the nCl,,'ork depending on lhe chcm i.,.~1 polenlia] of lhe SOI .... OI and the •• lenl of e general IMrrnodynam;c relation. for nClwork- rol"cm systems. followed by a d;>(,uss;o" of isolrollic , ...-.:lIing OJr nc1worh in Ihe second sectiun. Til;; lopi. h". alrcady Ix..,", 'reatoo in cia";'book,'·'. and II>< ....~der i. referred 10 lh.m for bru:kground infam,alian. In Ihis chapler. we discuss more ~nt impro,-erncnt< and "pproa.Ile,. and newer ex periment. ill Ihi. field. In the lhi,,! """,ti'm of the chapter. "''' will describe th" "fTeets of an exlernally applioo deformation on networks immersed in sol,·cnt. Again. 11K study of deformalioll of immersffi networks goes back !O apl'",ximalely Ihe mid.194&J·<. and only more recenl J.yclopm.n" wiUIK ind..ded in th is chapter. Also. we will consider in delail. in lhis chapler. the swel ling or non ionic networks only. The err""" of ionic groups on network cha ins are discussed in the following chapter in rdalion 10 criti<:al phenomena "nJ phase scpll rat ion in . ,,·olkn networks. We c..,ndude with" diSCUSSion of sorpl ion of linear and cyclic diluents inlO nelworks. Also c.wercd is their e. lract;on and. in Ihe case of Ihe cyelk moloculcs. th";r IraJlpin~ within Ihe net"·ork !1ructure.

6.1 Free Energy 01 a Swollen Network T he fr<:c energy ch,mgc Ll.A from s""U ing all amorphous undermmed nelwNk wilh a solvclll is gi'·en as I he Sum of Iwo Icrms: "'''. the fr<" ."ergy of mi,ing
"

5-4

STR UCTUR ES AND PROPE RTIES OF RUaaERLI KE NETWORKS il. A _ 6A ... ,

+ il.An

(6.1 )

T he c~p"'l>ion for t>A ..... is ",aolily obtained from the Jail;';'; theory of polymer solutions, ( Rigoro usly. ;t ;s the GibM free encrgy t>G"", Ihat should be u>Cd for Ihe mi.' ing "f the polyme •. instead of the Hclrnholt7. ffee ene'gy. T he differcn.cc is incons.cquemiaL howe"cr_) The result is'

il. , r... , = il.A"""b+ A ' '"' R'1'(~ l ln., ~- ~l ln', 1- xu, '::)

(62)

Here. t>A,...., and A~ denote the eombinat<>rial kod Ihe re,idual free energy. n, and a re Ihe mo!c numbers of sol"~nl and polymer. re\",,'Cti,ely. a nd " and I', are I"";r volume fra"tions. respeclively. In cq, (6.2). t>A«>n>b" kT{n , In 'I + H, In "1 ) is oblained from Ihe gcncrali'ed ideal mi. ing law·. The residual frec energy is A~ ~ XIl ] ' I ' ,,-he re Ihe quamily X is the dimensionless inloradio" parameter for the solvcnt_polymcc and i$. in general. a fune. tion of concemration. T he quantity kTX is cqualto the difference in energy of a sol""nl molttulc ImmCr1<:ol in pure pOl)"ner oomp.'n,d with a so"..,m molce,,1c in the pure Matc. T he ..,.;dual f"'" energy ha. acquired a ""~lr31 role in more rettnl studies of pOlymer;oI.'ent or pOlymer- polyme, miu ur('S bloc,use it~ COnttnlra_ tio~ and tcmpcrdture dC1"-'nc<:n ",ritlen for dilTerc nt models in Ihe previo". chapters. n.e discussion of . " "lling here ,",'ill be directed to Ihe wn_ strained.junctinn model. in gene",l. and will be given in lerm, of I~ fu nctions 8 , and D, (dcse.ibcd in Chapler J). which wt: wri le below [or convenic""":

n,

.y,'em.

~ ~ {k1'L{(~; -

,

I} + (/./0\8 , + D, - In (8 , + I ) - In (D, + I HI (6,3)

wl>ere I equales 1o x .)\ or z. An 3dvanlage of adopling Ihis general expression is Ih31 il simplifies readily inlO Ihe e.pression for the phantom nelwork I:>y equating Ihe lerms in the square bradct~ 10 "&-'ro_ liy ta kin g 8 , .. ~; _ I and D, .. O;n "'-1(6.3). On~ oblains Ihe e,pn.. ,ion for lhe affine model, T he , Ii p·,i"k and the eOnstrained·chain model' may also be dc'SCribemp"n~nl,. 1 _ (."I_,)I" E_ FOf uniaxial defOfmalion along the .,' direction. Ihe Iwo laterdl componcnts of lhe e"lension ratio in e<j . (6.4) become ~, = .\, ~ !'"I ',~ JI ' f2 '"' (.,'/",)"',,-11'. in which )' , _ -\.

SWELLING OF NETWORKS

55

0"", Ihe free energy of Ihe systc m is ddincd. sc,,,raJ uscful lhermodynamic and moxhanica l rel.tionships rna)' be ot>lained for the swolle" nel ..-ork. Relow. "" prescnl a few of lll"sc Ihal arc "[lcn ,,>Cd in dar-4elerilin8 and studying such maleri:.I •.

6 .2 The Solvent Ch emica l Pote ntial for aPl Isotrop;cal'V Swollen Network T he chemical pOlenliaL 01' , '" 1' , - I'?' of
,('A' _) iffl,

(6 5) T"

The quamil)' u, in lhe f,rsl line "f '"
(6PI ).., The faclor ,elalion':

(6,6)

X LII eq. (1).6) i. relaled 10 lhe inleraclion param.lc, X by lhe following 1

l),(R

X" :R1vJ" 11,;;0(1', .0 i)), .---a." v.)

{IU J

"' ~, t ~,", + \ )·l+ .. where th. second Ii"" of eq. (6,7) i. oblained by ditTerenlialing lhe Ihird lorm of eq. (6.2), T I>c th ird hne of "1/. (6 .7) repreStnlS lhe 1"<: pol}'mcr 'cfer~nce ;nlcradio" s;le modd' arc So"'e of Ihe lox hni que~ by which lhc illl,me'; '", pammcter III.y be ,ei"ed 10 m"l<."Cula' von· abies in a polymer ..... I'·em S)"l"III The cl'''ite p"rl of Ill< ....-aLl',,,1 dlcmi",,1 I'
56 STR UCTURES AND PROPERTIES OF RUBBERLIKE NETWOR KS

(aI',)" .. (a aA,,) (") 8). r, 1m, r

('0',) (31-" ""0" ")

- ~ r,

The first faClor on tilt right.hand side of C<j. (6.S) is obtair.c
(ap,)" ..

.

(Da~1

Rg

r,"

~: [I+ ~ K().)j

(6.9)

'11>< second factQr 0" the right_hand side ofcq . (6.S) is obtained first by wri,ins).

). .. c;;.)

t/l ..

e'

V,

ttl',,,})

III

(6,10)

when: V, is 'he moiar ~ol~me of soh..,nt, and then differentiating wi,h respect to ") L V, I V, ( 'm, r= 3 vl" 0't /' ~ 30''\}

(6

II)

Suhstitu ting C<j'. (6. 11) and «I. (6.10) into «I. (6.9) kads to the reduced soh'ent chen,ical potential due 10 lhe tiaSli<: activity of ,h. networ~'

(aj"),, =

~ [I+ ~ K('lj

(612)

when:: K('l is given by «I. ( S, 13), and

(613) Here, M , is the molecular weight of a network chain, p i. the network density and the quanlity x, i. the number of rcpo:at unilS of lh. nelwork chain, the ,'olumc of each being taken 'IS «1",,1'0 tho volume of a solvent molecuk. Subslituting C<js. (6.6). (6 .7), and (6.12) into «I. (6.5) leads '0 the total redUttd chemical potential of so)vent in a ,wollen network : (6. 14) Whe" lhe nc''''Qrk is ;nllnerscd in sol" onl , il swells by taking up !KIlo.,,1 unlil an « Iuilibrium is reached be,W«" Ihe e"tropic force.; thaI encourage Ih<: S(J lvrnt molo<:ul., 10 mi. into th<: net ....'ork. and the clastic forces of tl>c strctched chain. thaI tend to prO"onl r urlh~r ,"'eliing. At condilion. of equilibrium, the no,i"ity '" of soh'cnl C<j"alc'S 10 unitY anJ Ihe S(Jkcn, chemical potential C<juat", to >cr.,. The "cti"ily " , w;11 lhu. "" taken as Unil)' in the rollowing

SW ELLI NG OF N ETWORKS

57

The solution of C<J- (6.14) for "l gwes Ih. C<Juilibrium degree of 'wclling AII .. nali~cly. solving 0'1, (6.14) for fJ ~nd u,ing lhe exp",ssion for fJ given by C<J. (6. IJ) lead, 10 die moleeui"r weight of a nelwork chain betw,.""n cros,-link. ~,

(6.IS) ",-her(; p denoiCS the network den,i ly during fonnation, Equation (6. 15) i, " general .xpressiolt for a perfect net"'"Ork of functiona lit y <>_ It i. also gc""rdl wilh ",S"rd to Ihe funClion 1\(-\ ) being left "nsJlI."d. By proper choice of Ihis function. one may "dol" ony modd ranging from Ihe phantom nelwor k model 10 lhe con,train,.,J·junction model II) Ihe slip.lin k model, a, described in chapler 5. Below. we give ,he e.plicit expr,,-.. ion for Iwo special caW" ( I) a I.,raf"".."iooal phanl"'" network. and (2) an amne "e,work_ I'or 'he phantom nc,wo,k model . K(-I} '" 0 and C<j . (6.15) lake, the form

M,"

~

"iI PV , (::..) '" In t i

,

(6,16)

' 1) + ' ) +X"i

For 'he affine ne twork model. K (A) _ I _ A I. "nd Pv t V"

M'=-- In(i

1/'( ",tl' - ~~ I ") - '> : -;'t:;c;:'·l)+ "l+;;f"

(6_1 1)

f."

Equal ion (6.17) i. Ihe well-known Flory- Reh" •• equa.ion ". widely usW determining the crO>.S-link density of a !lClwork. As >JlI."prialc_ 6.3 Thermodynamics of a Network Uniaxially Strfj!chlld in Solvllnt When a nel "·ork at "'"elling cquilibt inm i. ""."lched "nia.ially, in e, cc", so".... nt. to a f,xrd ,,_" cn.;on ratio A. rquilibriu", ,,·ill be n,es,ablishc-d I'll' the uptake of ~".... nl by the ne'work Since Ihe len gth i_< Ii_.rd. , ,,cUing will ta ke pia", along the lateral direction" In this sec' ion. we giV
.'"It"""

58 STR UCT URES AN O PR OPE RTIE S OF RUB BERLI KE NETWO FI KS

), .. <:IV, ' I) )', "' ),l '"

The reduced 50Ivent

c~cmical

(6.18 )

vi/' (.')-'fl,,-'"

polenl;,,1 at constant

Icngt~

i. (6.19)

Irad;ng In

il.i', ~ In (1 -

,'j)~. •j + xli +tl)'j'; [l

+z

K (),,)] =

0

(6.W )

Tbc d stal. is parlly due 10 the soh"nl cnlOring Ihe ')"110m upon Ihe appl;c"';"n "r the len sile r"ott. a nd p;
(6.21 )

The pressure r may be idcmifoed wilh Ihe negali~e of lhe Sire.. along Ihe Ir"ns,-eriC dirtttions. T he finot term in «t. (6.21). designated '1-. represents the coeffi. dent for the ~o lume change of the 'y'tem "I fixe
.,- (m" ') -Q\ n ),

T~ .. ,

- ),,,~

(0,)

(6.22)

-

11)'

T, .-.. ,

when:

",- ~I (") or T.'",

(6.23)

i. Ihe isotherm al compressibility"t fI.ed Ic"glh . The """ond lerm in «t. (6.21) is tl\<: contrihution from the clt"ngc in composit,on and "alljshes for do«d by 'I" i. deri,·cd from the chemical pol.nlial of soh'em a •

., - (~) ,

On ,

•

(~) r,,,,,

r ,) il ln ),

("",,) T~~, / ,(0'",) i)ol, T~," iIIn),

(6.24)

"

The o,motic compressibility .... i. ,i mpl)' Ihe inwriC orthe bul k modulus K•• and is defined as

. '("") 1' Y,

1:..''''''.' ~>'-;:

(6.2 ~) , ~)

SWELLING OF NETWOR KS

59

whert: .. ... - til" / V, is Ihc .."molie p ....'«Ul'\" uamcl~> Ihe e,tr. pressure rCamc nloe in'ide ,md ou"ide Ihe n.... work .. Su"',ituting cq . (6.15) into cq. (6.24) yield s (6,16) Equation (6.25) gi,'cs the bul k ",odulu.", r,.<:d length. l'
-, K, , = I<..

11.1'[

or

" , - 2.A - J\ ,.,' + /J (:D A~ (z) Kpj)J

~-- 1I,-1 - . ,

"

(6,27)

when: K(Al ) - JK(Aj)/VAj. In mOSI •.• ~rim.nlal work. Ihc o;;n\o(;c wmptcssil:>il ily is measored in the ''''ollen bm undiS!orloo ","(e. Ih~ 1 i•. " = I. Thi, i, con.'cnicnl ly achicwd by dCS" 'clling Ihe n.,work by br inging Ihe s"'olle,, sySlcm inlo conlaCI wilh a ""Iu _ lion of known aCI;v ll)" Ihrough a semipermeable membmuc " . The line"r dilalion mlio Alhen A _ A, .. A, _ A, '" {'1<1, , )'1'. T he expression for Ihe 001_ "ent chemical polcmial si,-cn by <~. (6 14) has 10 be used under Ihese circum Slanccs. The ... sulling .'pr-cssion is'

bccoJ,,,,.

(6,28)

T he o;;",oti. eomprc-ssibilily .,f a n",work dcpcllds slto ngly on th. in'ocae l; on para meter, as is seen in figure 6.1 (Iaken from Ihc lilerollure " ). where Ihe con· strainM_junction modd was adoplrd wi,b fi '" 5" II} . , 1< " 1~1}. and RT / 1', = 25 N mm I, T ho solid curve show' Ihe result from C<J . (6. 28) for " .. as a funclion of \ ,. where X> .. 0 . A croSSOver from high compressi bilities for "., < O,S 10 vcry low eompres;ibiliti.s for ~' J > 0, 5 is clea rly seen. T he dashed cu"-c is oblain<:d fo"" .. 0,)81. a ' pcx;ial value chosen ,,, oblain a dis<"O" ~i nu ily in ",molic oompre"ibilily when .\ , = O.S. Inasmuch as " ..' i. proponionallo (he $NOnd der;,·.t;,·< of Ihe cias(ic fr<x energy. Ih e poim of di sco",h,ui,), in figure 6.1 ma y be regarded as (""responding 10 a spinodal point. This will he diwu,scd fUrlher in chapler 7. Tho bulk compres,;bilil)" of a s"",llon nel"'ork i; also sc nsilivcly dependenl on Ihe ikgr~'e of ~,,ccausc of changes;" 'he X p'or.mClcr. a, well ,,, bee.u..: of chall!!"; in 1he d,·g ....'C nf ,·tos._ link ing. Fig" ... 6,2 .; h " ".,. tc, ,,h, of <,pcri","n!> on poly(v inyl aco,"le) nClwork< of ditTer.n , <"'''_l ink 'kn sil ies. , ,,,,,II"n in ace1<..""

60 STflUCTU RES AND PflO PERTI ES OF RUBB ERLI KE NETWOR KS

10"

N

c

E

,.' \.

Z

• 0

~

,.' Ill "

X. _o

---~-~~

\_-- ...

0.'

-._.0.'

6.1 The <»mOlie compr<$Bibilily of. Il
>

•

N

>

-1.00 1 " 2)

6. 1 Resul" of "penmen" on poIYl\'inyl .< .....IC) ...., 1"0.... of dilTCTeo!)". Cirreocnl dat. of Zrinyi. oJ tl<".koy". The solid CUIV< i. obtoinro frum «I. (6.2 S) fo •• ph. nl"", nol_ t"i~ • ••

work. wilh X _ 0.4l7 +0.1.,."

6. 4 Ela st ic Acti vity of a Sw ollen Netwo.k T he difference belw<:c" tile chemical polenlial of a swollen network and a n uncr",~.linl:r:d bu lk polymer ~lthc same volume r.M·li o n " sho"", lhe cl~Slic con · [,ibution of lile Chains. The elaslic cOR1.i bulion of Inc nct"'M k '" the ellerllical polential of thc 5Oh'cnl is gi",n b)' eq. (1'> . 12). The aeli"ity a , of thc '''''''''Ill i. gi'-en

SW ELLI N G OF NETWOR KS

>.~

.

61

K =20 00 0

'"• ~

10

"

0 ,~

m

c

';i

,.. •

0

0

~

~

0

,.

>.00

0

0

,. ,. ,.. ,. " a," v·,

0

0

"

t13

H ~ II" 6.3 Activity fl • • nd for ,he sa"", polym<' In lhe "nC<"pIlh. r;ghHnost one

.1,'·,

"b.. ine
,n''''''''''''''''I$, Thi, rWtt.m""

"'=

p<>int "'... obuine
equilibrium ,wellin! mea.u""",,,"' .. The ri&h,.h. nd or«l in the figll"'_

Ihrough the relation In", .. til' t, If "Ui is the aCli,ily of the solvent in Ihe net· work and Uli. is its activity in Ihe un..:ros,-li"ked Ii""., 1>oIymtr 01 lhe ""me ",-,ncentratio•. lhen eq . (6. 14) may be ",nile" ti(ln", ) .. ln ( "'") .P [,

"11.

,\

(6,29)

The aCli"ity difTeren ..'CS ti(ln", ) h'" been measured for the 'lSleln rubber-ben zen." a nd ror poly(dimelhyl"loxane) In various ""1,,,nIS'" _ Releva nt c>peri· arc given in figure 6.3 by lhe opcn circles. The menIa l d"ta from Gee el experimenlal points .how a sharp maximum_ T hi. fealure i. also !;<'C n in the subwquenl e.'peri",,""," of Eichinger el aL " ". performed with more ",,",ura1e instrumenlS, Although lhe lheorc,;.:,,1 eu"", show m,,,ilna, Ihey are muc~ Ie.. pronouncro and a' higl>er dilation, Ihon IhoSt" given ~y "'perimcnl, The SOure<: of 'he diocrepOnd 10 c.,pcri. men,.1 Obsct1.'a l;Ons ha, nol yel be..,n cSlabli,bc
.1."

62

STRUCTURES AND PROPERTIES OF RUBBERLI KE NETWORKS

6.5 More Recent Tre atments o f Netwo rk SweUing lntcrprelal io~ of swetti~g dala in lerms of Ihe conmained·junction model has led 10 the conclusion" Ihal a high I)' s""olten syslem may be Irc,ned .,,.nliall)· as a phanlom nelwor •. Ncu".:>n scanering cxpcrimen« by llaSlide el al.~ have Ihown Ih.1 Ihe transfann.lion of eha in dimension. in ,wollen nClwOfks gives results dosc to. and somelimes below. Ihose pr.:dicled by Ihe phantom ""twork model. Comparison of neulron scallering dala wilh prediclions of Ihe constrained'junelion model has indeed shown " Ihal Ihe nel"'or. eh~ins deform less Iha n is pre. dieted for unswollen networks. In orMr 10 .<Xount for Ihe low degrees of ddonMlion of 111<: chains. a Iheory of d isinlcrspc",ion of netwo rk. has Ilttn proposed by Il:tslidc el al."'. This approach is along Ihe lines of Ih~ c' Ihw ..... m proposed hy de (knnes'. a<xordin~ 10 which Ihe swoUen coils e~dudc .:.ch oilier from a ,'olume del ermined by Ihe cr",,·link points. and Il,e chains arc essentially forced inlO conlact al these poinls. The ideas hve b<.'Cn discuSS<.". A tre.tment of posilively ionizc of polyelectrolyte gels ha s been con· ,id.r<;tress induces" nemalic order wilhin Ihc network, and (2) bdow Ihc lransilion l,mperalu," of the solwnl. Ihe gel colla","" 10 a deniC slale and the so""nl i. eeentra.~ on .tudies of Ihe mochanica l propcnies (modul u, and I'oi,son', ralio) of imm.~ and deformed networks. o,molic .wellins and de,welling "f nelwo,h, and ,<",,,II' angle neutron and '·"'''Y "".tlering front , "mltcn nCI"·ork,.

SWELLING Of N ETWOR KS

63

6.6 Sorption and Extraction of Diluents The 0'" ufmudel nctworh lui, gi,'cn a 8",at deal of useful information leading to " bener under, landi n8 of the pro l",rtie;; of cI~'tomcrk "'"leri~l. al a molecular ""vel. Much of this has 10 do with mechanical prorcr1ics: for e.• ampie. the evaluation of the molecular thcoric.. th" in''''ligation of 'he eff""'ls of nc,w(>Tk chain· length di,tribution , and tbe eff~I' or nctl"ork imperfections sueh liS dangling chains on uhimate .'trc nglh and maximum cxtcnsibililY. The present application i. di fferent in thai il in,'ol'es ,well ing as the claSl<>meric defonnalion. and .rccifocally addrc,>e' qllC"io ns related 10 the rale al which diluent a bsorbs into II network of known and controlled " pore size," and how rapidly i, •• " subsequently be eX lracled "-'"". The "'~Ie of absorption can be used 10 e.timate diffusion c<:>elf",icnl~. as can be the ralc of c"ra"l,o,,- e,.t .... ctio n ef(,e,cnck, eart he used lu oblain infonnalion on the extents of reaction in the end.linklng p""",dure used 10 form th. nelwork. and the deg= to which lhe e. tt'~ctabk chai n, arc en tangled with the ndwor k eba ins tlult iml>ec<SS

6.6, I Line", Diluent' 0 ". way of obtaining" nelwork .wollen "'ith d ilucnt is to form the network in a first step. a nd Ihen to absorb an unrcaCliw diluent into it . Ahernati>'Cly. th,' ""me diluent can be mi_1h Iype' "f nctworks can then be e.tracled to determine the ea", wilh which the ..ari"". diluents can he ,."moved. as a (unClio" of .II. and ,II,. S"me typkal resuhs ob'airu;c with inc ...","" in M . and ,,;th cd t<> e;.t imate v;, j"c. of the oIiff",io., coefficient V. cither in sorption or in c\tradion, Such c"emden" Can also be estimated ming

• L _____.'.',_ ___ _ .'.:,_____.'.:,_-'

IQ··'.... I;S"m·' ,'OMS I>OIwon., ",Ia,ive to the amount

fi,u .. I>A Amount of diluent .,Iracled f",,,, origi ... Uy P""'""t .• hown a. 0 fu"<:Iion of til< >qua,. <001 of the <Xlroction limo (n<>rm.lizcd by , ...hick""", .... 'h. samplc)"'. In the .. ill"mal;'" ,,,,ults. at room tom· 1":011",". the diluent was pn;«nl during ne''''ork fOmlalion and the ..,Ivenl employed r", ,be ""raclion "'.. lol""ne. The dilU<1l1 had. m<>l«ol., "'
%00,

1 . . . ..

li 5 Ti .... Jof'.""lc.",. of lh, "",igl" ,,.01ion ~'. of dim
'''~A<:1.d

«, _

"'
SWELLING OF NETWO RK S 65

--"~ ~- - ~ - - - -- - ~- - _ ._-

•.

::_----:-

01

-- --- -- ------.:. -

"

.

,

•

",\-____,.:-____,.,____,.,___,"_J '0-0 M, ~'jlt1l'" 6.6 DitTc",""'" in c,,,acli<>" cfflciency f..,.. I~e '''''' tn... of PDMS 1IC'''urk •. a. dc'amil>«l {TOm reinl" .""b a. ,OOse sho,,'n in r.g"", 6.S; p«1),mtr end·linked ..i'h
nc,works a, .welling "!Iuilibrium and ci,her i""rea.ing 'he dcgr"" of s"'elling by "re,ching the nctwork 6 or decreasing th. degree of ""elhog by compressing ,(. It is also possible 10 usc pulsed gradi.n, nuckar magnClic rc.... nanee ,NMR) 10 estimate diffusion oodlkicn!s" . In an)" cas<. ,her. is particular interes, in th. difference. in v.lues obtained for linear chains and for Ihe ~-yciic, of ,h. same mok:.:ula, weight, as describe<.! in ~tion 6,6.) . 6.6.2 Brallcbe<J Dill",nt, The eXlraCli<)1I <)f dilu.nl, ,hat are branched has considerable polential impor. tance, For example, in the prepardlio" of network, by radiation cross linking. tllere are pre.'Urnably large numbers of soluble rn"Io<"le\ f"rme<J 'hal are hiJ!.hly branobM (be<:aul'< of 'he cs,",ntially unconlrolle<J (om,alion and coupling of freeradical sp<.'Cic,j"J. ,\ n
66

STRUCTURES A N D PROPERTIES Of RUBBERLIKE NETWOR KS

deliniti,'" e,periTtl<;nl$ could b<: carried out uiing model branc~ c<1 molc(:ulcs which ha'" been pttpiln-d by conlrolled syn thetic methods"'. and then purified and charac1cri7.cd by sepa ration te.hniqucs such as gel ""rmcation chromatography".

6.6.3 Cyclic Diluent, [)iluen" crm,iSling of cyclic molecul .. have """" cxtcnsivd)' studied with regard to diIT".;on. in part bec~uS" of Ihe llcnermion of such 'pecic, in muny polymer_ izalion''''·''. Aga in. , uch cyclics can be sorbod into lhe network s after Ihe end_ linking process. or they can be p",,,,nt during the process. In the lauer Case. $(1m" are perm~nently trapped. a~ will be d"",ribed separately la'.r. making difficult Ihe calcul.tion of diITusion coefflcienls {) from Ihe ,,'traclion dala. In Ih. former ca",. is ",,,dily calculable. I'or bolh the linear .nd cyclic chains. /J in~r<:a ses wilh dettasc in M " or w;lh iner<:ase;n .II .. as C,pecled. The cyclics ha,·c ,·.Iue.
00-..",,,,. {)

6.7 Trapping of CVclic$ wit hin Network Siructures 6.7. 1 Experimenlal R"'uh. As already menlione.!. if cyclic molecule. are p"""nl during Ihc end_linking of Chains. "'>Ole of Iht>o will he Impped beca use of having been threaded by Ih~ linear chain, prior to Ihc laller bting chemkally bonded ;nlO Ihe n!imaie<1 from sol,,,nl .x"aClion SIUdies. SS>ibly significant increases in I"w-deformalion modulus have, in faci. I>..n ob
SWElLING OF NETW ORK S

67

'00

•• •••• >-

~

~

.. • u

>

u

~

" 240

2110

DP (cyclic) ~i~u .. 6.7

fur.iment,,1 ( 0 ) and theoretical t,"ppin~ efficiencies. u'in~ an MrntJ/c'u/<,. lG. COI')'righl 19\17 Arne""'" Che,nical Society.

6.7 .2 ThM.clica l Inlerprel.li,,,,, The lrapping pr""""" has been .inwtll.d u,ing Mom. Carlo mel hod. based on a rolalion~1 isomeric "ale model'" for lhe cydtc chains". The ftn;! ,Iep "'"' gen_ eration of a ,ullieicm number of cyclic ,'hai", h.wi"g Ihe hOI''"" geometric fca_ lu'e. and conforma,ion,,1 p.dcn;nCC$. and Ihe d.:.i,,;d degree of rolyrncrization Thi. was done for a large number of linear ch ain~. using the method, of matrix multiplica,ion .Iand ard 10 'ol"liona\ isomeric ."". l""o.yM.l' . Up", thi, [1Oinl. lhe melhod i. idenlical wjth Ihal u>Cd 10 gener.le di"ribu, ion funelion, for a non_ Gaussian approach 10 rubberlike d'_lticil),. a. d=ribe<: scheme is illu.,,.'<.><1 in fig ure 6.8 " fQT a chain labeled 0 alone end '1,\(1 N at the 011><: •. The coordinate. of each "cyclic" chain Ih u. gcnerau:d "-ere >10.-..><1 for detailed e,aminalion of tl><: chain', configuTOlional cha ....cleriSlics. rar,icularl)" lhe ,ilc of Ihe "hole" i, would prescnlto. threading linear chain . Of particular imeresl is Ihe sire oflhis hole in comparison wilh Ihe koo"'n diame'er. 7- 8,\. of II><: I'DMS chain. The qualily of Ihe selS of cyd,e confom,alions Ihu; gener'dled was lesled by cvalua'ing Ihe mc:,n -"l"are unrc"",bo..-d ".dius of gyra lion (I l,. for each ring si ... " . ",ing ,I><: malrix multiplica'ion melhod ..... '. The same quanlity was Clymcri7;tlion_ The mult •. in lern" of 11,e lo~"ri'hmic dependcn"" of (; )0 on Ihe degree o[ puly",cri~-'Hio". are shnwn in fi~ u re 6.9. The linea' relalwn,hips oh",,,'ed eo"[,,m previous re",llS for ;hQTtcr PD:"' S chain" T h,· r.lIio of
68

STRUCTURES AND PROPERTIES OF RUBBERLIKE NETWORKS

1!'! I SI801_ Cy~lIc n , "" 6.8 Scl>oma,ic o>f ,he cri ,mon chain" .

(Of

"c'ydi""io"" of • i"""",,«1 Monic C.. lo

cyclic< i. shown ill figure 6. 10. and is "",n 1(> approach a v"luc of 2 al large degrees of polymeri>;alion. as expeclC<J u T ile lrapping proccss "'as sim ulaled using tWo model,", tile more r~al istic of which is described here. As shown schematica lly it> figure 6 .11. a '0"" ,,'a' ""_ I give a good rep'esc" 'al ion of the e.. ""rimeni,1 ". ppin g cfficicnM. It is abo possible 10 interpret these experimenta l !"\."$ult , in lerm. of a po"'.' law for the trapping probabilities and fractal C'05' seclion, for Ihe PLJMS chain,u

• ,

A

"V• <

•

• • •,

,

•

In ( OP)

,

•

f """, 6.9 L..oprilhm;o rop"""nt",ion of the d
SWELLING OF NETWOR KS 69

.. --, , ...

0

A

"V•

••• ----<'-----------------------------------• • • • A •• • "• V •• • . ~

.~

..

~

1.0/ 0P

.i ~ "r<

6.10 R.. io or lhe m<,n""lu.", un"<,,urb«! ",diu, of pr.,ion of lin,,"r 10 <)ocl"

"DMS as. f"lI0 in"", .. dcgt .. of ~ym<'m"ion' . Roprinl<:d wil" pmni.· ";OD

from DeBolt. L. C . • nd M .. k. J. E. (1984). M""'QmQkrok••

Z(I.

Copyrighl 19\11

American Cb
fta.r< 6. lt S
""th,,, • <)"et" " •• " unit Cree of oonfticu. ,hu, POll'll"" "' .he c)"elk ' 0 1>0 ,... ppcd . Ileprin'..! "'i." ponn,";"n from DeBolt. L C . • "" M",k . J. E. (1984). Z(I. Copy,i£h. 19\17 Amenc..n Cb<mic.l Soci<.y,

M~rromok<~k,.

6,7.3 Olympic Netwo rk1 II m"y ~I"', be: t",>sible to usc ,hb ,""
70

S T RUCTURES AND PROPERTIES OF RUBBE RL IKE NETWOAKS

"Olympic" or "chain-mail" nell..'or k"". Such maleriJls would be very interesting IOpologi""II}', a,..j ..." uld 00 similar. in some ""f><'CIS. '" Ihc ClUen"nc"!; and rolaxancs Ihal h .." long been of intercsi 10 a varie1y or sciemist. and malhcmalioi"ns"" ~ " . Comp ulo, simulations". in parti~ular. <'(>uk! 00 '''ry u",(ul wi,h regard 10 eSlObh,hing Ihe condi1ions mosl likely 10 produ"" 1hese no,,,1 structu res,

References (I) Flory. P,}, PriMipks <1" PQ/p'''' CM"'islry. Co,,,,,11 Un;"'or.ily Press: Uhac.. NY. 1953, (2) T",lo,u. L, R. G. TM P~},.ks <1" R.M.>, f;iwlldIY. Jrd «:I. O ••• t>. IIhac •• NY . 1953, "'l . (10) on p. 502. (?) I'lory. I'. J. D;,04I. r",,,,/ay $or. 1970. 49. 7 (8) de G. Cornell Uni;'~rSi,y PreMo t,hac •• NY. 1919, (9) Sch~>
(2 1) No.""r1l<"r. N, 1\,: Eichioscr. Il. E.

M"'~ul<-,

1988. 21,.\060.

(22) !'lory. 1'. l. M"'''' ''''''fflIw, 197"!. 12. 119, (2l) lfall. R, c.. 1'.<1",.rd •• S. F. PoJy",,,, 1919. 20. US7. (24) B.II. R, C.; ~:d"'·,rd •. S. F. ,1/oc,omoJe1... F. M~c.l" 19%. 29 .

..

(1) 11 <>01""". H. II .: fLok
lJ. IQ%. (:(2) Ild",n. S.: It"""",. II. It .: 81>"'h. It . W. : I'rdu,n"'. J. M. J. {',.. .... I'h)'• . 1'190.

91. 2061

SWELLING OF NETWOR KS

71

(l3) lt arl.oJ. R. S.: " rud·homme. R. K .. Ed< I'<JJ,"<"I«/rol,·" (;,,1,. Amelian Cb<mica l S,,-. pm;-",. Sci. I9\1J, 109. 123. (36) l'!Iilippov •• O. E,; Pi10. 195. (49) II«h •• A. M ': Ilorkay. F .. (irk.y. f, ; bi"ri. M.; Geissler. E. ; lI",bl. i\ , M.: Pr"""I, 1'. r"ly",,' 19'91. 32. 8JS. (S I) f!orkay, F ,; lI"'hl . A. M .• Moliam. S.; Gci"ler. 10, ; Ro""ie. i\ . R . .l!acromokckh>3 19\11. ~4. 211% (S2) Z,inyi. M.: 11..,. ••. J. ; Itorkay. F . .If~ r"""okc"t., 19\13. 2~ . J697. (53) Horby. F., 8"rchord. W., G, 19'93. 26. 1:!%, (54) lIorb y. I' .• 8"rcb.,d. W,; lIech,. i\ . M" Ge;.,.lcr. E, Macromokck'~' 1'/93. 26. 3375. (SS) fl «h(, i\, M : SI.""'y. fl . R.: ~ ..k,. E.: I l or k .~ . F.: Zn"),i. M l'oiJ"l'" em",,,",,,. 1993.31.11194, (56) Gci"Ie', L Ho." Y. F,; Hoch •. A. ~1. Ph .... II... 1... 11 . 1'/93. 11. (>.15 (57) Mar\; . J. E.• Zh.ng. 7_· M, J. /'01,-~,. S'-""', Ph,•. Ed. 1983.21 . 1971. (58) Garrido. L.: Ma r\; . 1. E. J. P,.!!,m. Sci. . 1'"1),",, I'h)'• . Ed. 19 ~ 5. 23. 1933. (59) Ma tI:, J. E. J . Appl. PQ!,.",. Sd .. S,I""'I'. 1\11\9.44.20'}, (60) Ma,... n, j ,; Leekf'<. ~ ,: Galandri •• N.; G. fur. I'ol)'m, J, 1995. 31. 8(13. (6 1) OA"', M. II.. On.rao. K.; Erman. 8, J, 1'01,.,... Sd.• r ill,.",. I'h,-" fd. 1990.28.

,,,,j.,,,,,.

n

CO";".,.,.

I'm . (62) Ganido. L,; Mar\; , J. E.; i\c\;enn,o. J. L.. Kon,..)'. 11. , i\ J. Pol),,,,, .'in. I'ulpfl. Ph)'" I:d. 1~~8. 26. 2367. (63) Dole. M __ Ed . 'n ". Radi,,,;,,,, Ch<mei",' ,if .I/"",,,,,,,,,,",'"b. Academic Pre»: Ne ... Y<>r\;. 1971: Yol<. I . ,HI 2. (M ) Pep!",'. N. A.: u nger. R. ~;mc~ IW4. 263.1115. (65) Co"", ..",)" W. W, I" I'hpjr~II'rop""~,, 4 1'01)'~"",,. J. E- Mark. A. h .. nbcrg. W. W. G raco>k)'. I.. M."ddk.rn. Ii. T. s..mul,' i.) . I.. Koenig •• r>, An""",,,n Chom"'al S.>Cloly; W.. h"'g'''" n c. I~J. p, 91, (66) I' "",he'.) M J. SCli" ... 19\14. 263. 171 1) (~7) Tom.Ii •. n. "'. S<'i. A",. I'N~. 1721~). bl

l.",

(6S) Rempp. P.' I' A{"",-m. Symp. 1992.62. 213. (~) fried. J. R. f'oI,mn Scklt<. mid T..,.hno/cI}', P",n,i", H.II: Engk..-ood Oiff• • Nl. ,~,

.-.ci."""

eM) Odian . U . P,i""' p/tJ cf PoI,.."",i:al;c" , 3rt! ed . Wiley·!n, • N.... York. 19\11 . (71) !\elnlyen. J. 1\. In S;Jox"",' P'>IJ'_'$, S. J, 0 • ....," and I . A. S<mt~n. 1'-<1 •. p"'"'i« lI.n , Englewood OilT" NJ, 1993; p, 13~. (72) G .. rido. 1_. Mark. J. E; CIa,,,,,n. S, J .. Scmlyen. J, i\ . PoI,..H. C""""un. 1984. 25. 218. (73) G.rrido. L ; Mark. J. E ; O • S. J.; Scr"lycn. I, i\. PoI,'n,. ClimItIun. 1985. 26.

".

....,n.

(74) Garrido. I..; Mark. J. E.: Oa...,n. S. J.; Scmlyen.

(15) (76) (77) (18)

"

j,

i\. PoI,'m. C""mlun.

1~8S.

2b.

CIa""n, S. J.; Mark. J. E.: Scml)'tn, J. A. Pol' ..... C""""",,, 19S6. 27. 244,

CO,,,,,,",,,

Clar=n. S. J.: Mark. J. E.: Scrnlyen. J, A. Pol,..... 1987.28.151. Ikllo!!. L C: Mark. J. E. Mac,omolHuk. 1984, :roo 2369. l')'\io. T. I.: f ri",b, It . l.; Scrnl)-.n. J. A.: O.""n. S. J.: M.rk, J. E. ,. Polym Sri .. CI.,-m. Ed. 1981. 25. 2503. (79) Hu.ns. W.: Fri",b. It . t .; II" •. Y.; Scrnlyen. J. A. J. Poii'm. Se;" Pol"",. Cht",. &/, 1990.26.1807, (80) Flory. P. J. S'Qli"ic"1 M O'rI,,,,,k. of Cltain MoIffu1O'3. !o1<.-..:i.""': Now York.

p,,,,..,,.

,~ .

(81) M'llie<, W L: Suter, U. W. C""/M_,.,.."ITIrror}, 0/ LargO' M,,",,""'• . Wiley' r-,..... York. 1994 (81) Ca ...... E. F. J . PoI,-",. & 1" p"" A 1%5.3.605 (83) Gali • ..",oo. V.; Eichinger. R E. Pol}·.... ( '",,"""", 1981.28. 182. (8-1) Mar •• J, E,: Erman. B. RuMtdik.. E!,/Slic-ir},. A MokI"I". P,u".,T, Wiky· In,<'><;.""': No,,' York, 1988. (8S) Fri!lCh . H, L . W._rman. Ii. J. Am. C~m. So<-, 1961 . 83. 3789. (86) Callahan. D., Frisch, 1\. L.: Klempner . I). Pol)''''' /:n8' & i. 1975. 15. 10. (8 1) Rigbi, Z.: Mat., J. E. J. Po/i'm, Sci .. 1'01>,",. Phy•. FJI. 1986. 24. 443 ( !Ill) ClaIWn. S. J. : Wonl!. Z.: Mork. J. E. J, /""'8. 0'1'''''''''''''' f'oIym. 1991 . 1.223. (89) I"'al., !'-; Oh,,".i. T, J. PoI)m, & 1.. Poly'" I'lIy• . fj/. 199), 31, 441. (90) ~· ri "' h. fI, l.. /'10k' J, C~m, 1'193. 17.697. (91) Ma,'. J. E. 10'"", J. CMm. I99J. 17. 70l. (92) Wood. B. R., Scml~n. l , A.: Hodge. P. Pol,....... 19\14. J5. 1542. (93) Gi""'n. It W.; IIh",,b, M. C ; IOn&c". 1'. 1', I'rot. Pol, ..... S
7

Critical Phenomena and Phase Transitions in Gels

Th. term "gel"" h,,, b«n usc,J in a wide variety of ....,,,,c.... and Ih.,.., have I",en difficuhics in ",achillS an all-indusivc. ,,'orkable defonition for i,' ,2 , Perhars ,!Ie simplest wa y tQ pr~ is to liS!
clastic bod)', bul ,."crall)' with. '-CTy small modulus. Ifit d~s show plaslic flow , then Ihi. occurs ab",,<, ~ Ihn'Shold valu.: of Ihe sims. with full =<>"rability below thi. limit I! lypically con';." of two or mon: romp'Oluon. mOr<" Ihan IhousanMol,t wh." immersed;n II suilabk: soh" nl The eXlenl \0 which , uch a ""Iwork will ~wcll depend' spedf""ally nOI only on lite dcgree of cros,-linking, bUI also on Itt.: ill1Cr~Clions belween the ch~ ; ", and the ...,1>'onl',I,', While Ihe dege« of ,'m.,-linking i, eSlablishc-d during thc p.ep.r~lion of" nel"o.k, Ihc extenl of Ihe inleracl;on of chains and sol"enl may be modified as desired , and thcrefor< lhe Mgrtt of swelling may be conlrolloo. A gel can be made I(} ,well (}r shrink continuou,ly by changing lhe quality of the S<)h'Cnl wilh which il is in conlact. ,\lte.nalivd)" il may go Ihrough trilical condition. and, in litct, can exhihit pha"" 'mMiti"n" depending on the \)'pe of the polymer-solvent inlcr4clion and Ihe exlcnt of cross· lin king, T he discrele s.hrink· age of Ihe gel. by changing Ihe polymer-oolvcnt interaction parameter, is a .oh,m~ pha..- I"m lition 'imil"' 10 the ga ...!i'luid lra nsition of a MnJcn,ing ga~. The pos>ibilily of sllCh ph" .. "ansitiom " 'as, nOlably. firsl JiscuS>Crl by Du.. k and c<>l1" horJto'" m,ny )'~"'S as" <'., The" trcal",ont wa, oonfinc'd 10 "on ionic n"",,,,k•. T hey showed 'hat networks p,crarc~l '0 .h"", Ihis cIT«1 sh"uld be

7(

STflUCTURES A ND PROPER TIES DF RUeeE RLI KE NETWO RKS

s),nl hesi>:<:<J in the highly di lut<:<.i slate. and the s,,-clling :\<Jlvenl shoo ld rather he ]'><>Or and h",'e an in'~l"dctio1\ parameler whkh is Slrongl)' con",nll"dl;on Mpcn_ den!. The ro<,ihi li t)· "r flrsl _order volume lran,ition. in such S}'Slem, was later shown by Tanaka' using Ihe s".. mng exp",ssion for no"ionie nel"'erks' . The general Ihermod)'namic condilions fer Ihe realil"lion of critical oondilion< and lran,ili(>n< "'eTC .ubs",!uen,ly deri",d by Ennan and Flory'_ 11 waS 5hown Ihcre Iha t ,h~ pr=nce of ioni,.abl. gr"up' on Ih. nelwork chain s " 'ould grcd,ly fadl_ ;tale Ihe Ir.. nsition. The large ",Iume of experimentallileratme on volume pbase II".
II"'. The similarily of Ihe transilio n phenomena in gel, to 'he coil globule Ir~usition of linear chainS can be obscn'ed in Ihe e.pcrimenlS of Tttnah ot at. on linear polySlyrene ehains)I ).4 _ More rcco:nlly, Ihe eiassica l lallicc II",ory (..,n which the mi,ing part of tfle free energy is based) has been mtJ
'0

7.1 T he ory o f Critical Phenomena a nd Phase Tra ns itions The re<,lmpi
CRIT ICA L PH ENOMENA AND PHASE TRANSITIONS IN GELS

6)1, = (I"

_ I'~)IRT .. In (1 - '~) + "

+ X..j I q [ I

+i

K()')] +

75

~~ _ In a, (7, I )

Ill" is 1he ,:<"'''ibulion 10 Ih" 'ot"1 ch~",i~,,1 po'en,ial by ,I>< presence of ionic group. on the ch.1in •. A, de>cribcd in ,Ielail In chapter 6. X dcpo:: nds. in general. On the oomposi lion . a. well as on tempcr"lme and p,essufi'. I< is cmpirk-.. lly e.pr<:SS<'e X I"'ramdcf on ..~,mpo.i'i"" is of .ignil;".n' importance from the ""im of view of phase Ir.n.ilion •. and Ihi, will be: discussed in mOre demil lalcr, For e., ample. the cqu~IoOn for (he symm polysI)"""1C in C)'c1"h.JUlne al 2S <.: i~ ~ '" O.l.(J5 + 0, 33., + o,29,'l. and for lhe S)"Slcm polyisobulylcne in ben,.e,,,, ,n 24S C il i, ~ .. O.SOO + 0 .)0.,) + O,.J(I..j . The linear add ilivily of lhe io nic polenti.1 lerm in Ih. c' p,<,ssion for lhe chemical potenli.1 in eq. (7. 1) i. ,'alid ".. he" the number of iOllic groups on each nCI"'orl cha ill i, , mall. In thi' case. lhe conlribution of Ihe i01,izablc gro ups 10 the el><mical potential witl be' w~rc

:::'" d RT ~ - ;"(1''/ v" N .. )( ,,:i .~} (7,2) This equation will i)r adoplcd in the foilowing dcriv;'tioll of the Ihwry. T he chemical polenlial and ;l~ f,rst and soc"nd deri"aliws, taken with rc"r«1 to oomposilion. vanish al II>< critical point or a swollen gel fo.
~ h,'-l+ · · ·+ IlM .... -

.' ,

(7.4)

and (7.5)

" 'ilI equale lO:rero a, the cri. ic~,1 poi nl, For a network ... ilb a 'pccilied 'trueture. Ihe critical condition, may bc fulfilled m ",n." unt pres,ure on ly a l • u"ique lemperature and c 1/ 3. T he fulflllmc"t of Ihe..: cond ition, fo r noni",,;" h"mo!",I),mer ,yslem. Ottm. dillk-ull .. hen one ","'iderS Ihe gc norally "bscmod ,."Iuc, "f 1he .\ parameler. Pre..:",," or ionic !' roup' on 'he "C( ..."'~ ( ham, ma ke. the f"lfillmen t of Ihc..:

1"',,,.

76 S TR UCTURES AND PROPERTIE S OF RUBBER LI KE NETWORKS crilic.ll..:mdilions easier, as .hown b}' lhC
a,

(7 .6) and

(7.7) where 6ii, is lhe reduced chemical potential of an arbitrary ,ubunil of Ihe nclwork . Wbcn Ih. soh-enl activity is unity. the oondilion given b}' eq, (7.7) i. equivalent to the integral

f:6[I, .,'d" , ..

(7 K)

0

s",)vcnl chemical polcmials for Ina, = 0 are ploued in f,gore 7.1 for Ih"'" non ionic networks ha"ing different" paramo"",. Cut\'e a ilin'trales !weUing w;th good solveni. ~u"'" b shows a critical point, and en"'" c sho,,', a triphasic equilibrium bo;t,,"« n phases with " _ 0, 0.030, a nd 0,157. Figu", 7.2 ,ho"",,, ","uit. of calculations (>f ph.se tran,ition, based on eq •. (7 .6H7 .8) fo, a
(

• • ,

•~

·V ' • ". 0.10

0 15

fl C- 1. 1 Sol,..,nt chemic. 1 p<>!enti.l. for for 'hree ""nionic ""' ,,"'Tk. lI,winS diffe",nt X para ..... t.r>, A coD"r~irltd-jun<1 L Qn ",, ' wQr~ mod; an':: 0.0,030, .nd 0,151,

CRITICAL PH E N O M ~NA A ND PHASE TRA NSIT IONS IN GELS

77

0.44

0,45

~

0.46

0.47

0.48L---,~ 0.0 0,\

.., .., "

•••

••

t1cIl' . 7. ! R.esult' of c"lculalion. of plio", Ir.",itio". baoeJ on <"
O,6)- (7,g) were oblain<:c figure is Ihe unique err"", on ph... Iransi,ions of f experiments by Tana ka" On • ,cmp",al ur.,.sensit;ve gcl. The Imnsit;on is reporl",1 10 he fully reversible. The phenomenon of phase Ir"nsilton. in , ... " Hcn gel, i. especially of imere", when Ihe WSlcm has a low degree of " r<)$S_li nl;;ng, such as in biological gcis tha , are only sl ightly p"sllhc Ihresh"I,1 point of gclal;on, In '''''h ClOses. lhe" p"rnmel", is large a .... the gel behavior is do .. \Q 'ha l of.be affine model. In Ihe limil K ,. co. II", funCl;on K (.I.) ta k", Ihe .,; mplo form K (.I.) = I

.I. I

(7.9)

7B STRUCTURES AN D PAOf>ERTIES OF RUBBERLIKE NETWORKS ,~

""

"

'"

"

-

-

•" " " " " .,

\- --------------- , -

1t E

"

~

'" ."'r

0.05

,

"

"

R""" .. of upDl:h '" tho cur", ~ '0 ,lit hqhly , ... 01· k10 1colLI,,""" "f'OII cooIi"l to 22 C. and the ripo [>OI1ioro '" ,lit <01,," Ibplo<'ll ....,• .

t1t_ 7.J

n.,~

."'1< ",. ..

Expanding Ihe lop "'hmie term up 1<:> ,he f<:>unh ordor in oq. (7. 1). 5<>uinl N ~ 0 for j > 2, and US''' !! (tis. (7.3) ( 7.5). <:>ne <:>blai ns the fol1m. injl i!IO:l of !'qual ions in lhe offille lim il :

a.,r,• _ (X, --~') -i ...( n) - --'I 01 -

" "

>, ~ "I -+:1£.. ' 2 4

-. ( ') ( ') ,

x.

a.,., _ 2 X' - 2 ' 1+ 3 Xl - 3 '"l - "l+j, ~ x, a.,._., _ 2 ( X'

- l' )

,

"'

", - (.'+2')/ x, = 0

~' " " ' _ 0 + 6 ( Xl- j' ) ", - J", - 9l -;':( 7.10)

SoIUI;on of nj. (1.10) for

x, . .\,. and "1.! ~

'"l for laoge

x, k ....h

["9"'" ]""

~,

ou

>f)

~

to

'I" (7, II )

XI"'I Xl '" i

x,.

T~ ruult>. whi~h bc<:OtllC e'act in the limil of infillilC an: indcp" IIdenl of iOll CO!IWnlr"tio " TI,i, im" ,i.,. IhJt, in the caS<: "r chuin. wi,h .mal1 ;"" ,,,n""'m,;,,n ("hlCh ,,';t, ,he ."un'!'too" of 'he modol d.sc.ib.:d aro,..,). ,onic ~nd no"'onlC nc' ''-orh ""ha w identic.lI} ;n the litll;' of infini,. ch,,,n kn~th

CRITICAL PHENOMENA AN D PHASE TRANSITIO NS IN GELS

79

More recent thcol1.~ical "naly... "r weakly charged p<Jlyelcclrol}'IC gels by Khokhlov and oollabor~lors"'''' direct attention to Ihe p<Jss;!>il;ly of fOrmal;On of mierodomain Siructures as a resuh of microphar.e SCI>"I'11lio".

7.2 Thermoreversible Gels There is another Iypc of gel Ihal undergoes a phase Im"silion. ~pocifically Ihose gel, Ihal are "Ihernlort,..,rsibk.. ' These male,i"l. a,e highly ,wollen nelw",ks in which the cro, ... I;nk. a re temporary or physical; for exampk. "')"Stall;,... 0' imer' sesmenlal aggregal""'" ''''. These types of gel' arc lhermofC'"rsihle in Ihal lhey reliquefy upon i"orcas.: in tcmj)l:mlurt , "fficic,,1 10 meh Ihe cry1talliles or break up II>< aggrcgate~ and then reform lhe gel up<Jn cooling_ The m'>'t commonpla<x e.ample of such "male rial i~ collagen in the form of gelal;n. which i. in lhe gelled form with ",·ater in ,o,ne food prodU"1O. Some olhe. """mpl,.. of biopol)'tllcrs Ih~tl e~hi!>il Ihis propc"Y are de>cribed in chapter 15 on bioc!aSlomers, A ,ariely of synlhelic polymers al"" form Ihermore,·e.,ible gels. and ""me or Ihem arc described in thi, SC(;tion. The one, lhal ha'" been sludied moSi oxlen· ,i,,,ly con,i\1 of poly;:thykne~""""!>O."·"· 'oo. and vin),ll"in}'lidcne polyme". such as ataelic pol}'slyrene"'" .. a nd poly(methyl methacrylale)". Mosl ;m'cstigalion> of loch gels ~ave empl"yed ledt"iquei ror char.ctc.i~ing !~c gelal;on process (and sUt>st:quenl liquelicali"n), the n"lure of Ihe ""regalion (particularly in Ihe """" of nOnCl)'slalli7.able polymers). lhe corre'ponding ph.se diagr~"'s. calorimclric eff""t! accompan)'ing Ihe Imn,itions. ",Id .Iruclure. and morphologies_ Some of ,I>< '''''hn iques used '0 eharac1cri~ ,h<.'S(: gels and Ihe pol)'mers rrom which Ihcy arc formed are: analytical and preparal;"e gel pernJea · lion chrOmaIOgr-dphy. rracl;onal precipitati on. "c NMR and Kaman ,peetro· scopy, viscometry. >can ning calorimelry. ,'i.ual ob .. rvati,,". of lhe initialion and CCS$3I;On of no .... ror the gelation pro,:es" ,mall.angl< x_ray and lighl sca lier_ ing. polarized light micros<:op}'. and ",anning and tmn,mi",ion el""\ron microscopy"_ Rolati"cly lil1'" ha, been done on 'he """hanical pro[>CrI;..s of lhe gels. and ..,me or it focu.... on dynamical m<."Ch."ical pro[>Crties. ralher Ihan equil. ihrium moduli"'''_ II is possible. ho"'",'cr. 10 oblain cslimal"" of Ihe moduli of lhese gel •. in ' pile of the fact thallh.ir high degrees or dilul;on make them quite fragile. One direcl "-ay of dOing th i,;, by adding kno"'n weighls to the surfa~ of Ihe gel and then mC"~,uring tbe change in heighl or the s.'",plc·'. An ahern"live is '0 enclo"" lhe ""IUlion being gelkd in a dilalomeler-,hal'.:d gI",s cell. imposing a prcs$ure "hove Ihe ~UTrac~ or I"" gel\in~ $y~l cm . and then measuring lhe change in hcigh' or" "'''''Or)' column conlacting lhe lo""r portion of tbe 1I<'1""·loo.1'hi. I'I1..,.,.u.e-pulsing approoch has the aolv~nl"gc or being al>lc to follow relali,..,ly mrld gela1io"s. and 'he tran'parency or lhe cell pcrmil> simulta,",,,u; Ka lleri"g "'ta,urcmems for monitoring ch~nge, in Slruel"r •• ''''en prior 10 Ihe ge!;Hion point . Some of tl>< results oblained by these lwo m"hods arc ,,<<o" cannol occur. The re,uhs arc ror clh)·lcnc hexen,,"1 copolyme", "'berc thc comonomer is uscd wilh Ihc elhylene to

90 STflUCTUAES AND PflDPEAT IES OF flU BB EALI KE NETWOA KS

',----------, ;

,i ,i •!-----;;-----,;-----,;------! • •• •• •• .0 To .. porOlU,. ( ' e l

I'll" '" 1.4 Critical gelation CQn<;l- I• and , .. ltICS of the mol% b .. ""hcs.rc 1.21. 1.21, .nd 1.47, """"",i,cl~_

in lroduce kn(}wn numbe", of b'-dnehe! (}f kn(}wn lenglh and slruelure [( - Cl~ ,hCH J]. The values of c' .rc gec of Ihe branch poinlS with Ihe c'ySialli~a lion proceSli that provides Ihe temporary c",... links required for gelalion, AI each molccul., "'C lim;1 as Ihe ery,ulli/,ui,m approach"" '1.' equilihrium value. T he e!TcclS of concenlration r on lhe \ow.dcformalion moo ul u. haw also been repclclcd for SOme polyc1hylenc homopol)'mers, Signiflcanl cUIValu"" is observed in the dependence of lrl on concenlralion il""lf. hUI ilS dependence on c' i. lincar wilhin e.perimenta l errOr.• 1 il Ihown in figure 7_6·'. The modulus i~ «Xn to increase sil\fliftcantly wich increasing r. and Ihe slopes of Ihe V'l ,... ,; cu"... increase asympla!ically with increase in mnlccul., weighL This is presumably dut 10 " de>:,ca.., in the number of chain ends. which ha,"C la be relegalc-d 10 Ihe interlamell.r regioo. wben cry.lalli''''lion (cross_lin king) occurs. Sucb cross·link· ing;. obvio",ly more eftidenl ,,'hen Ihe number of such chain ends is minimired .

.,-----------,

, ,•• "

• •"6

•• ~ • •• i• • • • •

•

-<>

"

.

• Tlmt

0

~

"

(m'n.)

. . .

FI,_ 1.5 Values of lilt mooJulu, in 'h" lim" "r ""'Y .,,,,,11 ! I a •• f""",ion oftime 'OO, The t<mpenllu,cs ""'rl: 15 (0 ). 21 (6). a nd JS (0). and ,he "",,,,,," . 'nI,ion ..... apflfoxlm,,,dy SOo;, higher ,h.n ''''' enhe.1 ""11«11"3Iion <••

f''''''

"

'.• .. Z

!; •0

-

• •

1 O· c~, g2 c m~

•

~lK.'" 7.~ Moou' ; of gel> 1''''''''.-00 frum p<>')'Olhyltn< in ,"" lim;' of veJy ...,all W"r· ,"";on.•~",.." ••• r""",jM of ,he "4 .... '" of Ih< 00""""''''';0<'1''. V.I,,", of 10 ""SN ftom 5.8 (1o~~'1 cu"..,) 10 800 (upre"",,"1 curve), Reprinted wilh I'crm,"';o"

' M.

f'om Li. Z .. " .1. (19"9). ,1/""''''''''''",1... 21. Soci
C'~')" ;z.h '

''''7 ,\,,,., ..... ,,

( ·~."' ..·.T

92

STRUCTURES AND PROPERTIE S OF RUBBERLI KE NETWOR KS

The? dependeno: of Ihc modulus is consiSlent wilh Ihe imporlance of pairwise encounlers between chain segmenls in Ihe gelation procc>$, More de lailed C()mrarison. of Ihe modulus in the limits a ,, - . I and r _ c· have al,o been carried OUI". If a v...,ighl.avcragc mol,;cular weight AI., Or v;>cos;ty·avcrage molecular '''''ig''l M . is used for Ihis ""rpose. then there is no correia · lion bet".-een the resu lts for unfractionaled pol~mers and fracI;oMlcd on,,,. although the lrend ......,m 10 be 'he ,"une for the tv.·o Iypes of sample •. Howe'..... , a common ~pendcncc dor;' .-.:,ull when the same valu,," of lr] .r. plotted again,t the IIOIIIIIx·'."",rage ntok.., ,, la r weight M •• as shown in ligu,"" 7.7. Specifically. when Ihe data are anal)'/.cd (,,' this ba.i •• Ihe same ,'"Iuc of lhe modulu. is obtained. irrc,pc<:li,'c of the S
,

E E

z

"-

• • ,

,

•

0

fl~ ...-..

• 0

,.. ,.' M" g mor' 7.7 I\-I<>CC"'''lion. 0

"

.-

.-

approaching lhe hown., a fu,>ction of II>e number... ,'.",,,, mol<cular "~[h' " Resul" on Ihe "nr....,o;o""led pOlymer< a.... h"",," 1»' to< 01'<" <"i,. ck>, and ",,"," "" , II.cprin'cd ,..i,h I"'" mi~,,,," (",m Li, Z., Cl . 1. (193\» . .\/(".'".~"I". n. Corrright 111'17 Amc,i<<
CRITICA L PHENOM EN A AN D PHAS E T RANS IT IO N S IN GE LS

83

M

--~--

~·ii."" 7.8 Sl:e!ch '''83CS!i"1 '1>0 "''''''"re of. ""1)"lhyk",, chain;n a go! prel""od in 'he vicinj,y of ,1\< C";lH,ol c'OOCCnor'l1OO c· , ,.. ilh lh. f .uch.
Ih~ 1

Ihe lamella r cryslall il.. acl as cms,·li nks of ,'ery high f"nclionaiily ¢ ' They Ih~n Ih~ n.u,, 1 co"alen! cTOss.l in ks, bUllh;s may nol be imporlanl al Ihc wry high dih,tions invol,'W_ The resulting calculaled valuts or lhe mol",u!ar ""ign t M , be!"'ccn cruss·links noa ke up a consider.ble fraclion of lhe 10lal nlolceular ""'igb! .II of the 1: ge1' ;Ire .. I
Reference s tl) Flo,y, P. J, f)u·c",,,,. f"IJI'Ilt/a r S
mo"

(l ~

(4) (5) (6)

(7) (8) (9)

(10)

199,,76,49. HoI)'. 1'_ J. (>" "";1*' of 1'9Il'm" C/"""'.' ''y . Co,,,,,tI Un;,'",,,)' p,..." Ith.CII. NY, t9,3. Ho,k.~. F ': M c K onna. G. I\. tn Ph"jc~/ of p(~J'Mi!" 1/<JJttibooIc. J, E. M" k. Ed . Amcri<:an I n"i t"'e of 11,y<: Woodbury. NY. t9%, Du.. \;. K,: r.Ile""". D. 1_ 1'91 .." . Sci 1%8. '\26. 1209, Du«k, K.: I',in •. W. ,1,1" I'QI)'I'" .ki. I%'I , ~. t . T.n.' •. T. I'h,-" R~ •. I... ", 1'i78, 4(1. S!O, Erman, 8. , flory. 1'. J. iollJ<'''""","u/" t9~6, 19, 1342, hn .... V. f·.: Rod,,~"c,. F .. ('011«" (' 11«,·""",Ikc.i<" 1'lBO. Il. 9 7 ~ Tana'a, T.: I"il'",o""D.: S"n S. f. 1'"I"d, I.. Sw" I ('~, Ii .; Shoh. " . Ph,.,. Rc •. 1~ ·1I. t9!!O, 4 ' . 16J~ .

""'1'<''';''

8' STRUCTURES A ND PROPER TIES OF RUBBERLI KE NETWORKS

(5 1) (52! (S))

T.naka. T, Sd. Am. \ ~S\. 244. \24, ll"",t y, .\l, Pol,.,.." \98\. 22. \687. H.o . .. J.: 11.",\;y. M" U\brkh, K.: Kopoc<:k, J. EIU. 1'01,.,,,, J, 198 \. \7.)61. l lav379. l la\OSly, M" H mu •• J" H• .,.li,:ck. J. PoIJ_T I '18~. 26. I ~ 1 4, Hoffma n. A. S .. ArraM;abj, A,; L, C, J . Cm/Tolld /(,1,,,,,, 1986. 4, 2ll. Him"u, S., l1i.ok,,,·• . Y.: T .... k• . T. J. CM,", Ph)", 1987. 87. U92. Amiya. T.' Ili'ok.,..., T" Tonaka. T. M~(Tf)mol~/" 1'187. 20. 1342. 1'''';1>. •• R.I'. S,; Cu .. k', E, l . C.... m. P"'l ' Sci. 1987. 42 .97. 1\i. I. : Sun. S, T .. Swi,low. G .: Tanaka. T, N~/_ 1m. 281. 206 , Swislow. G ., Sun. S. 1.: Nhhi<>. L: T.n.k •• T , Ph" •. 11. ... I.n<. 1980.44.796. Sun. S, T ,: Nishi<>. 1.: 5wi, low. G .: Tana ka. T J. C;M",. Ph,.• . \ 980.13. S971. NiShi<>. I.. S,,;.lo'O', G .. 5un. S. T .. Tanaka. T "'a,"" 1982. 300. 24), Pro"". M ,; 1100\><', H. II.. Pr-au'nilz, J. M. ,f/Ch£ J . \ '189.35.803. HOOf>C'. H. H.: 8a~ <,. J. P.: BI.nch. II. W.; 1'<'''ni1>, J. M. Mal:TDm<>kc"". 1990.23. 1()96. Onuki. A. J. Ph."" SIX. '''PO" 19~8. 57. 699. Li. Y.: T."" t •. T. A"". Re•. Mme<, &/. 1992. n 243. Otah. K,: Inomata. II., Konno. M. ' Saito. S. J . Chem. Ph". 1989.91. 1345. Kat,)"m • . S. J. Ph)'• . CMrn . 199Z. 96. 5209. Osad • . Y. Ad.. ,I/o'..... 1991,3,107 Hi,. i. T" Nomolo. II .; Il irai. M.' H')1I"'i . S. J. App. 1'''''-'''' Sri. 1994. 53. 79. y .,.hOda, R., IIIno. M.: ll u. y, M IR"'-.oI. SJmp. \ 995.93.125. Osada. Y.: Oku.zoli. II., Gong. J. P. TII.IP 1994, 2. 61. Kureki . V., Sekil1loto, K. J. ClI<m. Ph". 1995. 1(12. 8616. Zhang. X. : U. \ ' .: lIu. Z.' Little,. C. L. J . C"....,. Ph}". I99S. 102. 551. Osada. y, : Ma'-'uda. A. N~'"" 1995. )76.219, YoshKl •• R" Uchida. K .• Kando. Y., S-o k.i. K.' Kik llChi. A.; Sa\:uroi. Y.. Okano. T. "'~"".., WI~. )74. 240 A .. no. M., Ho,le, K.: Yam.,hi, •. T. 1'01,''''' G,/. Nn.', J995. l. 28 1, O""tova. T.: Sulcim.nov. I., Frcnkol. S. Pis/I'm. C;,h NN .... 1995.3.387. NaK"'.\:', Y.: IIc,",-,,,,,,., Ii.: K.,~ . M.' I( .tao" . K . •If,,,,,,,,,,,,I,,,,,/•.• I'!'N. 27,

(~)

Tarur k• • Y.' K.gami. \ '.' M.1>ud •. 1\" o,.da. Y M'IC"Htwh"'"/e< 19'1,. l S. 2514.

(II) (12) (13) (14) (I~)

(16) (17) (1 8) (19) (20) (2 I) (22) (2l) (24) (H) (26) (27) (28)

(29) ()(I) (ll)

(32) (l3) (34) (35) ()61

(7) (lS) (l9) (40) (41) (42) (43)

(44) (45) (46) (47) (48) (49) (S
Don,.

Hi,.,... \'.,

,.,

CRITICAL PHENOMENA AND PHASE TRANSITIONS IN GelS

AI_""""".

85

S,,,,,.

(SS) T.... b . T 0..-... .• AI~. IM1. 1GJ11. Il ($6) M. u ..... £. $., T • ...u.. T. N"".u 1992. lSi. 411. ($7) ZlIana. V. 0 .: ·r .... k... T .: SlUba,.. ...... M . HIN_ (SI) ... ....... .. M.• T .... k• • T. No'"", 1992. HS. 4.10. ($9) SulUU "'.: ·r.""b. T. NQ''''~ 1990. 3-16. J4S

1\1'92• .J60. 142.

(60) Kohf.lI. L zto.o .... v. Q., T. ... k• • T. Hili,," 1\1'91. 1$1 • .101. (6 1) Osada. V .• R...... M".pI>y. S . II. Sri. .I.... 19'\11. M8(j). 12. (61) li. Y ., It " . L: ZIlan,. X. : lit"',. C. ~ . Mort •. Sri. FAt. C 199$. 2. 221. (6l) l"'l<"bo. T.: Opoo ........ K., Vam ... k' .....: M.wok •. T., Mw.octoi. K. Mam'",oJ",..to 199$. 28. 6$79. (64) Uo.$l. (61) K. ....... ki. II ., N. kamura. T.: Mi)"llmolo. K.. Toki.... M. J. C"' ... l'Ify,. 1995. 10J. 624I . (69) s...uki. ....; V. ....... i. M .; Kubiki. Y . J.~. 1996. 11)1. I1S I. (10) Allb..... I). J. C_ " '" 1996. 11)I.4.109.

PIo,..

(7 1) lu1lan.. T" Ni",". T.: IbyaohO. T. P",,,,,. J 1996.21. 1!>9 (12) Shiboyama. M ., M .......IIi. S.: I'"", .. ra. s.. 1996. 211. lO19. (1)) K"""'M. II , Shi""ya. T .' Y....... p. H.: Ando. I. P""",,. J. 19'\16. 211. 10. (14) K.... ~.1. R.. V..... ido. M.: II, . . ...... . S.: lip..... y , yono
M"""",..,... """

(76) ".,bin.., •• II K : Fit")'. P ,. 1'....... f"'ada,!>«. 1961. 64,

(77) Flo
v.:

P,",. . .Sci. 19S7. 25, 429.

~)J S .

Kboltbl,......... R. JftJ('"",,,,,,",,,I.. 1992. 2'. ) 114 .

l)onnid"",,,,,... E. : ErulkhkJ~ ..... R. AI'''''''''k~ Sirmtl. 199oI.l. 6111.

N.

n..".,

,,,,.

(80) s..~th, 1'.: Lcm"'". I'. I.: Pijl"'r>. I. P. L.' KII)'I" Sri .• l.m, &I. 1982, 20. In. (&2) Obbe, M ~ I..,....,.. M.; M.~. II . J . .I,.". Pol'..... Sri. 19115• .10. 471S. (IJ) R. "'lomo. R.: F.d ..."rds, C. 0 ., M.noIdk...... L. M _ _• 1916. 19. Jill. (&4) Sa ..... ,.ri. E. . Ohm ...... T _ ~Ia ....... M . Pol,.... J 1916. II. 741 (IS) KoIuiJr.o. 8 " Kn ..... ,,_ Loll. M.' 1Ia/ PoI~,. S E. K~ ... 'h, R. L Mil .... . .... J K. M..... FA. 1'1c"um 1'.-: 1'1 .... YorI: 19117: p. 119. (88) MeK.""", G. II.: G l>1fm.n. II .. U,.n. G . .tJlg...... Clt.m . /"'. FA. I"gS. 27. 902 (92) M
!)om..,..

c.,

rom -'

s,.,.. .... "'mcrico"

86 STRUCTU RES ANO PROP ER TI ES OF RUBB ER LI KE NETWORKS (9-1) Klein. M.: B,. I«k¢. P. V. D .• & ' ~ n "n •. II . Ma"",,,,QJ. S,>"p. 1990.39.59.

(''') Q""""•.I. ·M .; Klein. M

(98) Ho,ton, I. C-: l)on.old, A . M . .IIa<.-omoI. $ymp. 1990.39. HI. (99) I'hise. N .. Y.ng. Y.; I';, Z; Yuan. Q.; 101 . ,k . J. E.: O.n, 1:. K. 101 .; Alamo. R. G.: Mandolh rn. L In S"'''''-<''. CIuJ'Mr.,;,,,';"". ami 11,roq of Poll'".,i<,\'r,.wb ami G,I. , S. M A~ao"Q1lL Ed . ['knum New York . 1992; p. 211. (100) h og, Y.; k hisc. N .. Li. Z. ; Yoan. Q.: M.,k. J. E.: Cbon. E:. K. M .. Alamo. R. G .: M.ndd kcrn. L. In C",,,p1ex f·luU4. E. D. SO"",.. D. Wei", T. Winon, ond J. I""elk"vili. Ed • . M at<,ial. Rcscan:h Socic1y: I'i...l>ur~, PA. 1992; p. 32S. (I(lll Smith. P.: Lem,!,... P. J.: Pijpcr. P. l . J. Pal,..,. Sd .. Poi,-",. Ph,'" &I. 1982.2(1,

1',.,,,

2229 (1()2) Huehhok 1'. L. TrM,/j Poi,.,,,. Sci. 1994, 2. 2n

8

Calculations and Simulations

The classica l th"",;,,,, of mbbcr elastici ty are oosed on the G~u ..;"n chain mooel. The only m(>ittular paralnclcr ,h,,( enlers lhe"., lhcoric~ is the mcan_square tn d· to-<:nd separa tion of the chain. constitut ing the nc\"·ork. Howcwr. there an: various arcU of iotemt that «
function for the cnd·lo-end ,-«lor. as well as for averag<:s of lbe products of sewral vectoria l qUdnlilics. as "'ill be c,'idenl in these chaplers. The foundation, for .ueh char.ctcri,,,ti,,n •. and ",,"(r.1 oxample. of their application<. are giwn in this chapter.

8.1 Spalio l Configu ralions of an Isolated Chain Scwral a"peelS of ru!>ber elasticity (such as the dc~ndene<: uf Ihe cla .. ic free energy on network topology. number of effective j unctions. and eomributions from entanglem.n!» are successfully explai"rd by theori ... ba""d on the fr«ly joimc"crrnoclasticity. rotational isomerilll iion uf'(In stretching. sl ..... in dichroism. local <e&n,,,,,tal orienl3tion and mobilit)·. and charactcri,ation of netwo rks with sho rt chains require the use of more reali st ic network chain modo Is. In thi' ",-clion. properties of rotational isomeric sta te model. r"r the chains are dis<us..N . The notat;nn i. baoed lar~ely on IIw; Flo. )· book. Stotl. lleal MeehtmicIJ of CI",i~ Mol~cule." . More recenl infom,"tion is ",.d· ;Iy found;n the literatun:' ' . Due 10 Ihe .impli<-;!)" nf i" 't rudute. a pojycth)"lcnc~likc chain serves a$ a colwelli.nt model lor di>cus>ing the SI.ti>II~,, 1 prop"rtie< of te,,) chain,. T hi, simpl icity can be ...."en in ftg~rc 8.1. which .1>0"·' the planar form of" small portion of a f'(Il)"cth)"\ene ~hain . llond Iond anglcs may be r<:g~rded

"'

88 STRUCTURES AND PROPERTIES OF RUBBERLIKE NETWORKS HHHHHH

\/., \j \j

~"y(,

/\

~1~....

'0;''/ ,~.,y(, ~.•y'

/~

/\ /\

III Til< planar fonn of. small portion of . pol)"'(h)'I,,,,,, chain. Til< arbon

corbon backbonc bond. a rc labeled from / _ I 10 j + 4. Th.c bond anglo> ... 1.b<1OO in pa .. nlll .... from j - 2 10 i + 4. The , uppl<monlary llh bond .nso, it .hown by 6,. T<>ni""al rOl.t;"" • • bou, lho backbone bond • • rc ,hown by II>< angle ~, for II>< ith bond, (I>< VlIlue ohov... 10 0".

"",,«pond'

as

fi~ed ;n Ihe Study of rubber elaslicily because Ihcir rdpid Huctuatio n• are usually in Ihe range <:>r 'mly ±< ith bond in figure 8.1 stalionary, and changing ';'i, di'place, the carbon and the hldrogen atoms On the right of lhe ilb ca rbon, and hence ~hanges lhe distance> belWttn lh. \"a.ri<)uI al<)ml in Ih. chain. hltcracl;ons between atoms resuhing rrom change, of inleratomic distances delcnnine the energy of lbe chain as a runelion of cnnformaiion. ROlal;ons aboul each skelelal bond ar. subje<;ll<) a poltnli~1. The polenlial for bond i, fo r cxam pk, i~ shown in figure 8.2 in unilS of leT as " funclion <)f Ihe rolalion angle <1>, . This potenlial or energy results primarily from tWO ""urees. (lj The first i, the intrinsic rolalional po(cnlial of each bond. which may be repre..,nted as being Ihreefold. For Ihe Ilh bond, for example,

£(';',) - (Eo/ 2)il - cos(J';';))

(8. 1)

Thi. function indicat"" Ihree minima of equal enerllY al 0· , 1200. and - 120' , referred to a. "OIlS (/) . 80"C/u> , (g ' j. "ad "ouc/tr- (g- j, rtSpttli"dy. (2) The Olher conl ribul;on ;n~ol,"CS the non,bonded in~ra~tion, be(w",," ~ario u~ alOm 1»;1'$ separa led by Ihree or mOre bonds along the cbain. The energy of inleraction between (wo such I»irs of atom, i. u, .... lIy approximated by the Lennard-Jones potential:

(H. 2)

90 ST"UCTURES AN D PROPE RTIES OF "U88 ER LI KE N ETWO RKS

well above lhe glass IconpIhcr, From oomputcr molecula , dynamics simulation s and Ihei' correlalion wilh various C~pcnlX of these Imnsitions. Ihe chains w<}uld be f",zcn "nd the network would be devoid of Hexibihty and def(mnabilily. The two-dimensional pro';"'Clion of an inslam.nceus ceniigurat;on of an unde· formed :zoo·oond pol)"Clhylcnc chain is shown in ligure 8.l·. fl ;s an example of

""II

.0

.-

.., ,"

'.

" .c.:. )

" .Og"~ 8.3 "The; 1"-<>-ond p<>lyc(h)'lo"" Chain'. Thi • • pa(ial <XlnflKuratjon ""mputer ge".

wa.

"'in!! k"",,'n ",I "", of lhe bood 1.'1,Il(h•. bond 'I1,Ilie<. rotati"". 1angle, ,boul , k.· "tal ""0'\'. ,mo"i (h, ""r""'p',nd,ng ""al i".al ;"""""'" >lat«." nd the <"""",,"i.;'y he,,.,",,n I"" .. p,.fcl'
CM''([

P""-'''''''''''

CALCULATIO N S ANO SIMULATIONS

89

,,,,

•

:;:

w,

f\

••

•

•

V

,~

8.2 Tbo 1'0I
V

• ~,

. ~1,.....

f\

ba"";, ~ ,.

,~

fUDCtion of t.... rotation "ni!Je 6.,

wlltre T i$ tilt di"~tK:e between the t",o interacting atom., (J i. the separation betw«o the atomS when E = 0, a nd t i, Ihe depth of the pote01iat wdl .1 the minimum of E. Supe'JlO,ition of ,he energies given by (8.1) and (8.2) re.ult. in a po'enlia t ",sembJingthat shown in ftgure 8. 2. In gene"l . lhe loca tion and the energy value, of t/lt minima may be di$pJaced fr<>In the ".Iues 0' . 120". and -120" indiealed by cq , (8.1). and there may be f",,'er or more than three minima. Thi, depend" of 00".-..::. On the local chemical st ructure of the chain; for e~ample. the Ottu rrence of a double bond adjacent to the ,ingle bond undcr comidcrntion. The w311s aroun d .he energy "'c1I~ • .,.,h as th"", , hown in figu", 8.2 are generally surr",ien.ty """1>'0 oonfin. the bond rot~lion' 10 values close to the minima of lho w.:lls. Thu,. the bond rOlalional angks. in$lead of ta~ing a oonlinuum of ,,,,Iuc. bel,,'ccn -11 10 + "-, may Ilt a llowed to reside onl y in Ihe three minima represented by the I, g ' . and g - state" Representalion of chain configuration, by contining the skeletal bond angles to iwmu;,; minim. alone is ,..,rerred to ~s the rotational isomeric .tate (RIS) appro.imation. Thu,. Ih. chain conftguration i, rCI""sented a. a seqU"""" of skeletal bond,. in various disc"'te rotational ,totes. This approxima. lion was firsl proposed by Volken't"in" and waS applied mOSI .xtcn,i,oel), 10 the analysil of poly",.. chain$ hy Flory and hi' ",search group"" '. Two atom •• Io"g.he chain are called topologically distant if the)' arc scparated by se,oeral olher alom, along the chain contour. Two atom, may be 'palially dose '0 each othe< ev"n 'hough Ihey are topotogicalty distant . Exe1uded ,volume dr"",,' a re .aid 10 Ix prescnt if Ihe energy contribu.ion of twO topologically distant atoms, gi"en by cq , (8.2). is included "'hen ""alu"ting the total energy of Ihe Chain . In 'lle ahsenec of c,cluded·,'olume eIToc ... 'he ehaill c,' hibits it" SQ. called "'thet"'" di mension, or unperturbed dimen,ion s, It i, now welt established tha t Iher~ arc no ". cluded."olutn(: cIT~1S in chains in Ih. "ndeformed bul~ StaIC' and .hal the chains e., himl thei r tl"'ta or unperturbed dimens ion•. A chain v,ith n ,,"'nds. each hQ\'i ng m ',,-,meric "at",. has m" pos>ihlc rota,ion.1 "ales or configurattons, For a chain in dilu'e sol ution . as well os in .he hulk ,t.,c

'
CALCULATIONS A N D SIMULATIONS

91

one in'ta,uanC<)us "",,"figuratio" of the chsi" out of a possible 3"'" other isomeric """nfigurations, This spali,l configuration was computer gcneraloo using lnOl'ln "alu", of the bond length., bond ""l',Ie" rotalion"l angles a bolll skelel.1 bonds, prefcnon<'C>l ~mong the corresponding rol"lion,,1 i... meric Slates. and the cooJ'Cr; livily belw""n th"'" preference" The ""'thod of gcner:u ing ,nch configurations forms lhe basi' of ille Montc Carlo ( MC) technique. "'hieh i. an efficient numer· ical techn;que for evalua'ing ,,",is,ical ""raKcS for polymer chains, c,peci.lly shorter ones. A configur.'ion of a ~h3;n is fiyste"" suell a, the O1'e labelnl XYZ in figure 8.4(3), Such. coord inate s}.tem is ",fcrred to as a ··Iaboralor)'·f,xcd S)'" tern:' The bollds oflhe chain arc numben:d fro", I to" and the backbone aloms from zero to II, T he li rst , kde'ol hond i, eho",,, along the X dirn::tion. a. shown in figure 8.4(a), and Ihe second bon,l is in Ihe X l' plane. Sdocrjng Ihe firsJ IWO bonds in this way fixe, Ihe posilion of lhe chain in space . 11, figure 8.4(b), Ihe local ooordinaIC "xes for jnlemal bonds 3re iltow'". with such a local ooordinale sy>!em d~finnl for each bond. TI,e Xi axi, is in the direction of Ihe ith bond; the y;
v

,• .. .' . '

4

,

•

,

2

'., ,,

N,

,

''l , ...,.", H,4

(a) 1'1,,,,,"on' of a

nc1W"'~

chain "' •

.y'..m. (I» Oro",." ... foaou"" or. '''''''·or~ ch",".

," '<

~'+I

,;~h '. h . ,t
('.t1<si." ,'OOI'di".,<

92 STRUCTURES AND PROPERTIES OF RUBBERLIKE NETWOR KS

in the plane of bond i-I and i, a nd ii' dircclion i, cho..n 10 trulke an acUle angle wilh T~ :: uis oornplole, a righi-handed syslem , In figun: 8,4(a). 9, rep"'· senlS tnc supplementa,y bond angk and ,j in Inc ith coordinate sySlem

y,_,_

b, (11.3) where the transformation matrix Ti(O,
Ti(O, ¢» '"

~" si n 0, c,," ¢i

- cosO,cos ¢,

.in9; ,," ¢>,

- co.9, .in ""

sin8,

[

(8.4)

and Vi and Vi"" rtprt$CJt! lhe SlIme 'OttIO' in Ihe ith and i + lSI coordi""l. syslems. Il"p«tively. Using thi, transfonnalion oremtion . II>< pOsilion " or II>< ,)h atom rclali"" to the labor~tory·nxcd coordinate system i, wriuen as r, - I,

+ T, I, + T, T, I,

... ,.+ T, T1 " , T,_,I,

(8.S)

Equalion (8.5) may be wriuen in cor.ci .. form as'

,

·-'I ,,

';~ 'X' L. .... 'j

(8_6)

"

Ti'

where ' I represent' Ihe ... , ial proouct of transFormalion matrICes. 'Iarl mg wilh Ihe lirsl and conla ining (k - 1) f"lors "" identified by Ih. supe"",.i"l.

8.2 Statislica l Averages of Configu rational Variables The sci of posit ion '-cclo.. {" ) for all N kockbonc alOm, uniqllCly proscribes a given in"antanNlUS co~figuratio n of ~ chain . The", is one-Io-one corres rondcJ>Ce be1ween Ihe ...1 {r,) and Ihc SCI (4) ), of alllOrsio nal angks of the ch.in, The prnkobility of occur",,,,,,, nf each such con figu rali"'t. as explained in more detail in appendi , D. il gi ",n b)' •• p( - Ell",) ,/kT )

P( {~),) = LCXP(- I:"({¢),/kT)

(8. 7)

The average of any Funel ion or Ihe ..1 {t,) may be calculated according 10 I"" relalion,hip

(f( (r,)) ...

f

L f( { ' ;) ) ...p( - E({ ¢>!,lkT ) III r, })p( (¢),) =

{~I, !:>_
t·( I ¢),Il'/")

(8,S)

CALCULATIONS AND SIM ULATIONS

93

""here lite ,ummation. arc carried oul Over .n stlS of ~onngur alioIlS, This averaging may be carried Oul in variou, "",aci <)r appr<)~imal. ways. [n litis ~hapler. ils appro~imate .v.luali<)n by Monic Carl<) ,im uiali<)n$ will be OUllin<:, as represemed in figure 8,2. Formulations for cbains with mo", realistic inlcrdependcm bond torsional energies an: deocribed else",·here, ·,·,G. " . The MC scheme is based on generaling" chai" configur.>lion whose proixlbilily is proporlional 10 the numeralor of "'I. (8 .8). The 'Ieps "'" al follows: (I) Fi. lhe Ii",t bond olon g Ihe X axi •. and Ihe second;n lhe Xl' plane of lhe loboralory_ fi.ed coordinate ~yslem. (2) ChO<>$( the "... lue of ¢>,. This will determine lhe dire<;lion of lhe Ihird bond in space. According 10 the rOlalion.1 i""meric sla le ~lure,lh< values all he energy minima "rc assigned 10 each bond lorsionalangk, ':',. A"uming Iram (I). galldw · (g ' ). and galldw- (g- ) , slales for lhe bonds, one defines Ihe Ihree Slati,lical ,.-~igh1< u,. ~. " and w , as

u, - exp( - f::,f RT),

a• .. exp( - E,- / RT} (B.9)

whore E" E," and 1;;;, arc Ihe energies of Ihe Ihn,. Sia l"" of Ihe bond. The three .latislieal ~ighls gi\'cn in "'l. (K9) define Ihe probobililie$ p,. p", a nd p. _ or,he three .Iale, 3. p, ~ 11,/(",

+ u,' + u, )

PI' '" " ,,

flu, + U, '

-t u. _)

P, '"

J(u, -t a,'

+ ",-)

II,

(8.11))

Th. Sial. of lhe bond i•• <signed "ccording 10 lhe probabililies given in C<[. (8.11). A uniformly distribuI<:c of Ihe subpopul. lion OVer which (h. a.eflOgcs are 10 be cv.lwlled . Each of lhe N configura lions gcncraled by ,hi. ""hem< obe~ Ihe prohabilily of occurrence giV<.'n b)' eq. (8.7), Thu" Ihe .'-crage (((, ,» gi"e" by eq. (8,8) may be wrilten as

,'

(f(' ,»" N L

.<.f( r,l,

(g.ll )

Here, f( " )' i. lhe funclion o""",,;:lIe'
94 STRUCTURES ANO PROPERTIES OF RUBBERLIK E NETWOR KS

T he sample program Ii,'en in appendi, G calculales: (I) the persistence length of a chain in Ihe coordinale system of figure 8.4. (2) ils mean-square cnd-to-end ,-ector, and (3) the orientation collfigumlion funclion nf its middle skeletal bnnd. (Sec chapler 11 for Ihe definition of Ihe orienl.l ioll eonfiguralion fUtlCiion .) Applic"lions of the method for gener'ling non ·Gau .. ian dislrihulions for the end·to-end 5C]>al"dlions of network chains arc dis(:clion . Thi s is followed by the use of such distributions in predi~'ling pendix F. Going from the usual "Slruclurdess" mlOlccula, Ihcor;.s of rublxrlikc elasli. city" to the prc'$ent dctailOO repr=ntalions parallels going from the Ihwry of ideal ga$<S to the van dcr Waals tll<.'O<)' of ",,,I gases. The advantage in bnth caSCs is a mo", ",.Iistie portrayal of the system. but al til<: loss nf uni,..,,,,,,lit)' (ill that additional information spccilie to the chosen 'ystem i. ""'luircd). Some Iypical distributions calculated for poly(dimethylsilo>ane) (PDMS) nelwor~ chains having n ~ 20. 40. and 250 skelelal bnncl, are "shown in figure 8.S" . The three so lid CUC"CS show how Ihc distributions change, mo'ing to lower values of r a, " increases. The Gaussian distribulion is seen 10 be a re!atively gO<Xl apprO,im"tion to ,h. MOniC Carlo dislribution at the la'gest value of ~. as expec ted. The effect, of foing to 'till smaller ,'a lucs "f F are illustrole
,

• I • ,I

.-," ~

•

I

, •

\ 250 I

\

I

\

I

\

I

I

I

I

••I

40

~

M

M

r/nl ~· I~ .....

S.! S lou ui.alo.J d i",ibulion, of .00·1"-<00 «:para'ion, for ~1y(<1im<"lbybilo , · a n.) ch. in" . s obtoin«! fro m Monl. ClIrlo "mol;, ;on," """'" "". ro .. hon.1 isomeric sWe model' Tho POMS cha in, h"" (he sr-cificf lenglh /, .n.irnalioo for n _ 2S<1 i. sIIown by ,"" da,ht<) hne, '11>0 Mon ,e Ca'Io Co"'" '.-presonl
i< •• plin< ~ .. to ,he dio.m. 'I",ulal;"" d,,,. ror Sl,l.OOO cMm" and each co",", i, "",mal i"",, (0 an of "Oily.

.k.ltl"'

.rt.

,.

•

", ,

• • , ,

'" " r lnl

Fi~.",

8.6

Monte C,,,lo dist ribu tl on,.t 298 K fn, ,,,,,,'I' ;"'d ('" each C""~"

ber of , h lo:,.,

""n
'C<)'

. I'IM' I'DM S ch"in •. 1110 num·

96 STRUCTURES AND PROPERTIES OF RUBBERtlKE NETWO RKS

f\

• ,

/ \ (,,____ _, II! \

f't..

: I: ".-\ . .1 I I, \/ / / \./ ,: , ,,, I

r\

I. "t' \ \ \ '. .f~'\i " '\

~iJ\.//

.......

.

\"

'-

.~~~~~~----~--~~~ 0.2 OA 0 .<1 0.11 , Ift l

FIR." 8.7 Mon" Corio di",ib.lion. al 4I l.2 " for.n n-alkon. chain b."';n, 10 , k.lo· 1.1 I:>ond." The >oM run'< "....,.111< ... ulU for inlond rO"'ion•. III< long-d.shed cun'< for in
curv~

for f,..., rota·

of the di,lribulion is, in fa£l. incr~aoed in going from an inlcn!cpendenl-,lale RIS modd 1(> an independeni-Slale RIS mode1lO a freely rolaling model (wilb no ,otational siales al all). This shifl belween mod~l. corresponds to an increase in chain flexibility. which eau$O.'S a marked Mood.ning of Ihe di.lIibution$. Ihrough incrca<ec PDMS chain. is seen 10 be shifted to values uf . / n/ sumcwhal1argr I'E. as would be e,pected from its larg.. val ue of the characlcn>!ic ralio' a""""jaled wilh il' unj>n.'. AI"", included in this figure ;011>0 Gaussian limil for Ihi. value of II. For SlIeh . hort chain •. the Gaussian di'!ribulion is seen to give a ,c'y poor approximation to the Monte Carlo dislribuliun •. T hi. is p"nicularly true in the region of la'ge r. which i. a ,cry important factor when interpreting Jimilc:d chain extensibility and ela'lolner mplur •. The disercpanC}' wa, found tu become wof"1( as n wa~ decrea&cd ' )·". Chains of sulfur ur selenium are known to have extreme ly high 8exibility bccau~ of the lengths of Ih.i,. bonds and Ihc lack of any chain SUbsli'ucnl. l. Distributions obtained for these chaiM a re ,b(>wn in figure 8.9. wh~h al$() , hows a corrcsrunding dislribUlion f(>r pOlyc'h}'kne and .he G au"ian li mit "''''. The ne, ibility of the S and So chain s i. dcmonstr"dlc-d by the shift. in the most_probahle "al,,", uf r (as located by Ih. ma,ima in Ihe cur,,",) lu values .ignificantly lo""r than tha, shown by 1",1)'elhylene a, the same chain length. Not surpri,ingly.

n

", ~

"

~· igu ..

• ,.

'",

".

8JI Compa,;..,n •• mon~ .he Montc carlo d istribution •• , 4 11.2 K r<>,

""I)'dh~.

\c"" (0 ) and poly(dimc'hyl,ilo,"",,) to) chain. ha,ing" m :W
"com"",.,,..,. of

"",

~' j~.... 8.9 Mo"'" Co rio di"nbuoio n, , ' 298.2 K for ,"Ifu, and «10· nium ,,-jlh polyelhy""e" for .hain; of l() , k.leta' Oo",,h". Th.c dailieJ is tnc Ga" .... . .ppw,im,uOJl ",. chain of 'he ",me r",,' _mo~n_>q"'''' ""d .ln-<:nd dis"""" .. Ihe S ,""-in.

<"""

98 STRUCTURES ANO PROPERTIES Of RUBBERLIKE NETWORKS

tile GaUlOian disuibution i•• bcttu aJ>P",.imation for these chain. than it i. fo r pol)tthyltne. Some chain meleeulei have •• tron, ttndt-ncy to fQml ""tical rcJi""~ ,,-ilhin a OOI1fonnotion that i. othervo'itiC ..~ndom-<:oil in rnu~rc. These chains ,,'ould ,,,,,,,,fore be: C' l""'l«1 10 h.,·. multimodal dimibulionl of Ihe cnd.l"""nd di$1ancc. partlc"lurly in tlk: . a", of "cry short Chains, One sueh chai n mo lecule is poly(o~}m.thyl""') (POM ). wilh rcpo:a' unit [-CH,O _ I,.l(I.21 The bimod.1 chanactOt of its distribution. at ,, _ 40 skeletal bonds ~nd al m.2 K. is >I>o" -n in 'he Uppe""...1 part of 6.",.. 8. 10"'. The """ainin' I""" ofl"" r'lUK~' , IK: COfImbu,ions of specific conformalion •. opccifically a I"'rI'ccI rod. a once-broken rod •• ,wiDe-broken rod . and 10 on. S""h ' nf""""lion CoIn Ji .... valuable insigh Ll inlO lhe oonfom>atlonal "",~cup of hclK:olICnic rei)'''''''' and !lIcir elaSlomenc rcspon5«. As c~pcc'«I. applicalion of an c~tcrnaL force '0 ,,,,, emls of Ihese chins "". fou nd In .hift Ih. COC~'Sl.ncc equilibrium ""lw.. n helic,,1 .nd f""ndom-<:oil 5<:
• • • • • 0

•

• •o

0

-

~

rod

, ...... ",n-

0

• • 0

• 0 0

• • • • • ••

3 .......

........ , ....... , ......

.. .. ..

,

R""'I., . .... or ,he: 1010'' . Caoo dl>lribullO ... rur poI)'(u. )'1nClh)'I.... ) !''OM I.' m .2 K. fOf h'''' "I40 . k.lcta' "., ...... '_ .n.. hrcoh ..--.fled r.,. .""~ ctI~lfil."'i,,n <""."pond 'u ~ ,n~. in ,II< mll< ....'''''hol'''''l t ha,"" t~

(

'. 10

CliP

,,""ift.

CALCULATIONS AND SIMULATIONS

• •

~- ~ ,'~ -

I.

,\ I \

•" ,•

I /" I

_li

, M

99

, ,

"

~

"

~

~

,~

M

"

M

r;~g .. 8.11 EIr".:, : I'OM chains de>erib: dOllc;l. ""lid. and dashed r«,;,,,ly. Rop.in'c;I wi,h pe,miMio" f,om CUffO. J. G ,. 0(,1. (1<)86). Mayrigh, 19\17 1I"""ic." Ch<mic.l Soclc'y.

Ia.,.

nificantly lhc coniribuliollS frOnt the hclic;J1 scqucn=. as identifiable b)' their larY'r values of the fractional eXl'alli'.alion of clastome.ic materials. as described in chapter 12. Montc Carlo eo"'pOlrr simul;,tions have been carried out on f,lksian characlerislics of the chai"s Of] the clastic pmpcrtie. of lhe filled .... t· works , h' th" . arl;'sl .""h I'ud),» '''. i, was assu,ned that ( I) ,ht filled poly=r network CQn,i!1S of " c
100 STRUCTURES AND PROPERTIES OF RUBBERLI KE NETWORKS

.W

--- .. ..,,...

-~

0.15

__ . _· Sll

_ • .. .• 0

0_10 ~

•

.~

'00 ~~

•••

.,

••• , ' r mn

F'¥ ... 11.12

Radial di"ribu,ion f""".ion.

"

,.

!'{.J fur 11-.. er>d-h, ·..,oJ

" .~tor

obtained by

M."" . C.rIo .\imul'I;"n •. ,h""!n ••• funcl;on of ,he relali"" chain ... o• .\ion , /,_, fo< poly(dimetbyl.. I" ..... j ....works'" In thi, figure. and in fiS.'" 8.16.nd 8.17. T .. 5OO K. ,lit cb.h,. h.ve n .. SO , k.letal bond. beI,....,n cr<>lS-linb. ,be ..-.ighl pernon' of filler i, •• ,,""'ihtd. or>< radiu. of ,lit filler ",,,,,,leo is SO nm .

partiele . The effecls of these ~h.nge ~ in the di ' lI-;buli(m s on the i""lbe.m, are discussed in Ihe following seclion.

sl~ lra;n

8." Stress-Strain Isotherms Calculated from the Non·G auss ian Distributions The Monte Carlo disuibution, Ihus obta ined can be used in Ihc st.ndard ·· throe. chain" model'" for rubberlike el.,ticily. Some typical resu l... in termS of MooneyRivlin ;""tm:nn. for networks made up of r UMS chains of ,'ariou' lengths. .r. presented in figure 8.11". As e.pecled. Ihe network con,is ling of relatiwly long chains (n = 250) gives the Gaussian result If'J/vkT = I. The upturns in [1' ) obtained at smaller n ~re very similar 10 Ih"", fou nd experimentally in bimodal nClworh containing la rge prorort;on, of very shorl network £hains"". Such ncl_ works having unusual nelwork chain ·kngth distributions are discussed in grealer detail in chapter 13. The r.sults in figure S.U also show Ihal th. shotttr the nelwork chains. the smalieT the elongation al which th. upturn occurs. The Montc Carlo simulalion. based on the 'Olalional isomeric stale model for tbe network chains have thus bttn very u",ful in interpreting lhese uplurns in modulus. It i, al"" possible to interprel Ihe upturn. in modulus in thcoe i""thenns using analylical expn:s,si,:ms. for .,am ple .he Fixman_Alocn modification " of the G .ussia n dimibUlion function. combined wilh tm: constra ined -junc ,i on thoo,y an d reasonable ,'alue, of lhe constrain, par"meter ~." Sim ilarly mcfu l dislribu -

CALCULATIONS AND SIMULATIONS

,

10

-

• :=.

-

->- , -

--

, , ,

,

•-

6

-

20

40

-

~

>

-

~

2

o

0.4

tOt

250 " ' " I 0 .5

I 0.6

I 0.7

.~ 0.6

0 .9

1.0

l /a .·iK_ 8.13 !doduli predict«i for I'DMS not ••,,,• • h".i", 11 - 20.40. ond ::10 . kel< .al_ of [1'1 -'" norm. liad ~~ ,II< G. ... ia" prediction for tbe modul .... vkT. whe,..,,, i. til< "umber of ".."""~ chain...d ~T hu tt.: ,",001 >ignif"",Dt"e.

bond,".

lion runcli'","$ ha,'<: bttn obtained by mean, of ;phe.ical hannonic ",presenlations of r<>ialional opera",rs"·'"'. R.lhcr unusual .I,..".,.."rain i$Olhenn. arc obtained f'" net",orks consiSling of Ibe highly nexibl. S or Sc ch .in,. as is illumaled in figure 8.14". The normali "-"sian net ....ork•. Thi' i, qui!e dillerenl from the isO that .he e' p"rimen!al uplurns reported in the literalure for natural rubber were morc likely due . train-induced ~r~S1al1i :t.a!ion than", limiled chain extensihili !y. The ;sothenn. f", Ihe POMS chain. in the vici ni' y of .. ndomly placed fille< particles are . howlI in figu.... 8. \6 and 8.17"". Specif>ea lly. value. of lhe slress j. reciprocal elongation arc .hown in ftl\ure 8.17. The . unstanlial 'oc ..... a..:s in Sire.. and mod· ulus wilh increase in f' llc< OO lltent and elongation are;n al leaS! '!ualila.iw agIC"C· ment with experiment. a. d=rib"d in "haplCr 16.

.0

102 STRUCTURES ANO PROPERTI ES OF RUB BER liK E NETWOR KS

rig . .. 8. 14 Compori",n, . m"ni 'he ......... "rain i""i"l
C

,.

" " •

= = ,• •

-• ,

...

. ..

." • , '.'

•••

.•• ••• ••• ••• ~

.;';. " U 5 Norm.liz«! moduli pnxlic'e<, chain' h."in, ,I>< numb
The", a", a ownber "r din.....1i..... , in ",hich such filler . imulalions could be extcnded . For example. one ,h"uld iMcs,igale difT.",n, extents of panicle.Sil."'" galion . and d ifTe",n' panicl.,.,i"" distribulions. It ",ould also be impottant t" mod. 1 physica' adsorp'ion or chains onlo Ihc fIller surfa"". IIsillg Slanda,d l..en ll;r,d·J(mc> inlerac,ion polenlial •. Chemical adSOrpli(:>n. on Ihe o,he, hand. could be moockd by randomly dimibulillg acti,·. pa"icle si le;. and then

~oo

».00

,

,"00

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,

» 00

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,"

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>

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.

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.

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•

'.

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dl!i!!~::····

FiK"'O 8. '6 Dopooo.o",", Qr I.... no, ..... I;,..... nominal fur tile ,"ndoody fill«lI'[)M S "",works"',

•"

, •

"..,.,.r

I~ I/r

on tho elongation"

. ,•

•

.~ $....... .....!mllnflltll .. •••

"' ". "'

1

tt

•••

It

f

,..

t1!l .... KI1 '"", "",mali"..t loro"" ;':" modulu' lr l/vllT, ,ho,.. n a,. r" ...'lion or ."" ,ecipro<> modornly [,\100 !'DMS ne,works'"

104 STRUCTU RES AND PROPERT IES OF RU BBER LIKE NETWORKS

interacting chains with them th r<)ugh a Di rac 6·function type of potential (with chain. at 1= than some .horHange interaction distance becoming chemioorbed). Of particular intereM would be .imulations for chains sufficiently long to partially adsorb Onto several liller particles.. and to modd chain-«>mour distributions between the bulk polymer and the nile. rarticlcs. T his approach could ",,,11 gi,-e importanl and moch.nc-edod insight into mo1e<;ular a.peet. of elaSlomer reinforcement. For otber MC simulalions of non·Gaussian chain configurations. the reader is referred 10 the work of Haliloglu and CQllaborators l .\l6. In these st udit$. wnlig. urati""s of pol)"t hylc"". polyoX)'elhylcne. and poly(dimcthylsiloxancj chains were generated. and segmental oricntation in highly stteleh""' . non·Gan"';an chains ",'as evaluated.

8.5 Molecular Dynamics Calculations The MoniC Carlo calculations described in lhe pr<:<:eding sections are single..:hain caiculations where the detailed behavior of Ihe single chain is studied and lhe behavior of the nelwork is predi<;tt
,-

(I) Flory. P. I . SlalislicoJ M""Mnic, 0/ CM;"

,-

MIJkc"I<~.

Inl ..",",""'" N... York .

(2) Matti<><, W L; Suter, U, W. Con/",,,,,,'iomJl T'MorJ 0/ Larg. MIJkc.I... 1M 1(ol~l;""'d

I , ,,,,,,,...

Sl~"

M """,I in M"",QmQI,nJ",

S}'SI~mJ_

Wil<:y' Ne,. yu,k

(3) Honc)'Culi. I . D . In 1'Io,"iroI f'T()p&'k~ 0/ ['()/),"'''' 1/wuJJxw/<. j, E. Mark. Ed. Amon,." In,''''''< of l'hJ"ic> 1',-.,..: Wood~"ry. NY. 1_: p. J9, (4) ll oe, II,.), In Ph,.,;""1 1""1'''';'' 0/ [,(>I)'_rl lla"~. J, ~;, M .' ~ . Ed . A.,.,,;c."

I""ilul. of Physics Press: Woodbury. r-'Y . 19%: p. SS.

CALCULATIONS ANO SIMULATIONS

lOS

(5) Kloczkow,b, A. In Ny.1N1 Prop"';', "I PoI,-,s IIrmJJ>oo1<, J. E. Mark, Ed. American In,,"u,. or Physics I...... : WOO<;'o= New 'Vork. 196],

(7) FIery, P. I Mac.""",/uul.. 19J4, 7, 181. (8) Flory. P. I. P,;nn CII..",,,,al Socie,y: W.shingt"'" ' DC. '993: p. 1. (!O) t1ory, r . J,: YOO", D. Y. J. CM",. I'hys, 1914,6), 53~. (II) Yoon. D. V.: Flory. P. J. J. Clot", Phys, '914,61. B66. ('2) Mark. J. E.: Curro, J. G. J. <-·M",. Phy•. '983 .19.5705 (13) CUlTo. J. G .: M.rl:. J. E. J. C""'"'. I'hy•. '984.110. 4~2L (1 4) Mark, J. E.: C urro. J. G. J . C~ . PhF'. 19114,110. 5M2 (15) M.,k . J. E.. Cu"",. J. G. In C"'"",,~,ba';,", "j'IIWhl}' C.""·V,,*-,,t Po/yr>wrs, S. S. Laban>. ond R. A, Diclie, f.d., Ame ...... n Ct.emkal Socie1y: W~ .hinston. OCt 1%4; p, 41, ( 16) M.rl:, I. E-: Cu"o, J. G. J. Potly"' . .\·(i .. P"'/Y"" Phy•. t:J. 19~5 . 23. 2629. (ll) M.,k , I . E.: Delloll . L, c.: Curro. , G, M I/Cro",ok,uI~, '986, '9. 49'. (IS) E,man. 8 ,: ~brk. J. E. A~n . Roy. P~ys . ehr", . 1989.40. lSI. (19) Mar • • I . E.: Curro, I, G, J . CINwr. Ph}", 19114, SI , 6408. (20) M.rk, I . E. C"",p. Po/ynr, Sd, 1992, 2. lJ~ . (21) Curro. I. G.: M.. k, J, E, J . Chrm. I>~,.. . I!I8S. 82, ].820, (22) Curro. J. G .: S<,;:o"" , ki, A.; Sharar. M , A.: Mark, J. E. Clrtnr. f:",. Sci. I~. 49. 2889. (25) Sh.rar, M. A.: KIoc,kow.k i. A.: M.rk. J. E. Comp, PoI,m. Sd. 1994,4. 29. (26) Yuan. Q. W.; Kloczk",,-,.ti. A .. Mark, J. E.: SIu"af, M. A. J . 1'&1>", $(i .. p&},,,,, PhY" Ed, 19%• .14. 1~74. (21) Sankara. 11 , A,: Je(hmal.,;, J. M.: " ord. w. T. Clot"" M,,,,.,. 1994.6. 362. (28) Sunka ... 11 . 11..: Jo,hm. ',ni, J, M,: " ord. W. T. In 1I.1·"'Ul O'~_;'-'I""'J""k CI)tnp;JJi"J. I. E. MOl' , C, Y·C t"". •"d 1'. A. IIi ,,,,,,,,,;' 10rJ •. Arnc:ri<:.n Chemical s.o,;"'ty: Wa .hill.([lon, OC. 1995; p. 181. (29) T rdoar, L, K. G . T,Q/IS. f"""w,' S<x, 1'N6. 42. 71. (30) M.,., J. E.: Erman. II. JilJbJxdik~ f.'/Ib,kily, II Mokcula, I',in"" Wil<}'_ Intorx;' ~: Now Vork. I9&!. (1) I'i",,,,-,,. M.o Alben, 11.. J . O,..,n. Ph,.•. 1~IJ, 5~. '553, (32) Ernl.n, It: Ma, •• I . E. J. Ch< ~, . PhJ ~. 19811. 89. 331 4. OJ) LIo"'"tc. M A.. Kubio. A. M.: " reire, J. I . MU1;'"" Sci, 1991. I. IS!. (7) G"o. J.; Wei!>.:r, J. H Mac",,,,rJkcuie> 1'1'l' , 24, ISt9, 0") Gao, J.: Weine •. J. II ..1/a,,,,,,,rJkcul,, )'191. 24. 5119 (J9) (;.0. J.: Wein,... J. H. ,If'H'''.''t>kcU/'' 1992,2;. lJ4S. (41') Goo. J .. \\'
106 STfWCTURES AND PROPERTIES OF RUBBE RLIKE NETWORKS (42) O re'l. O . S.: K"""",. K.: Dlxr;ng. E. R. t""'p~y•. Uri. IW2. 19. 19S. (43) Gr-.-.l. G. S.: Kr=K'r. K .. Ducring. E. R. Phy. iro A 1993. HO. 1993. (44) Duc,ing. E, R,: Kremer. K .. 0 ..... G . S. IWJ. 26, 3241 . (45) I)"".ins, E, R, : K"' .... " K.. G .... , G . S. j, CJ",m. PI<)" , 1994. 101. 811>9. (46) Kremer. K, : Orcsl. O. S. In ,uoo" Catlo """ M'lkcular /),'namk . Sim"lali"", ;" K Ilinder. Ed, OdOtd U ni''''-';ly i'ress: O.fOtd. IW5: ~ , 194,

M""",""",""""

"oJ,.""" *iM
9

------- -----~~~------~

Th ermoelasti city

The importu,\l po configuration to be independent of tempe,oture. Undor those cireumstances. the dependen'" of stre", on tcmpcwlUr. is mi~ingly Simple. as .hown. fM examp le. by the <'
((r\)< -',

f . - ,·k"/" V IT);

(t - <>

(9. I)

that charaCleri,." a rolymcr nctworl in elongation where. it should be ...."<:.1I1ed. (,.2):12 i. proporlionalto th~ "olume of tile network. This additional a"umplion that (,.l). i, indel)endcnt of te"'rc"'turc would lead to the prediction thaI the ela\lic Stf(:!.S determined at ~onstant volume and ci<mgation 0 is dir«U}' prorortional to IIIe absolute lemper.ture. Such netwurk chain, " 'ould be akin to Ih. particles of an ideal go •• whi<:h wn ntri bu\lons to !J.l:.d , il should be n"".lIed lhal deformation of. rolymcr nct"'or~ resu lts in an inc".,", in Ih. number nf lho>c chain conf'gum,;ons of relatively larS<' end·lo·end separation that. in .<>nfonnatioru.1 term,. T"'luircs " redistribution of rolational S,"!CS ~oout lhe ho.m,h of 'he cha in h"c koonc S",c< in"anably oo"'e. or "II. of the rotational ,'a''''' "c~~"ibk 10 the , keto!a l oond, of a chain molecuic JilTer in energy. different chaill conligumtil>ns gc ... m ll )' d" ditTer in energy"· and. cor",·

'"'

108 STRUCTURES AND PROPERTI ES OF RU99ERLI KE NETWO RKS

.pondingly. <; ). generally depend. qnite signiflCanlly on t~mp"ralur., As a con_ !o<'1uencc. c~perimenlal "alues of Ihe ciastie force or SIre.. are genecolly found nul to 1><0 proporlional to the temperature. Most importanl, as shown latcr in Ihis ch"pt~r, the c.tc,,1 to which the experimentally obscr'iC-d relationship I>lccular theories of rubberli ke elasticity rtt(>untcd <arlior. namely that the retracti ... furcc i. essentially entirely inlr~molccut"r in origin ,

'''i,

9.1 Theory The primary goal of lhermoclasticity sludies is Ihe quanlitati". resolu 'ion of the cb.tic foltt inlo iii cnlropic and ">erg~lic OOmponcnIS" '" (9 ,2)

AS di=.scd in pr~ing .bapters. the ."t,upic contribution /, is generally ."peeted to be large and positive. ari,ing from the dccreawl enlropy of the .t"'tched chains. The energetic contrihution!. may 1><0 presumed to 1><0 of either sign. depending un th~ confom,ational chara<1cristics of the network chain,. and thus may either augment Or diminish/" For oonc",lone,... the equaliuns characterizing the StreSl- temperature rel.lionship:; for a pol),mer nctwu rk arc pr... nted mOSt clearly b)' consideration of a p:;rticular type of 'train. Unia , ial deformation (elongation or compression) is chosen for this purpose here. since this Iype of defonnation is particularly simple to visuali'" and eloar.ctcri"". and il is by fa r the mosl widely siudied experimentally·· ''' ... (The thermodynamic "rgu monli leading to the ''
<3"'.

f,

ii

(fJJ::I {)Lh ,.

(9.3 )

witb Ihc 'ymOOI, h.ving their u,ual mean ing" Its fr"ction~1 contribution to 'he total fo"-,, i. given, withoul appro.in,ation and witho ut ",Iiance <>n any parlicular model of a polyn",' ncl"'ork, by llIe thc'modynamically exact equa tion'

THER MOELASTICITY

109

(9.4) This imponant ratio is th us IlCen 10 be directly atla;nable from force-temperatu re mt'asurement; at cOII,tant length "nd vntum •. , The vnlume of tl\(, n.,twork may be: held constant. as cralure S<) us 10 offsct c~acHy Ihe efT""1 of thonnal expansion""'. Allernatively. Ihe network nla y be studied , 1'101. len in equilibrium w;lh a diluent or diluenl mi , ture~ in which the ehange in pol),mor--$Olvcnl ;nteraction, wjth temperature cuetl)' nulli fy the """'" t"'rmal efleet •. In order to minim;,e •• perimenlal diffICU lties. ho,"",,,,,,. thermoelastic measu",,,,,,m,, are genera lly carried oul wilh Ihe network> al conslanl pressure ralher than al con slant volume""'" In'erpretalion of such isobaric Ihermoelastic re,ults is nlosl cO""cnienlly .cco",plishctlthrou~ adoption of a 'pecif", elastic equal ion of Slat., The uSC ofth. thcoretical equalion of ,tate derived in chapter 2 fo r nelwork, in unia_
1.11 ~ - T [btn(f / T)/bTl,,,, - fJ T/(a' - I)

(9_S)

for data obtained at conlt.nt lenglh and Pf(Cssurc.. wher<; fJ _ (b I" v IbT), is Ihe lhermal expansion ""efficient of the nelwork .' Sm,il.,ly. for oondi.ions of oon· .Ianl elongation (or compression) a"d pressu re:

f.I/ '" - nil I,,(f/ T)/(J1l"" + ,~ T/ 3

(9.6)

In the case of studies c~rricd out at constant leng. h. n1CaSun:mentS in Ihe vicinity of" _ I ineur large uneerlainli"" due 10 Ihc form of Ihe denominalor in the difference tenn /Jr/(a) - I ). Hnd the difficulty "f ohl.ining sufficiently precisc val"", of " under .hese condilions, The efT""l of Ihis Ie"" can obviously be minimiled by carrying o ut lhe (nree- temperalure mea,un:menl. al f(Cla.;""ly high ddorm"lion •. Then . P T/ (Ql - I) approaches ~ minimum ,'a l"e of ,ero in the C4>e of elongalion (a> J) and - fJT in thc casc of compression (a < I). Under Ihe alternative cundilions. Ihc elongalion or cumpression" is ilSdf main· tained al a coml~nl value by appropriate adju.tment of the lenglh of ,he deformed ... mple 10 offset e_",cily Ihe changes in the undeformrd lenglh Ihal accompan)' the changes in temperature:. rn sueh .,perimenl' al cuml.nl pressure. Ihc rc<juired term is seen 10 be simply pro. independem of o.§ T he above aller· nali,..,s by nO mcans e_,hauSI .he variely o f le<:hniquC$ .hal may be employtd 10 oblain thermoelastic d3ta . Other approaches would ;nclude the e'lr'delion of ,lreSS I"",pera'ure i"fonnalion from a .. ries of . tres<----Ction

mo-.., "I""'''''

tA, ohmo·. "' oftaI """.F '. ,n. (u= ~ It
"0< co", 0( ""''''''''. """ ..,.i.Sdil...,. , ... ,"""""I """"""" ~nt "",w,",,, ;" 0(""" ..... ""{",n,.,,,,•. fOf ,,,n,,* t ..""", 'K ......" l',,~ ,he """olaF 0( .wood,.,

th,,, of ,... ...·<>Ikn "",~ .. l §Some "' ... , 'IP<> of

'"""'Iy "",10 dHlh,,,,,,, ",m, d,p.~""

110 STRUCTURES AND PROPERTIES OF RUBBERLI KE NETWOR KS

of teml"'rature at conslJnI force". It, any case. determination of relia ble '·alu"" of acquir«l at conSlanl pressure does require inclusion of such lenn! effcc,ing the conversion to Ihe thennodynamically rcquirod condi. lion. of con,IIIm volume. sin"" Ihe faclor In· is typically of Ihe order ofO.2-{U. Implicit in the basic ["mulate of rubberlike ela'licily is lhe assumption thai in Ihe deformation of a polymer nelwork. any energy changes ar. intramolecular in origin. [n Ihiseasc. the theoretical e"u,tlion of Slate dcri,oN in chap'" 2 may a['o be usc
1.11 from thennoelastic data

f,lf

so

(9.7)

Tdln (f' )oldT

where (f'}o represcn," the un1""rturbcd dimen,jons' or the net,,"or. chains. Thi. ~uation i. of considc .... blc importan<x since ;t 1""nnilS the comparison ')·" of rtsults of thermoelastic meaSurementS On pol}·nt., chains in the bulk. in nctw<>rx structures. with re,ults of vise",it)" mcasurement, on chains of the same pol)·,ner. e.... n1ially isoblcd. in d ilute solution. It .Iso ""tablishes the r<:l.t;onship bc:twcen the t;'urtly thermodynamic quantity . f.1f and its 111<>lecula. counterpart din {r')~dT. which can be imerpret«l In term~ of thC rOtalional isomeric State theory"- of chain oonfigura';ons.

9.2 Typica l St ress-Temperature Data Detailed discussion. of the CJlperim.ntal techniques emplo)·ed in thermoelastic ,tudie, may readily be found in the origin.llitCf,tUrt. w~i(·h i, extcn,i,..,ly cited in the f"llo,,·ing $/Xtion. It is important to emphasize hert. howe.·er. that u,,", of th< thennodynami,· relationships pn:slycthylcnc nelworks at c,," .. an, prcs.sure and at conSlant iength ". i. sl,own in f'gure 9. I. As "'as done in thi' study. the el.slic force is typically di,·id«l by the erosS-....,liona' are. <>f the undistoned !.ample at a <"nvcnient lCm1""rature in order 10 facilitate C. in the calculalio~ of the diITerence term /1(0) - I) . As " gcne .. lly the casc. the stress is found to ,·ary linearly with tCOll"'r"t"re within C' 1""rimentai error. thus indicaling thaI f.lf i(.<;clf does nol v,,,y ~ignifocantl)· with tem1""r""lfe Stress teOl1""ratu", relationships ha,·" al.,,,n detenninod for a number of bio..-ia.1<>mc". a, described in ch"pler 15. Suffiei~nl ,hain mobility for cla,(omeric Ixha,ior in nelwork ,tructures of Ihi, type ge nera lly r"<1uire, the pre,.,n,,, of rci;Itiwly large amonnts of low·molecular-weighl diluent. ·I"hey are Ih
br

UO ST RUCT UR ES A ND PROPER TIES OF RUBBER LIKE NETW ORK S

"" z

5

10

IS

lO

~S

30

3S

40

~ig .'" 10.7 Comp.ri""n. be'"",n tIM: 1'1<>1)' number N. and 'he number or - for I'DMS dillin, as. fu"",ion of ,h.
Ihal in Ihe mOil commonty O<.,-,,,rring range (}I vjl'• . N p i, always iar!!"r Ihan Thus. 'oc number <.>f tnla nglcmcnl$ resul1ing r'mn Ih~ d;lpe~ inlcrpenolr.llion of chai"s in Ihc cross-lin ked sial. is fa, gTrater tha n the 11'., .pedfIC localilCd points along thc chain defmed by explicil reference to lhe plateau moo ulu s. A. shown in chapters 2 and 3. Ihe mooo l omphasi,ing 'he g
II'H.

G= (I _ 1/1~) ,4<1" [I +...l..II' ,,'(( 1+1 +,,)'~')l Y. 1/> - 2

( 10.3)

whcre Ihe 4uanlilics ha"" 'heir us".1 signifIcance T he C(}rrrsponding expression ror lhe l oca li"~I _cnlangiemenl "'00.1 i. G = G"" + G,

'I ~ =

A(It"" I n, T,) RT

{~;o

( 1O.4)

whore 1 ~ i, Ihe enlanglelmn' "lrapping faelor."' tJliliauion of "'I. (10.3) require, C(}mparison, belW<:en uperimcnlal and Iheorelical "Ilue. of lhe: modulus. ><, as \0 oblain "a!"'" "r lhe: p"ram'l~r N. Those ,.,,!u~. would lhen 1Ii"o insighl inlo lhc Iype of <"()nSlr~in(. Ih.l "'" ol""r~live. Pm cun'plc. Ihe rC$ull 11' .. I C(}rr""i>Ond~ 10 Ihe Flory_Erman C(}nSlrainC
112 STRUCTUR ES AND PROPERTIES OF RUBBERLIKE NETWORKS

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9.3 Illustrat ive Thermoelastic Results A, was poinled oul obo"" •• 'peri,",,"la l studies of network lhermocbsli";ty mo}' be pcrformr:alues "r f.lf di,cclly from eq, (9.4). a thermodynamically exaCI equalion , Values Ihus oblained may thr:reforc be used to check resull' obtained at CQn~lanl prtSSure and in'<'1'O:led Ih rough ' he u"" of cqual;on~ such as eqs. ('1.S) and (9.6) which are based on the 'Ia' i$tirul. molecular lheory of rubberlike elaslicity_ Ikca"", of e.pc.imcnt.1 dirr."uhics asw
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~'il~'" 9. 1 Some the","",I"lic f,,, .morpnou< polyelhylene ntl"'''''' ;n ,he "n,,,-.,lIon "".... ""." •• t Ie~th, n .. SIre» i, «pI'<...... "'Ial;.... t" 'ho ,"\dd",me i, char· aeleri"'" by the ,'.Iue "f tho OIoo",tlOo <> at It.: .. "", ttI"per.. ture. R<print«.i "'ilh pcr· mi,,;on f,o", Of.... ,. A_.
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tomarily studied in Iitc highly swollen Siale, Some Ihcn-noclast;c dala obla;ned on elaSiin. a cros,-linked prOiein described in ~hapler IS. at'(: pr=nled ;n figure 9,2", Thi. malcri~1 was sltldi«l in Ihe l."al ly ~m.,rphou" state. in s,,-elling equili. b,iu m, al oonSlant p'=u",. Each line . hown gkes;, good repr=nlalion of dala taken by either decreasi ng or increa,ing Ihe tempera lure. Ihus illustrating dearly th. high deg= of reproducibility "lI"inable i" ,hern,.,daslic ;"""I;g;" ;o,,, The use or . compress; .... deform.l;m, in for<:e-temperd tu re measurements i, illUSlraltd in f,gure ~,-'. which perl:!ins 10 all utn(>rpho~! polyoxy<:lltylcnc nCI' "-.,, •. ,,,,.,lIen wil h " constan l amou nl of n.,n'-olalilc diluenl a"d ,tudicJ at con· ,Iant pre~u",JO. Th~ data ,how" wen:: oblained ;nd;"",lIy fron, slt\.'SS-slrain isotherms de'en-n;ncd for a comprcSSl:d cylin(lncal sample al a number of tem· per"tun::. , The isolhe,ms had been repre>ented ;n ,uch " manner as 10 yidd lhe "'lui,-alent of Slress--tcmperature daln at eon"~t\\ lcllglh, As a final example. som.. Iypical Ihermoelastic dala ;n tors;on are shown in ligun:: 9 .4". They pe'tain 10 a natura l rubber netw",k studied in lhe amorphous unswollcn ,ta'e at constant lenglh and pressu"" In th" In'" "r deforma,ion, " nctw",~ i. cha' acle,;>:e
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STR UCT UR ES AND PROPERTIES OF RUB BERLI KE NETWOR KS

11 (

The mo,1 CXlcn,ive Oflh"",,"re the siudies" 18 of Allen and coworkers, who have ctnplo~cd Inc melnOO of variable nydroslalic pressure 10 nllilify changes in lhe ,'olume of the Suld be c",,,red_ In Ihesc sludies. carried oul InUI far on networ's of natural rubber"·"*. pol)'(dimc( hyl>i\ox.nej", and polyisobuly· len."'. Ihe ""-luired oocflicicnl (fJ!I(}1'l...,- was generally obtained in 1,,"<) inde, p!urc. wilh appropriale control of Ihe nel"'or' volume. as "Iready o:kscril>cd_ The K«Ind im'olvC<J mcasu",mC""t of >·"IUI;s of Ihe c,><:fIiciel1ls (al j a'rJ'_r (ap/ a1")"" .. and ({}!lap )t,I' and uSC of Ihc Ihcm,,,,lynamic idenlily: (D/IDT ),., ,, '" (fJjji)1"),.~

+ (DI'IDT)'s({)fIDp)1.,T

(9.S)

Th= 1"'0 approaches gav" vailles of/,ll in "'Iisfactory asrttmen!. Thc network< which have be",n mosl cXlensive\y .ludic"l in Ihi' rcS,,,d arc those or "alur"1 rubber and I"'1)~dimeth)'biloxane). ,md Ihesc results are thercfore su,,,,nar;"',,! In lable 9.1' . lneluded for purposes of compan",n arc Ibc 1n",1 reliable of the cOrn"Sp'cr a narro,,' range in lempcralu,c. t Less comptelc da1a are available for pol)'isobUlylc,," nelwo",_ bU1 the rC$ults al ""nSI"nl ",)tUnIC and at ",,"\\anl pres,,"e a rc "I", in gOO<1 menl. givins Ihe average ,'nlue/,// '" ~O ,06(±O.().t) "·"'""'. Using 'he I<"$s direcl approach fo r obtaining Ihcrmoda$'Oc da'" ,,' <,:<>n"a nl 'olun'c. Sakurada"" anO$i!ion. al which ,'olume change< we'e .igl\ificanl and Ibc,cf(>rc had 1o he eorrttled for by usc "f "'1_ (9 .5). of the "ali,lical theo!)'_ 'n,e 1\\"0 Iype' of =,,11' "'''''' ogain found 10 be in S'e oomparison, slrongly indicalc Ihal re liable 'alucs of /,// may be obtained under Ihe rdali",ly simple conditioTl of constant pressur~ . ralher Ihan c'Onslant vol~me. A. is also shown in lable 9, I. a nd in more cxtcn.i,·c oompila\;ons"·". 'he res" l\; lanl pressure do "01 dcpend on Ihc choic<: of !he olhc, eons"ai"l (""n'lanl L (>. ' " j). (}' on ",helher Ihe 'YSI"'" i, Ihern,o·

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I), ViJCQt'I(UI;" /''''PO'''~' I)'n!. xi. 1987.15. 0\9, Ronn". N .• 0"1'<'''''' 00 , W, Co/I. 1'0/)'1'" Sci. l'>91. 27(1. $27. E""an, B,; Flory. P. J. M,"'"",,*,'~'" 1%2. IS. &'16. Flory, P. J. Slati"kal M,~hanlc't' 0/ p,~ymt'T' , MoG'aw Hill , y o,k. I~ . (84) Ma ndol,em, L. In ("""'I'"Ioo: o.,rord, 1989: p, J63, (85) .\1"• • J. Ii Macront<>kc.k, 1984, 17,2924 (86) Cl • ...,n. S. J. : G.li .....",. II . 1',,1)',.,. C""",'WI. 1986.27.26(1. (87) 0;0. D. S.: Suo T.·K .; M>rk. J. E, MO<",,,,,,o/,cu"', 1911, 10. 1110. (88) MOIk. J. E.; Eiscnl:>. P n . .<;"clli"8 C()r>CfPIJ in Poll'''''' Physk•. Cor ...n Unil"C";ly J.....: It"""'. NY. 1~7'I, (92) Mar k. J. E.: Zhong. Z,·M . J l'oI)'m. S,:;. /'01)",0 . I'h)", Ed. l'll! l,l'. 1971, (93) Langk,·, N. R.; Dickie. R. A: Won,. C .• Fo"y, J. I), ; Cit;, ... " R.; Thirion, P. J. PoIJ'm. Sci .. /'a" ,1 ·1 1968. 6. 1311. (94) Yu, C. U.; M .,~ . J. E. Mac,nmo/«uie> 1974, 7. 229, (95) Hocv•• C. A. J.; O'lIricn , M K. J. l'oJ)""" Sci. Pml of 1%3. 1. 1947. (% ) 000005. A,; 1\<,\Oi!, H, ,\IU('Tcnwlrt~l" 1971. 4. 279. (91) F,i'm.n, F~ \'.; D.<.Ii." ny.n. A. K . V"Ql.OfOOoi. SM1ir:, 1%Ii. 8. IJS9, (98) Naga;' K. J. Chem Ph!", 1%7.47.469(1, (9\1) Gent. A. i':. Mac,,,,noJ«.ito. 1'169. 2. 262. (100) h~i~"w •• T .. Nagai. K. J. Pm)'"" .'Wcl.. Pu" ,j.: 1%9. 7. 112l, (10 1) IsIoib,,·• • T .. ".ga ;' K. I'o/rm. J. 1970. I. 11 6, (102) Genl . A. N.; Ku"n . T. 11 . J. 1'6/),,", Sci .• /,,,,, ..1 ·1 191 1. 9. 927. ( 103) SI<1n, K. S.: Hons. S. D, J Md. 3S I..l51 (1(19) Flory. 1'. J. }'01)'""", 197<1. 20. 1317 (110) Flory. P. J., I:"nan. fl. ,I/rluumo/"rlk.' 19~2. IS , WO,

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.1 ' A

,.".-'mqooo'
,('0:)'0'1 ) i 1'0

,/(O'O"J) O'U

F'

,, 'J

,(ro'o,)6I'0

'I'~

h:(ro~) sr·o

.1 'A

,(;O O'f) ~L 'O

c-

.N

,.{v)'Q'f ) '! ·O ( 10'0-;) 61 ·0

, I'"

.(<~)l ' ·O

,/ 'A

~ro'O'f) L! ·O

1/1

(""""I ~I" ~''''" ' ~ )'10..1

'I'~

<1" ...

·,'"1

,;qqru ,...... 1'1 . ~~

£P/'«,) "I P.i. - .f/'!"" ' '''''UlS'''n '"'." •.(polUJ'"U. JQ '''''JJ,1 1"6

sn

A.l,IJj.l,S ..... '30~ij3H.l,

""'J.

116 STRUCTURES AND PROPERTIES OF fl U88EfllI KE NETWORKS Tobie

~, l

EfT""l of Cr"",· lIn~i" i Condilion, on f.1f

s,'" <>loom"'" ~"""~

/,If

« .,... lioHn,

N.,."I,.t>i><,

- Ml (t O ,~ )'

Onmt«l. ",,....llioe

~ O. 4 l (-'-0,O·W

UIIOt"i
.mo'rI><>"'

00<.,«1. "'",,pho.. (}t;rn,cd, <.,...11;".

In ""'"'"'"

O.ll ('w,04)'"

I" wtu,.",

O.Il U.Omt'

U""'''nlod,

• T. . ..

r,_mn_ ".

· T .... ' _ _ . ~ °T.., . , , _ , , _ ..,

, ~ ••",r_-.... ..

O. Il I~O,OJ)·

O.IS(1.M3r O.ll (loO,UJ)' O. 14 (±O,04)·'

U _'«I, OIn""""",'

In ooIo,ion

•• T .'" f,_ t
~ O. J()'

U"""'" ,«I. '~r'><>o' U..,"""tcd . <,}".II;".

' ''''''rI><>''

0.20 (1.0,00)'

O,lS (±Q,QlJ"

• T. ... , _ -.,,,,,, )0.

•T.... 'r""" "'_ 'l. rom -...""

, T....

,..-)0.

, T.... '''''" .... , T.... ,_

..........~"

1.4·polybuladien. IJ " . ......,... .. and poly(dimclhyl'iloxanc)"·' ........ AI>o. c'S;.cn· !iatly idcn!iC"dt vatUe! of f ,/I secm 10 be ohwined for S~'lcm' cros.·lin keJ b~ chemical mean. and Ihose cross·lin ked by high.., nc. This inscmi · !;vi!~ of h ll!o cro,s-tin king <"Ondi li(,", i~ of con~id.rabl. inlere'l. ~inre change'S in .uch condilion. are Ihoughl 10 h,,,'e a markeJ effec1 on the topology of Ihe resulling lICiI>.-ork . Cross· linking in lhe prescnce or diIU<"1 . hould yidd. n.twork or rdali,'ely fcow pennan.nl enlanglen!cnl" as .ho~ld also lhe prior. tcmrorary OrienUlion or lin ing up of lhe pol~mer chains. eilher Illrongh chain cryMa llilalion or by mea," of a mecbanical deformali"n··...... ' , T he p"".iblc dep<"nden"" of f ,lf on Mg"'" of cross.hn king i. al ... of imere'l. but dala ,e\e"ant 10 lhi . poinl m. somewhal limiled by the fac l lhat. in g.:ncral. only rcialiwly narrOw r~ n g<:\ or degrc<: or cross·lin king arc sludic-d in thonno· el.,lic experiment. I), Net".-orh of rclati,'ely low degree of cross-hn king are "er~ ~t(>,,· in re""hing <:quilibrill",_ a nd are usually insufficiently .lable fo, mea,ure· me"" owr Ih. 'equin-d range of tcmp"ra1ur~ and I; me inlc",al , "--q ui 'cd in ,uch oxperimems. High degrees of cross-linking. on the olher hand. pre"en! al1ainmelll or high dcf"nnations. which . of course. h.,'c lhe ad"anlage of ln inimizing uncer· taint ies from the diffc",nce lenn (J(o' - 1) I , Table 9,3" ... m",,,ri,,,,,,, >orne resull. from lhoi<' In,;,'al lIK"01""I""lic -,!udics rell !o be ",osl pc'linenl Wilh rega,d 10 the possible dependence of!.If o n degree of cross-linking, Valu"" or the effecliw dq~rec of c,,,ss-link in g. ~ / 2 V. ci!C-d in lhi, lahle arc in ""i i' "r mole. or cfO,,·links pcr cubi<.· cenlimcler or nct work. They were eakula,c,1 [ro", .-alue< "r If''J ror the un,wotle" nelworks. e.' l r"pol"'c~IIO" t ~ 0, thai i•. f"'''' Ih. "hmne)'·lt i"lin

THERMOELAS TICITY T . .... 9.3 Elfoc' of l>ogre< of C"",.I.mkin,

0fI

117

I,ll

----~-

Pol)"".

( W'~/l"I'

/.If

O.j2l<

MI (,,-O.l"1 - MJ (.1O.Ol) -O.44 (.1(H)41

,.

1'<>lY<'~11<"'"

1. ~ 7

N.,.,., Nbb
'"'

0.11 (10.01) 0. 11 1-,0.01 ) 0.11 (.10.01)

HJ

'" '" I.U Not.", ,."""'"

0 .14 (~01)

0 .11 (-'.0.01 )

1,1 1

0.16 (-'.0 01) ",11(,,0.01 ) 0.17 (.10.011 0.17(.1002) 0.'4("-0.01)

H3 .~

,... 12.0

G •. l .4'r<>I)'"" .. dit""

6.10

0.07 (lOJll) 0.0\1 (",'10' ) 0,0.1 (±n.Ol) 0.0.1 (-I.O.ol)

w,

lU 16.3 r"..,..I.4' r><~';"""""·

,,,

0,04 (.to.Ol) - 00\1 (to.o~) O.IO (.lO.O)) - O(ll;(io.o)) 0 ,11 (1.0,04)

9 .• ' 10. 1 11.3 16 ,.\ • " ...... ,............. """ ... _ • T .... f.... """"'" 11.

... "" • . • T • • " ''''' " ' ' ' _ /S, • Tal .. (,......

r..-..,. ... "'"

oonslllnl ",',·l. ' ) 2e, .. vlr.T/ V, Altho"gh. 5cvcmlfold mn~" or ,'/2V i. COycred ;n each of these ,toSs_lin~ iog. Result< of iin, .l _ II>.X and 5.4 -16.5. n:specl;~cLy. haY< been analy,,,d oi"'",,'hcrc and fou",1 10 be co nf;"n,,'ory of ,he above condus'on .t Relal;,,,ly lillie ,,,,,k ha. been ,lone on 'he pOs.si hlc del1"odc""" of hi! on ne'''.-ork c~ "in·leng' ), dislribution, I,d""" nppe.r. ]',,"'Cver, ,hal pOly(dimelhylsi10,"ne) clas1Om<:rs h''''ing'' bimodal di'I,ibuli"" of en"in k:nglhs hH''C ,,,lues "f hll lhal art oOly aboll' half 'he "aluc c.
,h., ""m"""

fl' """ "I~ n. " . no .·Ir;. ' ~" 'Il«< ,," ,,,,.j"-~;" w~~· ~ 1.11 """, t.,,," t, .. ,," '~<'I~. """~. • ... I""l'("",'hyl "'" ",~·",I ..,) ,.-.J .... u,-.I

''''*'''''.

118 STRUCTUR ES AND PROPERTIES OF RUeeE RLI KE NETWORKS

Ahhough mosl Ihe,modaSlicily siudics have been carried oul on nelworks in uniaxial elongalion'). s.....·e"'l ! "nd on (lOlyo~yelhylenc""" in dongalion "nd in romprcssion. and on Irans-I.4_poly_ isoprenc W""" ) in elongalion and in IOr5ion. show equally good agr«men\" b<,"""eOn val~ of 1.11 Oblained using differenl lypes of deformalion. T wo n",rc-~nl im'estig'lion. on a ,'ariel}" of d"SlOmer"l in lorsion and in ciongalion. howc,-er. h",·. found 800d agreement bclw«n ,·aloes off{(f. in general. bul poor agrecmem in Ihe cas< of some polybuwdienc elaslorners " '. Con.idcrablc eiri. of =u ll ' rde"anllo lite possible dept:ndcnce orJ.II On exlent of dcfunnalion, There is now an abun-

T . .... 9A Err"'l ofTy!", of DcfQrrna.ion on !.11

Not.,.I,"_.

Hiunplk>n

f.11 Q,ll(t(I,02)"

Comp ~

0.I& U .0.01)"

,.-

O. I1("-O.oJ )·~·

f.i<>npl "'"

o. ll (i.M l)"

c"",~,..,;o.

O.14 (io,Ol )' O,IO (fOm)"

To,,","

lOIonp'"",

_ 0. ' I (*1).(14)(

'r",,,,,,,

~ O . 14 ("'O . (I4)'

0.16 (tOO))' O.l l("'O.OS)'

F.lonp'''''' ~".,....

O.01 (->O.OI Y O.06 (~O.OI )' 0.20 t,,"0,(14)' 0.0 (01000))' 0.1(1.0. 1)"

£JonP'''''' CO",""",,,,, I",,,,,,,",

· T .... ' _ _ " .

' r .... r"",, _ , .. • T .... r... rd"" .... Jl • T . . .. " - _ ... , 1'.... ''''''' _ ....... ,. ... '"

'T.... ' ''"" . - ') • ' .0",

'ft.. --. ,""

• ........ r,_ ".... _..:

.~~U ; O&

, y..,., f _ ..."""".,

, r.... f_ ..r.....,. ..,

.... ,- '"' ........ ''''''' ... . " " '''' ''' '" """""" .

' y.... r_ _ ..,

,,

- ~, ~

. "'eo '"'" ...."....~ I'

,

Tl-i ERMO ELASTICrT Y 121 Tab!<

~ .6

Eff«1 of I ~1"lion on f.lf [);I .. ",

J>olym<'

- ._--

,-

l'oI)'
l>Oelh>'IIo<,yl . ", .. ,<

...c..II.,

".e " lI..•

-

1'0".,,1 ,ubbor

'.00 O,80-0.JO

••

~ O. JoI

' .00

...c,.l!,. ...c,.II"

O.9)· G.J' 0.6.> Rl/i

•• ••

r.,.ffio ";1

[:>«Olin'

N""" I-Chloron>p/1 ,h.l
.... 1.4.pOl)-""' ... ;.....

o·

'00

....

0.43 (+0.110)'

0.171 ,,0.0))" O, I8 (±O.(I.O)' O. I)(.I"O.ot )' O . 19 {~ O ml·

O, I'(ro.o2)' O.l! (:W.(I.Ol"'~

O.ll

o 14 (±0.0 1)'

•• 0.14

0. 10(,,0.0 1)' 0.011 ("-0.01 J'

,-

'.00

••

_ 0 1U (,.(O OSI'~

0,[1

0.11 I M.Ol)' - 0.20 (,,0.(101)'

'.00

- O.03 (-,-O.Ol)'

P... ffio ",[

[>«Oho'

" ,~.O ...

,.

_11.

" .:l'I

• v-.. ,.......... ....,..... '" n" ""_.

. _ ., ...""'br.....

, T. ... ' '''''' "",...,."

m. .."",,,

, r.... ''''''' ---.. ... ...

'r.... ''''''' ...... _.,

OA2 (.iO.OI), OM IH .lOr

u..... (; ••

P.r.ffin ";1

• T.... '''''''

!.II

• T.t", f""" ..

-O,O)(i O.Ol)' O.1 9(±O.0l)· O.lI (±O.O})' 0.16110 ,OJ1

O. I¥(±O.Ol)'

' , .... f. - .. _ SJ. , Ttot ,.. f"," . - ,. • T .... '''''. ""'"'-
',.t.. m.. ...r.n_ •.,

'' '".fl, .... ..

dC"ialion. ha, been found (0 be markedly ,lcllCn<.Iem On each of lhe variable.. invesligated in la bk, 9.2 Ihrough 9.6$" . The nosul l. lu,,'eyed here th uS ,ugg<"S1 lhallhc ela"ic fo,ce need nOl be """,Ived inlo it> scl'"T"~te contribution! fro m leI and 2C, in thennoel~'lic ana ly"""

9,4 Relevant Calorimetr ic Studies of Elustic Deformations

Dirttl ,a lonm~lnc mr~'urc nt"nlS carn ..1 om on " polyme, nel",ork duri n!\. the deform,tion procc>s h;lve also been ,,'cd 10 delermine v;lio<"S of f,/f".$<~'- . In Ihis approacb. Ihe heat" of the defo. mation I"OCCS< i. me"""ed in , ""mili,," mic.<><.ionme,c,. ~nd Ihe " ccomp:tll),illg wor , ". in Ihi: proe<:<.< " Ohl~in"11 from Ihe dill"'.nti.1 4uantHy lelL in tcgral<"{) fro", the in Itial to Ihe fina llcnglh of the ';,mpie, Then, ~c<-...,r"i"g I" Ihe I,,,, l;iW "f lI'cnno.;!),n:,"" ••.

122 STRUCTU RES A ND PROPERTIES OF RUBBERLI KE NETWOR KS Tobit 9.7 COO'lI"'. i50n or Calorimelric aM Th"'moel.,,k .... Iues of1.1f 1)1'< of d
PoI)m
N.,""I ",bb
C.l<>ri .... ,;,; ,'01<1<>

''' ' ' 'Uon

O.l3lttl.o.)'~ 0.2II(io.02I'

O,11 I:t<:1.(1)"

lou"""

~:Ioo,.,ion

0.11 liom )"

O.121,,-OO2t' 0,10(,,-0-02)'

,'""","

O,151-'-o.~f'

0."(0<0.02)"

• Tio ... , _ " ' - '" • Tio ... ''''''' - - . ..

· T..... f_~)L , T • • ""_,*,-,,,~""'

'T.... ''''''' ........... ",

• T..... from

.,.... ,..-,.-...1)

n.",mo
me..... Sl.

... (9.9)

where Ihe energy and enthalpy clLanges are esscnlialty identical ,ince tl,(PV ) ~ 0 ro' any dcfom"lion ItOI c;>rricd Qui al eXlraordinarily high pressures. For don· galion slUdies. Ihe deri~ali,'C (Oll /OL),.., i, oblained from Ihe cxperimenl.1 value< of tl,1I pl(>1led a' " function <>f t., Vah".. of [.(f arc Ihen cakul.ted rrom"·...·7<1

f,lf " (O£ (D L) r.vlf ~ (fJlfIDLh .,If - PT /(ol _ I )

(9.10)

which is an ob'io us "nalQguc 10 cq. (9.5), Similnr '"r.--cn 10 be in good agreement wilb Ibe'moelastic result. ",ported for Ihc ,,'me IWO pol)'mo:tS.

9.5 Relevant Vis.;osity- Temperalllre Results on Dilute Polymer Solutions Conwrsion of the thermoo)mmic quantity!. lf 10 Illn {,l).ldT _ 1.lfr permilS comp"ri'll)n of values of the temperature "odlicitnt of the unp:rturbcd dimen· sion. of thl: nelwork chains JS obtained from thermoclJ"icity mca,uremelu,. with value. oblainc'
,,.

{'11- lim {(.Io-ot - Il/el ~

(9.1 I 1

,,·he ....h-ot is Ihe relati,.., ",:!Cosily (Ihe ralio of Ihe \'i!lCOsily of a solUlion of Ihe polymer .-.Ialio" 10 Ihe vi:l<:o.ily "f the pure soh'enl al Ih< same Icmpemlurc). and c is Ih. oollCCntralion of Inc polymer in "'eight per unit ""Iume. Iypically g dl- ' ACC
THERMOELASTICITV

123

in "hich 4' i, a ,:ou"a llt ~nd,, ·~ {(1)/(..'lol'l' is a chain C~p',n.ion (aclM characterizing perturbation, due 10 cxduded volume cfTN:I,. I n a" ideal or 6-solvcn t. ,ueh perturbation, arc nu llified by polymer-wlvml interactio",. () i, unily. and Ihe unperturbed d imcn,ir>n' {I>. a~ d,ottlly calculable (rom Ihe in trin sic viscosity. Corre'pond inJ(ly. viscosi ty mcasurem~"' < in a serics of wl ... "IS having epoinl$ ovc r a ,ufflciently "'ide range in telllper~III'" yield directly the coefflcienl dln (ll.ldT, It ;,. ho,,"'ever. generally req\llreJ thatlh= e-solvenl' be of very ,imil ar chemica l , 1,,'clur(!'J, Thi, is ,,,.-c,,,,,,,ry in orlvent. e' -.:n a t lhe 1'lO'ipc<:li,'c a·poi nt•. I() have a large elTeel on the rdali,-ely &ma ll t~mperalur(! c<~lTtcienl of (.-' ). Viscosily- temper.. l",e mea,,,,,,,,,enl, in a , mgle sol",nl. u,ually ooe Ihat i, a thermodynamicall y goud Wlvenl f->r th~ polymer. m"",io " f~C1or is not unily. and il is imperalive to ta ke into :lccounl ;1; ,'arial,on with tcmrcrdt ~"" ). The requin:d value of Ihe lemperat"re dependence of <> is g.""r,,1)y obla inen th .uri('S g,vin~ Ille r.lation,hip bctw«n ,J and th,- Ihcrmodynan,ic p;,rumcters char"cleri7.in~ lhe inleracl ions NI"'«n s"h'"nl moic<: ulci and pI.lymer $C8",,,nts'. This "Pproach is ,-on, ,,luably simphfieice of a SOh'enl dosely 'imi lar in tlle",ical strueture 10 thai of Ihe r;->lymcr; for ", ample. Ihe choK:c of ,,·hex;ld.,:a nc in If><: siudy of polyethy· lene '. Such solvent-solute combinalions mi. ",,"lIlially alhermally. will, a n aliendant si mpliflCa(ion of the soh'lion lhermodynamics "nd lhe as.socia lW cstimation of dln {l)ofdT"'" ". Table 9.S" Ii,,, '''''IItS Ob1 ained on Ihe 1".-..", poly"'."" tha, have hc<:n siudic'd most ca",fully w;lh rega,d to th~it viscosily tcmp"rmure coemcicm" Vuluc, of ,lln{l)o/dT oall;"I"led Iherefrom a", oomp"n:d wilh Ihe corresponding ,'alues obwined in Ihermoelastic im'l.'S,ig.nio ns ' hrough """ of "'1 . (9. 7). There IS e ~""lIent agreemcnt belWee" 1h. IWO se lS of 'alues of d In{l)oi{rr Ihus oblained. as Ihere i. in Ihe Case of '''''1:ral olher polynlCr!t studied "'ing both "f lhese lech niques". The c~""lIe"t agn-cmcnt obtained NI..-""" ,'alues of d In (.-')old1' for the ",m., polymer chain> un'! in dilule solu lion. Mosl importanl. i( is inconceivable Ih.i such agn'Cment wo uld boe oblainc m:o t<>8r.
IN STRUCTUR f S AND PROPERTIES Of RUBBERLI KE NETWOR KS T.I>I< 9.11 Compor;"'n or Values or dln (r'l./JT D.duced from ThormodaSlic M."UI<m on 1..,I.too Chain,

.-; - 12 (JoO.2)'"

119 (10,(101)'"

_u 1111.1)," l'o>Ij{o-","".,..l l

~

Po¥oobo,)lm<

0.-'"'(1 0 ....)

O,Sl (-HIM)""

_0. 19( "-0.11 )

- o.U(,,-o.OS)'"

_0.10 ('10,0-')"" _D.4 l'Q/yg')'<1n)k""

,,--

Pol )tdi"",,") I~"' .. ..,.)

0.23(",0.02)

0.2 (,,-0.2.1"

0,19("-0,14)

O,jl ("-0,10)......

. . ....-...Iho_"" .. .............,... ........... • -r. ... (_,ho ~

("I"

'J .

' -r. ... _ _ ". , " ..... _

...... _

' h b. _ _

·

,oo.0( . "", ... tt, "...., .. ",,'.. , ~

, T . . .. (""" . - .0,

~""""

...." ....... ,,-

. ,'. . .. ( _ - " 0 >, • T. . .. (roar _ , . .

-,- "'. I

To' "'_~"""""'" ,_ ._ ".

" To ... (""" _ 1 ' . • T.... , _ "<,,,,",, 1l,

I ""'" ' .... _ " 'T. . .. ,,.. ,,'''''''' tOj.

of nlOlecu(ar order .... ilb dilution". l'inaUy < optical anisotropies d.lcnnined from the dopulari1ation of (ight iCallore.! from undiluted amorphous polym'''', or from Inc Slr~in bircfrin~cncc of both di(ulcd and undi(mcd "mor phou, polymer net· ,,'ork>, arc comp;".,.ble in magnilude to lhe "'>rn:s p<:rnding qu"nlil"" mea.urcd for (o""mo(ccular-"'cighl liquids. and me "cry much .maller Ihan would be C>pcclcd from n'.leri.1s containing significan l inlermolecular ordering'''. It should also be l"C'CalkJ Ilta ! Iheoretical "rgumCnl! p""",,,,ed in chapler 2 "'ad 10 Ihe expeclalion lha l excluded , 'olume inleracli",.... lthough a",un'dly p=nl in un,wollen or swo Hen nel work•. Ita'" no effeel on Ihe end-Io-cnd dimen,ions of lite ncl"'ork chai n$.t Ihc relaliwly high pul}'rner con<;<;nlralions ".,iSling in $ucb sySlcrn,. Tha' is, 'he chain mot"",uks . hou(d be in Ihc~r unp<;f1urll<'<.1 $ta'es. char· aClcrizcd by the dimcm;on (r') •. There me 1"-0 impo'I"Itl pieces of cvide""" that bear On Ihis poin l, Ringn' mi"uI"C'S of dC"'cr~lCoi and undeulen",'d p"lyme". gi,'c "alucs of Ihe radii of 8ynuion II ... "re Ifencrally in good agI"C"",eltl willt \'alilC' ohl"in<.'d u.ing dilu'e ion, of I he ",me pol),n",.-. "''''. The polymer ,'h"i " , are

,,,It,,

THERMOELASTICITY

t25

Ihus sl".,wo (0 be in their uop;rlu,bed co nfigurati ons in the undiluted ~morphous a finding whIch a forlior; directly contradicts Ihe p,..;""nc.. of order in suc h systems, The basi" post" '.,.. or rubber cla.li<>ic moments. light scattering intensit i"". optica l aniso tropics. and eyeli""tion equilibrium con· stants. or by e"nfonnation~1 e"Tlergy calculations). This "pect of rola tio~ "1 ,SI>meric statc thoory '. cMere
9.6 Rotational Isome ric Slat" Interpretation of StressTe mperature Res ults The energetic COnirobulion J.. to th~ lotal fo"'" I .fisc.. from the fact that deforma· tion of a pOlymer n<."i"'·ork requir", a oone'r><>nd in8 change in the oo"!igu~~t;on. and dimension, of ilS constiluent chain., and t hat dilT~rcnt chain e""figura tions ha'·e. in gcner.. l. dilTe""nt ,onfurma ,io"al energies'···.... Such dilTcrences in wn· form"tional energy ~re also the o,isin or the depend.,1Ce of lhe unperturbed dimension, on teml"'ralllre, and, a, already , hown. dln{,').ldJ' has the same sign asf.lf and is directl y propOJlionalto it . It is now appropriate to comider the interpretation nr these quantities In termS equenccs arc there!'o", characterized in terms of Slatistical weights embodYing the ~ppro priatc cIT,,';':n!s, A h"cr ou!line of th. use of rotalional isom¢ric "ate ihN'y to calculate {,J). and ilS temperature cocflicicnt u ." i, 8"'''' in appendi, G . Rotational illlmc";'; Siate c"kul"tiolls thu$ serY" to make quant it.!i\'e the obviou, qu. lita1i,·c ~,,"n..,ti on tllat cham mok'Cul"~ in which conformation. of high 'I'a tial o. tensi,·n arc of higll ~onfor"'atl"'lal ellergy h"'" pOsiti,-e "allles of J..1f. ", he",,,; a Cr><>IIde"';c ""t",-cell higl, e,"ell,ion "lid 10"" eo"rorm,,,i,mal energy gi''C' ncg'il"'~ "al"",", The rorm;:r ~"sc "orrespond" of course. to chain,

126 STRUCTURES AND PROPERTIES OF RUBBERLIKE NETWOR KS

T.bIt 9.9 (l'p
1'1."",,,

PoI)
"n"

!.II --<) .~

(CII , CH ,I 1S;(Cll,),O] ICH,CIlCIlCIl,j

1'01)(d;mrtb,' ..lo.. . ...)

1.'·PoIy""" d;"" ~

"-

1.... 1'01,;~

R" ... ,io"..1

lo'dlD I" IJ ff

(;'O_Oj )

- 1.Oj {.iO_IO)

O . 19 {~ .Oj l

0 , j9(iO_I ~ )

O,ll {J.O»! 1 - O.1S (*
O,J'/(±{I, 101 - 066 (+0_11)

0.11 (*.0))

MI! {i(l.OI]

O,lJ liO.(!'J) -0.I1{iO.161 0.23 (H.OI)

OJ,,* (±O_OlJ O.2:Z{±O.
O.(!'J (,,0-01) O . ~ (.J.(I.<>.I)

O_'l (.J.O,Ol) O_:1O {iO_OlJ

O.l«±O_<>.I) O,n (±o.oS)

1<;1< ,<:(<;H , I(:II<;H , I

'"'

,,-,

---"(l.0!> (.to .OJ J I{CH, ),Oj !CII (("~ 11 ,1(:11 ,1

PoI)'O.)
;""""i<"

.,""'" Pol" .. """" .... '1

lell (...:.:, II, 1(:11,]

""".."je

""""i<

(CIICK:II ,J

PoI" vin" cbiOOde)

--, 'r od.,...,,,,

E

PoIji..,butt"'" PoI)"( dUn<'h)w' ... 'I»· .....) Pol),\, """"hy):",, 0.*) PoI)'(I<''''''''''''r_ o.ide)

_

Elo" ;. '

"01),( l- ' )'<1'''-''),<' h, I """ h"I)1>" )

' ... ..... ... "',.... _."

I{CII, ), . CUCII, CII,J oc;{CII, I,CII,j (S;{Cl!,),CIl ,] I{C H, ),Oj

- >' (t o.l )

---"1.0 ("".J)

--<)_4J (.iO.ljJ

- ,.)

_ 0.06 (1.0_(1])

O.01 {-o.o.ol)

I(Cn, ).Oi ICIll\COl<' lj (C(C'I) ).cooc"O' «: Il, lCll,]

• ..."K'

0.01 (,,-.lOl) _ OA1 (±O_OS) O.16 (±O.!I9)

.

"

(.iO_S) - 0,19(."" II ) (1,:10 ("".:10) 0.1lI1 ±OJ") _ l.ll (±O. IS) 0.85 (iO_N )

,..

"",bo_-

,_ < ""(] . .. .

~y

, ,.,..".." . ""~ . .... . ,...."""~ _

r_.......

' " - . .._

.... , _ ... ''''' .. .... L _ _ _ ...,.

• A ....."._>
, T....

,0)<,

So. .10001..", ..

<s.

o T. _I.,. . - ,os_,. - . - . - .... f, ... to. R" ,

_t 101 po/)'C_'", _""">"ot)' , bot .... o. •

having unJlCnurbed dimensions which increa", with increasing'lemJlCrature. ,,'hil. in the lalle, Case lhey would dc<:n:aSl;. Tobl< 9.9" prc.\Cn l$ a sur"" y of thormoclos1ie = u)1$ ..-hich haw been in\crprcu:d. 10 date. in terms of rolalional isomer;'; Slate theory, It is Ihu. now poisihle 10 proviM a simple phy.kal piclu,. of lhe Ihcrmoel""h. pro"""i<:$ of lh..., and reiated polymers. Som. of llti. informal ion is 1''''''.''100 below in outline fom,. In the case of pol)'clh ylel\<:, • &«jucn,:c of ..-hich i~ shown in figure 9.5. lite nega tive values oblainetl for dln (r' ).IJT and f ,lf are primarily due 10 Ih. fact lhal ''''''S "onfo,maliona] ",ates aooul , keletal oonds arC of .ignifie
TtiE AM OELASTICITY

127

~' ia:~" 9.~ Sche"'aI~' di.~ta," of. ,,,,;(ion of. pol)~hy"'ne ,,~.in. In ,hi, fig",,,. and in Iho ",mai.inll 0""" jo> 'hi. chap'.'. ,h. choin. are , hOl,'n in ,he plan",. all _,,,,.. , ""nf~r_ ",alion "i.h the , k,,"'I.1 bond. in (1oe pl... of t"" I"'r<"", Ilon.1 ..... nd I~"'" aw.y from I"" rndcr by da$hro lincs, 'he .nQ"', .p<eiry the , kelela l bond. or ,he repc" uni( obou, ,,'hocb ro'a(ion. may occur.

;ncrca.. in IC'"IX"a ,ure dccrca .., (,2). due '0 (he incroasc in Ihe number of rdalivdy <x>mpaci KllutM 'tat~, Simil;c.;.,u,;c of Ihe increase in Ihe numocr of Iranj SlalC!l of low confotlnat","al er.ergy T he eha;~ molec ule which ha< httn moll cxl~nsi,"Cl y ,tudia.! "'ilh regard 10 il~ <x>nf'l!:ur~tional char~clcrisli<'$ is tht scmi.i1lorganic polymer poly(dimClh)"I'ilo~· ~h.) (I'OMS)" '. "hkh i. iIIu'lr~l<"SCd lig"ro, of very small end .to"""d di' ta""". T his relati.ely low energy or Ihe COmpacl "aIlS .Iates in I'DMS chain. yield. a po,ili", ."Iuc for 1-11. the opposite of lhat fo und for polyethylene. The cis and Irun" form. of 1.4-polybmadicnc and IA-pofyisoprenc h.", also been in'"C"igata.! in delail··~-"-" . Their .InICW"" are shown in flgure 9.7. in ",-hich the choice of II for R yield, polybUiadicne and lhe choice of CH, yields polyioopren. , Th< c;"'I .4 forms of polybu
/r"",

1211

STRUCTURES AND PROPERTIES OF RUBB ERLI KE NETWORKS

(.,

/'&,,/~~'"

!

'

!

(>, ~"ur< 9.7 [)ion. poll.""
pfl:CCd ing bond i, a l 180°. Ihu. providing a basi~ for Ihe ncg,.,ivc value obtained for 1.lf for these polymers'··...,;I , A number of othor. miscellaneous pol}'mers haw now !xcn it udie<J. bolh e' puimontally and tho"rttioally. We e"nsider first Ihose having posil i,.., values of f.lf. In these chai ns. lhe higher energy conformalion. a", more .'Iended Iha n th ose of lowe. energy. T he po~ili ,'c value of 1.lf rcporltd for polyoxycthyknc (CI I, - CH , - 0 - \ ,. fo, example. ,la nJs in ]kI'licularly imeresting coni ... " 10 Ihe ""gali ... cocfficknt of polyethyknc. 11 i. due 10 th. faci Ihat in Ihi' molecule ,r"•• " ateS abo ut CH, - CII, bonds arc of higher Ihan Ihe g'IICrtasc in teml"''''ture therefore itICreases the number nf Ir an. states of high spalial .".n,ion. The po,itiw "slues fcporled for hlf for isolaclic and alaclic poly(botonc-l) (C II(C,H ;) - CIL, - l, and po!y(.·l"'nle~e·!) (CH(n-C, I L,) CH, - l. Of. Ihought to ' ""o lt from the facl that .'tended I'UM. '''".. ,U tes about CI,! , ~ Cl'IR - CH , pai" of bo"d~ arc of relali,,,,ly high ~"" rgy ill Ihe'$<: chai"" because of steric repulsions belween lhe chain backbone and the articulal ed side groups R \ - Ctl ,Cl,t, and Ct I,C H,CIt ,. respectively)"-·. The following po!yme .... li'led in table 9.9". an havo negali"" "al o"" of/.lf. In th"!o< chain•. the highe' enefgy conform,,,ion, a rc k ss C~lende<J Ihall Iho!o< of lower energy. III Ihe t'a«; of pol)'(lclra mclh),lcne o~idc) {C H , - CI I, e ll, 01 , () I•. Ihfcc of Ihe Ii"" skeletal bond, exhibil a pfcf.fent-.; for !.",,,, ."lIe,

.""'IID'

TH ERMOELASnCITY

129

o""r the alter nat,,·. gauclt~ suu,,. (which bring inlo [,ro., ;",ily Ih • ..clati""ly large CI-I , groups), The increase in the pOpulation <>f oompacl gaU<"lt~ Slates aooullh<Sc: OOnds. upon increase in lempemlurly(vinyl chloride) ICHCI - CI I, - I,. or [,rcsumably high syndioI8C1idty, .rl: ,yndi"ta<~i<' form "f Ihi, pOlymer. Th. variely of chemical and stereochemical ""I""n= tha i o(",,,r in <"'polyme";c chains such as Ihe l)"n'e"!.'· makes it imp...ssible 10 givc " bricf. qwllilalive des<:riplion of 'he ",ol""ular origin of Ihis oopl:>lymcr's .-.egali,'. lemperalure roclftdenl of (;)0 (and Ih~, of .f- /f) . One in'ponanl oonlnbulio" , of course. would come from lhe ethyl .... S<'q lIenees. in which the low..,"crgy co~ fonna lions a", Ihe IraNS SlateS.~. already described . Another eonlribulion would arise from iW'actic pol}'ptopylcne ""I""nccs. for which lhe low..,,,.rgy conformalions ~r<: length L. ;n !>'lTalltl 10 Ihe !"Clic h"havi or. In Ihi' rega rd. idealilY of ela'lOmer! i, fund"n'cn lally dirICrcnt from that of gases. all of "'hich bebav<: ideally. of course. in Ihe lin,il of intinilesi",,,lIy ,,,,all pre,,"ur. , Th" •• Ih~ only type of nelwork th., wou ld he ide"l in the sense or c.hihiling purely enl",pic cla.ticity would be one made up oi" "h"in~ in which.1I conformalions were of th-e .. me energy, This highly ideali''''l oon.lrud ,s. or COu....,. Ihe "fred)' rolaling chain':' which served as one oi" 'he " mplesl model. ;n carl)' .tudic,!; of chain configuralion,. s.,'crdl ",.1 polymer chain. 31 least approacb Ihis idealized model in Ihal Ihey are ""nformal ion"l ly highly random wilh rolal;on3 1 Slales. when accessible. of almosl idenlica l conrommlional ene,gy. Polyisobutylcne· ··. ,how n in hgure 9.8, i< " po lymer of Ihis Iype . III the sim_ plost , 100>1 qu"lil~ti"c analysis. rolation, aOOul skelelal 00"'0.1. would merely interchange . kelclOl e ll , group, and p. all,t Ihese arc known 10 be n",,,l y ,denlical in !hdr int.-amok..,,,la, interaClions'" As ,hown in lahle 9,9.1./1 ,s fo und to be dose to ""ro. as would "" "~P""led fronl II>< lack of"rong ",n fnrmalional pn,r"rc"",; in Ihis moil."Cu lc. The actl",1 conformational all. lr';' of IhlS chain i. nm"h mo,,' l'omplic.",-d, howc~cr. partly lx"Cau>c of >lrong ;tc,k interaCli"n' belween meth)1 gro"p" along 'he .'hai",')''',

132

STRUCTURES AN O PR OPERTIES OF RU BB ERLIKE NETWORKS 04) Chen. R. Y. S.: Yu. C. lJ .: Mark. J. E . .I/"".o..wi«wk. 1971. 6. 741>

(H) liar';". J. A.; Standen, I . l'oIJ'1"" 1%7.8.97. (36) Fl<>ry. l', J, Pu" ApPI. Ck ..... M"",ontoi. C"'~, . 8 1973. 33. I. (31) .\ t i"r.li, F: Volpi". I).: Garibaklo. G . fl.: Cifcrri. A. J . Ph;'" Ckm. 1911. 15. 142, (38) Yu. C. U.: ~ h rk. J. E. Poly""', 1915, lb, 326. (39) Ctot. A. N,; " uan. T . II , J. PoI,'m, S. (SO) Sakurada. I. : Nakajim., A.: Shibatani. K, M ak,umol. C","",;" 1%5 , 81, 10J . (5 I) EIa,h.,,·. J.: Smilh. K. L Jr. J. I'dly"' , Sci.. 1'"" 11 -2 1%8, 6. 1/).11. (j2) do eo",lio. 1'.: Amclino. L: !'ric<. ( ;. J. 1'01,..". SrI.. P,.. , A· , 1972. 10.975, (53) flcder. R. 11,: Yu. C, U.: Mark.J. E. P<>i,.m. J. 197~. 1. 2}.1. 154) Price. C: Allen. G,; Y",h;mur•. N. Pol}""" 1975. 16. 2<>1 (SS) Hory. 1'. J.; Cif.rri. A.; Chi,"&. R. J. A .... ("hem . So<, 1'161. K3. lOIS. 1023. (~) I'rieo:. C,; Aile". G.: "" C. nJ, •• F,: Kirkham. M . C .; Sub",mani"",. A . l'oJ,'m,,, 19JO. 11.486, (57) Mar k. J, r~ J . 1'''')''''' Sci .. I'd" C 1970. 31. 97. (Ill) Op.loroor. A,: Prin •• W. J, 1'<>1)'110 • •,><-;•• Part C 1967. 16. 1095, (59) Rehogc. G.: Schafer. E. E.: Schwa .... J. II"~'~' MaJ:romoi. Chern;" 1971. 1~! 17. 231 . (6(1) Zhang. Z.·M .; M.rk . J. E. J. PoI,'m. Sci.. 1'01,..", Ph)". ~:d. 1982.20. 41l. (61) Wen. J.: Mark. J. E. Poly"', J. 1994.26. Il l. (~2) Orofino. T. A,: Cifcrri, A. J. Ny. , C/t.t",. 1%4.68.31>6, (63) l' u!4 {641 M..k. J. E.: Flory. P. J. J. A",. ('/um ..'>0<. 196$, 81, I ~ IS. (65) En¥"lIcr. A.: M uller. F. It. Kolioid-Z, w, Z. /'0/,-'", IOiIS. 1S1. M. (/.6) Dick. W.; Muller, F. It. KoJloid-Z .. ~ . Z. Pol"". 1%(1. 172. I (67) GorjlZ, I).; Mullor. F H . KoJloid-Z .•. :to I'al,·... . 191). ,51. 619 (68) Gent'", n .: Muller, F, It. Kolicid-Z. w, Z. "eI,-",. 1913. HI. 892, (<>'I) Price. C.: " ' "n,. K . A.: "" ConJ;a. F. "al},""" 1973 . 14. 338. (7(1) Allen. G.: I,""", V",himur• • N. J. ("hr ,,,. S« .. fa."",,}' n".,. / 191$. 71. 548. (71) ~lory. P. J. J. (""'",. Ph,.>. 19-\9, 17, )(I) . (72) FOA. T . G .. Jr.: I ~ory. p, J, J . Ph}" , CIwm. 1949.53. 197. (13) (11"""'0 ..... S.• Mark . J. F~ : Flory. P. J. ,\!ac,,,,,,,,kruk, 1974. 7. 325, (74) Cikrri. A. Tran" fa,aday S«. 1961. n. 846. (75) Mark. J. E.: Flory. P. J J. Am. Chtm. S«. 1%-1, U. I) ~ (76) M ..k. J. E,: Flory. I'. J. J . Am. eN.... •'>0<. 1%5. K1. 1423. (71) M"k . J. 1;.: Thom.,. G . R. J, I'h,.,. Che,", 1%6.70.3'88, (78) I' lory. 1'. J, /'uri ,jppl. CN.". 1971 . 16. JOII. (79) flory. P. ), 1914. 7. JKI.

p""

c.:

M,,,,,,O>H>J«,,/,,

130 STRUCTURES AND PROPERTIES OF RUBBERLIKE NETWOR KS

/"",-, /'8.,.-0'""', /' ~' \w, ~ \ CIIo C~ \

FiJI . ..

~ .8

CHo

The polyisobu,yknc ellain,

The semi·inorganic polymer roly(dimelhylsilmClhylc"e)"". shown in f'gure 9.9, should be vcry similar '0 rolyisolm\)"lcnc in Ihis regard. sinee ils ac«:ssible conformalion~ again differ in energy onl)' '0 rhe exlenl that C H 1 "nd C II" group, differ in their inlramolecular ;nteracliom . As ,hown in lhe lable 9.9, ils value of /,// is also ,wy ne,,,ly 'l ero, A nnal example of a rul chain ha.ing SOnIc char· acl(riSlics of free rO'alion i, ro1y(lrimclhylcne o_'ide). sI,o,,'n ",hclllati';ally in figure 9, 10"", Gil"''''' Slales aoonllhe Iwo C - C skelelal oonds bring CH, groups and rda!;-"Iy small 0 al"m, in'o pro, iIni,y and are of lowcr c!\Crg)' 'han l.a", SIal""; lne preference is. howe,,,,. reverse.! for the lwo C - O I"",d, in lhe repeal unil since Kaud,t' .tal.. aooul 11><:S!' oond< bring a pair of relativdy larse C . I, groups il110 eonOiel. Thus. lI"uciw .Ime. are preferred abou' Ihe f'r>11wO oond, in Ihe repeal unil. and ,r"nJ Sla leS arC preferred aboul lhe "''''aining IWO. n'aking lhe chain conformalions hignly ;""sular upon .,,,raging OWr lhe emir" repeat unit. Siretching :l poly(lrim(lhyltnc o~idc) Chain increases. of c"ursc. Ihe lo.al trans population. but gallCh~. ,!>.tr~",. lr-..nS;lion< aboul C - C oond. increaS!' lh. energy wherea' sueh transilioni a boul C - 0 oonds d",fCfl'" Ihe enerS)'_The '''"0 effeels would lhu, be CX[lCClcd 10 Off""l one anOlher 10 • large exlen •. tn agreement with this expectal;on, poly(lrimelhylcne ".ide) i, f!>und 10 be the "'[lSI nearl y ideal elastomer in lhal ils "alue of /,// is one of 11><: smallcst of any reporled 10 dale and is. in fan. tero wilh;n experimenlal error_ As clearly dc",!>nSlrated by Ihe examples ciled in lhi, sectiM. lhcrn,,,,,jaslic mcasurc"'tnrs and their inte'Pret.lion hy means of rolalional isotnerk .\talC

/"':./~.»"':./,"",

.;;; \ ",

H ~. ", ~ .9

The

Fi~ur~ ~.I O

,~ \~

pol)~dime'hywl mc'hyl.",,)

chaiR.

The poly(trimeth) koc oxide) ellain_

,,: \~

THERMOHASTICITY

131

Iheory Nu.idc a g'~at d~1 uf in"ghl int\> bm h thcnn odynamic .nd molecular "Jl'CCIS of rubl>crl l ~e ciaslicity.

Referern;es

6/ 1'0>1,.""" C~mjj"J. Cumel l Un,wr>ily p,..,... Ithac... NY.19H n ory. P. J. Troru. fa,a/myS.",. 1%1 . SI. ~29. Flory. 1'. J. 11.""'-. CIo<,.,. rcc/uwl 1961I. ·11. G41. Mo'k. J . E. J. Am. Clo<m. ~. 1910.92. 72S2. Trtlo.r. L. R. G. 7.".. Phpit-s of /It/N..·, Mi-<1kity. 3nl oJ, <'""'",,,don Prcss.:

( I) Flory. P. J P'il!("irlts

(1)

13) (4) (5)

o"rord.

197~

(6) I'n...:. C, 1',,,,". /10)'. 'ilK. I~,",k",. $e.. A I'n6. JSI. JJI. (1) M.,k. J. E, . Erman. II. RiJJhiI-<.fa, Pri"...,. Wiky. I.ror:scienoc: New York. 19~8, (g) F"'ry. 1', J. S'oI,ulicQ/ M",/uiniel 4Clmi~ J,f,4,'
(9) Mallie<. W L, ; Sutor. U. W ("",!!""","i"",,, 1'Ik"Of"Y of 1.I"g~ M<>iI-<.I... The Rma,i""'" /s""..,it- .~·I"" M<xkl in M",,"~~HJI,,~lat S)~IC"". Wiley: New Yor ' . ,~ .

(10) (II) ( 12) (13) (14) (I S) (16)

FIo,y, P. Jo, Iloovc. C . t\. Jo, C ifcrrl. A. I . Pohm. S<-i 19S9, 34. )37. Flory, 1'. J.: Cifc'ri . t\.: Hoc,,,. C . t\. J. J . l'o>IJ"'~. Sc. C A. J .• 1'I0ry. P. J J. A",. ChmI. SIX'. 1%1. 83. lOIS.

Mark. J, E, H.b/x>, Clorm. 1fChnoJ. 197J. 46. ~~3. Mark. J, F. . M"",-",j()J , /I~. 1916, II . 115, Godo'~k y. Y. K. ,t,/ •. I'oi."n .•'i
(,·Ior",. 1976. CIS. L Ed, 1~7S. D.8S1

P~I·I.

(20) G<e. G . I'd,""'" 1987. 28. 18to. ~ 21) Do)"'" R. F: Mill... R. L. /,"'y"'" 1987. 2&. 3'i\1. (22) Vilgi<. T, A. In Elasr""Je,it- P'''I'IIW' N,,"wk$. J. E Mark II Erman, Ed .. P",n,ie< Hall: Enskwood l..1ilf... NJ 11'92: p. 32. (13) Smi'h. K, L ;" In ~''''''"",,"'i< 1'01,.",..,. N",,·6.t •. 1. E. Mark ,nd II. E,,,,.n. Ed. Prenlie< Hol l: Englewood Oilf,. NJ 11'92: 1'. 116. (24) Ambadu:r. I I.; Kili.n. H , G . In ,it-I'oIy"',, N~,.y"b, J. E. ~h r k and M.

."d

E/ru",,,...

(lS) (16) (21)

(28) (29) (30) 01) 02) OJI

Erm,n. F.d" ''''''nlie< 11,11: t,ngle,,'o>'C. C. A, t ,; Flory. P. J. J. 1'"1),,,, Sd. 1962.60. ISS. T,...I"". I.. K. G . 1%'1, 10. 191 110)>"<. p , 11,: T",I"",. I•. R. G. /,,,1)_, I~ ·ro . 11 .2 1. Cl>cn. ·1'. y, : It;"",•• 1'. : Shon. M, J .11'4''''""", Sci. ("Iorm, 1973. 7. 889 \t.,h,;n, M. A .. T",r"",. L K. G . 1'(4)'1"'" 198'1. ~~. 189J

p......,.

1',"),"'"

THERMOElASTICITY

133

(!Ill) Abc. Y.: flory.I' . 1. ,Yarr_"kc~l" 1911 . 4. 2 19.

Abc. Y.: Fk>ry. 1'. J. •lf4(-._ , >kc~I,·, 19 11. 4. 230. ~hrk. J. I;. J. Cl><m. Ph, • . 1 ~12. S<>, ~ 51. ~"r~. J. I!. J. C"",,,, Ph, • . 1972, 51. 2>11. Mark. J. E J. PoJp". SWkrnI<. 1975. 8. 874. Su"",,"""aj.n, I' , I/. Coo,,,,,, P"/ym. Sci. 19'11. I. 18 Ta.,h.sJoi. Y : M.r\;. J. E. J. Am. C/oe'" S",·. IY76, 98, J1S6. (~I) WfIOd. L. A.; Roth. F. L. J. App/, Ph),>, 1'M4. IS. 781. (92) G,,,,n • . A.: Smilh. K. J .. Jr.: Cif. I%S. 39. ~917. (9') Clro";, A. M"kromol. Clmmr 1% 1. 43. IS2 (100) Chian,. 1( . J. I'~,.• . c"-,,,,. 1%6. 71), 2348. (101) Nat.ji",•• A.: Ha"",d • • F.: 1I.}'a ,hi. S. J, Pol,''''' 5<';" I'~" C 1966. II . 28S. (101) Oohdanccky. M. Coli. Cm:h. ('1><",. ('_m"". I%.'!. 33. 4,91 (103) K Olo~" A.: Onda. N.: s"il o. T. H;,p. 1'(>/),,,,, Ph,s. '(>1''''' 1914. 17.1S. (1 (14) [)cOol" L. Mork. J. E. p'~JMI!' 1'187. 28. 4 16(1M) l)Qrring'''''', K. I•. , McCru", . N, 1/. , l'oJ},,,,,, 1977. I~. 1 12.

(81) (82) (83) (1l4) (S5) (86) (81) (88) (89) (90)

c.:

f'<",.

G., W,,"'". W.

10

Model Elastomers

10.1 The Oeper.dence of the Stress on Network Structure 10.1.1 (i<,nt, al Approach Un1,1 4uite ,«"n lly. lhcrc ".~~ rdali\'tly little reliable qua ntilati," ;nrormal;nn on 1m: "".,ion,hip of 51"," 10 i lruclun:. primarily btt'au", or 'he ul1<Xlnlrollcd man_ ner in which c1aslomcric networ k, .....,.~ genemlly propared'- ', SegmontJ close IOSC1hcr in space were linkcd irr"spee l;'" of their loc,,\ions along Ihe cha in 1mjeclorics. thus resulting in a hi ghly random nelwork structure in which Ihe number and Joe'lions of Ihe cro",·lin ks wcre essemiall y unknown. Such a Sltuc lurc i~ shown in figure 10. I' , New synlh<:li< techniques al'(C 1\(>1'0' available. howe,·cr. ror Ihe pn:par.lio n o r " model"" ""I)' mor ne1wo,k. of known ~ lruclU"'. MQre spe<;ifi. cally. ir nClworh arc formed by end linking functionally term inated chains in.tead of haphazardly joining chain segments at random. then th e naturo of this ,'ery specifoc chemical ",~ction "r<wides the d",ir«l ,tructur"l in fon n. tion ''''~ ~'. Thus. the functionalit y of the croSS link ' is the S.me as that of the end_linking agcnt. 300 tM mokeular weight M , betw""n cross·li nh ,,,,d the molecular weighl dist ribution "re the .. me as tho .. of the starbng cha in' prior to th.ir being end. linked . An exam ple i. the reaction .hown in figure 10.2'. in "'hioh hydroxyl-terminated chains of polyjdi",ethyl.ilo • • nej (I'[)MS) are end-li nked using tetraethyl o'tho-silicate_ Characteri,ing lhe una network ,h ain.le ngth distributiun cquallO thaI of the staning chain" and cross.lin ks having lhe (u"",iun.lily of the cnd.lh,king agent. It is al"" J>O<.c
MOO El ElASTO MERS

135

FiX.IT 10.l End·hn\;mg by. 001><1<""";.,,, "'o.,io" bclw«n hyd,,,,)'1 group • ., ,I>< end. of. pol)'mo, cham and II>< olko< numbc,•• ""ragc m"locol" Wmos ,he cri'i<.lIy ,",(>O"al1' mol<;;"l., ""oigll' .If, bc,,,fto CfOS$ link •• and tho di"ri_ bu,;"n of M• • 1", char"" .. ril<"S II>< d,,,,ib",i,,n or M, (i.._, ""'Wo>!' chain_kngth,),

Ilccau>c of lhei. koo"'" '1r"C1U,n, ""ch model cia,"omers arc now Ihe p.eferred mal~.ials for Ihe quanlilali"e cha'aclen7.lUion of rubbc.!ik~ elaslicily , Such ,'ery specific cross· linking r::.lar Ihcori.. of .ul>be. elaslici,y wi,h rcga.-d 10 ll>c: inere,,'" in non-affinen ... of 'he ne,,,",,.k deformalion Wilh inc.ea.ing elongalion. Some of lhese rcsuhs ".e shown in f,gure 10.3'. The '"Ii" 2Clf2C, dec.ea","" wilb increase in cross·lin k func~ion.li'y f.olll Ihee.: 'n fou' I>ccaus<" e,,, ••. links ecmncNilli\ fou r cbain. are mOre eon'l.:roul;lll "ooUI by lhe OUClu.lions Ibal are enh.nced al high deformatio" " "d g)"e the deformalion ilS ""n·.ffi ne cnaracler. a. desc.ibed in chapler J, The dc-.;.e.,,' in 2("1/2C, will, dec",,"sc in ""lwor~ chain ",,,I,"
136 STRUCTURES AND PROPERTIES OF RUBBERLI KE NETW OR KS

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Ihe ero, .. links are embedded and lhus Ihe deform"lion is already hi ghly nonaf. fine. ev"" a l rcialively .n",1! defo"".,ion •. A more ,h"rough in""'!ligal,on of Ihe em"" of ",o".lin\: f.. nClionalily ""'1ui",. use of Ihe more '"<,,",nile chcmirul reaClion iIlU'1ral~-d in figure ,OA"'. Specifically. "inyl'lenninaled rD~1 S chains arc end·linked using a mnltifunctional silane·'. In ,he ,t udy ,ummariwJ in figure lO,S' ~. Ihi, r.action wa, used to prepare I'D MS modell"'l","orks h."ing fUnCliol\.lil;e. in Ihe range 3- 11. wilh a rdali,'ely un,,,e· ~" "cbic'-. a funclion"lily of 37, A. ,bo" -n in 'he figure. Ihc modulu. 2C, increaSC\i wilh inc,eas< in function., ily. as CXpecied from Ihe increase<S-linb. a nd as pn.-dic.<:<\ in chapler 1_ Simi' a,'y. 2C, and its v.11K relallve ,0 2C, bal h dc<·re",.,> for 'he "'as"n, de!lO,ibed in lhe di"''';$ion of f'gure 10.3.

10.2 Tho Issue o f Entarlglement$ Suc'h m,,,1d nel""''' ,na)' .Iso be ">cd to provi,k a ditcl;1 Ie>! of ,"olccular pn;diclioll ' of Ihe 'Hod""" of a nCI"'ork of known dcgn." of cross·linking. Some

MODEl ElASTOM ERS

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10.5 ",.,.,ri,.,.",,,1 d." •• ~ow,n3 ,he .ff<:<1 or ",,,..,.Iin' fU""'ionali,y on lC, t. 01<0<1,,10,.). 2C, . • 00 2C,!2C,." Repri"'cd .,i'h penni. · .ion from.~su," of ,he hiM~.o.ro,m.'i"" npcr; menl~ 00 model nClwork. ····"·..·',·.. ha,·. ~ven values of .{he elastic mooulus in gOO<J agrcemcnl wilh theory. Olh.,.."· l.""."." haw given values ,ignifica".ly larger (Iw, pr•."diCleJ. a nd ,i>c inc r"","", i" ",.>c lower righl porl;oo of figure Ill.l. Tner" are disagreement '. and {he ;s,"" has nol )"01 b..'Cn O1."WI ...,x!. Sine< the ", I.tio" ship of modulus 10 SIClurc i. of s'lCh fundamemal ;mro""".'C. Ihere ho> I>ccn a gre" 1 deal of research "ciI,oily on Ihi; area. l)i\<"ussion$ and "" rr"'ponde,1C<: " """'S {he Ll Ll lho .. of (hi, boo k "nd Ihose group,,'" p"rlieulorly im'Ol,w wilh Ihis i"tIC ha'" g'-lIcraIN a number of helpful $1!ggc'lion'. sew!",,1 "f whKh "'" C,pIQ01..,t in grt. lor 'lemil below. A.pttl~ of s"'aloM ;mporlaoce ""re dwllo be, (I) studyi"S Ihe CffCClS of ero;,.linking in solulion. (2) Studying Ihe eff""IS of swelling 0" "dworks ~mss linked in Ihe bul k (dry) S{aIC. (3) building on In,- demo"."a,io" by Vilgi. ,,,,<1 E.m,n" Ih,,1 Ihc eon'lmi"l ",,,,leis " n(1 , Iip. li" k ""Jdd, h"w "",~h i" con",,,,". (4) .. (;(I),i"g the off';"I ' " f cm". link IunellO""IrI)" ",,<1 ,lc£,,,,, "f t·,.,J<, .linking. (5) ., ,,,,ly,ng " V"ri· CI )' or dh'o01cri<- 1.... '1.·11'"". P:I nle" ""I)' Ilr"'" 1r""inS \"Cry dirferenl ,·"h,~ of Ihc 138 STRUCTURE S A ND PR DPERT IES OF RUBB ERLI KE NETW OR KS plateau modulus". and (6) gen • .,h";ng rubl>cr elasticity model. to include viscoela .. ic effects as well. Alt~ough a number <:>f experimental studies haw been impQrtant for providing a beller understanding ofthe>c i.s~. il mU>l be admitted al Ih. out",t that these problems have slill nol been unequivocally rcs<:>lvcd or "'cn unambigu<>usly formulatN . In an~ case. it is "",ful to oonsider the illuSlr.lti,'e studies of Oppenru.nll mid Rennar"}' on end-linked r OMS networks of dilTeront functionalitie•. since son>e of these issue, arose in Ihe detailed and careful anal)'si, of their experimenta l data. For example. Figure I(1,6 s~o..-" their ' -alu .. of the sm"II'strain shear modulus of end -linked PJ)MS networks with pentafuJlC1ional junctions as a function of ""twork chain den,;ty. The solid curve is obtained by interpolating through I~c • • peri,ll.ntal poinlS. and the dot d.,hN and the da,hc-d lines wCre obta;nc..! from the afli,1C and pha ntont network mooel'_ respecti,.."y. The data asree with pre_ diction, of {he molocular theories .t larger values of {he ch a i~ or junction den_ sities. At lower jllllC{ioll d.nsi{i~" howe.er. the "",a,urN moduli arc sig>,ificalltly larger {han the predicled OtICS. I'art of the ditTerence. but probably not all of it. is duc to problem, in bringing such lightl y cross-linkN networks to clastic equ iTi!'rium ;n the unswol!.-n state . Finally. Ihe mooul; arc s«n to tcnd to ~.o ".> a. the junction density goes 10 ,.oro. The"" dala ar<: r<:pr<:scntativc <>f othcr P D :>'l S nct· works. as described;n the paper b}' Oppermann and Rcnnarl). O,ta for ",<\domly cross-linked polybutadicne given by R.nn", and Oppennann ,. alw lie ab<)\'e {he theoretically predicted ,'a lues . bu{ do not follow a pronounced ,igmoidal ,h.pe as ~M~ -'- !!!It. "",,_ ••• "" •• I .. . . • >. , , , ,.j , , , , , , I" ' ., " " ,"r'" ' ''-. ./,' •• • G., " " - " ," V,, ' lO "'0\ ., ..,1 F"",.. 10.6 Sl>car n""',, {n, 3 f"""tion of tho chom,""JI nc''''Or~ dcn"ty ~rlI fot <1kl_linkoo PDMS nOh,'o rko h."inS • ju""'io" r,,"<,tonatity or r,,,," TIIC 'cml><~"nre ror the cr"",.]inkini'OO thoe ",,,,,h.nbl "",.."rc",OOO to th< oh"",,'1>.1 ",I.non,hir ( ). pn.' "",,",.;on.(,· ..) MOOEl ElASTOMERS 139 do thc data for the ~UMS systen,. T hu, . structura lly dirrerep' polymers can gi", networks with "cry dirrerent d"pcnd""",," of the mooulu, on degree of cross· link;ng. The Mparture or tho: upcrimcnta l po;lII , fro'" ,he .i",[>le ",olc<;ular th"ol)' when the ".twork cltains arc long indicatcs the presence of additional contrib utions to thc mooulu •. The reaso", behind thi' clcp"'tu re 4~. (..-)" ' -"' ) " V (T . "l lCo ,vf " Q (10. 1) where (I). IS Ihc mean_"lttarcd e,,(I·lo..,nd distance "f a nelw"rk chain in th~ unpenurboo:i. N •.. The I""er i. defoned as the number of chains of moleculal' weighl M . in" ,\Ctwl>rk chain. and varic, linea rly with lho: inve .... of 'he "-I""'" N. ~ MM'=L(-'-) . M , V. , (10,2) Figure 10.7 c"m r ,ne, "a ltlCs of .v, aml N, I", r DM S chum •. f"r which the molecular wright .11. he, ..·,'Cn cn(allglclTICn!S;$ 8 100 II mol '-" It i. vcry striking MODEL ELASTOMER S 1~1 10.3 Interpretation of Ultimate Properties This seo;tion focuses on thc di""",s;on of "nfill ..~l d'''tmncrs a t high ci<:mg'l1ion" with a n emphasis ds 10" significa nt toughening " f the daM"me,. lis molecular o,igin. howc¥or. has b,."<:" the source of ..,mC ~onlro'-Cr1ygo. It had 'ottn widely a l1ributed t" Ihc "Iimiled c",,,,,sibilily" sign a 7.cro probabilicy 10 ~ c<>"figu"'~l;(>n unl.:ss il~ en,J -I(>-< nd >eparati"" ,is infinite. I-Iowe,cr. lite increase in modulus had ge,,,,,,,lly 'ottn obscr¥cd only in ncw.orks I~"I c<>uld underg(} s, r.in-induceci cry-_I,4_polybutadicnc net_ worh' ·"'·I'. The dah' indicated tho higher the leml""al"r., the lower lhe e. tent of erySlall;,,,uion. and thi s was found to diminish the "himale propertics. In facl. the upturn~ diminish and cventually disappear upon incre.", in l~mpo:(al"rC" '''. The efTC\OI. of i""rease in ,we!ling were f"und to paralld 'ho", for increase in temperature. a. ",as ".'pc<;tcd. si"". dilu~nl also SUPP"""-'S nelwork ory5t. lli1.a_ lion' .. •... , • Fi~ .... IUJI • S,re» ch'"g. ';<>" Cll"'C 'I~'~''''g ' ho h,sh-ck>nsau"n uP' um;" ",<> 142 STRUCTURES AND PR OPERTIES OF RUBBERLI KE NETWORKS .....n-. ~ 1\l.IS8E~ • '! < ~ ~ '" ''" , •• l",---'---",---'---ot.---'---71.---"----i,,J 0.2 OA W; OJ:> 141 ."' .,~" .. 10.9 R<pr=ntalion of t .. uplU'n in modulu. for nalural rubbe," in ""m, of tho Mooo On the other hand. in th()5e cases where the upturn s arc due to limited chain c.sen·cd" ·"·". howe,·.,. by ,Ite use of SOme of lhe end·lin ked. noncrystalli7.lt ble mooel POMS ne!wo. k$ de,,-,.;be<j abo,·e. Thes< ne!wo,ks ha,.., higb "",ensihilili,,,. presumably bccaus< o f lheir \"~T). IQW incidence of dangling-chain nelwork irresula ri!ies. T hey ha'''' partieu. ,hort larly high cXlen,ibili t;es when they arc prepared from a mi"ure of chains (a«", nd • few hu ndred grum. pc. mole) wit h relative ly I",' g ch"ins ("round 18.000 g mol '). giving a bimoda l distribution of flCl,,·ork chain lenglhs' ·". Apparently the very ,horl chains in such flCtworl s arc impona", hccau.'I<) of lbeif l;mi!Cd e"e"$ibil;,i",. and l/tc rdatively long chai n. are iml)()'l~nl beea..., of their abili,)· !o relard the rupmre prClCCSS . Such ··bimodal·· ",odd nelworks are discussed further in chapter 13. A. will be documenled in Chapter 13. stress-strain measurements On o",·h himod,,1 I'I):"IS "e,work; exhibited upturns in modulus lhat we re mIlCh 1."$1 pronoUllCCd lhan 'hose in cry"alli,.ahlc pol)"m~r "etlmrh ,uch", n,numl rubber '" d,. 1.4.jK,I>·bu lad"'"e. Furthermon;. lhey arc independent or tempe,,,tur. and h.,·. '''ry MODEL EL ASTO M e ~ S 1(3 are not diminished by ;nc!)rporatio" of S<)lv<:nt, Til""., characteristics ar~ ,,-hat is to be txp<:<:too in the case of limitoo-chain cxtcn,ibility"'l'", Thus. these results pennit inlCrprelatio11 of properti" .uch as lhc elongation at the upt urn in lhe modulus and the clongat,!)n at rupture ("ma"'mum cXlensibility"). 10.4 Some Oth er Unusual Networks 10.4.1 Da ngling-Chain Nelwork s T hese same very spo.·.,ilic chemical reaction. enn also he used to prepare nelworks containing known numbers and l~nglh5 of clangling-chain irregularities , This i. illustratoo in /igur<: 10, 10". If more chain end, .r<: present than reactive groups on the end_ lin king mOlecule •. then dangling end, will be produ~. and their number is cli ...... tly 1oichiomclric imbala,,,,",, Howe,'Cr. of ncc<:ssity_ lhe sa me lhose of lhe elastically dTc'Ctiw: chains_ locir lengths as shown in parI (01 of Ihe figure , This conslra int can be . "",o,-ed by ..,paralely preparing monofuJld '!laching them as shown in parI {bl· Since da ngling chains represenl imperfection. in a ,>ctwork strtlCture, one would expecl their pre!iCn<:e 10 have a delrinltnlal dfect on Ih. "ilimatc proper_ ties. (\-, and if I A-),_ of an elastomer. This expectation is confi rmed by all exton- .re. a. H •• , .I O. 1O T wu ,""I- lin~i " B t'"""~I''''' fur p.-.:p",ing ""'''''''~' ,",'"t, In'''''n """,,,<,, .nd k"8'"' of d.ngJm~ cha",, '· i<. "'i'" 144 STR UCTURES AND PR OPER TIES OF RU BBE RLI KE NETW OR KS .. , , ,,• • 0 ,, i• , I , ..; o. • --,.....'',..' ,- •• ,- ,.. , • • • FIK.... 111.1 I Valut> of the ultimate $t",nllth ."""'" ~ •• function of the mole,,"l., ",'eight ,\1, b.>-lin k. for (un!ilkd) lCtrafuoction.1 I'D).{S net",'"r", at 2S 'c"'. si,.. series of ....'Sul l' oblaincd on I'OMS networks that ho d been tetr.functlonally cross· linked u,i,,& • "ariet )' of Ic:chnlques. Some pertin~nl ""ulU are . hown. a$ II funClion of the molecular weight bel"""n cross-links. in rogure 10,11"". The large.l ". Iues of (JI A' ), "", ob,"incd for lbc netw",k. prepared by ""leeti""ly joining funCiional group< occurring ei ther ., chain ends or liS side groups along the chain •. Thi , is to be cxpecU,d because of the rciati,'ely low incidence of dangling ~nd, in such nc1work<, (As ah""dy dtscribctl. Ihe elTc:c .. an: pro· non",--ed when , uch model networb arc p",paf<;,1 fmm II mixture of relali,.. ly long and yel)' ,hort chain,.) Alw .•s expecled . the lowcst values of the uhlmalc properties generally occur for Ihe nelworks cn",..! by r.dialion (U V lighl. high_ ~ncrgy cle"Clron<. anJ -r-r-ddiati onj90 T he peroxidc,cUT",1 nelworks an: gcnc rally imennooi"te 10 lhese Iwo c.tI.c mcs. with the ultimate properties pn"umabl)' d.""nd ing on whether or nOt Ihe free ""lie~1$ ge nordtW h)' !he pefO,ide arc snmcicnlly ",acl ive to eausc om'", chain ;ci""io n, Similar results were ob,"incd fo. Ihe maximum exten , ibilily"", Til"",, ob>crvation, are al leas, .... mi-<ju"mi'ati,,, and ccrtain ly inlere.lin g, bUI they are $Omew ha l deric< "r model nelwork, P"'p;li1:d by end-linking vinyl-termlna lcd I'O MS chain''''', Th o lel ...~ functional end.hnkin g "grnl was used in varying amouut, sma ller Ihan that cor,l<,ichion'etric bala nce hel"",n it> aclivo h)'druge" "10m, and the responding chains' term;na l ,'illyl grou",. T he ,,11;mate properl;"" of the""" IIelwo,k< , wilh hown numbe ... of dangling ends, "" ro then compared with th""" ubt ained on "''''",,,1;,,,), l<'" MODEL ELASTOMERS US nelwork, pre"io",ly prepared SO a, 10 ha,'. negligible numbers of Ihese irregularities"', Valu ... of Ihe ullimale ,uenglh If/II '), of Ihe HCI"'orks conlaining Ihe dangling ends were found 10 be lower Ihan Iho«' of Ihc more nearly perfect oel,,·o, k•. wilh Ih. largesl difference< occurring al high proporlion , of dangling end, (low 2C,), a, .. peeled .... T hese result, thus oonfirm 'he less detinitivc "",ull, ,hown in figure 10.11. The val..es of Ihe ma.imum c. lcn,ibilily ,1>0"'.'<1 • ' imilar dependence. as .'peeled, 10.4.2 Nctw(>fks Olnlaining Rq>lali ng Chain< End·lin king funClionally Icrmi"alctl chai". in Ihe pre",,,,-,, of chains whoi<' end. arc inerr yidd, IlCt""orks Ihrough which Ihe unallache,cl> un.lIached chain. C;tn be ••11110100 from an cI;t;lomcr a. a function of thei, Ienglhs and the dcgrte of cross-linking of the nct"'·"rk"". This subject waS covered in chapler 6 in 'he di;cuss;on on .wel l'ng. 10AJ N.,worh I'rep<\red in Solulion or in" Siale of Slrain Two l""hniquc"S Ihal may be ui<'d 10 prepare nelworks having .i mpler lopologieS"." a", illUSlr~I<-.:I in figu re 10.12·'. lJa,ic"lIy, Ihey i""olve ""paraling the chain$ prior 10 Ibtir cross-linking, eilher by strelclling or by di"",lul;on. Alier Ihe cross·linking, Ibt .Iretch ing ro",e or ."1",,,1 is ",mo'-.: sIre", slrain propc'I1CS in elollg~lion, Some re,uits oblained on I'OMS IIClworb cros ... hnled i" $<,)"t;on b)' meanS of ..,.. r.dia· ,i"R n." ,h"",-cd thaI Ihel"(: was a .""Iinual d"".-..a>c in lhe lime I«juired 10 re~ch claslic equilibrium, a nd in Ihe e~ lenl of sIr"" ",la'''lion , upon deerea ...' in the volume fraClion or polymer presenl du ring Ihe cross· linking. Alro, al high" dilulions l!>ere was a d""rease in Ihe MonllCy- Rivlin 2Cl cons,anl "' wel l. The«: observation, a.-.. qualilalively explai1'ed ;n lem" of nelwork oon"<'C, ivily and topolngy. If a nelwor k is cros,_hn k<~1 in ",hu;"" and Ihe ",,,-enl " then rClno,-..'\leh " way Iha' !he", is <1.'dI""",1 O\'erlap ;n Iheir configuralional domains, II IS primaril y in Ihi. rega,d, " amely decreased <'hain_j unClion en,,,n,~i"g. Iha, rolul,on..:r"" _link,,,1 ~mplc~ h",·e >impicr lopol. ogie" "'ilh corrcsl"'ndingly si mpler cia''''mcri. behavior. It i. appropriale 1<' commenl al Ihi, POinl 0" Ihc oppoSile:lOn of e'perimc n(. Ihal is. cross.linki,,!: a n.lwork;n 'he undiluled slalC and Ihen studying il' "ress."ain i'olher ms in Ih. ,wollen SIal •. Sud' a diluenl might be illlrndu""o"c) . 1(;\ li n 1 ''''''anla);,' in ,he inlerpr","I;"n of .1",,, ,Ira;n "'olh ....",. 146 STRUCTURES AND PROPERTIES OF RUBBERLI KE NETWOR KS \ 1/\ ) ~~ <>-.'._"",....,--- t'i,.... 10.12 Two l«hniqu<">l ilal may be u"" 10 prq»'" .... 'WOf~ . of >i mplcr 'opol. "",'" Roprin,eJ wi,h pormi>Sion from Mark. J. E. (1993). The Rubber Eh>!k S..... io M"k . J. E...t .1. (hh.l. I'IoJ.k~1 f''''fH'''~' "f l'oIJVH<"Ts. l,ld e~ .. W..hi ....1Q". n Co Am.ricon Ch.mi<~1 S<.> 10.4,4 Network, Cont~inins Unusual Diluent< There i$ • oomplioation. h",,'e"tr. which (."n ottur in (he CaS<: or nelwork< "f polar pol~mers at relali,'ely high d~grces of '''·elling..·... The obscrvati"n is lhat d ifferent sol.'om •. at Ihe degn.., of swelling, can h~''C si!'l'inca ntl~ different elTttts on Ihe ebslic fon", This is apparenlly due 10 a "spttific S(>IWnl clTttt" On Ihe unlICnurbc'Car in Ihe ,.",iou, molttular forms on the da'tk rquations of ,laic, Ahll"ug/l fr~".nlly "bse..-.,d in sludi<. "f tile solulion proporti.s of unCc ma)" depend ;ignirlC"mly on lhe degree "f 1'(1)" n>o: incrc",,-~I hy the pre"'n<" of some diluent" """Ie MODEL ELASTO MERS 147 References (1) Flor)' .... J. I'.i"",rl",vf P,>I_..,..., C"'m;,/.)'. Cor .... l) U"'~""IY P......, hh..,~ NY. 195). m Trdoar. 1.-. 1(. G. 1'1>.- l'h,'.kJ of /("""", H/""kily. )Id "d. Clarendon I'n:ss: O,fOld. 1~ 7~. (3) MOTk. J. E. J . C~rm, E,I. 1~81. 58, 898. (4) M.,•. J. Ii.; Erman . II. HuhIx!rliJ... H"-'lidIJ'. ..I .lful.,wm, !"~, W,leyInle,...,icl>C<: N.... York, 19~8, (5) Iohr\:. J. F. ; lir,mo. H. In I''''~' "Mwl •. 1(. f. T. SIOJ>!O, J:d Black", Ac:otkmic. Chaprm,. & 11.11: Glasgow. 1991: on press. (6) Mark, J. Ii.; Sull"'o,,, J . L J . CMm . Phl". 1917.66. 100!> (1) Rcmpp. P.; 11m" J. E. A"g" ·. ",,.!.;.,,,,,oJ C_k 1979. 76/17. l73. (8) Mark. J. Ii. M~kromM, Oemie, Suppl, 19 79.1. 87, (9) M • • k. J. E.• I(.,halk.r. R. 1( .: I. I.. J. Cltem. Ph),>. 1979. 70, 1794. (10) Meyers. K. 0., 8",. M. L; Merrill. Ii. IV . .I/,,,,,,,",,,,,,cwk, 1911(}. I), 1(»,. (I) KMflJm. Bull. 1911(}. 3. 6(lJ. ( 12) L1orenle. M A.; M.rk.J, E- ,11 1J("Q..,..I,..,u~·. 1980. 13. ,,",I . (13) G""I'.b. M.o Ma"",. ko. C. W.: il<', f:/wtlicil)-. J E . M" • • nd J. ""'. IOds, A"""k,n C hern..,,1 So< .. l)': W.sh;"glon. nc. 1982; Y,~, )9): p, 329. (16) Mork. J. Ii. A<'C, C"",,,,, Re< 1985. 1&.202. (17) Sl . dlor. II.: J.>eobi. M ..\\. ; W. i\I,Ik'QJ,IOJ, Ck",., Iffipid C""'m ... . 1983, 4.129, (18) Sung. P..JI ,: I'an. S._J .. Mark J. E.: Chan~. Y. S. c.. Llokey. J. I, .. Kennedy. J. P. P{J/J"'" /Ju/l, 1983, 9.37S. (191 Llck<)·. J, E" (:hanll. Y. S. C : Kennedy. J, I',: Zha~ Z.·M. ; Sunil. 1'..J I, ~h,k , J. E. P{J/y"' , "0111. 1'184. II. IY. (26) J'an¥. Y ': M:lfk . J. I, .. Ch"ng. Y S. C. Kenncdy. J, I'. /'{J/)'m. JJul/rlin 1984. II. 319 (21) Kennedy. J. 1'. J, Appi. r~I)m, Sci.. Appl. I'vI)"", Sp"". 1'184.39.2 1 (22) Enid, .. S, G o. E.." "",.. Y. V.. Gor>h kov, A, Y. Ad•. P"'pH. ~;. 19~6, 76. 129. (23) !.anyo. L. c.. K.Illnn. S e;. 1981. 3l, 2449. (2S) M,II<" I). /I .• Macosko. C. W. J. PO/),m, Sri .. PQ/),m, Plop, f.'d. 1987. 25. 2441. (26) Sharar. M. 1\ .; Mar • • J. " . Milk..,...'" ('ltemic 1989. 1'10,49, (27) ~on;lh, T. 1.. .. Ibid ... Ii. : II<~ ric k . J. ~ . R"'~ 'hem, 'fN"Iuw/. IWO, 6J. 256. WI) Cia""", S, J .. W.ng. Z, ; "'", •• J. 1: . ,.'.", l'oJ,-m. J. 1'190, 26. 621. (19) Komfiekl. J, 1\" Spl<S>, 11 W.: Ner,g<". It. Ibye". II .: Ei",nbach , C . I). M",·"",IOJ",..k, 1#1. 24, H~ 7. ()(I) ~jjva. I.. K .. Mar.,J. C, : 11<",,00. ~'. J. M"h"" ~J/, C"",,,,k 19'11. 192.499. (31) lI.nyu. A .. Slc'n. R. S. ,llac,QmIJJ, Symp. 19'11,45.189. (32) lIol.nd. C. M.: 1I,,rcnl~, M ". : M.,l. J E. /·,,!,-m . H,dl I'i'll. 2~. IOJ. (36) O,~a .... . II 1',>1),,,,,,, l'Wc, .1 <. 1116. ...,. .... S.Ii'''''. (,roo,." c., 1018 STRUCTURES AND PROPERTIES OF RUBB ERLI KE NETWOR KS (37) Sharar. ),1 . A. J. Mam"nq[. 5<:i., Mam"'lO/. lI,p. 1992. A)(l. S3. (3S) 1I .,., u",u. E. E,: 83)'sal. B, M . PoJ,.",,, 1993. 34. SI~3 (39) "ald. S. K.; M.lone. S" Cohen. C .. Gill.,or. J. R.: Colby. II. H. ,\I =~'''''' \ 992. S24 1 (40) Ska~dr. M. A. 1M. J. Po1J_#~ IoId"'. 1992. \S. 2J7. (41) Fioel>n!""'. S. L., Stein. J.; Zumb,"m. M. A. J. PoiI'm, .'>mr, fA. 1993. 31. 2697. (48) Sh.rar.),1, A.: Matk. J. ~, IV}", .. , I9'J.I. 35. 74(), (49) Stlarar. 1.1, A.: M.,k. J. E.: Aj.(Jh a:J~ l . A. A,·A,}, ,{prl. PoJpn. Sbm. So<. 1919. 101 . 7lH. (64) F. lenJ ... J. A .. Veh. G . S, y" M"rk. J. 1;. Maa<mr<J/e,ul<-, 1979. 12 , 1207. (65) l)j;ch, S., Finkcln"nn. It.: Ringodorl, It., SIiC liS7 a. MODEL ELASTOMERS U9 I), ViJCQt'I(UI;" /''''PO'''~' I)'n!. xi. 1987.15. 0\9, Ronn". N .• 0"1'<'''''' 00 , W, Co/I. 1'0/)'1'" Sci. l'>91. 27(1. $27. E""an, B,; Flory. P. J. M,"'"",,*,'~'" 1%2. IS. &'16. Flory, P. J. Slati"kal M,~hanlc't' 0/ p,~ymt'T' , MoG'aw Hill , y o,k. I~ . (84) Ma ndol,em, L. In ("""'I'"Ioo: o.,rord, 1989: p, J63, (85) .\1"• • J. Ii Macront<>kc.k, 1984, 17,2924 (86) Cl • ...,n. S. J. : G.li .....",. II . 1',,1)',.,. C""",'WI. 1986.27.26(1. (87) 0;0. D. S.: Suo T.·K .; M>rk. J. E, MO<",,,,,,o/,cu"', 1911, 10. 1110. (88) MOIk. J. E.; Eiscnl:>. P n . .<;"clli"8 C()r>CfPIJ in Poll'''''' Physk•. Cor ...n Unil"C";ly J.....: It"""'. NY. 1~7'I, (92) Mar k. J. E.: Zhong. Z,·M . J l'oI)'m. S,:;. /'01)",0 . I'h)", Ed. l'll! l,l'. 1971, (93) Langk,·, N. R.; Dickie. R. A: Won,. C .• Fo"y, J. I), ; Cit;, ... " R.; Thirion, P. J. PoIJ'm. Sci .. /'a" ,1 ·1 1968. 6. 1311. (94) Yu, C. U.; M .,~ . J. E. Mac,nmo/«uie> 1974, 7. 229, (95) Hocv•• C. A. J.; O'lIricn , M K. J. l'oJ)""" Sci. Pml of 1%3. 1. 1947. (% ) 000005. A,; 1\<,\Oi!, H, ,\IU('Tcnwlrt~l" 1971. 4. 279. (91) F,i'm.n, F~ \'.; D.<.Ii." ny.n. A. K . V"Ql.OfOOoi. SM1ir:, 1%Ii. 8. IJS9, (98) Naga;' K. J. Chem Ph!", 1%7.47.469(1, (9\1) Gent. A. i':. Mac,,,,noJ«.ito. 1'169. 2. 262. (100) h~i~"w •• T .. Nagai. K. J. Pm)'"" .'Wcl.. Pu" ,j.: 1%9. 7. 112l, (10 1) IsIoib,,·• • T .. ".ga ;' K. I'o/rm. J. 1970. I. 11 6, (102) Genl . A. N.; Ku"n . T. 11 . J. 1'6/),,", Sci .• /,,,,, ..1 ·1 191 1. 9. 927. ( 103) SI<1n, K. S.: Hons. S. D, J Md. 3S I..l51 (1(19) Flory. 1'. J. }'01)'""", 197<1. 20. 1317 (110) Flory. P. J., I:"nan. fl. ,I/rluumo/"rlk.' 19~2. IS , WO, (7:2) hrry, J, (13) (74) (7S) (76) (1T) (18) N.,,' "oIp"" G,,'i.,.. ~ , I'''~ 'Y!'I"'" .. r W'OJ ",,,,,!)S .l~l .')J~~." ,(./, \" ,It '( '\')~~!I' 'U"!I"UUUJ"I' 1r.!xr.!Un '0,1 UJO 1"110$",1' "II """""d.,, S"!P "" ~ "I r J(""~I "O!!1"UJOJ~P J~l UO'II!W'Ol3Q Jel n~(lIOVII L'j L 'UO!I~lU.'!JO Wlu,w3JoS ,oJ ~nb!uq:">1 ~.>SU!JJ:U!'l "'II pu • .(dooll"" J~ld~4' "'ll. . ,NUl:! P"O: ,mSuN JO jU3W le3J1 """e,,,!, <>lOW "'II .(<1 ""."OlloJ '~u!l1no '! UOII"I U~!J<J lulu~wjJoS JO .(.0:,," ~ldUl!' "II .~. \ I UO!l;»S 01 ·UU!lew. upl' '''In'''lo," JO ,,"I' "'II ..q!J"~p J~ld"4' "~1 JO "0'1""" lSJY >11.1 ~u!Jn.~~w 'uOI'''1 -U '!-I0 IUIUJ,,'':!;). JO ~a,p ''11 ',,!,W,,),P ~1]U"U !UlOI~d SUO!lnq!J\,!!' jUUO!WUlJOJ _\100 '"In''''low,,,,,,! AJ"'Iqru '41 U! '1'4'" '''''In"",<>," SUlJoql(ll,JU P"Poo "P"'jJ U'JII\I"'I iu! ldnO;) 'Uln""IOluJ.>/U! .(q "",,",pO! "II"""" " . ~JOlI\lJl1 ""'I "\UIS,\".o w IOA'I IU]U.)wil>s J~] ]~ UOUU]","O "".1 '~ ]""""""OJ ~'o... '"u J~] .('1 f"'Sodw! Jfqn< ',,,,,]OUU 01 UO!]U,"\f~U<X> :)<10 WO'J oil 0] WOp""'J II"J J~] ""W4 ~JO'" pu JIJ"'UO]S~I' lI~ UI ""041 ' UJWJJ ~11J~d JlI 'IinJ ;:uu 'P!IOS JU!IIR)W' '0 .\,,,"!\{ U! SU!"4' ""' ,u"'~4M ·..' W.(IOO .("~IS JO ""'!IlRI ·ill, H! 1"41 ,uruJ .(1l'U!'S!P ""JJ!P ' ~ JO." I:)<1 .u>q<tru "! UO!l~N'!J0 1P.'u,W~"S "Jldu4' '!41 JO ,n"'J "4' ' ! UO!'''WJOpP ,!dO""OJJRW JO '1"." 1 II~ Ie UOIW,U'IJO I~IU:>W~"" JO uo"d" """p 1."'Ww'~,"W '4.1 '4"""" JO UU!P,J!P '41 ~ UOI" ~ 1"~~nl_"J US!IR SlU~W~"" '~JU,"]JU '4] JO ~"!4J).1J], JO =iI,p 'Imyul JO ""~J "'''''''IX' ~1I.-'!1:>1l1od!i4 ''''1,0 '41 " I ·s."I ~u" ,"0411." 's"o!]RJOilyuOJ "Iq"sod II" .)~u]J;>pun AU'U lIl"4' '41 ""' ""'""Il\dUl1l1SI" ~JO'''l"U '4' l3U'4 pUR 'WOPU"J ;\]~ "D,wil"" ulmp JO ,UO!,nJ u"!JO '41 " lm, p.1U01S!pun "41 "I -p:>UlJOpp "1I~,,!dO:lS':UJRW '! ~JO_"P" '41 ""'I," ' (>!"4'} ~JO"'l'" JU SUO!I . 8Jn~ uuo, ''II JO U01UO'''P '41 WOJJ "lns'J uou",u.uo IU, u""'~"" 'J'ld "4' '!4' JO 'OlOJ '~1 >q II!," 4'!4'" " ~'O''''JU """'In, P'4JI;\]I' " IlU!'''!U" JU "'", '41 UI ' l\"'~. I"U'JI.' "'''lIS JO ~UI]U'IJO '41 01 Jnp '"",d, UI SUO,,"," 'UO IU'w~"" -U!"4' JO uOllflqU]SIP -,!doJlos!u-e ~41 0] SJ.'\I'" UO!I")II '!JO '~lfl""l0W ,0 IC)U 'WSOS ,.0,"I'U """IS ]"".u' uOllelU alJO lelU aw6as n SEGMENTAL ORIENTATIO N 151 l is Ihe ralio of lhe stretched knglh of the rubbo:ry sample to il$ ondcformod ref~rellCC length, The first clement along Ihe diagonal of the matrix repre""nt! the e.tension ratio alo"g th~ din,eti"n of StCCtch. which rna)" be <:<>nvcni~ntly identified as the X axi, of " laboratory_fixed f,.. me X YZ. T he olher two elements ref.r 10 Ihe deformalion a long IWO laleral di=ti",lS. Y and Z. The homogeneo". macroscopic deformatIon of a network result. in the deformation of Ihe nelwork chains, des<:ribcd hy the microscop;c deformation tensor, as A'. (~) [:' ", ::l 0 A' '" " 0 0 0 (I) , , , (:l) {7}; = 0 &~ ( I Ll ) (.')" Here, fI ", '\" and A, denote the in the defol1l1ed network. T he angular h,..~c kcu refer It. the enll<:mhle averages ,,"cr all chain', either in lhe defo.moo stale or in the "ndeformed ",f.rence slato, Tbt lat,.r is indicated by Ihe 'U"""ripl "'" appended 10 the "ngul"r brackelS, Nelwork chains in the rcfercntt Slate will be assumed to have Ihe '<>-Called UnfX'rlUFMd (limen;;.ion,. charncl~riSli<; of 1h.·IU colldilioll')' unlCSlO the nel,,'ork is iSOlrOpically swollen prior to defoml.tion, Similar ({cfinilio", apply 10 the olher two direclion,. Th-< av,,,,!,,! depicted in c«. ( ILl) may be evalualed aflcr lhe ehoitt of a molecular model of Ihe network. Results for the am"" phantom, and con.I ..... ined.junction mood , will be dc""ribed in Ihi ' _""""ion, In the affine nc\"ork model. "'I , (1 1, 1) 13k .. the form (11.2) ,i n", Ihe chain ends lransform aflinely. For the phantom network model'. lhe mean chain dimensions Irans[Orm affindy tml 'he HUClua';ons'ro independenl or macro"""pic deformation. and thcn.~ fore Ihe d.formalio n althe mok>.,ular \c,..,1 i. less lhan lhal gi"en by c«. (11.2). Wriling the a."rage squared .~-conlponenl of Ihc end-t(H;nd vlXlor in Ihe deformro nctwllrk as ( 11.3) and ",ph,cing Ihc avcr~8C chain d,n'cn;;i"n, " nd tlucluations fwm the equalion, in appendi. to lead, 10 152 STR UCT URES AND PROPERTIES OF RU BBER LI KE NETW OR KS (11.4) The s<xoond term on the right_hand side of "'I . (11.4) i$ an identity. The ratio depicts the ratio of the _~ oomponcnts of chain dimensions in lhe defonnoo and u"deformed .tatCS. Substituting eq. (1l.S) into "'I. (l 1.1). togelher wilb similar lerm, for the y and: component., kads to til. molecular deformation tensor A' a, A'.. [ ( t - 2f¢,»,1 + 2/¢ 0 0 0 ( I - I f¢,».-' +2f¢, 0 o 0 ( 1-2N) ~- '+ 2/¢ 1 ( t (5) In rcal ncl"'orks, the slate of microscopk deformation is oblained by <XIn,idering the fluctuations of junclions from their mean pOSitions. as was fit'll dim. by Erman and rlory' _According to the oonl1 modd. the ra1;0 shown in "'l.- (1 tA) can be appr,,_,imatw by Ihe ralio of floctua1;ons of junctions in the dcform«l and undeformed St.tes. tllat is. ( I (6) The r~lio on the right-hand side of eq. (II .6) is gi".n by I + B,. whe", 8, is given by eq. 0.36) for tilt: con.trained-junction mooel. and by cq . (4.29) for the constrained-chain mooel . Substituting into cq . (11 .6) a nd then into eq . (11 .4). togeill., ".-ith similar term, for the y and : components. leads to lhe molecular deformation tenro, A' for the eonSlraint models a. " . " (I - I f¢,)>-' +(2/ot»1 1 + 11(>-)1 0 , (I - 2f¢,»' -' 0 0 + (2/4»11 + lIt), ')I " ( t 1.1) ( I - 2N)>+ (2NlI l + R(>-- 'l) The constrained.jullction model assumes thaI constraints acling on junctions 'pringlikc forces. Th""" forces result from lhe entanglemenls of Ihe j unctions and th<.~T I>"ndcm chain! with Illc surround ing ellains. Thus. an additional Slrain field surrounding each junction ha~ to be defined. ACCQrdingly. the slate of deformation around a jU'lCtion. resulting from the distortion of the ""n.lra;nt •. i. tnken c~cn '0 ~ SEGMENTAL ORIENTATION e'. 0 0 ((.t... •• )') --"'«l'''',.) ) 0 0 « & ,,)1) ({a .,)l) 0 153 (1 1.8) which may Ix: wrillen e'. [ I +t~) I+/)(r') 0 0 ",-here Ihe funcl;on I)(:\') is given by cq_ 0 0 1+ D{:\. ,,j (11.9) (3. 3~). 11.2 Segmenl al Orientation in Network Chains: The Simple Pictu re SegJl1~nlal oric o,ol;on in chains "f a d~f'm'lled nclwork is characltri,~ by Ihc u . rigidly affixed 10 one or more bonds along spal;al oricllialioll of a given Ihe nelwork chin" ' . a, ,hown it! figurc II I. I'or .,a mple. Iran,ilion momcm veclor. u. rigidly allached 10 each repeal un;1 alollg lhe cha in. are considered in Fourier Iransfonn infrared dichroism mca,urcments'. provided lhal If><: di[lOle mOme'lll change implied by a given infrared·aClive norm"1 "ibralion ma y be ascribed 10 Ihe rcoricnlalion or u. The X a~i, makes an angle X with u . and €I dellOI .. Ihe angle bclwcen IIIe chain veelor r and lhe ~ ' ax;,_ T he angle=1,,, , , ...... J B o Fil." 11. ' Spolia ' on<"""io" of a '"«'0' .. n""ly am",J ,,, """ Of more booo.. alOftS ....'wo,\ cluin. The bru ~.n ",Tid line ~o ing fl(>m jun.hon A 10 junclion 8 rep,=n" • rltem j, iJcnti!\cd by the "'~.. xrz, ·nrc X • • j; '""'",.n . nllJ.c t ,,·jth ... _n<1w.xn the eh.in ' ",,10r ' " nd tho: X ",js, I~' e tS. STRUCTUAES A ND PROPEAT IES Of RUBBERLI KE NETWOR KS show~ ;n the figure. repre..:nl, 1"" angle bel"'""n u and r. ond ;s used ;n eakula· l;ons of oricnlalion as will be oullined below_ In uni.xial deform"l;on. the X •• is;s con"c~ien1ly ;denlificd "';lh lbe d;rccl;on of th. applied load. which may be lcn,ile or comprt$siw , The orientation of u w;lh re'l""'t 10 the X .. i. is c~rrcssed in lerm. of the orien1alion function Si x ). which; s given by lhe sa:cnd Legend« polynumiall',(<X>i X) a s S{x ) " ( I',(co, X»" (1/ 2)( l (<X>i' X) - I) (l Ull) The O\""bar rcfers 10 "''''''~ging o"cr all C(lnfigur'dtions of Ihc chain. subi'-'-'I 10 Ihe C(lnditions imposed on 1M positions of Ihc junclion, A .nd Il . Fo, the ,in,plesl cue of lbe aITm. network !!,,)(iel. for e"ampic. lh. JUMtio'" a", assumed u, be fixed in ~p"cc. Th;, model forms lhc starling I"'inl of the formulation, in lhi. chapter_ The aWr'd8<' (c",l X) ",.ul1< from twO '( i 2" ~ ",he", [)o i$ Ih~ +- ")11 (, I). (1 Ll I) configurational facto. for "'!I,nental oricntalion given b)" ( 11.12) For" given chain Slrueture. the """ond-order moment! {~)~ and (~ CO,l 4» . apf'C.ring in th. front f.clo r f)o may be: ca\cula 1c-d hy variouslheorelical "'hcmcs, among which is the rota(ional iwmcric state formalism "·- " . and by Monte Ca rlo een ,hown 10 be in,'crs.cly proI"'rl;onalto chain lensth. "nd is C(ln',-,n;emly expressed as D. _ I/ n. where n is the number "f bonds in lhc nelwork chain . Fur a freely jointed chain wilh N bonds. or for a real chain approximated by" f"",ly joint«l chain of N <:<:tlli".lenl bond. ( II' :S 0). the front faewr is I /{ SN), a. " 'as originally sh""'11 hy Kuhn and Grun " . and latcr ctabor~lcd by Roc and Krigbaum" and larry and Monnerie" A-xordingl)'. lhc fronl factor of 1/{5N) may be: in1erpreled as Ihe orielllalion modulus of a Kuhn cha in. Del'<~rturcs of rca' network beh.,';or from the ,impk Kuhn formulal;on have bttn di'lCIlSsed by Erman and Bahar" . The enscmhk a"'''~gc of <:<:t- (I LlI) O,"Or all n,,,,,'ork chains in Ihc deformed ,late le,, ( I I 13) SEGMENTAL ORIENTATION Su,,"liIUlio1\ of '"'I. (1 1. 13 ) into c'q. {II . IO) le,d, 10 Ihc funelion: segmenl~1 155 orien l,tion ( 11.14) Equalio" ( 11 .1 4) mdY be exprc ....oo in lerms of lhe macroscopic doforon.,ion Icnsor by ,",,"Ii(ul;ng from '"'I. (1 1.2) for Ihc anine network modeL aod frOnl cq (11.5) for the phantom n.,w" (I l.lS) Segmental orienta liO n i~ 'hus conl"n;"nlly >n funCliemalily and macroS<."Opic deformalion. The ""!llncntal orientation fllllClion for the con.tra ined-ju",,'ion model in uniaxi.1 dofonMtion follows u' ( 11.16) where ~ ;s a coefflC,ent measuring Ihc cxlent of coupling of the sesme nt 10 the cn,iro",ncnl . Th; s furmul"lion is based un "u earl;er treatment of bi refri ngence in ronst ra inoo ·ju nel ion networks"'""·. In Ihc case of nl'lworks ,w'ollen prior 10 ddormatiOl' " lid/or cfOss_[jn ked in solulion. eq. (II. I S) is replaced by " ffin~ phantom ( 11.1 7) wbere .u i$ lhe: I'olume f"'''lion of polymer Juring cross_linking.•~ i$ in 1"OIumc fmOion during the .,tretching expe,imen l. ",Id ,, ;s IIw extension ralio rdalive 10 Ihe swollen sl,\le. 11.3 Higher-Order Approximfl(ion lor Se gmental Orientation Equation (11.1 I) "'pr~n for relaliwly long chains. In a SI riel se" .. , Ihi s e'I'rcssioll and lhe rcsulti ll ~ relalionship of S(;d to deform"li"n. giw" by cq . ( 11.1 7), arc "alid for "",,,"ork. ,,·ith relati,'cly long chains "beying G;ouss;,,,, $lat i,IIC$ a nd sllbicx·too 10 only small Slr~ in •. EQuatio" (II. I I) con,ist. of th. first ,er m I>f a ser;es "xI'" ".,on in powers of I /~ (where n hn of Ihe c.' pre"i"n C 156 STRUCTURES AND PROPERTIES OF RUBBE RLI KE NETWOR KS orientation in short chai n.". ~kr •. we g!'" Ihe seri"s expan.ion up 10 .he S«:(}nd· order terrns:t S(x ) = HDQ(~' - ~ - I) + f), (~' + P - p _l)-+ DI( ~' +i ~' - J ~ -')] ( IU8 ) "'he", DQ .. 2'n + 14'1< + 126'10 (1 1.19) 30'128, 18 (,' )~ V, .. - - ( 'I< -!- IS,1o - 10'128' ) - 1 S (.-'}o (11. 20) S4 (,oj. D' ' ' - ( 'Io - 'Ilg. ) - - (1l.21) (,.» ! 7 The ,'ari",,< """meionls in "'Is. (11.19)-( 11.21) me dcfinc'll 'h E W (Ill. - I ) ( 11.22) , =..!.. [(J(,.' cos' ,~)o _ 1) _, (J{"cos' 4» . I< - 20 (r'). 7 (rl)~ (" ). ) (,..,)~ 1 ( 11.23) "')0) (r')! ( 1124) ( 11. 25) Equal;on ( 11_18) i. derived for" nelwork who .. junction poi,ns transform am · ncly wilh ma«:" oblaine< ","'110....... " ''''''".,." IJ". I), .• "~ V, 10,.-", ""'" ~''',", '''''''''''ely... Il,. Il,. , rod Il, ;" "., P'C"""' SEG MENTA L ORIE N TATION Sex ) _ ( l jSN)P' - I / A) + (l j I SN')(A' ~ A/ 3 - 4/ 3A') -\ ( 1/ 2IN')( A' ~ 3.\'/5 - 8/5A') 157 ( 11.21» C<>mpa ri son with c.... . (ILl8) sbows Ihal Ihe roefficienlS Do . V ,. and I), for Ihe freely joinl .... chain are giwn by Do " I/(SN). 0, = 1/{ 15N'). and I), _ I /P IN'J, 11.4 Experimental Determination of Segmental Orie nt~ tion in Rubbery Netwo rks be,," Slrain birefringence experiments h""o Ihe m'»1 commonly adopted tcchni_ que for delermining segmenta l 0' ifn'"tion in n"worh~·'''' ··n H o"'~v~r. due 10 uncertainties in bond pot",izabilili.s a nd Ihe presence of large contribut ion, from intcnnolccul.r intemclion~ Ihi' I'''hnique cannol reliably Ix used for quanlitalive Mlconinmion of orie"I.li,," of specific ,',:'clo';31 qua"lili.s in ~ nelwork chain . For Ihis ","""n. deleonin'lio" of ""gm~n l.1 orienlalion by birefringence "'ill nol be lreated he", in del"i!. Tho inloreSltd reader ma), refer ", Ihe pap<'1"$ by Eonan and Flory"". and 10 lhe refercllCes ciled I""rein. for cilher expcrimenlal del.ils or Iheorclical interp"'lalions , Polariled Fourier Iransfoml infrm"d (n ' IR );u-" and dculerium nuclea r magnelic teSonaO<:SI to Ihe silualion wilh bi",fring<:1ICe e. perimenl.', Ihe a nalysis of "'~'enla l orin wilh lhese Icchniqnes lead, to infomtalion thai i. "'-""""' ''SUfaIC, in precision and i" dela;). w;!h l'CSuh. from Iheorelical approach"" 00,.,.1 on Ihc rotational i",,meric sta le model of chain Slalio band, associaled with Ir~n 158 STRUC TURES AND PROPERTIES OF RUBBERLI KE NE TW OR KS csscnlial1y a f>clil iou! and somewhal ambiguous emily. defined as an axis of cylindrirul s~mme1ry wilh respeel 10 u . If one dcm'!1<:S Ihe angle btt"'",,n u and Ihe ~xis of symmetry by n. and lhe a ngle belween Ihe axi, of symmelry and tbe d irection of slrelch by oJ. lhen Ihc segmenlal ori.nMion S( X) nla y bt ",Iated 10 100 oricnlalion funcI;on $ (") of the symmwy .. is by S(X) = S(")S(o) ( 11.27) where S(<» ~ (l/l)(Jcos 1o - I). and S(") is gi,'en by"n expression s;m;l.r 10 eq, (11 .10), Equalion (1 1.27) was inlroduced by Fraser '" and applied 10 or~n\a· linn in ddormed pOlymers by Rc~d ~nd Siein'" _In lhe laller work. Ihe anglc 11 h.s betn rtferred 10 as thc "nglc Del",..,n Ihc streich din'Clion and the "scgm.:n IS" of lhe "h. iO$_ For incidem radialion pOl~ri=l along Ihc direoliun of slr"lching (ideHlified abo,·c wilh Inc x axis). lbe .bro,bancc of u may be resoh'ed into Iwu compO' nents: a~ par.. tlel tu Ihis .xis. and a l pcrpcndicular 10 il. In ,erm' of lhe angle~, "l a nd "1 can be cxp,,-'UCd as " I = lu i' cos' X "1 = ( 1/ 2)lu l' sinl ~- ( 11.28 ) The f"cw. 1/2 in "' (1 1.29) where he X) i. II", dislrihulion funClion for th~ X a~gle>, SubstilUling cq, (11_28) inlO .~ . (1] .29). and u,ingcq. (11 .10) then lead , 10 T h< compOm,ms " I and " -'- may ailernali"ely be ~xprcsscd in le, m, of Ihe angles (>. :l nd 11. In this case. the dichroi<' rat io lake> Ih< f"nnJ~" I + (2eol' 0 - I)(cos' d) R - I t (COI'o _ 1/2)( 1 (cos! d») (11 .3 1) wh;';h. u';ng Ihe expression fo. Sed) . kad, '0 2 R- I S(Dj " Jco.1 n _ 1 R ~2 (11.32) Equalioll (11.32) i. implicitly b"",d on Ihe prcscncc of a c~a;"..,mbcdded loo:;al axi" ,,-i ,h rC SEGME NTAL ORIENTATION 159 11.5 Theoreticellnterpretation of Infrared Dichroism Measurements of Segmelltal Orielltati oll ill Rubbery Networks SIr.in·orientalion ,lola n'~ y $uit'l bly be inlerpreled in t<:rm' of the reduced o rien· la,ion. IS). define.! as~' _{{I ~ n ') W}n. Phantom (11.33 ) Affine Do Define.! in this manner. in anall>IIY with Ihe definilion of Ihe reduced force. IS) may be seen as 'he orienl.1llional modulu, of lhe nelwor ~. In the firsl-order approximation . given hy e<j. (II .33). Ihe reduoxd (>ricn tation is independenl of deformation f(>f both the affine and 1M phantom nctwor~ modds. Rc>ulu of polari~ FTlR c.(l POM S networ~. from the work o f Be,beIl ot al .' are PfCSCnted in figure 11.2. where the reduced orientalion is ploned as a fUnClion of in,..,r:>e o"en,ion ralio. Allhough agreemcm belwoxn experiment and theory appears reasona ble, Ihe data poinlS art; always .'. . ' • • • 1--...< ........-........._......................._.......... o.o:n o o 0 ~ 0 ... _.... ___.... _..!l........ 23.000 0 .ooo.'.:----:.c,-----:.c,-----:.'.-----.. ::-----!,' II), r;~ ... f 1.2 Result. of pol.rim.! Vri ~ e>perimen!> on ""I)tdhne,hyl>ilo .. n<) PUM S ",,',,"orh" . Th<: ml uotd orionl.I;OIl i, sho ..n .Iong II>< ordinate a, . funclion of in, ..... eXlen";on ,",io, Th<: cird. o fermenl <>p<s·hnks, l "be hOli",,,13) d.olled lin ... "'prpcn l p ....-d'oc'ioo > fo. Ihe p/la"",m ncl"wk '''odel 1'.,1l> Iho -"",<>nd of I «t , ( II .J J). ,",';'h <> ~ 4 .nd 1'0 obl.;,,,-.] 'hoor."o,II)' Ir, .. , «t ( 11 ,11) (IJ. 1.045 for .If, ... B.ooo. 2,405 r01 M, 2 1\I,OOO~,nol -I). "If ,,"J 180 STRUCTURES .. NO PROPERTIES OF RU BBERLIKE NETWORKS ""'w: abo>'~ lhe phanlom ... Iwork pmJlo;tion. E>.pnimenlO on other .rsl ....... e:.hitMlN lhe same chanltlorillio,:'lUJ. n,., dq>3r1U", from ph:In.om nclwork bt'ha';or ;" le""rally auribulN 101M dfccis of eman~mc:nu I"''' conSirain flOClu;U;ons al II>c: IIlOl«ular Ie",,!"", This ""plan'lion hal bt:en """'oily "orifled fur ""'t'1."1~1 Orichlalion in weU·dor.n<:d I'DMS ""Iwor". ,,,,,a,un.",, by polarized I'TIR . The cu"""" in "gure 11.2 iltui!ralc the pn.-diclion. of lIoc conSU"J;nc:d .. junelion model or rubbfr oI"tidly, applied 10 Ih. oegmcnlal "";"'IM;on of 1M POMS nelwor" wilh .pociHod molecular .... .;&1\1 and jU~l io" f'uncliona1;IY'. It should be: nolN Ih.al lhe prediclion for lhe conslrainod.junclion moion. The dep he equal 10 17 (i.e., n/ N _ t?). following 1M work of Flory and Changli, Thi, oomparioon show. Ih,I .. pc:r;'""ntal dala yield "," ..It. doser 10 lhe rw.nlom ... I.... ork model. aDd lhal lhe Kuhn model prooicl$ much higl>er orienta tions. A crillq"" of 5(""onull oncnlal;oll based On the Kuhn modc:l has bttn ~nlod el_hc-~ 'l. ..r---------~--------, •• •• • ." ' ' '. . 0 . . . . . , .....' j • o.ag.",,~!::::~----.••;;-.-------;.d. II , .1t._ IIJ 1><1"_ _ of ...,....... 1......... """'" <WI d,.'R 1rn"I>, 1.... Iarimi ~·nl1 <>rcnm,k •. rho Kuhn. ~m"". and phan . H,O! ,..'n >1,.,&It' I,...". 1'1'«1;'"",.".,..... by'''' SEG MEN TA L ORIENTATION 161 Tile ""uils of Ihe FT1R mcasu rcmen1> on POMS ncl"'orks prcscnlcd in lhe prooeding para,raph are also in reasonable ag~menl wilh Ihc l l-{_NM R data of Deloche el aL' .U.l ' . All hough prNictions "flhc Iheory " "" ..,d on Ihe conStrainN-junction model of rubber claSlit;t y)9 agrre rcas<>nably well with expese contribulions are u<.crtcd 10 persisl . ",'en in the swol len nct ,,·ork. but Ihcir effects should vanish if the nelworks h~d hecn formed in Ih. highly diluloo s!ate. O ne mighl Ihcrefore S<'paralt the contributions from inl~rmoloc ular SOUrttS into twO parts- a local a r.d a" c~lallglentenl component- and express Ihe cffccli,·c con· ngurdtional factor a$ ( 11.34) wher<: ~' ",Heels the ~onlrihuli"n< from local intef1nok,euI3f correlaliOll! and is due 10 Irapp<.oJ cnlanglement •. Sirong conlribuliMS from IJ,"' ha,,, been at .~,-~- -~-~----, .~. - . I ord..- ......._ , ... "'- .= ./ •.= \ .= \ Ii... """ ....... 0.015 •• ••• 1_'... .., •• ,,' •• .. , 162 STRUCTURES AND PROPERTIES OF RUBBERLI KE NETWO RKS reporled for Ihe OriCnlalion of p"lyiSscr.ro fo r ,mall alnoun", of ,,,,.. lIing as is shown in the ~gurc: Ihi, could nol he Iheorelically rcprodLllXd ,,"ilhout a modifICalion in Ihe configur... lional fronl faclor. Thi. d~",ase would liJCn be :' ltributed 10 Ihe di"'ppearalJCC of intennolecular "(fccls upon swelling. The 10""sl theoretical curVe and the indicalc". Calcull ii on. performed for I'D MS nOh"orb having v•• • '.0 O. OM -00 • 0.81 om , PDMS 0.56. 1.0 1 1" "'grn<,,'al orknto,;on for poly;><>p"'n< (1'11') .M N>MS ""'''w ks. l1>e uprcn, <.pcrim1"\. "Hte < u,,~ In roug/'> ,~, poin .. Wa. <>I>t.iooJ u>inS ,he cons'rilill In ,he >""olk,, ""'0 . ,,-i,h ", ~ O.~8 .1"<1 0.81 • of poin" i. for • '" ,I..,,, n by ,he 01"'" dl'<"k' and tri.n~"". ""1""C1i,-cly. TI,,' 10"", .lj! "~ 11.5 Eff""", of .,,-.lI in~ 11.0>."'" II".. I'llM S cham. 1)1'1 '<, SEGMENTAL ORIE NTATION 163 various soh'onl co~le~li duti~g cro.s-ti nking, Ih~, I,. wilh different "2.. values, al:><> yielded good "gr~,<mcnl hetwccn Ihcury and expenmen".I'. confmn ing furlh.r Ihe negligil>ly wrak eff''''1 of '"'ermol"""I., interaction. on segmental or;o",.,io" in Ihi, roIYn\<:r. References (I) Nagai. K . J. Chem. P~y•. 1%4. 010. ~I~. (2) FIo,y. 1"'. J S""iJ:riro/ ,\I "'~lmj", "I elwm Mol..:ul".. Inlcr<: New York . , ~ (3) Flor)". I'. J . 1-."""",,. of l'oJ.,"'." C"'·,"I, ,,y. Con>c11 Un;,·o"';l)" l'r= 1~5J; p. HI . (4) Flory. P. J. Pm.;. «"y. So<. /.<>NJ.m. A 1976.151,351 (~) Enna". II.: Flory. I'. J . .II,~·"",J 1 ~8J , 16. 1601. (7) a., 110.0"'... L ., M o",,,,rie. L . JIU("'''''''NffW'<J $.>.r"". Z.: Erman. a., ItJh ... I. .Ifdcr""''''H'~/''., 1995. 28. 5 ~ 2 (~) 1Icobe<. S.. (6) Bah.r, I., Erma". I~"o h",. Ilh""a. NY . 199~.:>.8. 225, L. : \10"""""'. L.: JJ.,h.,. I.: Erma". H. A/,,,,,,,,,,,/,,""" 1995. 23. HI. (91 Flory. P. I . ,Ifoc,,,,ookc..!,, 1~74. 7. 381. (10) Mark, J. t,;.: liml.n. /I. «-M<,/J (2 1) (22) (H) (24) (H) (26) (27) (28) (29) 1l0) (31) I nl e~."",, · Now Y,,,k. I~~g Erman. 8" Monnerie. L. 1985, 18. 1~85, Erma", D. : lbh,,,. I. A/I"r"mo/,eul... 1988. 21. 452. Enn,n, H, : ftaliloslu. T, : Ilah.r. I,: Mark. ). E. A/a. E,man. B.; Flory. I' I . M Il'.. ""'wI.,.u/~, 19KJ. Ib, 1607. ~:tm . n , II.; Qu<slel. I . P. In 1.;"')'I: O~rord . 1915. Q...,lel. I . I'.' Erman, 8 .; Mon""ri • . I.. .Ifde'OIlW1Nk/-' 1985. 18. 1991. Amrum , K . IIokob"". L., Q ..... lel. I . P.. M"",,",ic. L. PaI,....'r 19%. 21. 817. _be<. S.; Ccrn>004, I , Ii. M" ~ . hi. An,.nean In,(i ,u« oi PhysO.."S: W(l()<j bu '~ . NY . 1996: p. 291. Duboull . A, 1J<1och<, II.: It~". I . P~y""" I,*,. 2S. 1405, Du bo ul'. A.; Dcloche. (I .. H~rz. I . M"",omoMcu,,"s 1981, .10• .1096. So" •. P.; Doloehe. B, MlKr'ffnoiw,w, 1?90. lJ. 1999, To""lIi. A. Ii. In Ph,.,,,'u/ P"'P"";'" "f 1',Jp .... ,., /ian,Ih,..,/<. I . .,. M., •. 1:d. Am«ican In,lil"1< of I'h)'>i<"'; Woodbury. NY. 1'191>: p. 21 1. )cloch •. H.: ()ubaull. A.: l' o,L. J.: I_'pr. A I.',,,,,f'hp. 1..11. 1'/lj6. I. t..."I. M,,,,,"""'<,I 16( STR UCT URES A ND PROPERT IES OF RUBBE RLIKE NETWO RKS (ll) II<sb<.. So; IIokob:tJl , L.; .\I""".-r;" L; II;Ih.... I.: Erm.n. D. Pol,.""., 19\13. :l4. 1119_ OJ) 1',",«. 11., D_ U. J, C/o<m, Ph"., 19~J. 21. 1511_ (J() Read. D_ L: Stein. R. S_ M(J(romQ/«.J" 1%8. 1, 116. (35) Jo""" D.: K""nig. J_ L J. M,,,,,,,,,,Q/, $(i, /.1"""",01. Chtm 1979, C17, 61. (36) 10"". D.: Ta.lSin. J. p,: Monocri<. L. P'OIf' CoIJ,>id ('oI",n. Sci. 19\1l. 92. g, (31) EntI"". ~ .• II;Ih.r. I.: Ildbcs. S.• IJokob,.. L,: Mon"ori<. L, Pol,." .., 19\13, 34. 1858, (38) FI<> (39) Erman. D.: Flory. I' . J. Moc,om. 19!1O. 818(2), 267. (4 1) I)cloo;l,,,, a .: S.mu l, ki. E. T, MQ"""""""'/" 1981. 14, 57S. (42) Du",,,I(. I\,: 0<:10<1>0. 8.: li on. J. M"..,on>oIN"I" \ 987. ZO,:>096_ 12 Networks with Semiflexible Chains and Networks Exhibiting Strain-Induced Crystallization O as.kal theories or rubber elasticity are b.l.«>d on models of flc,ubtc polymer chain. thaI arc .ulT>eicn lly lon g [0 •• hib; ! Gau.sian b.:h"vior a, described in chapter I and in appendix P. Additionally, {he chain. art phanl()mli~e in lhe sense thal1hcy do nOl jole'act wilh one mhc. along their ront"u",. The 'heories described in chop!« 2 were ""sed on this picture of tbe individual chain. In thi, chapte" ...., describe the elasticity of networks thaI de!",,1 subsmmially from th ose "dJressOO in the classic,,1 theories. The depart""'" may resuil r",n, lwo $(>U1ttS : ( I) the chains may be only sorniflex ibk. a, a result of which lhc scgrocn(, of neighboring da;"s compete for in the deformed ""work, and .hoos< ['referentiall y orien led c<>nfigu""ions, ~nd (2) the chains may fonn cryslaltile~. upon deformalion, a, a result of which lhe homogel,eou , ,(roc(u", of the classical network model rna)' be transformc-d ;n (o a nonhomogeneous One having micro· ph ..., of ery'(alline and a morphou, regions. The subject of er)'Scali ;,.t;o" under deforma l;on, for network, in general. i, relalively old, and ha, been !feal",l in SOme delail in previous scudics '-', For Ihis reaSOn . CI)', latli""t ion Hnd ""me of its eff"'" "'ill be rev;ewed only briefly al the end of {his chap te •. T he main e",phasi~ wili be given to nelworks wilh .. n,iflcxible chains'. 'P""" 12.1 Netwo rks w ilh Semiflexible Chains bamptcs of nc'works wilh ""millexible chains a re Iho,", in which Ihe ehai .. , haw rodli k. seg.mem. ,",parat,-d by tle.• ibl. space .... or Ihose where Ibe chain, have bond angJcs appre..iably larger than lelrahedral, ",c'>tporalion of the"" chai ns inlo • network Slruclure r<:suhs in materials Ihal ""hibil segmen,al o rien'alion. signif,,:antly largtr than tho .. shown by du,sical nelwnrh. Specific . xamples would include network. prepared f,om ",,,,nal;'; polyamide chain, ' -' or from chain. con,"ining Tiqoid-<:ry>I.lhne ""I""n,.", "long Ihe ,Tir"'tio n of the b.ad _ bo~c' 1• • Ik."a o"" or their unique chain S(tuelm,",. lhe",' nclwurk, are ca,ily oriemabk. allhe mulccolu lewl. hy m"c"'''''opic ,leform"liol". The uricnlationai 166 STRUCTURES A ND PAOPERTIES DF RUBBERLIKE NETWOR KS Iransi!;o", may easily be con lroUed by a ppl ica!;on and removal of a niSOl ropic a ••;de-ch.in liquid· CTyslalline nctw,)rks. also display fealures thai may be described by Ihe Iheory!(> be pn,senled here. Such liq u;d-crystalline nC'Iw<>rb ha"e "",used wider ;nterest Ihan Ih<>", of Ihe main..,hai n type . perhaps due to the .ase of Iheir prcparalion. eharaCleriz.al i<>n . and handling l f "". "" <>ro recent work in this area co,'en; a ,'a';"ly of "r""lurcs. wilh a number of n"vel physical properties" "'1 The idea Ihat mecha nic;,1 51""" applied t<> a cross-linked nematic polymer m!]' indl>CC an isotropic-nematic ph.", lransition was first formulated by de Gel,,,e, . and waS followed by sc,w.l .ddilional Ihcorelical lrealmcnts"-" . The ba.ie molt<:ular cfTrel in defurmalion of networks wilh s<miRc~ible chains in"ulves the facllhal lhe chain, are embedded in a lanice which defo"", wilh maer"",opie Sl min. a feature Ihal i. abs.:nl in Ihc phan'om ne'wor ~ murie< predicl increases in scgmenlal orienlalion d~e to local in,ermolocular coTTelal;on •. However. ' he !<'Suit. do nol show a dtpondtnce on the .1;ll'n= of I!>. chains. ;n ,p;le of ilS eJlpcc1ed importance a, " parameler. The Ih"QTY rs introduces , he 001""1'1 of a scmiRe.iblc ",·o,mlil:c chain in lo Ih. calculation of the ei •• ticily of a networl: . while Ihe tbNry of Erman and collaboralors is based on the semiHexible cbain lattice model of Flo,yo·" T he prediction, <>f the tw<> 'hoo ri"" are quite .i milar. The la\lie<: model i. ebosen to illuslra'e work in Ihis .rca. and i, described in dClail below, In Ihi. secl;on. w. descri be the nClwork as a la\lice into which Nl freely jointed chain, a ", embedded. t<>gc,her wit h n, solvc1l1 molreul ... The characteri"i. ,i", uf a lattice site is laken as Ihe ,i,. ofa sol,',," molreulc. A chain "'fcn; to a nClwork cbain link ed 10 Ihe nc1work at ;1. 1...-0 ends by acti,'C jnnClions. Each nelwork chain contains m freely joinled segmenls. ",ferred to a. "Kuh n segmem.:· of length I. D<:P<' nding on Ihe degr"" of OIifToess of Ihe chain. the length I may be signifICantly larg<:r Ihan that oblained for ord inary flex ibk cha ins, nlc imporlanl quanlity for Ihois lheory i. Ihe axial ral;o x. denoting II,e ralio of'he leng,h of a segment 10 ;1. latcral dimen,;on , T he lale ... 1dime",;on., of the rods are assumed 10 be of Ihe size of soh'Cnt mOJoeules ur lallicc siles. Figure 12. I(a) displays an iIIusI""ive nelwork chain con,isli"S of 20 Kuh n .. gmonts (i _Co. frocly joinl"" rods) belween IU'O lelraf"nel;onal junclion< A and I.t 1'0' ,impl;cily. Ihe n. l,,"ork cl"l;n, arc as>um,,-d to be monooi,persc (i_ c_. compo ...-d of the s",,,,,c n~mber m of rod<. with all rod. h~ "i~~ idenli,'al ,·.Iues of Ihe a.ial ralio xl. Following an carl)' trealment by FloryH'. Ihe model n1"Y SEMIFlEXIBlE CHAI NS AN D STRAIN-INDUCED CRYSTALLIZATIO N a 167 b 11.1 (a) An illuSlrati ... nel"wk nal june''''n' A.nd II", (b) The: orienlalion of . gmenl a. d<6n«l by 'he lwo Euk, "nsl<1. V. "nd l "hc ~ dif"«I;On i. ll>c: p",forr«l direction. thal is. lhe: diroc1ion of ",etch. (e) i\ ""l""""" of y • ...<sin ';', .ubmokp)'righl 1991 Americ"n Chemica l t~ .. "'* IV'''" .. Socicl~. easily be adapted to lhe ~ase of more than (me lattice ,ite ortupi(d by either ""I,'cnt mole-cule< or Ihe rodlikc "'!!l"cnIS in lhe later-dl direction . Th. orienlalion of a !p"cn segment . hown in figure 12, li b) is defined by Ihe two Euler angk>. "', and ¢, . llte Z direclion in In.: figuN indicates the preferred dirocl;on. also rdcrnod to a. the do"",;n axi" ]" simple c~'cn,io". Ihi. direclion coineidt"S wilh lite direction of stretch, The "Ibscripl k indicates that the oricm,nio" of the ""gmem lies wilh;n Ihe klh solid :tnsle, In t'onforlnily with the lanice lrea tmenl of ~lory""JO"', the accommodation of lhe rod wilhin li,e I"nk<: is achieved through its repre!iCnlalion by a '""Iuellcc of ,., "' .Tsin Wk suhmokculcs, QC(;upy;ng xi,.. sites and oricnled along the prererred direction 3S shown in figur~ 12, I(c), Thus. ,., ChantCleri>cs lhe orit gi,-.,n rod. II is cxpresscd in terms of ¢, and ¢. as JO ( 12, I ) for the rod exhibill"g lhat partit;ular orio"',, 'ion. ,\oconli"g 10 thi' dell"i1i"n, th~ ,'alue of,.. incrca""" as Ihe rod I><>.:omcs diro,;cnted, and is Iherefore rererred to a. the "di;orienlalion index:' The junclions of II,e nel .... ork al tl>t extremities of ".dch ch"in arc assumed to displace aninely with the m"«(>!.Copic deforma,ion. This a,sump, ion, which i. also employed;n Ihe dassit; pap.,r by Kuhn atld Griin" and Ihe affine ".Iwork model of Wall and ~loty·', i, 'lOt exactly corrc.::l but siml,lifics Ihe problem eo",iderably. Imprnvements in the 1heory to inc lude non -affine displacements of tl>t junction, ma y he c""$i,lc,,~l. h"",:v(·,. in " mcan·field "pp"'.th. 3S ha, hccn done pre,'i 168 STRUCT URES AND PROPERTIES OF RU BB ERliKE NETWO RKS 12.1.2 The Partition Function and the Free Encrn of Mixing of Network Chains T his diIKunion adapts the Flory lattice treatment of IKmiflexible chains 10 net· worh. The reader i$ "fcrrtd to the literatu,,'" for further det~i1s oollcerning the wlvent molecules in a lauice model. For a syslem of~, networ k chains and lanice COnsisling of 11(1 .iIO$. thc respecli"" volume fractions arc giV<:1I by n, T, ., nlxn,/II(I r, n';l1(1 ( 12.2) EO The eXpeCted number Vj+I of ways for adding Ihe (j ~ l)st chain molecule to n. lauice si tes "din mxj of them already occupied by j chain molceuk.., can be obtain<: "1+' ., (: - I).... ' (12 .3) wh.re , is the coordination number of the lat\ice. an j ",,-' - f::Lnl~'" n,m ,t k diwr icn t~tion ( 12.4) Here . n,~ indicate., th. numlxr of segment' of thc IIh chain whose orientation I~, within Ihc kth solid angle. and }., is the corresponding diso6cntation index. T he index k in the last summation denole., all sets of..,hd a ngles available to the rods. The computation of <:;'' enables us to calculate lhe combinatorial pan Z""", of the partition funclion for the lallie"C from ( 12.5) ( 12.6) where "" is the kth fractional range of the solid angle. Tbe lotal configuration partitio~ (unetim' of tile mix ture i. fou nd f,,,," Z .. .. Z..-Zoo,., ( 12.7) In the original t"atment of Ihc network prOblem". Ille additioll ,1 cont ribution 10 the partition fUllct ion resu lting from the juxtaposilioll of the ends of chain. a t Ihc sitts of junctions wliS not e·onsitrople tmn,ilion. present",t bclo"'·. The term _ In Y",,,,,,, . wbich will bc uS<." SEMIFLEXIBLE CH AINS AND STRAIN -I NDUCED CRYSTALLIZATION 169 simplifiM fonn by sut>slituting "'I . (12.3) illlo "'I. (12.5) and ming Stirling"s approxim.lIon: - In z.-b - '" In. , + n, ln(' d mx) - (n , ~ n,,,,y ) In ll - + "l( m,' - I) - ~,(m - .,( 1 - Y/ x )) I}ln (z - I) ( 128) Similarly. _ In "", ~.. is "b,~in,d fr"m "'I. (12_6) as ( 12,9) Equation, ( 12.M) and (l 2.?) may to" ,ut>slilH!ed inlo It.. -t. Tin 7... 10 oblai" tbe free energy of mixing. c,pre~sion Il. A .. _ 12.1.3 A Sct of Nonlinear Equal,o", for Evalual ing Ihc Orienwlional ))isuib",ion "ndel' OcfO""'''IO'' Characteriu.lion or Ihe " tien'ali,,,,.1 dislrib"lion f unctio" nl~ is lho first slep ror ..,I"ing the problem o"tline'n r"'-lllim lbe drterrninatio ll of f"ur Lagr.nge mulliplier, 10 be e,'alualed from a "uolinear eq ualion. The forma l ""IUlion i, outlinM here and Ihc results for lbe approximate closed-form solUlion 10 the li"ea ri, ed eq"alion arc presenlcd in seclion 12. 1.4. [f IheK arc n" conSl r~in'< imposed on Ihe '}'SICm. Ihcn a n eqlli[ibri ,,,,, dislributi"n of rod' among dilTerenl orientation, is ohlainM by minimi,ing the frC(: energy of the syslem wit h res!,,"01 10 ni.l<' T he imposition of eXlerna[ oon"taints require< Ih~ ~se of Lagr.ln~ "'u[lipli"", while minimi/jng Ihe frc:<: energy. [n the lreal",enl here, the cnd -to.cnd yeclors for c~ch of ,he n, chain mo[ecul", arc assumed 10 be he' L n# . in ';) sin ( [2.12) where. '~./ . '>,/. a lld ' 2./ arc lhe X. r . and 7. "Om!"menI S of 1m: eoo'lo-cnd """lor r for Ihe jth ,"Olocllie. "lid I is the lenglh of" rod. II should be notc~l that 'he con,,,,,ints giwn by eqs, ( 12.10)-( 12.12). where ~ach carlesian com ponent of the end · to.cnd vecl", i. kept fi ><"<.1 . are d,lTcrcnl from 'hose of the Kuh n and Grun Ircatmem o,. In the ['ller. o!l[Y Ihe component along Ihe direclioll of Ilretch i, fixed, The di,,,ibution of "I~ b addilionally . "bject to the oonslrdint ( 12 13) 110 STRUCT UA ~ S AN D P ROP~ RTlES OF RU eSERLI KE NETWORKS The equilibrium di'lribulion of rod. belwc<:n dilTere nl oriemalion. wilh con· maiM' gi,'c" by cq •. (12.10) (12,13) i. ""Iermined by dilTerenlialing lhe Helmholtz [n,:. energy expression and equaling 10 Urn: ~ In Z"'.... -+ " L,. L,. "/)' cos If'•• -+ ., L "i .... · . in V.' cos Ct· + ~ L "i~' sin v,· ~n 9") ,. ....,..". _ 0 : . [In Z_ ~ " j)' 1- tl ( 12.14) .' Her., n# .. w. expl- a)'; +<> - I + fi co> v·, +.., ~n If•• co.,,;, + ~ sin ¢, sin 0.1 ( 12. 15) where (I 2 16) h should be nOled Ih"l Zwmhdcpoc nd , implicilly on "J~ Ihrough y. and Ihe partial deriva!i," oY/ Onj) = yoinl'" has I:,.:< n used in deri"ing «I. (12.15), T hi, reli.,. on Ihe rca«>nablc approxim'linn Ihal Ihe dislribulion nf n,icnlalion, fo r a given rod can be equaloo 10 Ihal of Ihc who le en,.,mbl • . Thi' 'pp",.;malion beeom.. rigo"'u , when Ihe nctwor k chain is .ufficicn,ly long. Equati"" (11. 1$) .... Ihe probabilily di"ribulio" of dilTerenl orienl'lions among Ihc compo,ing Ih~ jlh chain. It r.suh. from IWO clTcelS: (I) nricnla,ion due 1<> strelc hing of Ihc chain f",m ils IWO end •• thai is. O, ienlaiion nf Ihe n,n;lalion, or fro'" local c.eIu.ioll of lhe ""llImo of O'le scgmcnt 10 ils neighbors, T h. laner effecl en'ers eq, ( 12,15) Ihrough Ihc lerm in Ihe expon.", . I" lhe absence ofilller· ",,,I,,,,uia, co,rdalio n•. the dilTerence bel"'ttn y and .~ inl'od~ by Ib~ IaUice model >".nishe •. Equaling jfx 10 un ily;n eq. (12,16) lead, 10" _ O. T he dimibulion gi,.. " by «I. (12.15) ' hen TcdIl<.."C.I" ,od, A""" "y, '[. -'"m'"'""41r ~,~,",",,, oxp(<> ) c~ p Iko< ,.., +"'1 SIn .p. cos <1>, + 6 SIn .;" Sill . 4>, I (12.17) In «I. ( 12.17). Ihe solid angle w, of e'l . (1 2.15) h" $ bo,:"n replaced by ( 1/ 4..}.in .t, J\·I,d";'. ,, - I in «I. (11.15) has b<."C11 rcpla«'I,aills may be ... ,iuen ,IS (tn: 1) of, ("Y)/ 1 = A~ ,~ \' ~~ 'uo"eWJopP I~!""un "',1 ~1~U:li :")l"'J'J~ ~4> 01 4IS""II"Uff:l\l1 JO 0!1~,J "41'~ l'Juff~P'! UOH= IP ~41 SUOI" O'I"J UOI'U"IX~ ~41 '! J~ I'''~ -UO!I RUlJOPP 01 " ,!Jd • JO IU~U("ltu"", 4~ .)lnni:><·U"J'" "41 ~! "'4·" "«I) rl f1w A,¥ ~ .q.')AI~ " (1') (/,IU/~¥ '" ·{y) ·,~ «(H I ) ... (1') (I d"'~~ ~ .('i')~~" (1') ~'1 "'~I' =J ~> U! :)SO'll 01 1~'1"1'" ",e J JO ~l"~U _<>.I"'I\I><·U"O'" "''II 'uo>leUlJ(lJ'p ""W" '"PUll SUI"4' p:llu!of 'I""'J JOo! · "oJdq '.m <'n (I fl'<-~) '" «nt) ""OII""U H61"Z Il ',Ix> u! f Idu:>Sqn.' "41 I." <,,( Y)"' f' f, t"(~r') _ f'z , -'! '"'I' '-';'0 I"A ".nl><_u"xu I<>OJ J!"'II 01 f /., put, .rA, _/".r, ilu"onl» ~'1 .U1»41:1\11 '" "1'"Ul SO UOII."'mldwI' "'1lm) V(ij I 'r I) -00 ~q ":),,,g SI II'!'1," '/" ''I "'"'II )0 4:)P"' 31UI'!,'IP hq (I n I) (6 I' tl) 'II» "'OJ) p:ll~U!W!IJ >q 'RUI U U!I ( Inl ) {'\> uI' l:~ u!S q 1 '''' S('" ".>"!S"- + 1- ! ",,"0' Ii t " l(> UIS··'/I ."I" 'i'1' '.(n_ )dX3 ~ .'I ''i'1' .,'f"-, , = "JJ ({\l'll) 1'9 U!S ''II "!' ,q + '9 ''''' ' ';' U!' L t 'Ill 'OJ# t v +! - ',(~ - Jd'~ )( ' , I UIS· - ...I' J'1>1' 'f"- "'-rx,, " I (61"<:1 ) /'" U!S '4> "ls >, + ''' ''''' "~,I U!~J. + >'I' SO"" I t Q + I - .,,, - jd,,,)( 'f' '.~ .,,'" ' ,"l U!$ ••,! ( ,\> uo< '4- "!' 9 ~ '", w:> '1< "!' J.. + """ "'''Ii + U + I (sn j) ~ Ll ,,~ UI> '1'1' 'of" ) '4>/' 'J , .,'I"t NOI.lv:znW1S Ati:) 03:::.nON1 · NII/IHS ONI/ SNIWt:) '" WI 311l\ X31~IVHS 172 STRUCT URES AND PROPERTIES OF RUBBERLI KE NETWOR KS Sub>tilUtion f (,W" / 1 = A(m/ J)' " ( 12.25) {,W"II ~ (;.. )'''/1_ (mjJA.,} ,/l ( 12.26) T he Lagr~"gc mulliplicrs ., and fJ are "'Iuallo cach other for uniaxial dcformati(>o. Combinill3 eqs. (1Z.2S) and (12.26) wilh cqs. (12.ISHI2.2!) in Ihis p;lrlicular case lead. 10 Ih. expression ( 12.27) , J" dO:> [. d1!J .io ", cos ¢ 1 (4), v) ;(;:::-cc;m = ' • 0 d4> J 0",Nf(':' ."') lm' I~ ,'" (12.28) J" where I(q,. 1") is Ihe orientational distribution function gi''''' by 1(1/>, "') .. ~i l1 ¢ ."p[- UA .in "-'I I cos 4>1 + I sin o;t>l] + {j co. " + l' .in ¢(co. I/> + sin .p») (12.:!9) The dummy subscript k has boi:n nmiue , x"" . . 1>9r J" d [. d¢ sin ,pjl cO, \!o) Equations (12.27). (12.28). and (1 2.30) form a .ystem of thrco:: nontine-~r equations 10 I:w: :iolved simultaneously for y, fJ. and "y. which are all function. of A. The evaluation of .Y. P. and ., enables one 10 c..,l " I .. I .\ L: cos ~'i " m- L:• ni l< co. "'. .. (3m) III m i~ 1 (co (12.31 ) and (12-.12) SEM IFlE XI8lE CH AINS AND STRAI N-I NDUCE D CRYSTALLIZATIO N 12.1.4 Th~ 173 Linearized CIO>e Solution of eqs. (12.27). C 12.2~) . and (1 2.30) ""Iuire. somewhat extcnsi.·c numcrio.,.1 calculation •. T ht equations may, t.owc,,,,.1>e solved analytica lly for small {J and "/ (and "Iso 'm~1I a). that il. , m.1I (J/ x - I), by ul"'nding lhe lrigonomclric function. in Ihe exp<>nenlial J"lrl nfeq. (12.29) in a Taylor seri.. and keeping only the lcnn, Ihat a", linear in co. /3. a nd "/. The ,lelails of Ihe calculations and Ihe results obtained ~rc describt:r the Lagr~nge multiplic,," {J. 1. and (J an:: (J"" p / nr)lf1), (12.33) 1 '" (J/ m>' '':!)t n ( 12.34) ( 12.35) 3 x. ~ (4/,,) _ I .., IO.'JS ( 12.36) In 0"1. (12.36). x. is the Icllgth-to-wldlh ratio of ..gillen IS aoo ... which Ihe systcm is lotally aniso"opH; . A ,·.Iue of 12.3S i. oblained for it in the Flory_Ronca model for solution. of ,odlike particles. Using Ihesc rel.tions in "". (12 .29) lead, to thc " vcmge (cos) ¢) which is used in II>c upr.!!..ion for Ihe oricntation funClion S .. (1/ 2)( 3(cos1 ¢) - I). The latler quantity is , __ , (>.' __ '_) [I+ 5M.,_l - "",Ix. ] _ S" +S, 5m ,\../ (.n--t!x.) ( 1l.37) where . -'a"[5 .<. ,,::,;,,~.,jx. ] S,"' M I (nli x.) (12.38) i. lhe inl~rmolooula, contribUlion 10 ",smemal orie nlali"n ari$ing from l-.>nlpcli_ lion for space of segmen1< in an " rient ..1 ."vironmen!. From "". (12 ..l7) il is seen th"l S, ,-ani,h". for x/ x. < l. In this ca ... th. problem reduces to orientatio" in a nelwork or phantom chains. The value of Si also decreases with increasing solwnt conlcni and equate, to lero itt the lim iI_ ing ca .. of " = O. a~ i$ ",.dily seen from eq (12.37). Results of calculalions for lhe orientalion fu"ction are tl\ agreement with tho .. oblaine.! by Wala",k" by .imila, .rgument, u;ins constrainl •. H i. 10 be noted Ihm lhe resnll prop<Jse 17. STRUCTUR ES AND PROPERTIES OF RUBBERLIKE NETWORKS The basis ror the inte ...~etion. in ~he Jarry-Monnerie thw ry iJ energelie in nat ure. whereas n ,fcr. is obtained by considering the liquidli ke nalUre of lhe Kuh n "'gmenls and hence is of cnlropic origin. I n ~hc pr=-ncc of Ihcnnolropic in~erac~ions b<:lwttn neighixlring 'epl:at unilS. Ihe orientation rnnclion ro, the dry network is round to be" ,. -'Sm (" -')1 l - "uj Sij ( 12.39) "I " ( 12.40) ~ . u iO l + and l' i. defined as" ' j-' '. 1-- (1 2.4 1) In eq. ( 12.4 1). k8 is . he Ilollzmann <·onSlan •. C is a <·(mStan •• 6,0 is .he moan opt",",1 an isotropy of all ..,gment!. " i. the di"ancc b<"""en . ubsc:gmcnts for dense packing. and " i. the numbe' or f,rsl neighbors .u rrou nding a gi,,,n seg",cnl. The de,i"ation ofeq. (12.4 1) i. givcn by Bahar el 31. ... 12 . • S The Relationship of Stress 10 Ikfonnalion and Orientalion The lrue Stress in unia~ial deformation i. obtained fro m lhe elastic free energy by Ihe Ihermodynamic relation <7 _.!. >.(O GII .. ) lJ>' - Y ( 12.42) r.r .. ,.., ..." ),,, whe", Y is the 100al vol ume of the ")'Stem. and .he related 10 Ihe parlil;on (unction Z .. by «I. (12 .7). The •.'plioil . ~p",ssion f", Z .. is obtai~ed by substituting the expression, for nj~ from eq. (12. 15) inl" eqs. ( 12.8) and ( 12.9). T he final form is _ lnZ.. I,. _ n, ~ In v. + n, ln (.., / ", ... ) n, In p - v,( 1 - Her)] n,{my - I ) - ", (m - I) In(: - I ) - mtl,ln(Z ! 4",) ( 12.43 ) + (jI,, ~ ";m!3 + hn,";",!3>" ~ z- r d¢ [ dt;f( ¢>,,,,) (12.44 ) and /(" , \ .) is gi\"on b~' cq . (12.29). Per(",ming 'he dilTc",ntial;"n indie SEMIFLEX IBLE CHAINS AN D STR AIN-INDUCE D CRYSTA LLIZATIO N 175 ( 12.45) Usi ng the identili •• i.!!nZ t»:"" - Oln? <Ja !.l In? /J{J !.lIn%" rr, a;- ;u: + Off" iJ,\ ... ----o:r iJ,\ ( 12.46) in cq ( 12.45) wilh &lnZ --a;;= - J', ( 12.47) lead. to ( 12.48) Subst ituting the expressions for /l and l' given by cq •. (12.33) and (12.34) ......., obtain the ';m ple e.pre .. ion for Il>c stn.:" in Ihe first approximation ,,,' ( 12.49) ,,,,ualion (12 .49) ,I,ows Ihal. in Ihis approximation. Ihe relalionship bet"..,.,n " reS> and deformation i, independent of the degree of rigidilY of .he chains. T his o~t\'al;on i. in distinel oont .... '1 wil)' 'he relmion, hip bet ... """ orienl"lion and deformal;O" ind;c.I«I hy ,he lerm S, in c-q. (1 2_38)_ The laller n.:lationshi p becom.-s discontinuous al " spcci l;c amounl of ool,-.nt below which the network becomes perfectly orient«l. indicating occurrence of a phase Iransilion . T he pos. , ibilily of a first.order pha.., tran, i,ion is diseus...J in the following seclion. Elim inating 'h e . , Iension ralio betw""n c' S ~ - '- ['1-t 5x. XV,/X. ]" 64 1 - ( ''':11-'.) SmO (12.50) where r. _ ".1ioli V ( 12.51) is the shear modulu , of the ""I"·ork . ) 2. 1.6 IsotropiC- Nematic Ph.", T.-; ... , i(io", in Defor med Polymer Network . T hi . = Iion descrihe. Ihe cffOXI< of * " rop,<,-""malic pha", 'r.n,ilion, in siretch«l '>etwork. ,,,,d is based on It.. " 'ork "f Kloc:zkowski el al. " . which i. the co nlinu~'ion of' he fo rmu lati"" doseri ""'! on '""'ti"ns 12. I.J ",Id 12. I .5. There are \,.-0 Il'crm<.>tlynami.: P'''"'""'c," whIch ma y eaSIl y be controllcxf during the rhase ,.-;,,, , nio,,, Ih~ I C"~ l h 0' the ""mp lc (mea,ured by the c 176 STRUCTURES AND PROPERTIES OF RUBBERLIKE NETWOR KS ralio ~) and Ihe applied force f. The ;;;otropic-ncmal;'; Il"dnsilion al oonstanl kn&'h occurs whrn Ihe free ~ncrgics of Ihe ncmdlic and Ihe isotropic ph.se are C<)ual: (12.52) The phase ,,-ilh the 10weSI free energy alwaY" prevails. f or alhermal "Y"lems, the isolropic phase is stable al low axial ratios x and al low concentrations '1 of the semi_rigid chain,. For sufflcienlly high axi"1 rali", and high values of '1, Ihe nemalic phase has a lower frcc energy Ihan the isotropic One_ There is a crilical v"lue of Ihe axial ratio '"en' below which Ihe nematic solution never exi,{, regardIe .. of lhe deformalion ~". The proper free energy i!.G,. for sludyill$ phase lronsi\ ion! al con,tan{ fo,,-"O [ i, a Legendre transform of i!.A,. (12.53) wbere i!.A .. is the Helmholtz free encfg)'. The phase Iransilion al conslam force occurs when the free energies i!.G .. of Ih. lwO ph.~.r<: C<)ual, (12.S4) One may also oonsiokr Ihe phase lransition al comtallt sires. u, as was done by War""r and Wang"'''''''. although it j, experimentally dimeull to k""p " COil_ stanl. FifU'" 12,2 sho"', Ihe efT«! of the eXlcn,i"n ralio ~ "n lhe force [(mea, urN in noV.- Ilk8T unilS)". The h(>rizontalline joins Iwo solulions which hac'. Ihe ",me 0.015,--,,--,,-,-,-,-,---, V2 " 1 ,0 m .. 100 0.010 x ,, 6 MOO ,/" 1 3 5 7 A. 'h. 9 11 Fi~..... 12.2 Relalionship bel".-cen lit< foroo fa nd "I"n< f'gure "'present< the nemalic ",Iulion . and Ihc 10..-0' part. thc i"'lropic ",Iu· tion_ 'Il>c ,'.kul." .. "''Ore I""rf<>nnoo for dry ,,olu,ion. ~'hich Ita,,, Ihe ""~ fm: .""r~y :l(; "I c'On".,,1 f""", f. "> SEM IFLE X IBLE CH AI N S AN D STR AIN · INDUCED CR VSTALLIZATION 177 r. free e",,'&y !J.G at ronstant foro: and i. <: process of deformation , Slartin~ wilh tf><: polymer . trip in the isotropic !Iale, fhinS one of il' ends. 3"d inc.easing the 1o;,d on the second end leads 1<> lhe h<>riwnlslline in figure 12.2, The system will spon lanwusly tran,· for m 10 Ihe ncmatic sta lc. which ha> lower free energy al wnsts"t force. Tf><: length (extension) of the syslem will in.",a s<: f this transilion to the more ordered phase. T he transition o,;curs al 1o",,,, )\ in .he pre,",nc. of anisotropic thermotropic effects, Fo r the con,tant·length ':>pcrim"nl, when both ends of the sample are ",<>unted in damps at a certain . ", .n,ion. the fn:e energy of the nematic phase will become lower than th,u of the isol'opic ph""", There ",i ll then be. phase tran,i. t;on al conSiam len~th. a<:Companied by a large drop of tho fo"", (rcp=nlod by a ,... Iical line o n thc pIOl ). In klme ca""s. Ihe fo"", may even drop to negative "alues, Ihat is, t~e sample arkr Ihe pha"" t"",.ition will be ,;ompressed, Pjgure 12.3 show.< Ibe change "f Ihc oriClllation funclion (or order pa rame ter) s ... (P,(c(>$q» fo r the mrehankally inJured phase lransf<>rmalion al COnSt.nt length". T f><: orientation function S increase, smooth ly wit h deformation and Ihon ab"'plly jumps upwa.d at Ihc tra nsi tion point, Similar Cur,es "'''''' <>hla incd in proviou. cakulation," for "et"'Or~ chains of", .. 2(l rigid segments with "afiou. axial r"tio" '0 ~ ~ ~ ~ .. , m.. 100 a." OA en 02 ~ U -" --- ..- 0.' 0.' .0 "2 - 0.0 ' , , ./ , , f... J , " F;,.u .~ U.J Ch'03< of Ihe "'Oe"'O'''''' 1',,,,<1i,,,, tOr order "" ,.meter) S _ (I',(CO!'~» fo, II>< n",ch.""",lIy intl"",-'>.l pi..,,,, ,,,,",Iormat;',,, " , <''''''''n' I,nglh ". All ",1",.. <>r 11>0 I"'r.melol> of ,h. m".lel arc ,1>0 >aln."' in fisur. 12.2. ,,00 Ihe ,to[!,'(II;ncs , h"w "",la· .,. hl< ...,lu,K"" f". h<.o,h plt,,-,.,,_ 178 STRUCTURES AND PROPERTIES OF RU88E RLI KE NETWORKS 0.002 0.0015 -=• (/ 0.001 i$OlIopi<': m_ 100 ,.S v2 - 1.0 00005 FiK...... 12.4 MOOulu. LFI of Ih< ..."..,,,. a. a r"!>Clion or invcri< < polymer .. mpl< unOCrS 12.2 S train-Induced Crystallization 12.2.1 General Feature. T he p ..... vio us SI.."CI;ons dcscrib.,d the .ffecls of chain nexibihtyon .... gme ntal oricn · tal ion in deformed nelwork<. Dy the u.., or a Iallice modcll"Cprc,",nling 11>1; liquid· lil< nalure of Ih. polymer, il lOla. found that when 'he Sliffnos. of Ihe chain, exceeds a critical value. 'ponlancou. orienlalional ordering lakes pk.cc upon deforming Ihe network, Thi. Ir~",itio" from the les.-oricntoJ 10 IIw highly o,icntoJ ,1,,10 lakes place irrespecli,," of Icmp<:r.Uul"C.•lId i, ~ssenlially an iso· lropic to nCIllalie like lra n,;t ;on . During .ueh a ".nsil;on. Ihe .... gmenlS a',umc an o.icntalio""Uy ordered sW le "01 associatc'< prediclion of Ihree-din,.nsional c.)'statlil"" di'lin<;,ly .... '''''''101.1 from the r... ",aining amorphous region •. A wide cia.. of nelworho, '"". on Ihe olher ha'ln. where wtll·defined eryslalliles form and grow during a" anisolropic dcfor""uilalli,"uio n will "" bricOy I\."v;,·"cd . SEMIFLEXIBLE CH AINS AN D ST RAIN · INDUCED CRYSTALtl2ATION 179 The firs! q uanmalivc descriplion (o f strain ·ind uced crystalli"nion wa! by Alfrey a nd Ma rk", in which Ihe behavior of" . in!>ie chai n (hut nol a IlCt"'ork) was st udied . This a" alysi, was impro>'ed a"d ."Iended by ~lo,y711 into" t""ory of slrain·indnced crysla ll ization of IlCtworh. "nd Ih i. trealment has scrv",J as the basi, for all later (heore(i~" 1 m<>del<. In the fnll<)wing =ti<)n. we present the b~!ic . """ mplion. and !"Csulu of th i. theory . AI «, m.nl ioned.ls Ihal "'Cre essentially COllSlrlldel, 12,2.2 Model. fM Strain·l nduced Cry,lalli'."i"" in Stretched Nctwo,h T he assumpt ions of the Flory theory of ,train.i"du(>"d crystalli".al;on in llC1\>wb a rc as follows: L Th. di,pl. ",,",e,,(,.,f ,he ju""(;" ,,. of ,lie "",w.,.k t"n,forn, amnely .. ilh macr.,. SCOp IC "",", 2, Tlw: ,..,,,'o rk d,.in, ron"" <>f 1I\'Cly jo.nled "'&mOn(... nd ;c ""grrK''''' ,tIC"""". "- S Only i""ipic"' cry".lIi'''li"n i. ''''.Ied. 6. Tben; is "" .igoif",.", ch.ng<: in "n L"'JlY . ,sociaoed "i,h 'he formaoion of ,he nuclei, Fi~u,.. U.S A ,mall """,ber of neighb"ring «gIl.e" ... I[smog I""allcl (0 ""e ",her (" form. er)'".I1". nLIC,,"u., 160 STRUCTURES AND PROPERTIES OF RU88ERLIKE NETWOR KS 7. The di ......~io ... of lhe ",)·".lI il<> coiociimpl'fy;ng ",ulllplion n. I."" removed by Allegra and coworke,.." ·" in • mo", romprcben,iV1O ",,".men. ,) 8. A oh,.i" 1>"'.... 'hrough • Cf)"". lIile 0T11y once. 01'1<1 010n8 lhe direction going rrom ono end of lhe eh,.in 10 tbe olbe,. (Permi,,;ng lhe cha,", 10 enter 'he c~'".U;tc .Iu"g the """... direction u "..II WO' ""n,idered .ubseq .... nt1~ b~ Smith" .) 9. The fomo.li"" of !.lIi! .. b~ ch.in foklina i, i~nored. (The posoibOli'l of chain fokling ,,'.s b,.r in 'etm, of. non·Oa",';"" modol b~ Ga~lord' .) '","cd Th~ poS$ible e(fcc,s or fluct uations of j~nclions From their mean I"-"ili(,", wer~ firs! introduced inl0 lhe model by Allegra". along !he lines of lhe Rooca.Allegra model", and lalcr by Kloclkowski "nd oollabordlOrs. using Ihe constrained.junelion modoJ'· and ,h. con.lrai " cd-<:hain model". The lane, tWo lreatments gen· erali"" Ille Flory theory'" \0 nMaffine disptaermenl of june,ion' 12.2.3 Predictions of the Mol""ul.r Thenri.. The Flory Ihc"Ory of strain·i llduccd crystallizalion e~pli citly e~pres""s the e/Tcct of !Irain on: ( I ) lhe cleva(ion lOr In. melting poinl T.... (2) the degr« of crysla lliniry w. and (3) the d a 'li< f"""'je.,hibilcd by Ihe nc1w",k. The"" relalion$ are W " I - {(3 / 2 _ ¢/).»/!3I2 _ O)} II' , , , ( I U~) = - - - - O( .I.} I ;' T.. hi '"' n", f "mper~lure = vRTf( .I. - .I. ') _ (611/ >:)111",)/( I - "'} {ll.S7) fUllClion 0 in cq, (l 2,55);s O " {!rf/Ie)C~-~) and Ihe strain fUllClion (12.58) ')In (12.59) In thc~ c'qual;"ns. hI ;, the heal of fusion per ~gmc" l. Ie is Ihe H"It~mann conSI"nt. and ~ !rd "r ;s Ihe incipient cryslalli,-"Iion lemperature of Ihe undc· formed polymer (wilh 'I b.:ing lhe enlropy "f fusion pcr segmenl). A. u,ual..I. is lhe etonwuion in simple lension (dcfincd ~s Ihe nuio of Ihe defonncd lenglh 10 the lenglh at Ihe SIal. of format ion of Ine network). " i, !he number of segment. (."ni.lj.cal links) per chain. and" is Ihe number of n,lwork chains p"r ullil "olume. Some resuhs of Ihe Flory affllle network model and ,h. Flory conslmined· junction model a rc pr=nlcd in figur", 12.6. 12. 7. and 12,8. T he del. and lhe T! T! SE Mlf LE X IBLE CHAI N S A N D STRAIN-INDUCED CRYSTALLIZATIO N ", "" . t- 181 '" "" 300 "" , , " l. .'i ~u " 12.6 "Tho dcp" .. n,,,,,Mure TMon clon",';on f.". ~ .. 10 for ,1>0 oon"",ined·ju,,<,i<>o n""kl.• nd fm" l~. Hory ">Ode! (" ~ 00)". .m".. junet;on model with ,,~ 10.'· C"r respond ingIY. ligure 12.1 gi"cs lhe de~ of crySlall;n;ly U a funci i" " of . , lens;on" Figure 12.8 .hows lhe \>chao;or of lh~ rWuccd nominal .trciS (modulus) U"jl " HT as a runCI;on of lhe in,·crsc elong•. liun -1-'." For lhe offm" li m;1 (" '" 00). lhe mod ulu. i. a monow,,;cally increas· 0.15 ,----,----,,--,-----, 0.10 a 0.05 '.00 '--_ _'--__...J,- , .'lU" , , l. 12.7 TI>o dc:~t<.... or c.)",," lh"" ) •• " {,.oc';'," ,of Olo"g'l"'" ~ lXl.hf 919HJmol ',"n or ,, ",n r... difr''''"l ,'"I .., 182 STR UCTURES AN D PROPERTIES OF RU BBE RLIKE NETWOR KS >> '"- ,. ,., ,., c ,. ,.. ,. " : ~ , ,. ,. 02 A;' " ,., F~ ... U .8 Redu<:e<1 nominal ing function of Ih. elo"g;lIion. It also beeom.. a monQlonieally increasing func_ lion of lhe elongalion in the phanlom limil IN ,. 0). ,,'hieh i. nOi shown in the figure. For intennecribcd below. lh""" fealurcs ha"c been ob-;.c""d in .c,·cr.1 expc'imcnl~1 studic,<>3,1I-OJ.. and thc 'Ire ...... train isotherms shown in figure 12.S are in ",Hi sfac'o. y quali'ali,,, agreement wilh .,pcrimen!. 12. 2.4 ne Effecl' of S'r.in-I nduccd Crystallization on Mochaniesl Properties Of direcl ",Ic~ance 10 lhe prediclion. of ,he previou, ""'lion. crystaUi,.able pulymer nc,,,·o. h such .s lhose of m,lur.1 rubber Of d,- 1.4·pOlybu'adiene .,hibil """' -Slmin iso1herms in which Ihere are pronounttC,ibcd in 'he following chapler. Also. for" crystallizable nct ...or~ . these up,urns disappear upOn suftieicnt ....elling. 3S illus'r~led in ligure 12.91>· ... As "' I'«ted. 'hey als<> di,niniSh and e'''''lual1)" di",ppcar "f'<,n incrc"", in lemperalurc..·... On lhc olher ha11d. in Ihose "hc"" Ih,' upturns due '0 limi,ed chain e~'cn>ibilily. inc",.", in 'em""ral"", ha. ITT.,ively lillk effec' on Ihe up,um '''"'''. AI,.,. i~ 'hese ~a"-". C""'. "fC SEM lf LE XIB LE CHAI NS AND STRAIN -IN D UCED CRYSTA LLIZATION ." - 183 •z! " 0 0.12 - • • • •• • I ., .-. M M M •·It u... 12.9 Siress-m"j. i""l!>enm fo' • hig/l ly '·l)"lol li",blo ru. 1.4·polybu, adie". nel· ,..." • • owol"'n wi," 1.2... 1i"~lorobcn"'"" 10 val...., of ,II< ,-olu"", f"""ion " of pol,.,...." of 1.00 (e ). 0.80 « () ) . OM (0 ). (1.40 (A ). and /I.2() (6,)"' lI.eprinteOrican elI,mical Socie'y . welling can "',,n make the upUilnS InOTe pronounced because of the already imposed .trelching or lhe chain. from Ibe d ilalional clTeets of Ihe .w.lling...... Two additional fcalu ",. in figure 12.9. how.,,,r. mcrit additional commen"". Firsl, Ihe initiation of Ih . .... i,,·induero ery"all i~.alion (as e,·idcne<:.; de".."ure of Ih. isotherm from linearity) i. facililaled by Ihc prese""" o f lhe low-molecular·w.iglu dilucllt . T hu~ in a sense thi s kinclic .lTeet aclS in opposition 10 lh. thermodynamic crrccl (which is primarily Ihc depr"";on of Ihc poly_ mer mching poi nt by lhe diluen l) . The second inlcrCSling poi nl has 10 do wil h Ihc downturn in the nH)(l ulus prior t" its increase. As shown schematically in figure 12.10,.·"". this is probably due 10 Ihc facl tha, lhe cry"aUites are oncnlcd along Ihe diowlion of Siretching (f'gure 12. IO(a). and the chain seq ue nces ,,-ilhi n a Cl)"slallilC are in regular. highly C>lcndcd C(>nform. lions_ T he "raighlcning and aligning of portion, of lhe netw"rk chains Ihus decreases Ihe def"rmation in Ih. remaining amorphous regio ns. wilh a n a("C(lmpanying docre.sc in lhe stre.. (figure 12.10(1,))_ MOlh the downturn and subsequen l uP'u rn in Ihe modulus are in 31 lea" qua lilalive ag",,,,,,eo l wil h 11>0 predicli",", of '••." 111. STRUCTURES AND PROPERTIES OF RUBBERLIKE NETWORKS ,. ~'iII "~ '" n. lo S".in-indUttJ «)".,.lIi..,;o" in. polymer ""''''ork ,h. , h.. bce.l don- pted by. fOKe ".,n, ,.,., ",,,i.e.1 direolionl"O. Reprintoo with I"'""i .. ion from M>rk. l. E. (1979). A.c~ , Cilmo. R~, .. 12. Copyright 1997 American Ch<mkal Society. entirely, dl>C to , train_inducetl crysta llization. [0 the case of ,oollCrystalli •.able model networh guch as those of PDMS. il is clearly due to limited chain .xt~o_ sibility_ and thu, the """hs on thi' system " 'ill be """,mely uscful ror reliable """lml1i o n of lh. va riou~ nno-G.uuian Iht<.lTic""S of rubh<:rlike elasticity . Such distributions are described in :,ppendix f. Some illustrati", resuh . on thc cff(>uhs .hown ror I'OM S ""tworks ;n Table 1<.2.... ). T.1!It 12.1 (."i.!_1.4_ "ol)'butadieno Network, .( Il ia!> EIo'W't ",n T .... "..,.'"'' , '" "• K_ rq a .. "rt"'" Uhir... " proro"'" M ... HI Hi . "' " ~, '"' ,".1 ."'.,";. LrJ (% ) 1'" S,.,.. "....,. " ., "'pl.'" . '" , 6,22 ~,I~ .;, ~ "·"""on (""" .. f_ (''''J). K_ .:..~ ~ ~ ~,." al. (r.... ' "" .... "' '''' '' ~' ~ ....,~" ' . ,""' ~, SEM IFLE XIBLE CH AINS AND STRAIN-INDUCE D CRYSTALL IZATION '.00 ,." O~ ,,. , o. o. • n.o en RO.l62 O.OJ.C 4.42 4.12 po~ (''''''-1 185 Ml.l3 O.Q)}(i 1,.;' "" "J .. of ..... ,,... _ , ..... "" ............. _ .. I., 1l<priMod .... , . . . . . - I, . . " "_'j"".... . . "_I '''I). ''''' _ _ F...""," ... ,,, "',,'.J. Il-• .. ... IEdo.). "","'oJ h<;-..l'" ..r .. W..... . -' <>C. A_'• • Socoooy ~ ,.., A""' .... o....Al Soooo" ~_ ... I References (I) M.ndclkem. L C,),.lalka/ion (2) M.nJoIh,n. L, In O,ford. of PoIym<'T>. C""'I"~I>t"'i... ~kGflI,... lIill ; N,>w York. 1%4 Pol!""," .'icW"U, G. A llen . Ed, P 1 98 ~ , (3) M,ndelkem . L. R./)/).. CI>t", Trcivk>/ 1'I. GM, (.) Magi ll. J, 11. RIIbIH. CIwM. ·I"rrlw>i. 1995,68.507. (~ Aharoni. S. 1.1_. f.d. I>)"",M,;". (.'I>.oT"'WrlZllIily",,'k N~'K'f). I/"pkl C• ..,,,,,,,,_ 1984.5.287_ G .. Rock. S,; Rederl. G, ; 7.<:,"<1. 11., .t{,,;'-mmol. CM"". M am. 19'/1, In. 2555. (2 1) tlarolay, G . G • O ber. C. K. 1""",' Sci. 19'13. 18, 89'/, (22) W.",er. M. l'~iI. T."",. /I."y . Soc. 1.i>IId. A 199.1 ,.l44. ·I IB. (23) f inhlmann. 1I.• 6 ,...nd, H. It TRI P 19'14, 2, 222, (24) Bo"'h. A. T .• V.rich"n. L, M II"""""I, Th<<>ry S,,,,"I. 19'14. l. 5lJ. (25) Br(5).41 n. 1'"" c. ...... 1',,""" 186 STRUCTU RES AND PROPERTIES OF AUBBERLl I( E NETWOA I(S (28) Shilov, S. S., Okrkr. H. W.e Z' v, N.: 1'1'«. N. A. AfIJd. SMn ....' AcaJ. Sri. Ii 1975, 28. 101 ( ~ J) Schw'<7_ J. Mui ,,,,,,,,,. C :Iorm .. Rap;J C()mmU". 1986,7.21. (44) W",,,,,, . M.: Gellin8. K.; Vilt;is. T, J . CIwur. Php. 1988. 8~. 4-108. { 4~) W.rner. .\I . In Shit- CIw;" Uq",d C'p'al ('oJ}" ...... ~, McA,Jlc. Ed, ~I. c kie '" Son" G I"K"w, 1'189: Vo>l , 88. p. 7. (46) ~rand. II . .1/ak,omoJ. Clwm .. &piJ Commwr. 1989. 10. 57 (47) l ;'man, II.; ~.h" . I.' KIoc"kc"'" 1990. 23. !i341. (49) W .,....... M.: Wang, X. J. ,llac,,,,,NJI«"kJ 1991. 24. 4932. (SO) W.rner, M,: WaRgo X. J. M""romo/«id.. 1992, 25, 445, (51) EnI'Ian, ~ .. lI'har. I. , KloI:,kow;ki . A,; M",k, J. E. In Bou,,,,,,,,k /"OIJ~Wf N"~",,b. J. E. Mark aoo 8, Ennan. Ed •. P«nlitt lIall: NJ . 1992: p. 142. (52) K"",ko"'.ki. A, ; Ma,k, J. E,: E,n,.n, II.; flahar. I. In Pol),,,,,, Soh w"",. SEMIFLE)( IBLE CHAINS AND STRAIN · IN DUC ED CR¥STALUZATION 187 (68) Mark. J. I~ In Ph,.,/nIII''''f''""k. '" l'aI)"I""" ~"" cd .• J f . Mark. A. EillCnb1."""11« • • F. T. Somubti. J 1... " ..... ,•• and G . O . Wi,ull. Ed .. A......,an Cb """fI" ,.f"..""""', ..... 6J._ ,M (n ) KIocrIr.,....ki. A...Marl:. J 0-«",;:;,,_. _ .E.: SlYnr!. M A.: Erman. H ID '~""/oQi$. fII,""""'.-I n..-..-,. <1{ '.,-.Ir FA . P\c1>.,. I.....: III.... VOI k 1991: p 227 (78) s...,'h. K L J, ~ 0 - A : Cirtr-r •. A. K_ Z ~. PM,..... 1\1601. I~. 49. (1"J) !>brl:. I It ....... /kf. 1~1"J. n. "'l (80) !>hrt. I . E. &« Sci. 191"J. I ~. 40'1 (81) :\.brl:. I . E..: Erman. II RJIbtrlik. FJ.,"""y . ... MoknJw "'__. Wil<)_ I'.,....c_. I n'~: Ne.. Vork. 19\18. "YI, .... (81) CI>ou. 0 .-0. Snti'h. K J . J, 1:' &, 199-1.).-1. 2\10. (83) Chi" , [). S .. Su, T.·K.: M •• t. J. E. M~I" 1~11. 10. 1110. (114) Ma rl: . J. E. /'01,.... 1:-" . .'N-' 191"J. 19. H( (SS) S•. T.·K.: M • • t. J. IJ.•I/~, 1971. 10. IZO. (86) Andrady. A. I... lIOrenle. M. A.; M.,l. I . .E. J. CIIm,. Phy •. 1980. 12. 2282. 13 Networks Having Multimodal Chain-Length Distributions 13. 1 Uttimal e Properties and Non -Gaussi an Effocts As was ,".n!love th. dependence of u!tintate properties on non . G~ussian effect< a,ising from lilnilOO.ehain cxwnsibilily, "s was already poinlW ou!. The following chapter pro,old"" more delail on this application. and others, In facl. the efTect of network ch"in-Icnglh di.tnbution"', is one '"'1"""1 of rubberlike das'i-linking Itthn;qucs traditionally used 10 prepare Ihe n«"..or ' .lrUOIU,", req"ircJ for rubberlih cia,· ticity haY<: been .. ndom, U""O"l rolled pr<>cc>sc •. as "'-as mentimlCd in chapler t(t", Exampks are vulcani7.alion (addilion of ,"Ifar), pero~ide thermolysis (frtt_ rudical couplings), and hil',h-cnergy radiation (free-radical and ionic ",.ctioll'). All of the"" lechniques arc random in the ",nse thai lhe number of cros,·link. Ihu, inlroowed i, nm known di r""tty. and twO unils dose togcthn in spa<.~ arejoincJ irrC>peCti,"" of thei r 1<>C~tion, along the chain traj.:ctorics_ The resulting network chain·length dimibution i. unimodal and probably vcry broa d, On the theoretical even beller. com",1 ne(wo,k chai,,-ilnaU deforlll"liun. a"J Ihen ae( "' ruptu", nll.lei. Another reaSO" is '" dctermine wl'elho, control or chain.length d;,,,ibution Can be u",'tIlo m"~i",i~e the Ullimale propenies of an ,hmOtuer, 190 STRUCTU RES AND PROPERTIES OF RUBBERLIKE NETWOR KS .· i~ ... 13.1 Skot<:h or. bimodal network in .,hich tl>< ,hort chain. a", a,bitr.,ily drawn in I><•• icr Ii,... than tl>< lo~ ch.in,. "hich a n: ""'''' I),p;<;-.llly associ. too ,,-ilh rubbcrlik ,., FIl"'"'" I .U fnd·lin kinB of , ide c",in, by. <. in each s,de chain. th. ""... Iinking chain, <culd 1>< cla>tlcally .(fecl i,,,, or> in loss ronlroll«l method. of crotlS·lin king. Figure \3 .3 JOO"'1 a schomalic di!ng chail1$_ The major difl"erence i. Ih. larl(e amounts of belh very 'horl chain. and vcry long chains in Ihc: bimodal ""Iwork. conna"ing .harply wilh thc: ' mall amounts of soch cloains in u Iypical unimodal Ji>lrihulion, The case .h",,-n here. when' Ihe shorl chai~1 pred"",i""" numericall y. il of I""ticular imerest and "cd to i!iu""". lh" iml>o"""1 dilT~,..,.ncc bel""'" number di>lribulion< "nd w"iglll NETWOR KS HAVING MUL TIMODAL CHAIN-LENGTH DISTRIBUTIO NS 189 As was deseribed in chapler 10, a varicly of mood nelwor ks ""n be preJ>llred osing lhe new $ynlhClic techniqucs Ihal d<>sely c"nlrollhe plaecnlCnlS of <:ross, links in a n~I"mrk Sl ruclu",""'o,,_ Thu ~. cnd.l inking funCtionally ,•• minatw chains, in'tead of hapha'lIrdly jo,n ing cha in segments al random, controls Ihc struclure of Ihc re,uh ing network . As already ,n.nlioncd, the (u""lionalily 9 of a cross· lin k is the "une as lhe number of functional groups on the end.linking agent, More importanl in lhe present conle~I, Ihc m<>le.;ul:t, weighl M , bel"""n eross·l inh and lhe mok<:ubr wdglll dislrih"lion are the same as In<)>;C of the sumin, chains priN 10 Ihei. being end·lin ked , 13.2 Bimoda l Ne twor ks lIimodal nelwork> prerared by llle", end-li"xtng ".'<;hniq lteS hal'. >"ry good ultimale propert ies. and Ihcre is currenUy 'Much inlcresl in preJ>llring and charac ter· izing ",och nClworb'·'· "''''''·''', and dc"eloping 1"'"<"".i,,,,1 intcrp",la.ion$ fo. Iheir properlies)'-'" I),2. I Materials and Synthetic Tech niques Most bimodal networks symhcslzed to d •• e h,I\'o been prepared from pol)'(di· (pnMS)_ reason for lhis choice is.1\c facllhat lhe polyn,er is ",,,dily available witlt either nydro,yl Of ,"in yl end g"'up~ and lhe rea<;.ions Ih.l these gt'ol>"cd hydroxyl-termi"aled clt"ins, ,,-hich are ""wily ob,.ined from the usual ring-oP'Cning polymerililtiom of Ihe cyclic ,rimer Or t.,ramer", The ends of the chains arc ","eled wilh lhe alko .• y groups in " multifunclional orl\"· nosiliCllte. such as lelr~elhylorlho-;ilicalC", "' duoTil><:d iu chapter 10. In lhe app)k•• ;on of inlere" he"" a n,hlure of lWO polymel'$ i. I>c;ng cross·lin ked. 01>< componenl comi'ling of vcry ,hor. chain. and the OthCT of much longer cha ins, The resulting network i. ,hown sehcma'ic"lTy in figure 13.1". An alle'nali>·. approach inw.lvcs 1toe addilion re(,clion bel""",n vinyl groups a l the ends of " pol)Iner chain and the active hydrojlen atom, on silicon alolns in .he (Si(CH,)HO - 1 repeal unil' in 'III nligomcric poly(,nethyl hydr()gcn silo'" ne) As" final uam rlc, a nelworl be cross·linked using a .. ifnnctional reactant carrying , ide choins le",ninal'"-c group ,uch as N H, .· ' Comp",~ing two of these N il , gro ups with a mc.al ion pm"id ... a ""I of addi· lional, .. the r shon. chains thai could"" ela slically effeclive . This approach i. illustrated in figure 13.2. These id"", <;'1n ob"iou,ly be e..ellded 10 higher mod· a li.i~ (ITimodal, elC .. "vCulu.liy "ppmaching an c",,,,mely hr",,trcmely ""usual, Knd ",,,ch differcnl ffUm Ihc u,;u"llInimodal dislributiou oblamed melh)"I'ilo~anc ) 0,,,, "'''Y NETWORKS HAVING M ULTIMODAL CIiAIN-l ENGTIi DISTRlaUTIO NS 191 I • 13.3 Nr~ in on infinitesimal inte .... l.round 'he ,pocificd .. I... orthe mole<:ular ,"'eight M ." 1\ broad unimodal di,tribu,ion is .ho"'n by ,he hatched 'rc., for ",f.",,,,,,, puf]><>Sd. Reprin ted wi,h (> 1991, Am.. ica" Ch"nicat Socie'Y, distributions. For ""ample. a bimodal network consisting r>r 95 ","'0;. chai n. of molecular "'('ight 200 g mol ' and 5 mol·;, of chains of 20,000 & mol - ' would have only 16 )0'1 % "f the short ~h"in'! Ob"iously. , nch pOCnliar distri h<>tion . can he ob,ained only by carefully "",nlrQIIeJ 'echniquC'S ,ueh a. the end_link ing rea'" tion, being described. OtIC of their use, i. to ""t • ,nolccular deseriptiQn Qf cla,tomer ruplure. as described in th. followi,,& ><:et ion. 13.2,2 Tesling of the We. kO$I- Lrn k T hc-ory T h. w.:akcsl-lin k the<JI)" was ICSled by preparing end_linked network s £ontaining increasing "mount, of shott chain •. Qf Ihe order of 11}-20 mol'!'. "", RetnHrhbly. there weTC no ';gnifr 192 STRUCTU RE S AND PROPERTIES OF RUBBER LIK E NETWORKS FIJI." 1J.4 Sketoh !howi", tbe ,'er1i<.1 roe~~011 b)' 'w<> !hort eloain'''', In 'he limit of an am"" dcl'orm more than tbe a"=8O ar.d tbe !hort ch.in, deform bo th.n 'he .... rag•. contradicts tbe ",eak...t.link.tlleory ~<s u m p tion that short-chain breakage occurs ~rst. well befo ... tile rupture point , Also, in an affine defofm"l;on. a decrease in tile number "f $h"rt chain. in a himudal nelwork ,bou ld de"rta.., Ihc magnitude of lhe upturn in modulu. but n01 the elong,uion at which il begins. E. perimcnts. howeveT. indica!< that the fewer the shorl chain. in the bimodal nc!work. the higl1er the elongalion required to bring about the uptu .trm. T he ..... k.st· link ;ssue ha,ing bc<.n """,I ..,d.;t bee,"", or interest to see what wouk! happen in the Case of bimodal networks having such overwhelming num· bers of short chains that they cannot be ignored in Ihe network's r""po""'. In fact. there turns Out to be an uciti ng bonus if On<: does put a large number of short chain~ into the bintodal n~t",'or k . SP'X'ifICally. thert is a synergistic e/Teet kading \(> mc-Chanicai pro[>Cnles that are ben"r than those OMainahle from the usual unimudal di.tribution! T h""" result. are the focu, of ">",1 of the remaining SIX'lions "f thi$ "hapt,r. ""ry NETWORKS HA.VIN G MUtTIMODAl CHAIN ·tE NGTH DISTRIBUTIONS 193 13.2.3 Elongalion Rcsuhs /3. 2.J. 1 Charac/u",,,i<m of Uplurn< in Ihe Mo • a FIi:~" 13,5 T)'pic>1 plolO coo.... f)( no>m;"al ,,,,,,,, .g.i"" clong.lIion (or (110''''011.") bimodal """""y P I) ~S ",,,wk. i~g ,,( loog ch,,,", (M, ~ 18.500''''01 ') .od V60 (0). .nd 2.."0 (e ) in 3 ",,,,. ,). I;act! ell"'. i, label«! " i Th 1"'= ",(>1% ofsho', cluin, tlul t!\e ""''''ork «lota in,. "nJ III< .... unJor ...h ... r.' • ..p... ..,." 'he ,uP'u", e".r~y (a ""'"'u'" or 'he of lhe ol. "ono<,)_ R'ptOd<>e«l with permi .. ,on (rom LiOITnlc.,\I A __ ,"00 '"'''''IV'''''''" 19 4 STRUCTU RES AND PROPERTIES OF RUBBERLIKE NETWORKS networks ~re plot\oo in .uch a way lh"t lhe area under a stre<.-strain iwtherm oom:'ponds 10 the energy requirc-.J to ruptu", the nelwork. If th. network consists entirely of the .hou-chain c<>mpo ncnt. then it is briulc (which means that the ma,imum c~\cn.ibilil y i, "cry ,mall). If the nc(work con.i.t. of only the 101lgchai n componen!. the ultimate .,,,,nglh is '"Cry low . In neither case is the maleria l a touj!.h ola'lomer. A, can readily be st-en from the figure. the bi moda l nelworb are much ;mpro"ed clastomers in that they ca n ha,.., • high uitimate strength without the nsna l d iminishoo max im um extensibility. This corresponds 10 high val"", of the energy rcquiroo for rnptnre_ ,,'hkh makes them un u,uaUy tough d ae"tal;on of some si milar results i. shown ;n figure 13.6). Relaled ;mprovemen1. in mechanical prope" ies ha,"e also been reported for poly urethane cla 0roIOI l<. SoI\oo1 I ()~ (,/10'., " " • ~ 0 ~ ~ " ro 0 0 " o. " 0' .-;,; ... 1l.6 Keprcs<:nl31i,'c " ..... " .. in i""ther"" for birnodal (un.wollen ) POMS ""'_ work. '" <1 1.1><1<.1 "" ilb II>< mole percent of .he much .horl.r ch. in, (Ur' M. z (l.!2<1 ... IS.S I mol- I). The oPC" cin:;los repm1enl II>< ",.1" ob13ir>cric> of ;""r< ""'<";biljly in the ;rolht"m'. c,'cn al these Io~' I""'pc'alu",". The .h"" e>1 NETWOR KS HAVING MULTIMODA L CHAI N -LE NG TH DISTRIBUTIONS t95 fnneti(}n,,'ily'-"" ". T hese ,.suils can be \I&<.".c the ;mp",,"emcnt. in the properties obtained . The syn,hcses and properlies (Of "".er.1 add";(},,,,) types of bimodal networh were . 1"" in'''''tigated. T he first of lhese i,,'·ol,·cs prereacl;ng the shorl chains prior 10 ; ncorpo,~tmg the long ullCS in lo the bimod,,' StrucIU"" ·... Thi, m" l:e5 the r.,ulting network spa ti.lly .• < wd l ;IS comj>Ositio n. lly lIete'ogen"" ", in tha i m"ny of Ihe . hort chain, will be segrega ted in to densely cross-lin ked donlain, "hieh are only ligln ly cross-linked to other su<:h dumaim . Thi, is shown sch-cm,,tically in figure 13.7'. -Il>csc ""twurks couk! "'".., a, models for inhomogcne'O u,'y .r,"","linked 01:,,101l'.fS. fo r u.mplc those cured by thermolysi s of a partially im miscible pero, ido_ Another typ< i. basc". Finally. it i, a)"" possible to r.infon;e any wP< of bimodal network with fillor part~It$. thereby further im pf(>.;ng it. mcehan~al propertics$><" ,. Meas"",,,,ents of the mcehanica l a lld optica l properties of these DCt works a, a fUDCtion of temperature a nd deg"'" of swdliPlg were used to test funher the cODClu,ion cited abo,,, that crystalli,,";on or other imeJmolceul.r organ;>ation was nm the origin of the ionproV\.~1 pr0P.abl. polymer DC( work. ,uch as natural rubber ur d s- I.4-polyhutadicnc. In another expe,iment. illustrated in figu'e 1l_9. tempera'",c "'·as founJ to ha,·c litll. effoct On the Mooney· Rivlin isotherm, for bimodal I'D MS ""Iwork•. as would bc expecled in the case of !;m;tcC(>ntinu;lies ur ~· ; ~U .. 13.7 Sketch or. bim<>d.1 nc"" ,, ~ lb., ,. '1"',;"lIy.' w<1I a, rompo~t;o< 198 STRUCTU RE S AN D PROPE RT IES OF RUB BERLI KE NETW OR KS t1gu... 13.8 SkClch or. "mult • ....,'" in .. 'P' .... ,""ing ",,,.,ork h"'ins a bimodal dhtri. btJ'ion of cha in knS,h., .'en Ihougit Mne of ,he .hon eIIain. a", >112lun: 13.1.j Reprin'ed wi,h po:nni»ion fT~m ~hrk. J. Ii. (I98S), .'1 «. C,",,,,. ~. .. 18. Copyright 1997. "n""K:.n a.omi",,1 Socie')'. o' " .l~ .. M S .... ' ..wk """toining 75 moW. or !lion ch.in< (220 B mol ') rombined wi,h much lonJl<' chain. (18.5 ~ 10' I mol ' )' . "",be"". di""",nit>le change. of slope. as is dC'",onslr~'ed in figures 13.10 and 13. II. respeclively). Simila,ly. <"'tiling can even make Ihe upl ums in "'odulus nt<m' p"'. ,,,,unc~d""". ~5 i. illusn.... t<" Jig"'" NE TWOR KS HAVI N G MU LTIMODAl CHAIN -LENGTIi DISTRIBUTIONS " ," .' •" " ~ , ~ 0., ... ' 90 Q· 2076 ;.,zn " " -0---0- 197 2.38' " " ~ ", " ." " 2."86 2 ,222 . <0 " ·C F i ~~ .. I,}.I O T )'pic>1 'ho,moda$!ic da'a on bimod.1 POMS "",w .. rh. wi," circ~ ... r... ring ,.. m 60 "'0' '1'. o · 2_222 " " 90 "'. , % 0 ' 2016 o I. ·c .,,. ... IJ. l t RCJ>'"><:n"';'., bi..fring<""" ,."'1.... ' ... telalion, on bimodal rUMS ""'wo,,,, "ilh filled ci,d", representing ... ,ullS "b,.ined '0 ch«k ror ,..vt"ibiTi,y' already bttn men,io'led in chapler 12. Ihe upturns in crys,"lIi""ble pol},,"cr nel works di>appear upon ,,,OieiCll' swdling6>,... A nnal c~perimcnl "r relevance concerns 'he sp'l1ially-hclerolt"''''OuS r D MS ne,work. men,ioned ea rlier''''_ Any ",dcring lloat m,ght ha,'e~" presen' would pre.umahly ha,'e been effec,ed by Ihis .h"oge in S[>atial h.'.rogeneil),. bul 'here was no di$Cernil>l" eff<"" on the measured .I."omerie properties, T1tc at""'e r,nd ings argue against Ihe P'="''' "r ,,"y ery,.a llif,m ion or othe r !ype of internl01"""",, Ol\lering..... ', and 'he up""no thus secm to be in!ramol._ cular in origin, In a,ldi!ion . infrared and Itam ,,,, spcc!roscopy indicate that bond· 198 STRUCTURES AND PROPERTIES OF RUBBERliKE NETWORKS " v.' \,00 0 ." ,., " oc-' 13.12 S"... -"", in i.",hmn. al H ' C for a I'DMS bimod.1 Tlor! Ch '''''&, Eaoh ll a, angle diSiortion i. o f rclalivoly minor importa,""", atica" in I'DMS networks"'. The O~M i"",eases in mooulus and ultimate '''englh the,efore haw 10 be due 10 lhe limited exteM;bihly of Ihe very .1I0r! ",,'work chain •. In qualilaliw I~rms. the Chain s $OOn exhausltheir &palial conflgur,dions. ,h,if ,nlropies plummet, and the elastic force rises correspondingly . II i. pO>.illlt 1<, character;ze Ihi. non · Gaus.!i.n limited cXlcnsibilily more quanlilalively in • "umber of wa)'i, $Orne of which are de"",'ibeJ below. The in,'ol'.... Ihe interp"',"lion of limil,,,,,,,,hain c" cnsibilily in l~nnS of the ~onlig"r.'ional chardctcriSli.:. of the "DM S chains making up Ihc nCh'mrk ,(rUClIlrcM Tbc f"" imr><>rlan. itcd...,ha in extensibility is 'he elongalion <>. at which 'he ;""rea s< in modulus tirSl bc<."~1C"S discernible. f,,,,, NETWOAKS HAVING MULTIMODAL CHAIN-LENGTH DISTfll8UTIONS 199 Although th. defonn.tion i. "onaffine in the vicinity of the uptum_ it i, JIO$,ible to pro,-ide al leasl • semiquantitative interp"'t~tion of ,,,,,h re,uits in lenns of the net .... ork chain dimen,ion,"'. At the boginning of the upturn. th. a,'eroge ""ten_ sion , of " ".Iwork chain h~ving its end·tc mnimum ex tensibilily occurring at this point in tl>c dcfonnalion . The values oblainw indicate that the uplurn in mooulu, generally begin,", approximately 60-7(1% of ma . imum chain extensibllity'·"'. Thi. is approx imalely IWice Ih. value Ihat had been cstimal ed prc,;ou.ly' from strcs.<;-- Slrai" isotherms of elaSlollle ... which may haw beOOc beginning of the uplurn wilh some th"" ... tic.1 results of Flory and Chang"" on dimibulioll func_ tion! for 1'01'01 S chains of finite Ie"gth. Of telC\'~1'>CC here ate Ih. cal(; ulated values of ' / ' .. at which the G"ussian dis"i!>utio" function Sla n s to o",,,,,timalo Ihe probability of eXlended conftgllrations. as judged by comparisons with the results of Monto Carlo simulati"", . The th,,,,,,,tl,,,,1 ,,'Sults ...... suggcst. for examplc. thai ,h. network of PDMS chains having n " S3 ' kelcH,1 ""nds, which was experi_ men,ally "udied, should show an uplum at ~ ",,11lC of r/ ,.. a lillie Ie .. Ihan 0.80. The observed value wls 0.77."" which is Ih", in .""lIenl agn"CnlcP\ witb theory. More qll~ntit ali,,' characlcrizal;<>n (>f thi, limited cha in eW:nsibilily r.:qu; res. of course, " non..(iau"ian distribution fuoction' for Ihc cnd·to-end scp;!r~tioll , of the shorl n.t .... ork chains. Ideal for Ihis eaSC is the distribution obta;nw by Fix",a n and Albon". and recnmmeudw p;!rticuiarly for vcry slt"'t chains. It waS found to gjvc a mud bell« app",ngatiOIl for bimodal POM S ""''''<>rh. Thi, i. _,hown ;n figure 13. J 3. The obsc .... d upturns in Ihe reduced Stre.. 1/' 1 "t high v~lue. of the elongation <> arc ",en 10 be .... 11 reproouC«1 by the calcub l~d re, ul,," Other no,,·Gaussian distribution funClion' ha", a lso b.,." " ...."'..."f"lIy u.oo for t hi' purposc. ". is ,how" ill tigu .... 13.14 and I). IS "'·". The expcnmemal isothl'rms can al'" he interprelw u,ing the van dcr Waals Iheory of ruhhcrli~c ela,licity"'·". as illuthcr "ppr<.aeh. Montc Carlo simulatinn'. ulili""s the wealth of informa_ lion Ihat rQlalional isomeric Slate Ihenry provides on the spat;al configuration, of chain molecul(;,. Some lypica l no:;u ll> on bil11Qo.I;ll nclwOr ~ , "'" dC'«:ribctlu, chap. ter 8, In brief. Montc C"rlo cakulat;"ns "",,"~I 0" the rotal;o"al i",meric stale approximation a rc u." "IIO ,imulalc s""li,,1 "",figurations and thus Ilis"ibu!io[l 200 STRUCTURES AND PROPERTIES OF RU08EALIKE NETWOR KS ... •• '.' • 'e e z , .. " " 0 , , .,• - , .. , • , , , , ••• • , '~, .... -,>_0 __ .~ ~oo. o- ~ "'.0--<)---1~ ~ ~ " '.' FI ~u .. , ••• ,,' •• '.' 13.13 MooIId;ol r[)MS ... ,work , (M, ~ WJ .0<1 18, SOO Bmol- I)". Tho hoo Ii,..,. "'p.... nt ••!'<"im,ntal dat., "nd the r,lle tbe ..:.olt. of ,h .. ,ic31 0" ". is l>.b 1"''''''' "1010 pIoco of the par"""'l"" >pc' cifoed r", 'he lh«>""ical Co"'., .... (ho .. roq"i"-'.! 10 appro.; m.,. ,Ile """,,,mj dc< r•• "" in rteeinl mod"le !Ie"" .... fu nClions for the ond_l I 32.J.2 ChaTilCI ..rio"'lo~ of l/if Rup""" I'rocess A second important characte,· istie of an elastomeric network i, Ihe "n"", <0_ of th. ciongatio n at which rupture at o<curs, T he corrcsp<,"Jing "allle,< or ' I' .. , ho", that rupture ge'lerally appro.imatel)' 80-90% or """imum oJ,.in e",cn,ibility'''. The"" qllantit.,ivc results on ,h"in dime n,ion. ore wry i'''p<,,,a,,, hu, ma y not appl y din,<,ly to "",,,,,m] NETWOR KS HAV ING MU LTIM ODAl CHA IN· LENGTH DI STRIB UTIONS 201 90·,. .. o o o • " .. . , " . ,. ---,. 0" 0 . • ~1o<>n<)·.Ri'· \ in i«>con,istin8 of o~,in' h.,il\ll ,·.llI<> of M. - 220 .nd 18.500 B rnol- I ,. Il:><:n set of np<100 ,,"'iln 11>0 mo", ",,""'"I <>f ,non en;lio, 1"."""1 in the "",,'ork . Reprintod w;,h p< 13. 14 Compo';"," I> tn .. m, fOf bimodal PDMS n 1997. Alt\¢nc.n CII.m",. 1 ~iet y, othcr nclwor~s. in which lhe chaills could have .'ery different conFlgura lional characlcri,tics and in which the eh"i" lenglh dim ibulion would l'res' '"'ably be qu ito J ifferenl fromthc ,'cry ullu,,,al bimodal distribulion inlentionally I'rodll« 202 STRUCT UR ES AND PROPERTIES OF RUBBERLIKE NETWORKS . .' .. • • • • ~ • " "" ~ .1 • ,. . n,.~ ... • • • / • " ~ ....."... IJ.I ~ ComPO';'Otl, t.:lw«:n cokulaled and " .,p<:~"","lal '·al ..... of III<: "',,,,... ""'f'iC Cmac,,,··) and !8.SOO ~ mol - I." l1tc ignaliOfl •. ,"p.... re,. '0 Ih. O'J><,i . men'" ""u'''. and ··=ar· 10 a ",vised c''''ul.,ion llul avoOl, II>c p"'. '· ... gin8 u>e ~.~. lilalion on ""Iwo rk chain lenglh dislribulion . As ,no,,·n in flgufC 13.17. d<,\:fC." in l""'peralufC wa. found 10 incrt." Ihe u tent 10 which thc '·alucs of Ihe uhimate "",ng,h of Ihc bimodal nelW{)rh exro...d Ihm<." of II,. unimodal one," . Thi. sugge.t> Ihal bimoda li ty facilitales "r"in·ind"""" cry,.. lIiza' io" . po,,,ibly Ihrough increaSC<J ori<."ntalion of the more c,,,ily cf)·stallizabl. chain, inlo cry".I· li'..([lion nurl IJ.l.J.4 Thermo. .. '.' In pf""dcticallcnn •. the abo,·e r<sohs ,lcm o~"ratc Ih~1 .hon chain, of limited c>.tcn,ibility may be bonded inl o " long-chain n«wo rk to impro'" ilS tough"" ... It i, also po6sible to ~ehicvc Ihe con",r..: erlce'. T hu,. bonding a .mall number of rela,ivdy long cla'!Orne'>: chai n! into. ,hort-chain POM S th.nnosct greatly improves I>oth its cnc:'g)' of rupture and impact ,esi,. lanoc. as i, illu,lf·aler the molccular ,,·eight,; in"oh'oo. Lower concentr.li()n, giw ,,,,.lIcr improwmenl< than can othcr.;i..: I>c ;tchine". ~nd highe' concenlral;''''S will g'adually ,:omcrt Ihc cornlX"ilC from a ,dati,'d}' hard m,"c· rialto onc that" mo", rubberli ke, NETWOR KS HAVING M UlTIMODAl CHAIN · LENGTH DISTRIBUTIONS 2C3 .I , , " ,,;~~ : ,., ,, , , ~~, o ." . ' ... " ~ I.t 16 S'''' .. -ot ,.in 'IIO,II<,m, fo, unimod.l ...d b,modal PDMS II<'WO'~' con· sistinS of , hOI' choin. (220 S mol ' ) co,"hineonding Ihw""cai """.. i, labeled ... ilh It.: "",Ie 1""<:<"1 of >hon ch.in, ~nl in the netWQ,k. 13.2.4 R<1uhs in Ollter Mochanical lJeformalions /J,},4.1 Briw'al t,:n~lt.Jion Thcre are nwner'O,, ; "Iher deC"nnalion.' of in,.re I), Some of Ih""" dcformalion, are c-on,i{kr.hly morc diffICu l1 10 study th.n ,implc clollgali<>n and, un[",lunately. ha"c th~refore nol been a. cxtott't:ly in"c.tigalcoJ . 20' STRUCT UR ES AND PROPERTIES OF RUBBERLIK E NETWORKS ' •• 0 ' ,, L' ~~: : .'\~ ~U.K' 0.' • U.K' o o ' .0 1.0 10 ~ Mn ' 9 rnor' 13.17 The "1Iim... " ",n,th shown . $ . f.""lion of Ihe molecular weighl rOT unimod.l (U) and I>im<> .II. ~ ,1/, !x1"""n ••.,...link. Mol 'I'. Short ChG •• • Fistu" 11.18 'lbc cnc.1.)' required for ,ured by ,he f.ll in'.... " I.. tj rup'"'" (Ii, ) .1Id th. impac< Slrc"8th (IS) (M mu· ,how".s. fn"",ion of composi,ion for bim<> network, in tho vicin; ' y of ",.,m «m[>Cr.,",.'" 11>< II>Olo "",n: 220 .nd 18.:100 I Some of IIr.c ea rli.sl Sindi" of bi'" ial .'Iension ha,-e im'ol"N Ihe direcl sireiching of a sample shc<:1 in ''''0 perpendicular d ireclion. wilhin irs plane. by Iwo indepc"dcndy va riable amou ots. Equibiaxial .xtcn.ion. ho~vor, i. "'1 uivalen l 10 comprc'lion, and Can onore co,,...n;enlly h< ;m[>O«:d by ;uHatio" or . hett' of Ihe ciaslomer'. A good accounl of •• perion"mal bia.i.1 .",cnsi".. ="1,, "n Iypical unimothl nelwork," has 1M,." ~i .... n hy Ihe simple molc, NETWORKS HAVING MULTI MODAL CHAIN-LENGTH DISTRIBUTIONS 205 a ' ~' ia~'" 1 3.1~ Rcpr=n,"'ivc "'.... -""''" ,.."1>:,",, for un imooal and bimoo.1 I'DMS "",w<>rk> in uni.,ial •• Iinial ",on.ion (ri8h' .ide)" Each Co",. i. I"bckd wi,h ,h. mol< 1"""""'1 of ,he shorl chain. pno«:n' in lbe ""Iw<>r~ TIle ore" circle> d.. ~ poin .. m< •• "r~ LUing inc"'asing C< to 'e'$l f'" m.~ ... ibiliIY. "'1''''''''' improvements being made at lower e",cnsiom upon usc of lite ronSiraincd.junc. lion 'heory'. Equibiaxial eXlen,ion re,ul" obtained by illfta ling ,he..,I, of unimodal and bimoda l ".,Iwurh o f PDM S arc ,lIu.lrated in figure 13,19"''''. Upt",", in the modulus are ""'" 10 ocr.r at high hiaxial cXlcn.ium. as cXp<'("tcd. Also of inle,.",I. hO"'"""cr, are Ihe pr<>nuuncW rna.ima pr=ding ,he uplum •. This represent a challenging fealure lu be c>pl ain ~1 by molecular theories addreso.ed 10 llimod.1l elaslomeri. ""I,,'oth, in gcneraL U.1.4.! Slwil' EI.'tOlncric nc:lwurks can be ,tudied in pu," ,hear by 'Ireldling a ,heel of the ma terial thaI has a l>rgt rJtio uf widlh lu length. in 'he direction perpendicular 1o lhe width. E.perimenlal result. on nalural rubber n.lwork. in shear'" arc nol well accoumoo r", by 'he .imple mol""ular theory of rubberli ke el.Slicily , The cunstrained'jurICtion Ihenry, howc...,r. was fourld 10 give ",cellent agreement with e, perimenl. a' was d=riheJ in ,haptcr 5. Result. of such shda l nctworks of rDMS arc 'ho .... n in f'gure 13.20" The bimodal I'DM S networks .howed large upturn, in li>c pure·,hear mooulu, al h'gh 'lrai". that were , imilar to thu,", rcponc 206 STRUCT UR ES AND PROPERTIES OF RUBBER LIKE NETWORKS .." • _ • 0 , '".. ... .. , 0 ....,P : .,,\ ' u ., \." , '" , 0 0 """,, . 0 0 O' 0 0 0 co· ",,> •, • 0 ~7---~--~--~---O 1.0 1. S 2.0 2. 5 ' .0 .ia .... 13.10 S'reloO_"ra,n ,~"bem'$ for unimo<\:l.l and bimodal PDMS ,.""o,k, ,n shear" , Locn cur...,;, ~.bcled wi,lI ,be mol< p,,,,,"n' .,r the ,no" tha,n, rr=n' ,n tbe "<,work . • nd ,ho ...,,,ic:. 1da.noo lin .. I<><->t. lbe ,uplU'" 1110 open ,i!'d<s "'p .... SCn' dot. poin" n>< f,lled cin:1es "'presont dOl. oOi.in«J ou, of ""lY"""'" I" ,..t fo, <'e>",..ito;l'ty. po'.". Ie, "plUm, in mod ulus, p=umably Ix..,ausc of diminiShed ,hain.ju"Clion cOlan· gling. 7,,,,,',,,, /3.},4.J Very lillie work ha s Ix.,," done on da,lornc<j ;n 'orsion (,"'i,(ing " cylindrical $;Imple .round ;IS long ",is) , T here are. ho,",,,,,,, •. SQrnc rosull. on Iln:ss-s'rain br:havior and network lhcrmoclasl;"ily'-'Ul . as was described in chapler 9. Mo", ","ul,s are presumably (o"heoming. parlicularly On bimodal ",,! work, and on nelworb (:(Im. ining $(>rne. of IItt unusual fillers described in ch,pt,r 16. T he ",me ' ype' of bimodal PDMS "ct,,'ork. ,nowcd r-~U>cr differen, behavior in lOr$io n"', SpeCifically. no unambigou' UpIUrn$ in modulus were o\>s NETW ORK S HAVI NG M UL TI M ODAl CH AIN _LE NGTH DISTRIBUTIO NS , 207 Fi~. .. 13.21 Sketch ~I' 'll< 'HI< of "'1111''' u>«l in !I.e ",,_r_le., 1es, " , in the ralio of ",,,1«;01., ""eights "r 11,,, short ~, nd 101111 cl,,'ins_ As is shown in figu", 13.22. lhe j ncrca~ in lea, e".rgy d id om serlancc in explaining the obscr\'oo IInpro,,,mc1\ts. A .ubscquem ser;es of shear ie'S!;" e,mblishcd ,h. depend.""" of Ihe lcaring propcrlie, on Ih. composili"n of the bimodal networks and lhe lenglhs of lhe chains used 10 prepare them. For exa mple. figure 13.23 shows Ihe depend."", of Ihe Icaring <"orgy <>n lh. amounl of the rl -<;h"in <'ompo"oo!. The n,axim" in lhe cun..,. oblaincd locates Ihe composilion giving Ihe greatest i",," lensile Slrconglh dc!">"nds On Il>c r'lIlio of Ihe molocular weighls of Ihe Iwo components. Th~ i""rcase in .Ircnglh wilh d=.'!C in II>" molecular we;~hl of Ihe .horl ch"in, mu.1 eventually become a decrca.., w'hen II>" chains become 100 short. It would ob\'iously be importa nt to carry o UI addilion"l siudics!o establish Ihe mok""ular weighl al which Ihis occurs, /J.1,4.5 Cfdic 1>i".liOll in Il>c bimodal networks. T )']Ii",,1 .... ult. arc . hown in figure 13.25. On< or Ihe i",po'tanl ob;;e,valions waS the dcpcndcne<: of c,)'slal linity On Ihe network chain-kngth distribUlion. Some mca,u'cm<·"I, h" ve bo.,,,n "",de on I""rm""<'nl sci rOI I'DMS 11C(\\'orks in ~o",prcs'i,'" ~ydic' ,Idor"'," i"n.~ As Call he =n fro", f,gure 13.26, Ihere apl""ars '" be I.;s I'c,,,,a,,,,,,, sc( N o":'l' i" Ih~ Ca'" of Ihe l>imooaJ Cb .. •• •• t - ~· iKU .. 13.21 Toan"$..:nc.gy nt:.... , CUrVC$ (o, monomod.l (unimodal) a nd bimodal Pl)MS ""' ,",'Qrh a •• f"'>Clion of tho '' ' ' ",.. " . lIo,h typos or nct"'ork ~>d an "'CIa", .If, of6.800 B mor'. and Ih. ",">ooai """ ."",",IN of 7{) mol% ch.in, ha"i n~ ~( E 930 g mol - ' oombi" , ~ 160 "> ~ 1 20 ,, < w " 80 ,OO~--C,'Oc-'-C'"O--'-o6~O--'-o8"O--'~'~OO WI. % Sho rt Chain, .""'H t'iJ: " f~ Il.U IxpoOOcncc of ' he < """""",,."on or ,bon cb.in, in bimodal I' DM S "",,,,·or " '" 'rile ,'oloc_, of ,II, rn, 'he shon ch.in,.ro gncn in Ill< fig. "re. and ,he for It..: long chain, w:" AI, _ 21 ,3 ~ 10' S",,,t ' Thc m."ma ""'"' at 'pp""imal ,-.1"" NETWOR KS HAiliN G MULTI M OD AL CHAIN_LENGTH DI ST RIBU TIONS 209 ',---,-~~~.---.:~--, M, . 21.300 ® 9.950 6.400 , •• •,• , •• , o !iii 4 .200 0 0 0.0 0 02 0.' r ;K." lJ,14 101.""",," le"oilo SU."~'h of !>om,,,,!>1 I'I) MS "ctW"f~ ' ,h",,"~,. f"lICti"" of the rati<> of ,he m"...,"lo, ""iglll' of lhe ,."" ch';n' 10 ,he 100& on ch.in. can b< "blain«! f " , " , the .b>o:i,,,,, ,h. Th i, is consiSlen t wilh some .. orly 'CIi" Il< for pOJyurdhane da"omc r~, shown in figure 13,27", Spe<~fica lly, cyclic elonga lion measurement< on uni,~ odal a nd bin,odal ""I "-o rks indicalrd th.t lhe bimodal ones sm"i,'ed as mIlCh as an order of magnit ude mote cycle, before the o<xourren"" of fa tigue f"il ure. lJ.2,5 RCl'ulis on N<>nmcc hani",,1 I'w ren ;e, /j ,1.5./ B;refr;/OKcne<' Hirerringe",e mCHsuremcnlS ha'-. been shown to be '-.ry .. nsi livc 10 bimodalily.• nd ha ve therefore .1", !x-en uIICII 10 cha.-aclcIres,)" , There is sc<:n 10 be a lor!IC dccrca><: in Ih i$ rocfiici/ Abw,,"",/ S~he"'.. Thi s type of e'l"'r imcnt i, ""sed on Ihe fact thai ",I"nl mole on ;.olYc nl molecules "onSlr~ ,n" d i" the 1'<""1 of PUM S d""(",,,,,,,, ga'" C"idcncc ••• •• ~ c .l!! 0 .• " •• I, C Fi~... 13.2S Tempe""u", d<""",k""" "f the d)n.mw-""",hani<:alloss l. n~1 fOf "";m· oo.ll'l) MS ",,,work•. m.~.ured.t I I II,,". 1'ho v~I" ... of 10 are 0.120 (e ). 0.660 ($), I.W (0). 1. 10 Itfifu tlCl ion.1 in.l~.d of ,,,,,,,r"nobon.' iullC';""') (V). 4.00 (ill. 1l.l( O )...... 18.S( '.'d. "0 ~ • "c 0.9 -- • " .2 ~ " 0.8 0.7 O!;----,2'0:----;'~OC--'6~0C--_,i80' "IT«. Time, hours 00011"'"';''' .,,,,,,, "" Fill"'" 1.1.16 of ")'Cll< "n, n,odal ( .t.) and bimod,1 ( e ) PI)M S ,,'~'o NE TWORKS HA VING MU LTI MODAL CHA IN· LENGTH DISTR IBU T IO N S 21 1 z .0' .0' of--.·~,. ,,~·~--.' Mo ( ~f<:m') fi~u", 13.27 RciatiOfl5hip bclwc., number of < n\Odulu, "I \0% Joro"" .. i"" for unimodal (fillcJ c\",I<s) "nJ bimodal (unnlJcd oi,d <.) polyur<:lh.D< n«wo"s". - • • U •0 • , • • ., ~ •• ••• '.' •• 0, n I.U 8 Slrain < ." """' Opl"",1 ' being considerably H ~", . longer (2Ux lol smol ')"', for several crystallizatLon temperalures. w),ich <;o"ld l>o indicali,-e o[ an unu,ual distribution of pore sires. T he efT"CI seemed 10 he moS I pronounced for ";modal n.. worh, " 'hich ar<; d,$Cussed lalor i' l lhi, chapler, /3.1.5.3 /I"/l/led /" " .."igUli(J,,", C~I"" me,,;c m~a"J rcmcn" on bim<>d,,1 roly. (ethylene ",ide) nct""",h indicale<;1 Ihal Ihe .hoil chain. ,,-ocmcd 10 deere.", Ihe amount of crySla lltni. y in .he ""'Irclcl,ctl .Iatc· ' Thi.,. an inlnguinll n:.,,11 212 STRUCTURES ANO PROPERTIES OF RUBBERLIKE NETWOR KS sin«. as memioned above. lhey inrrellu the exlem of cryslaUiza lion in Ihe , lrelchcJ sh,le. A ,imil.r siady on a differenl pol)'mcr, poly(lclr.lhydrof"ran). did nol ,how any decrease. ho'.,e,..,r". As ed 10 ohl.i n i"sighl inlo lhe d)'namk. a nd slruclure of bimo(bl elastomers. 13.3 T rimoda l Networh h.,.., As menlioned abow. (here been difT.rc"liai scann ing calorimel ry measure",ent, on soh-enl molecules .'On'trained in Ihe pores or a va riely or PDM S ela._ lomers_ So:Jme resullSOn nelworks having a Irimodal dislribU lion of """wo rk chain lenglh, arc presenl<:<J in figurc 13.29"' _The se,"crai cry""Uizalion Icmlle,.lures ob""rv"d for the benlene in Ini. nC1>''Ork could possibly be used 10 obl. in addilional informalion on the pore size< """"nl. Although Ihere h"'·e bee n allc"'pt. 10 eV"JIUale lhe mcch:on ica l properli", of I.imod., .la'lo"",,,. lhi' ha' nOi been done in any orgaoi,ed mann ... The "",ie problem i. I!u: la rg<: number of ,".riabl .. involved. specifICall y Inr« molocul .. weighls and IW •• .. ~, . , ~' , ... " .... - - ., u Toomper.,_ · C •• t¥'IIo," 1). 2'J T)J>ica1 I)S(; ""an rOr" trin>Odal I'IJM S DClwork eon';'lin~ of 13 ","" ';' chain, ha,'ing. value or M. of 41 7 t mol '. 7S mol % en.,,", h",',"~ 2.7W ¥ ""~ '. ",,,I 12 0101% eh"i", ha'ing 1~ ,9 ~ lolgmol "," ·f h, >... cllin~ ","-enl ""11<",,, elJ",.IIi,,"<>1, ICmrcrdlu,,-'S were ",,,nilo«o "' .. , ben,cne. NETWORKS HAVING M ULTI M OOAl CHAIN -lE NGT H OISTR IB UTIONS 213 invol,-ed arbitra ril y chosen mole..: u".. wcighl~ a nd comfXJSilion •.,.. '.'···. NO! surprisingl}· . only mod .. t irnpro'·cmcm. have been obtainrd over the bimodal materials Some =n! COml,"!ational studi"",". ho""""r. indicate Ihat it is possible to do simulations 10 iden lify those Inoiecuiar weights a nd composition. Ihal Should muimize funner impr(»"cmcnlS in mechanical propertiC$. Such simul3lion, a~ being eXlendC th"t charactcrize phantom nct",·o.", in general. with more =nl methods Mvdopcosilion. a '" kllo,,"n ,00 A randomly gc'",r~led n· fu",,'ion.i network thai has Ihree Iypes of ehaius ""ill ronlain .,,,,,,aJ combina_ lions of Ihe InrC<' l'h'lins al each n.func!i,,,,al jUlleti,,". As 3 firsl approximation, each combinati"" can be ch<»"n ra nd"n, l)" at each Junclion, Once Ihc I)']><S of l'hains emanating fro", a j uneli,," a rc e'labl .. hed, th.;r mc,,,, ro", work "" to obta in Ih< variance of the mean chain dimensions in the cn",ntble. T hi. appr""ch can )"icld a nelwork whose mean chain ,'eelor:< arc known . Generati on of Ihe network should hoc lenninalC 0""" 214 STRUCTURES AND PROPERTIES OF RUBBERLIKE NETWORKS I .'t u... 1J..lO NOlwork chain.10 n"mber or cll.in. in" n i"r."il<,imal inlc .,oa filloo wilh diago".1 'Ia.hes.• bimooal di.lrit>ulion by Ihe dOlled a",.s, .,nJ an ." ... "",Iy bfo.d. ps<:u-unimodal d;,,,ibulion (oblainal by combini"1! "'1"''''l< unfilled oroa. lI"I""inled "ilh permimon fmm Mark. J. E. (1994). Arc. Cium Rn.. 17. Copyright 1991 Ame";"'n Chemical Socioty 13.4 NelWor~s of Very High Modality allr~cli"e m«:h.1nieal prorcrli<:s of bimod~1 nelworks h"s hoen in terms of 8 "delegation of rtSpon,ibilili.-s:· with the $hon chain$ ..... ing in one role and tlte long chains in another. If Ihis is piel.re is true. then il would be im.resting to preJMre and sludy nelworks h;,,;ng c",raordinarily broad mo1t:cular ,,·.ight distributions. in that Ihc", would be netwo,k chains "f all con""i'able knf!hS, available for an)" conceivable m«hani,m that would improve properti", . Pol)"mer pn:P'If«l from slOgle pol)"meri"lion " 'ollid ,,<>t ha"c " hroad cnough di'tribution, bY! the combination of a ...,i" of ,ample, of gnodually 'Iiffering a..."age molecular weighl could provide Ihc desired brood· nes" II should be possible 10 achieve a polydispcrs;ty index.of SO or higher. Thi, is illustrated in figure [),30. An cla,tomer of thi' type has apP'.rently neyrr b•."e" symhc>i1ed. aoo might well ha..., u"""ually allrael;"e mechanical propcrlies, The inlcrprol,uion of tlte 13.5 Elastomers Ihat May Have Been Inadvertently Bimodal EiaSI"nlCf$ cured with ~ulfur frequenlly have imp'''''oo me<:hanical propcnics when lhe curing condilinn, arc cho...-n to give cross· links that con,i't of ,'hoi,,, of ,ulfur atoms'·' . II has bttn ,uSl,,<:,ttd Ih"l if ,uch polysuU;Jic crv«·link, Can th<;m""I.-e, :tel "' da"ol""r;'; nclwork chain •. thel\ a bimoda l nelwork i. produccJ. albeit inad""tenlly'·'. This is ,h",,'n sdwma,;cally ;11 fiSure 13.31. NETWORKS HAVING MUL TI M ODAL CHA IN -LE NGTH DISTRIBUTIONS 215 1><1"""". figW'< 13.31 Sketch 'howing the dilr.",,,,,,, "",,,,,,,ulrodic crou-link (Ionj• • rod a polysulfidi< Ont (rigil!). In lbe I."er '""'. ,he ,·""on. O)/',"Ifur a'oon. ,-"" poss;t>ly 3<, as addition.1 da"ic.lly err..,li ,·•• I>oin' in ,,'ha' i...",n'J>lly. biont>d>.1 no''''o,k (S« Iogend fig"" 1J.1.) '0 v ( Fi~ ... lJ.31 SkclOh illu"ratin~ oow i"""'tplo,.".... of rc;K,;on in ,he Ca5< of a "inyl hydrosiJa'ion <"d.li"k;"8 can 1000 '0 ,dd;,;o.ol ""t,.-ork chain' Th. oii$ofl1<'ic pol)'. (methyl hydrogen .iIOA""") chain. ort 'hown vcn,,,,,lIy. and ,he "'JIIl bet,.-",," ,<>ct on ,hem po;$ibly contribute d.~tic-.ll y 1).1. Calculalion. conduclc'sible bimoda lily ga"e ""ults in good agrttmenl wi lh ~xp"rim."I'o, A similar situation may occur i!l Ihe case of ""Iwork, e1ld.linkr.;l using Ihe add i';on reaClion in"ol "iog ,·jnyl cha'n end. and hyd'og.:n ;tt"m! in an oligoOletic pol}'(Olelhyl hydrogert .i louncj 'G) ,... In Ihe case of inrompletc reaclion •. Ihe segments bet""«n tilt reactod silicon atoms on the "Iigomer may be long enough 10 ~cl a. elastica lly eIT«ti'e chain, in ;, bimodal struclure. A sketch of Ih i. possible siloalion ii shown in figure 13.3) . F,n~lly. a bimodal chain· leng,h di~· lribUlion loa, al"" heen I"oposed 1" «plain 13.6 Other Materiels in which Bimodality May Be Advanta geous T here appear to be ", Irer Ca>C:. wh,,' e a biOt",I" 1 di,tribulion "f ch"in leng,h M ... me olher phy.i<:"! properly ca ll he "d'ama~c'()u,. again pos,ibly Ihrougi> Ihi' idea of a del'galion of re;ponsibililiN. 216 STRUCTURES AND PROPERTIES OF I'IUBBERLIKE NETWORKS For example. in Ihe area of Ihennoseu. Ihere seems 10 be an improvemenl in nloch.nical pro""rlic! when the pol},mer being cu red has a bimodal distribulion of molocolar weighls'OO. In this cas<:. Ihe improvements may be due 10 diITcrent morphologics ...-so iling rrom the fact tha' the long chain. in " bimodal distribu· lion oould ha"e considerably lower solubililies Ihan lhe short chains, Also. il is well known Ihat the 1I0w Ch araClerislics of a polymer during pr""""sing' " can f"'<.Jocotly be adjusted by Ihe "ddition or a small amoun, or polym", of ei ther ''Cry low or vcry higb molecu lar "·cigh, . Anolher example is in Ihe a rea of rubber_ loughenc si,,,, rnO,I'" "'. References (I ) M"r~ . J. E.; Artdrady. A. L. Rublvr CMm. hth""l. 1981. 54. 366. (2) Liorcn'<. M. A.: And,.dy. A, L.: M .. . . I. Il. Cell. 1'01,..". Sci. 1981. 259. 1(156. Zh.ns- Z.·~l .• M "r ~ , J. E. J . Poly",. Sci.• 1'01,''''' Phy•. t:d. 1982.:ro. 4n. (4) M3fk. I . E. In Har,,,,,,,,, and 11"""", £/aSlkil}". 1, E. Mar •• nd J. L.I, Ed •. A... rica" (,·l\<:m;';'1 Sociely: \\, ..11;"8100. 1)(;, 1982: p, 349. (~) M"'~. I . E.: Eisenberg.. A.: G, •• "ley. W. W.; M.n m 16) (1) (8) K""";8. I. L.: Wignol" G. I). Ph,..lroi P,o"",k. of Pol"""". 2'hl «I. A",en.c.n Chern"",i Sociely; Washi"i,,,n. I)c. 1'193. 1'10<)'. J'. J. P,;.t;p1~, "f Pol",,,,, C"'mi."y. Co">ell Uni,.... il)· "'e"" Ithaca. N Y , 1953. Treloar. L. R. G. 1-'" I'h)";,,, of RuMer HI",,;,;,}": ),,1 ,-d. Clarendon I'reso: O.foN,197S. M.ri(. I . 1:.: Erm.n. 11. lIub1...,liA. /:"I,,,,klly. A .lIokta/,,' Pr~', Wiley_ In'.""ic:ncc: N.w VOlt, 19M. lluoche. F. J'/oJ·,ka/ l'rop.rllu< of I'o/Y""'''. Wile)'. lnlo"""'n",,: New York. 1%2. M3f\:. J. L J. Clr (9) (10) Il l) Gott lieb. M.: Maro>ko. C. \\' .: IIonjamin. G. S.: Me)"'1'S. 1(. . 0.: ~t"rill. E. W. .lI"'.-'*'"~~. 19~1 . 14. (0)9, 112) M3fx. I. Il. Adv. PiA!,,'" Sci. 19&1,44. I (I)) Ma,\:, J. E.: Lal. J., Iierican Cl>crnic.1 Society Washing'on. IX. 1982; 110>1. t93. (14) Q"",lel. J. 1'., M ..k, J. It Adv. I'(>/p". Sd. 1%4. bS. 1)5. (IS) Ma'l . J Il. Au . C/wom . lin 1985. 18. 201. 116) M3Ik. J. E. 1'0/)',", J. 1985. 17. 265. (11) Malk. J. R Bdl. l'oI!,,,,. I . 1985. 17. 144. 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M""""",I«"/£. 1'Ill6. 19. 1212. Kilion. H,.(l. Coil, PoI,'m. Sci. ''IllI, 2¥.1, 1151, Kilian. H ..('. P,OCU, I, G.: I. E. J. Chtm, Phy" 1984. SO. 452 1. Sun. C.·C.: Mark. J . E. J, PoI,·m. Sci .. 1'01,-"" I'h!". 1'.<1. 1987.25. W73. H".yu. A.: Stei., R, S. MuJ,:ronwi. CIo>", .. M""",,,"". S),"'p. 1991. 45. 189, Dba ... '1',: Kawabat •. S.: K. w.i. H, J. Pol)''''' Sci .. I'"" ,4 .] 1970. g, 903. Xu. 1'.: Mork. J. E. J . Pol)''''' Sci .. PaJ),,,,. I'/'y" F.il. 19'91 . 29. )55. Xu. !'.: M.. k. J. E. PiII,-",,,, 19\12. ll. 18·H. I\ i,lin, 1\, S.. S."nin' /1"0.'''. J, M. ;r'.<:iskr and F. W. G, F•• ron. r:Iy"",,, //k. j, E. Mark. Ed. "memn Inllituto of Ph),>", Pres>: Wood .... ry. NY . 19%; p, 5lS. Kuhn. W, : i' M",'. .1("""",, '" (94) Jack. &itM" 19'92. 256. 1425:.nd I'<"i""nl ref.re...,.,. citro Il>ehrk . J, E.; Sun. c..c.; Dod!!",n. K. 1:..... I',~y"" J. 19\12.28.823. (98) 1.:1:.""n . S. J .: l>lark.), E.. S<mly NETWORKS HAVING MULTIMODAL CHAIN -LENGTH DIS TR IBUTIONS 219 (101) N .... ' . M.: nh. G. K. f;w,. {,oI,·m . J . 19~8. 24. 733. (102) Sh,.,f. M A.: M >r~. J. Ii. J . .\1",,,,,,,01. $<1.. MIK'''''io/. lI~p . 19\11, 1.67. (lOl) Sh.rnf. M . J. 1'00ymnic .'ofOH!' . 19\12. 18.217. (1001) sh .... r. M , A I, Mac.""",I, S<'i" M"".. M .. Taka ha, hi. K MO A'.,. !..ct,,,". ~39, (Ill) (114) (11.1) (116) Chen . T, K,;Jan. Wen. J.: Mar l . J. Wen. J, ; ~.. ,~. J. Wen. J,; Mar~. J. y , H.J . .'of",,,,.'iri, 1992. 27. Ill. E. 1Iub/x. C,,",m. T,c/""A, 1994. 67. lI06, Ii. Poly",. J 1995.21.491. E. J , Af'PI. PoIJ"fn. Sd. 1995. SS. 113.1. 14 Small-Angle Neutron Scattering Small-angle ,",ulron 'lCalicring (SA NS) experiments from netwQ,k. " ..". inilialOO by Benoit "nd coll" oomwrs' ;n thc mid-1970s. Currentl)', SANS is an ,,,.(>Orlant major lcdnique used in 'looying network !(nlCIUre and behavior. Us impOrtance lie'S in its king a di,ocI method with which ohscrvation, may he made at rhe molecular-length scale .... ithout the need for" theore tical model for imcrprcling the data. This feal ure makes nCUifon ""Illering a valuable 1001 r<>, testing ""nOU! mokeula. theories on which current understanding of cla'tomeric networks is ",,'" T he general fealures of Ihc ltthniquc nre c. sect;,)" 14.2 by a review of rdevan' c_,perimen,"1 ",o,k. Sec';o" 14.3 lh<:n describes clifT.rem thoories of neUlron sc~ne'ing from nClwo. h. and compares Ihem wilh "xperimemal re,uUs. 14.1 Gene ral FealUres of SANS The l",-,hni4U' of neulron sIving n," source are firsl coliirn,,'ed SO Ihal lhey atri,.. atlhe ... mple in Ihe form of plane wav,",. Figon: 14.1 ,huw, soch an incidenl ))culrQ)) wa.-.:: un IWO <ealtering ""nlers j and j. After inlen,eling wilh the "",Iter;ng cenlc ... Ihe nculron. "N'" ;n "arious di",-·t ion •. In " tlCu",," SCJl1ering experimenl, the inle",it y of thc scallered ne, ,," SMALL-ANGLE NEUTR ON SCATTERING Inddent Neutrons .. ,.. 221 I 'I Neutrons Fig.," 14. 1 A neutron "'.'" ioci "0" "Y" e~pe r;rnc"t. an; I"trt i.;ularly u,.,fuJ in " ud) ing ,h. dy"ami~.. of a $y:\'em. suc h a. relaxation or diffu,ion. Scattering is callc~1 eiaJljr if lh. energy change " 110" ",," tlering is and lhe ",agn;lMcs of ko an,1 k are equal: ",ore ,pccifk,dly. 1'41- Ikl ~ 21 '.c'" ( 14.1) For tnc claslic case . rnc m"gnitMc of q is ,~ , ~ 4~. 0 3Sln2 (14.2) as can be obtained fro m the geometry in f'gure 14. I. The two ""altering centers ,how n in thc ~g ure '""y be labded po;nlS along a nct.m rk ch"in or they may belong 10 different cha ins. depending o n the type "f experimen l. There are ..",,,,I ",h'antag<;< of SANS ""cr olM' kind;; "f ;callering s~c h a. light . x· ray. or .Ie<:tron """ltefi ng" '0. Fi ..... the wa"clcllgtbs of the ne~"on. arc .u it"blc for investigating the cha rac'eri,l;. d,mon,ions of polymer mole<:uks. If tl>c seal · tenng angle 0 is less than a drg"" .. vbjeclS the <;>.1: of a polymer mok"<: ule ma)' be studied with read iJ y avai lable neu tro n, ", il h the pro!,<,r w"vcLo"Kth<. T he length scale to Ix inves ti gated i. ~pproxitnaled by the inverse of I'll gi,..,n in eq. (14.2). Thu,. di" ,,,,,,,,. of I ItJO nm n>:,)' t..: [Jf\Jb<:d witb ",:ut"'". having ",avdengtl" between 0.2 '''''' O.~ if _nc ~,,~Ic 0 i, equa l h' or less ,han" degree. The adjecti..., "'mall-angle" refe" 1<> :t nglcs below one degree. " ,"",md ,,!I, 222 STRUCTURES ANO PROPERTIES OF RUBBER l iKE NETWOR KS ad'·.nt"g<: of neulron SCOllering owr ,·ray ""allerins is Ihm Ihe energy of ncul""', i, m""h less Ihan Ihe energy of oI""lron, of the san\<; wawknglh T hc",fore. Uperi""'''IS may he performed wil h much damage 10 the sampl", Ihan wilh .·r~y ""allering. NeUirons inleraet wilh Ihe nudei of alom,. "lid Ihe Slrenglh of ",allering i. dcscrib«i by the lenus """altering leng(h" or "scallering cross I<:<:tion." Some """lei. ,"ch as Ihal of deulerium, "'atler m""h mOre Slrongly Ihan olhers. for .,ample hydrogell. Thus. ",me hydrog<:n atoms of a polymer molecule may be rep laced by deUlerium atoms 10 "chic'.., "contrasl." 1l>c proc.:s, of replacing hydrogen wilh deuterium i. rcfeTred 10 a. delller.,ion or labeling. Chemical dif_ fertnces. helw""n dou'er.,ed and undeuterated polymer mol«"I., are ,..la'i,·ely smaU and Ihe t""hlliq"", may con"eniently be used "'ith only a small nllmber of deulera'ed chain,. There arc two di'tinel form, of """tiering by neu1 I."" ti"" (1 4.3) Here . b, i. Ihe ",aUering en)!;. S«lion of the ilO scaltering CCnler and . # is the , .....10r from Ihe ilb 10 Ihe jth "..,nler. a, ,hown in figu,.., 14.1. T he doubk summa _ tion" earried over all ""a ttcring ecnle" a"d Ihe angular hrocke" denote thallhe inlensity is from "II ~ib le configurations of tbe ""allering cenler •. The ratio S eq) '" / (q)j l (O) (14 .4) of I (q)) 10 tke inlen,il)" 1(0) obtained al '.ero scallering angle i< referred 10'" Iht scatle, ing law and i, Ihe basic quantily reported in experinlental ,,·ork. In Ihe Slud}' of uniuially deformed nctworh. il i$ ron,..,nie nt 10 """h-c th. scattering ,'ector q into a componem1lj . parallel to the direction of .irclch. and anolher. 'iJ - perpendicular 10 i\. One can lhen wrile S eq) in lerm, of Ihe two .xperimenlally measured componenlS S (q. ) and S (qi)' T he dynamic. of enain. can be ;tLidied by qua.i.elas!ic ""Hiering. for example. in fhe spin-ccho foehn'que. which probes Ibe mo,ion, of chain segments of length ""ales of" few nanomelers . T h;, technique is p'",icularl y usef,,1 for lhe study or the dynamics of junction II,,<.-lu"tio n>. "nd lht n""'t"~I;"'" of ,h~in segments in the ''''lwurk. SMALL·ANGLE NEu rlWN SCATTERING 223 14.2 Experimental Studies The first studies of "n"lI·angle neutron seatlCflng Were mad~ on am(>rphous un· cross-1in~ed polyme,.. by Ki rslc"·". Ballard cr al.". and COlt'''' and colla llora. tors"" '. The"" in W.' ligations led to lhe impo'wnt ~(>nd usion lhal lhe dimen.ions or polymer chains in Ihc bul ~ Slale arc idemical wilh lheir unperlurbW dimen. sion s" in thela S1 ah,·a}"". fall b,""'-cen Ihc prediclion . of 100 affine and phanlom modds. In e.>;t><s. although ellipses a,e pr<" and in\"oh'<:<1 m"a~u",mct\'" or quasi..,la.tic inoohe,,,nl SCat· tering. Their result . ,"dieale lhat lhe netwo,\: chain segments d iffuse a,ou nd in lhe nel",·o rk. wilh a le",poeralure dependence (If I~esc "'"lions rorrcspondi ng to "clivalion n>, and ,""n'!","."on of lhe re.uits wilh tho"" from fn,,, end·laheled chains. The ,,<:ulroot spi" ..."ho .echnique i. essentidlly a quasi. daslic neutron seanoring t""hniquc lha! dir.:clly me.1Surcs Ihe time-dcpoendent ",.nering fU nClio", S(q. l) al ' u!;>-m(ll<:<:ula r icng.b ",ales. The results of I;;".. n a nd colJaoordturs .howed lhal lbe '" "I><' of IIlLclUa(inns of junctions ao-e ,,,hSlan' lial. but ",mewh"1 Ie,," Ihan the IIUCluation, of" phantom nelw(lrk model, Their re, ulls al", irnJica lw tha , 'hc mol"", •• ,f Ihe june"o", ,"" dill·u.;,'. ami arc ,imilar 10 Ih"'" e.jl<.'C1~'« from lhe Ro"", ,,,,>'-'::. the mo.i"", of ""y 224 STRUCTUR ES AND PAOPERTIES OF RU BBERLI KE NETWORKS the deuterat«l junctions wc~ fou nd to be much slower than Ihose of dcUlerated free chain ends. as cxpcc!c'd . 14.3 Theo ry of SAN S from Netwo rks Th«> rirs of SA NS from rolymer ""tworks n,cal W{),k in th i. llcld w~s by Elenoit and collaborators ' for a ""twork chain whO>(; SCl!J'lCnlS Iransform affinely. !)caHering from a chain in a phanton , network ",as wter fonnulatoo in a "cry concise and clegam way by "'~rson·'. and this approach is p<=ntoo in the followitljl Stttion •. The derivation is limiloo to unimoda l phantom nelworks w'ilh tetrafunctional junction,. in the inten.. t of cmpha,iling Ihe basic fc~ turcs of sca ttering from nC'lwork •. Dcril"dtio ns for other junctio n functionalitie., bimodal ""tworks. and polydisl'Ct"Se networks haw Ilttn given by Kioc:>:kowski 01 at ... .. and B iggs and 8311". 14.3.1 The S.d ttcring law The bask 'l\l.ntily of intertSI is Ihe sca lle,ing law S(q). defined in "'I. (14.4) as thC ratio of 5Callcring inten,it y at angle 6 to that 3t () ~ O. For • networ k chain conlaining n scaUering cenlers, it is S eq) " / (q)ll~ "' ( ~ + l ) ' L l cxP (iq . • , ) w (roj ll/ru ( I~ .S) " whcre i .., R and W(ril) is Ihe dislrib ution function for the """tor ril co nne<:lin g the points i and j. T his fonn of thc scattering law fnllo".. from "'I . (1 4.3) by assuming all sca[[(ri ng e'n.s-Sttlions In be idenlic.l. Ihal is. b, ., bj " b. The intemity of radialion at t he scallerins angle 0 '" 0 follow, from «t . (1 4 .3) as 1(0) "' h'(n + I)' ( 1~.61 Dividing Nt. (14.3) by eq. ( 14.6) . and aver.. g;ng over all oonfigurations of the pet ir of scanere.. ; and j indicatc'd by the integr.lion after multi plying with Ihe di,l,ibuti o " W ( r~ l give! Nt. (14 .5). The term W (r.) "'prcS(n\S lhe ;n ttamoleeu l". pa ir ,"(ItrtIMion fu""tion if Ihe $<;;auering o;" ters d,·,ignalL'd by i and j belong to Ihe 50,"", chai n. If labete<J chain' arc w<:ll $<;;pa ra,ed from each o lhe'. then thcre are no conlribuli"ns fro n, inlcrchain interactions and lhe intensity gi""" by Nt. ( 14.5) represents scaueri ng from a single chain . Th~ scattering law &ivcn by eq. (14.5) wa , first detiw SMALl ·ANGLE NEUTRON SCATTERING 225 14.3.2 Scan.ring f",no a Phan lom Networ k Chain As deseril>ed earher. a suffic;ently long ~h"in may be rcpres<:ntoo as a freely jointed chain. Son ... of the IQedI conlig~ralio ,,,,' IcalUre. of Ihe chain jsuch as Ih. ori~nla'i"n of a ' I"-..,if", Vttlor .ffi.ed 10. bond). ho,,"'ever. are lost in 1M<. arc based on the freely ioimed lo,,~""hain modd ,",·hoSt end_lO_cnd di5\ribution is salisfaclorily reprc5cnted as Gau"ian. Furtncr informalion on 'he problem of scancring from phantom ",,,work chain. may be oblained f"'nl the original " 'orks by rca""n" and Ullman". 111c ve<;\or ' y conne<;ting a bilelcd p<.>int i On • freely jointoo ,.etwork chain (0 anOlher lahek<J poinl j on Ihe 'kImc chain oheys Ih" Ga u"ian distribution fU'1Clioll as thai given for Ih~ .nd·lo.;:n,1 ''eCtor. Thus. "",l>C IV(,~) .. (2n(4).) 'I' n p ( - 2:J,)J ( 14.7) wh .... (10)0 is the mca"·.lqu",,, distantt helwc"~ points i andji~ (he undcf"rOlc-d state. idenlified by the subscript ,..::ro. The distribUlion funeli"" Wjr.) in the deformed stalo i. ( 14_8) oi). where (4). and (z~) are lhe mcan«ju"", component' of Ihe ,·eclor , . in Ihe deformed I,:.to. SUbslituting Iho e~prossi(On for 11'(' .) siven h}' ( 14.M) inlO cq. (14.5) and performing the inlegr:.t,on ow. Ii •• = Jx. d.rij,!:.lead, tl) .'1. I", 12 ') II S(q) '"' -, L. OXp( - q, ("~)f2 - 'hv .)f2 - 'I, (:.,)/2) (14.9) ~ 'J , where n + I i, ... pb<>..-d by ~ fo r 5ullieicmly long chains and 'I" qy. and 'I, arc the compOnents of the Squaring b"'n ,id., of ' he abo,·c ~'IU "I;o" and 1:0 king Ihc ensemble a"""'ge Icads '0 (1 4.11 ) .inee Ihc cru:;.s_prodnd (. '1 - Ll.r.) '·~ n;shc •. in".",uch as the mean vol"", and in .. antan.ous Ouel"" li"n, a ... un""rreb,ed_ This rciatiun folio,". f...",.-ding 10 226 STRU CT URES AND PRO PERTIES OF RUBBERliKE NETWOR KS the mood. mean chain ,-eeton; lransfonn affindy with macroscopic dcfomlation and their AuclUalions are independent of defonnati,m, The!\e fluctuations are given b~ (xf,) = ~!(x})~ ({~xfi) - « a..r~)')o ( 14.12) with similar expressions for thc J' and, componcn1s. Here. ~, is the x component of ,ht principal defonn,uion gmdicnt tcnSOr 1: ( t4. 13) and rel'res<:nts the ralio of the frnallength ofthc network in the x dimetion 10 lhe corresponding length in the statc of refacnco . The x component of eq. (14. II ) may be wriHcn as , ~) ~- [,'x + " + "• ) «a..-..-}' )o] ,J.) {-'t}o .•• ( IJ. 14) ,,'itn ,imila r eXI'TC>sions for the J' and 0 componenlS. For a fred~ jointed chain, Ihe ",'crage quantil~ (4)0 for xij is related to the awr6g<" (~}o of the "'hok chain " wh.re ( 14. IS) f) is lhe fractional d istance: ( 14_1 6) " ~ I I-jl!n The ...,Ia'ionship het",""n « Do';' )') and (~). is derived in appendi x ~: a~ {(~r~) ') = (t, - ! ,fj (~). (14.17) This ...,Imionship is ,'.Iid for HUe!Ualion, "r the di'lance 'q hcl,,·ttn Iwo points i and j belonging to th. same chain. Substituting this relation into eq. (14.1 4) and using the resulting ex pn-s,ion in ell. (14.9) leads to th. 5'Cal1cring law: , L. S{~) =;;I where e'r i-l-"I[ I (! - ~" )'1/211 ( 1418) '" ( 14.19) ( 1420) 1'0' ..::altering pam llcl "., the direction of e.tcn,ion.~' .. ~I ' while for 5'Caltcring perpendicular to this dir"'tioll. ~ . ~ ~ l '" I/,;x:.. Equation (14_1S). flr! t obtained b)' l'car.ron·'. mOlY he SMALL· ANGLE NEUTRON SCATTERING . ., L:: /W -ill'" 2L::{n - jl/U) " and con~erling 10 an 227 ( 1421 ) I'" inlev~l. T he ",.,,11 " ( 14.22) Afler integralion. the $Callcring:; parallel ami rcrpendicular to the direction of stretch 3re obtained, resrcctiwly. as (1 4.23) ( 14.24) where <> _"' _1 (1 4.2S) /l .. 1 -.l.1 ( 14.26) ( 14. 27) F,{x) .. ~- ' J", .e dl (14.28) and erfc denOICli Ihe colnple"'''hlary eTror function. fiy ma king use of I"" series and asymptotic expansions of P, and f j . Ihat i•. =' ( 14.29) , ,, ( 14.30) lin' f 'o(x) '" -'2x .• 4-, .- , ~ .() I"H~ , X • '. . "" _+ 2x 4..... 228 STRUCTUR ES AND PROPERTIES OF RUBBERLI KE NETWOR KS it can be ,hOlm that S t. ) may be simplified t<} the f<}ll<}"ing limiting fonn,: I Smoll- "bleh oo' .... pond. to tho "'.1Iering fron' an unp<"urbed G........ " «HI" 2. 1'<>"J.",.""lt"8th li""t Iv - . O~ c'q. (14.23) V.'os" _. !,mS(ql )= 1 __,vl~'/4 f )/~] (14.)2.) and ",. (14.24) V"" ( 14.32b) wher< ~ i. the <>ten",," mti<>. 3. Shpoctivdy·'. ~( 1 _ ~' /4) ( 14.)3. ) S(qJ) ~ ~( I - 1 /4~ ) (l UJb) " - lim " 14 .3.3 S<;attcring from an Affine ,,"elwork Chain An .ffine network chain ha s Ihe prol"'fly thai i" end points t"tn,fom' .ffincly wilh macr'-'S<'opic deformalion and the points along the chain arc phantomlike, while no reslriction, are imposed on them, As was stOied earlier, the amne net· ",'ork chain i, obtai ned from a phanl Om ""twork chain in the limit of infinitely large j unClio" functionality. To obtain the ...'allering from the am"" network chain from the phantom network chain result. Or.c therofore has to e' pn",-' S(q] for Ihe phantom network ch~in in term. of r.ct,.,ork (unction,,!ity. The .. cxprossiuns haw bttn obtained by K!OC1.kow, . i et aL""-40, for the ge, .. ral functional net,,'or., T hey are: 1, l.on8-..·.""l lim,' (~ _ ~) : li..,S(q.) .-0 ,,"'he .. k indie.1"" ei,ber tbe ~ 1 _~[" (1 _.!.) +.!.+.!.] J2 ~ 2 ~ p;l .. lId (1 4.)4) or th. p<'p " ... ,oh di...:IiOOl. [,., ,Ite limit of i,.,C,nitoly i>'F ? «I. (1 4.)4) r,"(l""", to ( 14,350 ) hmS(¥ d - 1 •• ," 1) '+ 1) (l4.3'h) SMAl l-ANGLE NEUTRON SCATTERI NG 229 2 Sm.lI-q limit Iv _ (0)' 4-"'] l lim S(q, ) _! {I _! [~ !!!..21+ .......,YY9. (1 ~. J4 ) "'hkh lead, in to. limit of inhtutely LuS" ", to ~~S(ql) ~~ {1 <(2~'- 1 ) } - limS1qd _!{' (14.J7~) (I07b) " Equal;ons (14 ,31 )- ( 14.33). (14 .) 5). aud (14,)7) pmvide c.<pre>.sions {("ialing the in chain dlln~n,;on. 10 Ihc macroscol>ic , laic or deformalion, T h. 1""•. meier S(q) ;~ prorOftjonal 1<> Ih< s.:auered ;nl eh~ ngc~ 14.3 .4 Scallcring fmm a ehai" whoSt Individua l ScgntCIII. Deform Amndy 111is mode of def" ..",lion of a chain applies 10 a chain in a highly re'lricli"e environment. "",haps when the ,ystem is \'Cry dose to lhe gla •• y slate. T he M:attering law ror soch a chain was obla;n<-d by Iknoit and <,<:>lIaOOralors '. In t (l4.38a) ( 14.38b) 14 ,).5 Sca llering I'rom a 1."I>< I.d Path in the Networ. T his mooc of seaner;ng i, oblained when a long. lal>I)' of SANS f.-om la\>ckd pa,h. in the polymer n<'work was proposed in 1978 by Wa'",,' lInd FA,,-..rd." . "'ho used Ih. replic;, formali.m inlrodu<.W earlier by Dea m and Edwards". The)' thus Jcrivcd an c~pression for S(q) for" randomly c",..·linkeoJ. lelrafu,,,,,i"'tal phanlom nelwork. Ulhnan " 'as the lirst 10 ""dy the S,\NS fro", laocbl cro",·hnh...ll'.ths (wilit ."""ific-d numbe" of e.-o,,· linh along tl", (Xtth) in " "n"",,,laL (;3u".ian . phalllOm. Qfu,,,,,ion.1 ne',,"'o,~ "'ilh treelike It,po]ogy " He dcri"cd the genen, I c., p,,,,,;oo for 230 STRUCTURES AN D PROPERTIES OF RUeeERLI KE NETWORKS the «allering form factor from a p; '" "J' J' (i i' '1 S(. )_ "l~~ ~ JiJ ~ 'I(".p - II ),' I~J+ O - ~I+(I 1 1 (I - >"1(2{9?(¢> _ 2)) + 41- 2 i«l -()+9(1 1> - (.,0 _iJ) _ 1}1o, · ·1 <~ 0 - 2 (OJ!. - 1)"",, - 0,1 q-1ni-",I)]) ( 1)10, 0;1 ( 14.39) T he exprc""ion for S(. ) (ro m er",,·linh>d IMths !!i"en by «I. (1 4. 39) differs ,ub· stantiall y from «I. (14.22) for ""ttering from end_lin ked chain•. However. in the limiting ealoe when n ~ I. eq. (1 4.39) n;d"';C110 c>q . ( 14.22), T he main rea son fo r this difference is that fot erc,,,_linl:ed path •. the double intcsr:lIion cannot be simplilied to a si ngle integral and can only be evaluated numerically.... ' . T his is i>ecau><: the variables 0 and ( in «I, (14.39) are not separable and cannot be as for end -linked ch.in •. AI.., . th. dQuhle ",placed by a sillglc "ariable 'I _ 10 summation o'"r diff"rent chains belonging to the path cannot be simpliftcd, and a 10 be carried out for . " ch member of a double sum. The double integration expre";on ginn by «I. (1 4. 39) is for scaltering from paths wit hout d"ng1in~ eha;ns at the end •. T he c.tcn~ion to paths with dangl ;ng eh.;ns would h.,'. to follo"' lho method introdnced by Ullman". h". 14,4 Typical Results from Experiments and Comparison w ith Theory T he transforma tion o f tbe radius of gyra tion of chains in uniaxiall y strelehed or swollen nNworh Can Ix "aleul'led using the lhroM ic. ! ex pressions &i"cn in the p~ing $IlOIion. The 1i .-$t e.' pcrimc of the 1"On_ st .. ined -junction theory"". Figurc 14,3 pre"'''t. the tr~nformati"n of the radius of gyrat ion of chai ns ill an i$Otropica lly swollen polyisoprenc network. The circks !iM Al l-ANGlE NEUTRON SCATTERING 231 r---r--: , .r---'-~--:-l ~ ,, " -a < , ••• &--- --- - ----- - _._- 4 ,, a < .'c----L --~c_--L---7c-" 1.0 1.5 ex 2.0 ~'i~u .. 14.1 R.,ul" uf ....... " .. '''""'' of ch .. tn J,men,,,,", by SANS on 1"'1)·i"'-'l"".... ntlwork."'. The ci,de. and lriongb "'I'rescn, .>perimen,.l data r.,.. ditT.... ", molecular ..·..sh".II, be' ..·.. n """,,·Iink>. The- d.,. in 'Oe: upper holf oro for fC$Cn' ""lculal ,.,ull. "'I'",.. nl da,,, from Ihe neUlron """ncring experiments of lfastidc et al." _and I~ ",,1id curv" "'as obl.ined'" from Ih. con!1r1lincx!-junction Iheory. The resul,s of ",,,,lfOn ~a'tc,i ng •• pcrimcnl' "rc ,,,,,ally repfCSCnled in ,he form of K ral ky 1'101", ~s ';S(q) 'c<su, q. Figure 14.4 ,ho"" ,u<;h piNS r", uniaxially deforn,.d a nd ,,,,ollen nCI",ork'''. The points represcnt exp ,. The .lUdy of II", dynamic; poin" alonl: a ell"in is an"lher line of """,,,rch Ihal has been ,\c,cI"l'ing in n.-o:"1 )""fS. It "as inili'"Cd by Ihe .he""'lie,,1 work of ur ••• ., .. ;--~---:;--~-----: 1.0 10 V I Vdry f"i ~~ ,. 14.J Tran" •• nd lhe solid ou,,'< ~'.. oboainoo'" r'om the ro"W.in«i-jUOClion model. )., '" 4.6 II -~ , o r ,.,.... ,.- ..,.--" -.-_ ................ - ... ......•.....- ...•.........-........- ....,._-_ ... _. •.<7_.' " ••. , --.- •••.•• I'iKU .... 14.4 1\ "'Ik~ [>l"u Ii S(q) ... 'II for uniuiaUy d ",,,b", n", Woo ""''''' "'P""<'" eAp,,,;,,,,,"'.' ""u,,, [," di"",,,,,,,. I"'~""'" .nd per_ """dieul .. 10 lit<: <"en';"" di« ;"'''''pic .. obtained from mr>k:>. Tho ..10, d • • hW and dom SMAll·ANGI.E NEUTRON SCAITERING 233 Warne?', alld li>c cx~rimenlal , ,,,di., of B iUin. and colla oor.'o,,/'o,., ".ing incoh.",nt neulrQlI scaltering Icchniqu." Neulron spin-echo measurement' of Oescr el al."'''''' c\early 'how thai network junction, an' nol frozen bUI Hoctua!. in time, a nd ,h •• patial range of these fluCluations is of the " rdcr of nct work chain dimension., as predicted by thc Janlc~Gull' Ihoory. Refereoces (t) lleno;" II .; [)upl.".ix. R. : CIbo', II .; <;0"00. J. I' .: !'arnon,x. II.; Jannin k. G. M""","""""ks 1975. 8. 4SL (2) Ullman. R, A"", Re •. Ma"', Sd, I'llSO. 10.261. 0) HiWn •• J. S. J. AI'pl. C,p', 1919. 10. 261. (4) M"""" na!6. 12, 8. (8) Wi!!JI.II. G, I). In I'hyli,(# I"d",,"i<'.< 4 1'"'y",.", , J, !C, Marl, A. Ei..,n bc'll. W. W. Gr"«!lor;c"n Cb &, m. A",.,;c.n I""i,o'o or Phy"'" P", ..: Woodbu'Y. NY. 19%; p (10) lI;ggin'. J, s., 11<""", H. Nr~".'" Sc"'",i"X I'~'" 1'01)'1""" Oaroooon l'r<>" o.rurd. I~ . M"",,,,,,,,I. Ki" .. , R. G.: Kru,.,. W . A .; Scl>cllon, J ( ·IIt"", 1913. 162. m, Kin ... 11. .(; .: Kru,"", W . A ; lhel, K , 1',,1)" "'" 19 7~. 16. 1::(1 llalt"d. I). {L: Schd"" , J. : Wignall. GIl. " ",. tn/y"" J . 1973 .9, %$, CO(l"n. J P.: [).xk«, D .. Iknoi'- H .. Famo",. 11.; Hiuin •• J. S.: Jann;" k. G .: Obcr. R,: I'l001. C,: dc, Ckli", • ..,. J. ,11..r",••"Iiff.I.. 19 74. 7. 863. (I~) Daoud . M ; Colton. J. P.: l',mou,. B. o )an","!:, (i: S '''~ a. G.; lie"";'. II. : Our .... i• . R.; !'icot. C .; D"C.. n"",- I', ,;, ,\IQ"''''wlec~''' 1975. 8. ~ (16) FIo,y. I', J, I',ind{'l" "11'"1,..,,,,, (""''''''''','. Cornel l Umversit y Pre .. , Ithoc •. NY . (II) ( 12) (13) (14) 1~5J . (11) llell,"n&. ~I : l'icol, C, : Reml"', " .. Ilea, J. E. MIJC'''''''''N:tI/n 1982. IS, 1594. (18) Bonoil. fI, : [).xl ... I) .: n " pl,,,i •. II : C .: Re,np!" 1'.. COlton. J. I'.• B,: hOlllnk . G .: Obc., K J. I'"J,..". Sc;.. I'Qlp". 1'Jo,-.. 1.;1. 1976. 14.2119, (19) Il indky, J .\ .. Iho, C . C : Moo<" 11, : y ", H. ,I/IX",,,,,,I,,,,/,, 1978. I I, 836. (W) 1I,,,id<. J.; 1"'-'01. C: Ca,>d.au. S. J . .II'h"""W', Sci. Phy", 19111 . IIIQ. 13. (21) 1I."i"o. J.: Ilupb"" 11. .: I'ico'- C M'''''Hmm.v:uicJ 1'l84. 17. 83. (ll) ''''''ido. J.: !lu•. J. E.: 110,,<. ~'. J . f'h ,·.' iq ... 1985, 01(0. 1%7 (23) IlaSiidc. J. I" f'h,-,ic< Fa,,,,,",. l!e~lolhe~. 19K~ r"",. .11"",,. J.: 1"'''<,]' , P~y.'·wa IgS6 ::7. 11 :<4. J. ; 110,... I'. I'h,.,ka 19~6. I0>4A. 2S 1 J .. I . ~'. I. In 1'" /,,,.·, .1/",,,,,,;,, \lorla~, Il;;rlin l~ g8, p, III 1<81 IINKIe . J . I 1 1. 124) lIa>lid<, (H) u.."i Ik~.'" ~L"'C"" ~rnn~or 234 STRUCTURES ANO PROPERTIES OF RUBBEBLIKE NETWORKS (29) &mi Pro>,. H.,.. 19 8 ~. (>4) (35) (.l6) (37) (38) (39) I. 637 !ioU<. I'.; ilastidc. J.: IIwic •. M,: C<>Iletle. Co: Lopp. A.: IleTl, j, P'Qg, C~II. Poly"" Sci. 1987.75. 152. Oeser. 11,, ; "'>"iok. I,; F.r.I!'>, 11,; Picol. C; Riung. M ,; 1/.r.<. J .• PM'''', C. Mac,omol«.l" 1983. 1 6.~. IlellZlrng, M ,: C: /l or,. J. M",""""01«./,, 1984. 17,663. OO"gIl.. S. R ; M.oollnachic. A .: Allen. G. M"",omol«ul" 1980. Il. 774. Yu, Ii .: Ki",..,. T.: Kim. C Y.: Am; •• I' . J.: Chaos. '1'.: Londry. MR.; Woo",n. I. A,; 11o•• C. C.; l<>dgo. T, J ,; Gl inka. C, J. I. AJw,,,us ;" RuhM, H",rI(11),. J. 1..<01 ~n d I. E. M.,~ . &h. 1' "","m Pross: New y o.k. 19U, p. 401. p""" /:"'"",»,,,,,,, ""J (40) Ennall, R MiN;romokc.it, 1987. 20.1917, (41) Alk •. G .. Hli.e,. P. N.; Goodyear. G,; Higgin',), S. raraday Sym". Cirmr. Soc. 1\172.6. 169. (42) Alkn. G,; HiW"")' S.. Wrish1. C. J. ~,,,,"'ay S)'mp. C"'m. Soc, 1973. 7. J48. (4) E~n. S.: Ric hIO•. I). fit.< •. R .; Ev.'cn. D.: Rich"'r. D., Faroso, II. Phy • . Re>. u", 1985.IJ(). I().II. (4S) Pca ll<>ll. D. S. Mac,omoll!11. 1(, C J . P~y. , 1'/88. 49.1785 (SO) Zimm. II. II . J, CM"', P~y" 1945. 16. 1093, 151) Ullm.n. R, J. elK'",. Ph,... 1979. 71.4.l6. (51) [lob),<. P. J. Phy•. Coil, eM'", 1947. ~1. I~. (53) W .."",..\1 .: Ed..-lIn! •. S. F, J. Ph)'" A ..' ,l/al~, Gnl. 1978. II. 1649, (54) l>com. R. T., F.d,,· ..J •. S. F. Phil. Trwu. Ray. Soc, l..ondon 1976.180. 317, (SS) lJllm.n, R . MacrOJmJJN'.I"., 1982. IS. S82 (56) Oeser. R.; Picol. C,; Hor.<. ) . In PQ/!,'''''' M o,ion lit /J".", Sp""'" Springer V",I.o~: Ilorlin. 1983, (57) Vilgi •. T. A.; Oou<, f, J. P~)'>." );fL. Pol},,,,, Ph)' •. FAi. 1938 . 26. 2291. ISS) fS,j"'ard •. S. F.; Vilgi •. T. ". R,I'. Pro, . Phy• . 1988.51. 243. (59) W.rn<,. ),1 . J. Ph,•. e, 1981. 14. 498S (60) Iliisin,,-). S. In n-.",~ ,\faiN"" S,,,"u ami T«Nr",,,ty-N,.,,,,,, .kim,,;",. Acodomic 1',....- Ne'" Vorl . 1979: 1'. 381 (61) HiW." J, S,; Nichol.., •• L. K. ; Ha)'tor. J It. /'01),,,,,, 1981. 22. Hi3 f''''''' "" 15 Bioe lastomers There are a vari.;, y of biopulymcric n,alerials ""hieh exhibit ",bberli ~c ciashcity' ". This i, perhaps 10 be CXPO:;I<;d when one recalls that mOSI biopoly_ men are randomly ooiled cha ins Wilh con,id.,ablc Ilcxibilily.• ,111 Iha, they arc frcqucmly covalent ly cross-lin ked or have sufficien t ""mbers of .ggrega1c'tl unils 10 e.. isl in network ' ('uctum . OI1e wry lar~ group of plant malerials. lh. polysaccharides. a re in this category. and lhey do "' in their fu nctioning. In ma ny of Ihes<; ~a$CS> ho"'~"".r . Ih. cro~.linking is thue primarily for a OOC Wilh ,,'aler or aquoous solution •. ,uch po lys.accharides fonn gels which do . xhibit the high oJ.formability and =ow. ability 1ha1 are the hallmark, of rubberlike da$!icily_ NOl surprisingly, however, ",htliwly rew mtthankal prop"rly n",asu"", memo hav<: been carried oul 10 ch'!racleri,., Ihe .truelu"," of Ihesc gels. The bioelaslom...-r1 ()o:Xurring in animals. including ,'Crlehr.IOS and m~mmals. howe'-er, " '" Ihere ,I,~n s 'poeifically dcsisned 10 function as mociaslomc". 15.1 Some General Observations It is useful 10 summa rize some grnel"dl information on bioela'lome", lh.t i, pre.. nted ct"wherc ". E,'Cn with Ihe lempOrary restriel ion to biocla"omef$ which are pro tei ns. lhere is an almost .taggering variety of iOlereSli"t material •. For example, Ihe", is clastin in "eneb"",~s (including mammals)'- _,0.'., ""'ilin in in=ls ", "bd ucl in in mollusk."", arlc,;al clastotner in octopuses" ". circulalory a nd loromotionol prolein, in ccphalopod;;~l.l), and Y;$<.~d .i lk ill spider ...-eb<;'" ~. Si""" they arc ma",mals, polymer .denti,,. ""d cngi n,..,rs who arc inlcrtslc-d in biod"'lomers hHC focuSc~1 hea"ily on da\un' Any m.lcnal, or Ihi, Iype, howe,,,r. arc wonh '" 236 STRUCTU RES A ND PROPERTIES OF RUBBERLI KE NETWOR KS sl~d~ing i~ lheir ow~ righl. 10 learn more aooul rubberlike elaslicity and biological fU!>Clion . Such studies should also provide guida""" on how Nature might be lnimicked by ,y"'''''lie ch"mi .. ~". to prOOu,."c beller nOn biological clailome",' $. Elastin i. a ch<mical eop<)lymer of o _amino acids which arC plactd along lhe chain b.,ckoone wi,h suffkiom irregularily 10 pre,,,nt crystallizalion entirely. no mailer how low lhe lempcrdlure or how I'igb the elongation . The chains art kepi a$ ft,,. The on,-'5 which predominate (wilh spc<;ificalion of Iheir .- R .ide chains) arc glycine ( - H). alanine ( - CH,). se.ine (-CH,oH ). and "aline [- CH(CH lhl. Sct:ond . Ihe sid" chains eho«n art predominantly non_ pola r. 'lbis w., Ihoughllo rulve Ihe ad"anlage of minimizing inlerchain imerac_ lion, Ihal could inlerfere will. Ihe desired chain mohilily a nd (au<;e sluggishn<:ratur(s) who,"" eb,lin mighl olherwise become glassy al low temperature,'. El .., in chai~s .ynthcsi>.cd in "ivo are enzyma tie.lly cross_lin k'li in a higbly specifIC manner. T he cr<>s,-hn king occurs at lysine rtpeal unils al carefully placed loc",ion. alonillh. chain backbone. The)- are fi." o,idi7xd. and 'hen fu .. d intI) an a romat ic-helerocyclic cro.. -li,,~ of high OIabilily'·" . Thi. mean, Ihallhe cro..links a rc fiXM in number ""d in 1""',lion along Ihe protein backbone. They mighl also be carefully posilioned spalially, by viMac of Ihe Iysinc repeat units being preceded and ,uCl.-eeded by .Ianine unilO. po .. ibly in a -hdical conformalio". As already ",tnlionM. elastin is in Iho highly swollen lIale "'hon il is function_ ing as a bioelaslomer. Si""" il is bathM in e",,:ss body fluids. it is at ,,,,,lIing equilibrium and lhe system is .. miopen. in Ihe Ihermodynamic """"", in that it "'Ill exchange diluent (but nOI polymer) with the $urr(>undings_ In such an Op"n system. "relching an clastomer cau'" illo swell more. and compres,ing il causes il to swell less'" It is Ihcrefore ob"ious]y imporlanl to ta ke . nch chang'" inlo acc<>unl whe" elaslicily mea,uremenl, are conduCled under open conditions. Failure 10 do so)!) has cau.. d considerable """fusion in some or the lilorature on clu'lin" . Altcrnalively. Ihe problem, of a c1oSl.'tl ,ysltm Can be a>oided. eilher by remo>ing the .,,-.;u solven' and then inllMning Ihe swollen ""'work in oil (10 a"oid evaporation of Ihe sol,,,nl if il is ""lalile)' . or by using a no"""lali!e solvent as Ihe swelling "8<'nt and Ihen ~ imply carrying out the mcasuremcnlS in Ihe open :llmo,pbcrc". Thc", c,pc,imen" arc de""ibc BIOELASTO MERS 237 fibers whi~h subsequently provide reinforcem""t for ,I duri"g ,IS funclioning in the body, When th.,.., collagen fil>;rs are C'''"Clc-d a. pan of Ihe typical elast;n purifoca tinn scheme. the etastin ,etains this fiberlih shape. Th.,.., "fibers" arc generally nOl in rego""r with onr an."her an,l thu. " xpericncr diffenonl strains ~. the macroscopic deformation p"M:<:ed, . The probkm can be avoided by work· ing " 'ilh stpar.... tc dastin fibers. but this invoh e. rather painstak;" , manipulati<ms uoMr a ,nicroscopell. T he question of the st<)fage of daslic .norgy i. of general importa,,« in the area of rubberlike elasticity. and assumes particular importance in the casc of bioela.wmers. The i","e i, perhap.l:>c>t illu,I1'aled hy •• imple example. Energy i. expended in defo,ming an elJSwmcr. and the elastomer store. some of Ihi' energy in the same way Ihat a comp"",,,,,1gas or "Mchrd spring Slores energy_ 60lh arc capable of doing work on the surroundings. Ho"eve,. only a small fraction of the energy •• pended in deforming an elastomer witl generally be rtt<>'"r.ble during its retraelion . This is because some of tile energy of defor",atio" is degraded into he" t by viscous off"", •. No, only i, Ihis ",a,,"ful, butlhe heal generated has to be n:tnovrd, "helhe, Ihe elastomer;, pan of ,In organism or part of an "ulomobile tin:_ M uCh work h., t:....,n carried out on bioelastu,ucrs to !Itt whether Nature has been more 'lICCessfullhan synthetic pol)'mer chemists in preparing elastomers in ""hid such "'hcat buildup" is minimilcd , ' 15.2 Chemical Aspects o f Pro tei" Bio e la stom ers t5.2. 1 0,,,,,,11 Amino Acid Compositio" Although "II oflhe ma terials to be ,listu>st<,I,n this .....,tion"'c protein •. they vary in their diSiribution in the animal kingdom. including ooth ''''rlcbratcs and inver. teb.-,Ile •. They can Iherefore be e'rected 10 vary considerably '" their ami110 acid ('Ompositi011 . The most important examp le. l"I1'licularly 10 people. is oIacc<11 ,. wl> Z38 STRUC TURES AND PROPERTIES O~ RUBBERLI KE NETWOR KS sq uid. and cunlcfish, clastomcric proteins OOC ur in lhe arleries" ". u·l\cn: Ihey ,,,,,,.., a d)'namic_n>«:IuoDical purpose oim;!", 10 Iha l scrv«J hy da"in in Ihe arteries of vertebra It., Snmc final examples an: somewhal diffcn:"1 from lhe olhers in Ihat Ih. da .. lorner fu~lions oUlSidc of Ihe body of Ihe animal thai produced il. They art several of the silks prodClCcd by spiders in m.king thc webs they use 10 caplur. prey'" .",,, ". The two t)"pes of greatest intereSI are Ih. vi ""jd ,il k in the calching spiral. and Ihe frame silk and dr~gline fibers. parlicubrly when swo llen wilh ","dler In di..:ussing Ihc chemical nalu"" or primary s"uctun: of SO"'Cof Ih"'" prolein bioc)"slorners, ;1 is im portanl 10 lirSI ""rulini,,, Ihe 0..",.11 .mino acid composi. ti on 10 look for genera l l",,,d •. and this is done in Ihc ten",inder of Ihis seclion. One can Ihcn uamine thc sequeneing or th"'" amino acids wilh;n the protei n chain. as is done in Ihe rollowing seclion . T he repercussio"s or lhese fealUre, on higher <>rder $truclurdl a'pc<:ls. induding chain flc, ibility. are discus$ed "hen..-er appropriate. Some or lhe mOSI rclevanl compo~ilional informalion fo r rcpr=nlali,'c prolein rubbers ate gi'"en in lable 15.1. As be .... n from Ihese results. gl}'ciIW; is parlicularly p",v.lenl in dastin. ""ili n. and abduclin. Si""" gi}'cine has Ihe .mal_ k51 possible side gro up. Ill<: - 11 alom, il ..-as orig.ina lly tho"ghl 10 be pr=m lltea",", or Ihe flexibililY il ,,' of ",Ie"an"" is Ihe factlhal ala nine hos the - C . I! side group. ser;ne Ihe - C lllO]·1 group. and valine lhe CI I(CII ,), grouP .• 11 or " 'hich .rc small (a l leaS! when com pared with Ihc ~ltcm"li,'cs) . Vel. in the prOleins sU""e)'l.xI in table 15. 1. alanine and valine arc m"jot "ompone nlS only in ela'iin. and seri ne is pn:sent in large amou"ts only in resili". T he "",," conclusio" ca n be rcach ..xI fro n, 'he co nde,,$ed data 1Ii"en in lhe firsl row of ,"bit 15.2. These resu lts imply Ihal Nature has ""vernl devices for gi"ing prolein chains lhe ftcxibilily and mobilily required for rubberli ke ela5lic;I)". and o nly 0"" of Ihcse invol.....! Ihc use of small side group". It was also Ihou ght Ihal thc no"polarily of Ihe side group< in daSiin is impor. lant wilh regard 10 ils elastkil y, ,iDec Ihc a~ncc of strong coulombic inlcraclions 1>,,,,,.....,,, Ihe chains fadlil"te chain mobilily. As can be seen from lable 15.2. c." Tobie IS. I s.o"", Amino Acid Compo,i'i" .. for Po""",' A""'" Kid 1la>1,. R«ili. Jl.J .l.l Gl",io< AI,,.;,,,, ~V,li .. "" U 11.3 ",,"0" II .! . A"'' ' ' ,__ ~ J ,. I l.1 U " . (,........ &""..1 Prol';" Kioela"o"",,,, in Mol. ..." .. Atod""';o ".', ,.,..I.~~_, ",. " ., U U " Vi",,,, ~T1c .., "" 0.' 1O.j BIOElASTOMERS 239 Tobllin R ... ~. '" ., " .~ 61." :2.0 lM ..'" , AN"",," ------'" Oill:' Vi>cid ~lk --.• 21.9 ~l "' 41 .6 I'.) !1.9 • """"'" ,<1 howc,·cr. rcsitin and OCIOpUS biod.SlomCI"$. in p"rlicular. haH high r.lios of polar 1(> nonpolar side Ploups a nd slill exhibi l good rubhertih daSlkily. Pnhaps Ihi. sbould have bttn e'P<"lcd. since Ihese pTO\ein. a rc 50 highly swollen when Ihey are elaslomeric. The nelwork eh"in' would be well separaled by the pr=n<~ of Wlvenl molo::ulo5. and would ;ntcraCI rebli"ely linle, e"en if Iheir side groups "'ere very pol; ... In facI , i'""cased pobrity could be one of the "I{ema{e m~hanjsms for ;mp""';ng rubberlike elas{icily. The pol.rily would incre.so {he swel ling in waler Ihul i. essemial for any of Ihese proteins 10 be "lastomeric. In any case. Ihe condilion. fOf rubberli ke cia"icily eerlainly seem 10 be me{ in (heso prolein s. In elastin. for exampl<-. a lmOS1"1I Ihe . ,pcrinl,nl"1 evidence indi· eate. a highly ,"udom. irregular ncl "'ork Slructure. coll>i'ting of chain, of VCr)' bigh mobility. Also. Ihe "proof is in lhe puddi"S:' of cou...,. alld lhe rubberlih clast",i!y of Ihese maleri"ls;~ ob"io"s to Ihos<: studYing lhem ;n "ilro (n..,,, if i{ i, not always oonsdous ly appr~dat~d by lhe anima ls "sing Ihen' ,n vivo). 15,2.2 Amino Acid SeQuencing The 100ni ""lIlCJlCing is not known fo> an )' of lhe proteins mentioned in {he pre"iou> SCp"t;ally for lhi> cross-li nk ing reaeli'm The pans of {be chain, well ,"way from lhe cross·links are rich in glycine (Glyl. aJanine (Ala). valine (Val), and proline (1' m). as ,ho,,'n in {able 15.1 In tnany pans. these four <>-~",ino add, are nol randomly 'rmngod, bUl apl"'ar in !hree repea! S¢quenC('S. Tb(St' ",.. uenee, are {he - Val I'm Gly-Gl y- lelrapeptide. the -V al~Pr<.>--Gl y~Val -Gly- pcntapcpl;dc. and Ihc -Ala~P",- Gl y-Val----Gly-Val- he,apcp,idc"" In all In".. of lh",,' ,-epcal sc .. ""nee,. " pro;;n" is followed by 3 glyci,,,,. and thi s i".iri"S cC or 'l'if"~1. This h~, giwn rise 10 "'hal i. 2 ~O STRUC TU RES ANO PROPERTIES OF RUBBER LIK E NETWOR KS called 11M: ··' ibrslionar· model ror Ihc elasticily exhibited in elastin""'""". In this model. lile P"Plidc uni13 in thc5e ..gmeniS are vlsualitt<.i as undergoing rocking or 'ibration,,' mOlions of rel. ,ively large amplilooe. Strelching Ihe elaSlin SI"'lehes Ihese II helices in a way thai damps Ihe amplitudes of lhese rocking motions . This con"rainl on the ch.in dynamics i. l'icwro as lhe origin of Ihe de"Creasc in enlropy tbal. in all other elastomcr$. has bttn allribuloo 10 Ihe simpl. "relching oul of chains in a tOlally random nClwork Slr"Clu",". It was IhuS proposed Ihal in elastin Ihe elaSlk fon'" has a uniquely difT.renl molccalar origin""'·" In spile of a large number of e>pcrimenlal ",udics and COnfOnnalional cakula_ lions in il< support. Iher. is slill 11""1 Skoplicism reg;.rding Ihis model. One objection invol,'es Ihe fael Ihal only a rclati'-ely small ponton of Ihe cbstin back_ Ixme consim of Ihe repeM uni" ciled, T he resl seems IOlany random and Iherefore pre~umably inC"~pable or forming any types of regular struelureS thaI would undergo tl>< dcscribtd libralional mOI;on,. Also. r-:M R ~Iudies of elastio ;"die.te thai ;IS badbotlC is highly mobile. ,,·iln corrcblion time, of Ihe <>rMr of 0.1 1". and Ihi~ IO'ould seem to argue against the p"""oo: of heli= of any signifICant ,Iabililyor permaneno:. Also. il is po"iblc to I",ui$faclorily (xplain Ih( Ihenno_ clast;c" and elastic" properties of cla"in using Ihe ,t.ndard rotali(}n.1 isomeric stale represcnlalioo" (}f claMin sequene"CS. wilhout III<: nc'ed for ;nlrodL>cing addi _ lio",,1 structural features of gr~atcr complexity. 15.2.3 Cross.linking Chemislry The cross-lin king of d aslin. and presumably most of lhe (}ther ela,'omeric proleins a, woll. occu'" with much bellcr cOnlrol Iban is u;"ally IIIe case with non· biopolymeric .,aslomc ...'·· ·. The crOSS·lin k. gcnerall)' apr""" 10 be co,·alent bonds. as e,·identtd by Ihc r.ct Ihat tht$(: InaleTials do not di.sol", ;n ,my sol"'nI';; cxttpllh"", tn" are kno\\n 10 cI~a'.. peplidc bonds. Some "flhe spider_web silks mentioned earl;';r arc c'ccpl ion. in thai Ihcy do d;,,,,,lvc ;n no"degr.d.lti", 501· ,·.,nt,. Thl.,..., ewss_links are 1herefore physical and lemporary. and are Ihoughlto be small cryslallil e?' If ·M. In bolh d aStin and ""ilin. unique cross·links a ", formoo by a !<'ri<s of Sleps. al lea" Some of which a ... ~n"Ymatically contwlled. In the sf'<'Cifie case of d"Stin. th. cross· links r<sult rrom the o"idali.", and fusion offour lysine 'ide chains 10 fom, an aromatic helerocyclic pyrid inium ring of consid.r~blc .13bil ily. T he cOlire cross.li" k .1",Clu re co",i.ts of cilher of Iwo i$Omers called desmosine and irode,mosine, They arc shown in pa'" (A) and (II). respectively. offigure I S, I' , Eacb such SI,,'elu ... ha" four allaehmenl ""inl$. wh ich m.an. it <~'uld. in principle. become a cro,,·lint of f"t\cliO/'alit)· cighl The aHa,hm""t ""ints act pairwisc. howe,",,,. wilh a gil"Cn pair a ltaching 1<> t,,·o ",bti"ely closely sf"lccd rc[l "",j<),,. BIOELASTO MEAS 241 ~ I I ~~ . 'I'" 'i' ~ c I 'I'" ~~ . I .'I~" .. J5.l SIr""'UI'" of 'hI'« 'YI"" "f c..,...link, ,""",u"inS in d."in' IA) is dnmo>";ne. ( h) i, ;.00..,...,.. .... a,od (C) is l\chyd,"I)',;n,.. "",.!e""i .... "'gmenlS bet",,,,,n lhe dosely 'paccJ cross·ltnh acl as effecli", networ~ chains, howe,·e,. the nelwo,k would have "n int e..... ting bimodal dimibulion of '1Clwo,k chain lenglh, ", as dc'SCriDcd in cha pte, [3 , The mole fmclion of Ihe very short chains. howe,'e r. "ould ..,...,sumably he much t"o small 1o h"vc a beneficial cffc""l on the ",echanical pro!"," i., of the elaSlomer, Ther<; are thoug),! to be Olh<;, ,\rucl"res Ihal possibly co ntribute 10 the cms.·li nk ing in clast in. One of these. formOO from two I)"i ne sille chain,. is ~hown,n p"n (C) off,g u'e 15.1. The e,,,,,-lin k, in resi li n at"<: .imil"rly highly 'p.:eif,e, but nrc fom,OO from the phe""lie side group, of t)"os,,'" reptat " " it" . T hey are lhouglt, to be lhe dilyrosine and "i'y,osine sl ruclo"", ,h"w n in pans (A) a nd (6 ). rcsptcli,-.,Iy. of figure 15.2. IXtail' of their bio,y nth""" are """,,,,ially enlirely un known. hen less is known about the "alLlre of the e,o<s-li nking in lite other prolein cla~lon"rs '. e~c:cpl for lite alrc.Jy mcnt,oneJ I""hablc ",I.: "f .m~1l om" ",,t, of crystallization in 1ft<: case " f Ihe ' I"dc.-wcb ,ilk,'" 15.3 Netwo,k Thmmoelasticily 1).3.1 G<:ncral ReI",-. ncc In prop,,''''1' ",,,,tel> f", Ihe cla'" '" ddo'''';lllu'' .. f" ' "hl>< rlt,~ III ,'tOI';;'1, Ilt~ III''''' "np"rtunl pic"", of ~'r<'n"o'tt~1 I" fo''''''li,,'' " hI'"~ ",,,ch of I h~ dil'lt~ for"~ i ~ ""'mpic and h,)", ,nllcl, i, "II~rl' U2 STRUCTUR ES AND PROPERTIES OF RUBBERLI KE NETWORKS B '" I HC I 0;, ..... 000" I I '"I HC - NH;; HC - NN,; 000" I 000" tlj;"", 15.1 SI'o informa'ion is obtained from Ihermoel"stic or force-temperature measurement, $il\<'<: Ihc bioel;tslomcrs dilTer >0 greatly in chomical Sln!Clure from Ihe Wmme. · cial non biological elastomers ",hich were u.oo 10 establish the foundation. of el"ticity theories. they h~,'e been much studied with regard 10 thei. thcrmoclaslicity. Elastin has bttn the n,os, st udi"d. 11:silill considerably Ie ... and the others hardly at all. As al ready menlioned. sludirs of Ihi' I}'PC are greatly complicaled by th. faci Ihallhese bioel~stomcrs have to be highly swollen to be claslomeric. Water is lhe preferred diluenl ,;ntt aqucous sol utions are us<-'" Iu ubtain the dC$irod swelling in vi,'o. Si.~ "'ater is relatively volalile. howc''Cr. water -.wolle. samples are almost invariably siudied in eonlact with walCT. which means Ih. elaslomer is at ,""cliing equilibrium. in a scmiopen system. Sintt Ihe degree of swelling depends on the e, lelll of deformation a. wel l as on the temper-.!Ime" , it is """,nli~1 to lake into aCCOunt soch changes i" interpreling the thermoelastic data. 11\= Cor""" tion. arc. of ~ourst, ~Iabo.alions of tit<; usual oneS which account for the faol Inat the measurementS are c'drrie'" oul al constant pressure ralher than <'<)",Ian! volu",. [\"'". Molec"lar model, proposed to aocoonl fot lit<: deformalion mocha,, · ism of tit<; .,ate.i al sludied ",'ould lhen ha"e 10 be eons;stent wit h the thennoclastic re,ulis. partic ularly wilh regard to Ihe relali"e importance of entropic and energetic e/Teets. .,,,,>5 15.3.2 Elastin The earlieSllbcrmoda.tie sl udi"" on elastin ,,"'" carrie'" out in 1934!" T hey ,,'Cre COnduCled On water-$wollcn ",mpk:<. in elongation, in a thennooynamically upen system. The enthalpy of deformalion "'" fo und to be n<satiw "nd large. and this ob""rvali,)I] "'as eorroctly c.plai,,<'" much laler 'IS "'ising frolll 'he eITcets of therma' c.,p.10.,io,,", Mo.e >!"-'C;r,cally, .,)<1,';" incre,,!." ils dey,.., uf ~quilihrium BIOElASTO MERS 243 s",.. lling wilh w~ler a. Us lemperature is lowered, The corrcclnc,"' of this ""plana. lion was verified by doi ng mcasurement s on claslin ' '''ollen in a "'.Ier a nd et hy· kne glycol solution ',f a composition so cho",n a, 10 bring the tl,er",al ""pansion coeffieienl of Ihe sw"lkn daslin network ~"'ntially 10 ~c rO. Under Ihesc condi. lions , theTC WeTC no large enthall'ic chan gel and . in fact. Ihe clast;'; foree waS found to be appro.imately 80% emropic in origin. lhat is./.II ""as appro>.i ma lel)' 0.2.'" Additional me"surem"nIS """i<;iSI~n l wi~h Ihe earlic~1 work in Showing Ihat large amounlS of hcat were ... Ieased whe" ebsli" was >1telched while in swe ll ing ~'i<>--hydrophilk inleracrions ",'ould depet>d On del i, . hown schema lically in ftgure 15.3"'". Since the rticvanl inlcr~c!ion, w(luld '>etur only al Ihe interf;,,:...,!> bel"''''''' the globules and Ihe 1""Ollooing wa ler <)r aq LI':O"S solulio n. Ihey would be at a minimum when Ihe clastin was in the u""re.ehed Sla'c. O<:fo<ming lhe bge,,'ed experimont ally. The oil-Jrop m",leI is mdie.,Uy dilTe ... nl from Ih" u,ual mood of a ol\C.phasc nelwork of random chains whkh greally imerpenClrale one anolher' ''''''''. In Ihe former case. the ",'aler i. ",munif'ormly consigned 10 lbe ~pa)"'tC 111 The oil. drop modd pn.~IXoh Ihal I~ rgc 2'4 STRUCT UR ES AN D PRO PER TIES OF RUBBERLI KE NETW ORK S FiI:"'" I ~.3 The ··oil.drop·· model for oI.,,;n' The orisi0.lIy _"her",.1 glob.,"" of tOe: protein ,h<)WII would be .;.formed into p,olat. cl1iP3"Od, by ,he irnJ><>OC enlhalpy chat>gcs should .C ilS one-phasc na'ure keeps tho in'cr~clio"s con"an' al constanl COlnp<Jsilio o In one slICh experiment. lhe .ystem ",,'IS kepI cl<>Sc<.1 by ,,,"dli~ g \be eI~>lin nelworks in a limi'ed amount of a vcry nMvolJtilc ""h·"nt. lrielhykne gl)"ct>1. and Ihen carrying oUl lhe lhermoelastic mcasuremcn\s on \1\0 swollen samples 1110 anomalous enlhalpy effl.'\" wcn: found >imply s""ounded by an incrl 1o be complrldy ab~nl ~nder Ihl.'scnl;a!ly zero. in good agreemenl ",i lh the lhermoelastic resu ll. in Ihe closed gylc<,] 'yslem . Valli'" of I he lhermoelastic quanlit y1,1f COIn "I"" be o hl.incd by C ga''''. BIOELASTO MERS 2' 5 !~\ ~,/"'/~ I H/ r. o " 0 I I /~y, C C" I / \ o ""'-.N /~ ! " i ........... ...................... -. •11.1'<' I S.4 Schcm3u" d,as,"," of ,II< pwlinc-,I)''';ne-un>l'0 '",f""" molccul", mechaniC' energy calc"lalions "" the amino acid resid"". li" ed. The calculotion. were ending. the ••,rond wilh .ccounl for local hydrogen bonding (be,,,,,,,,,,, adjacent rcptal uniIS). and the la" wilh acrouGl also for longrange h)'drug,n bondin g. Thi. 'H)< of long_mng. h)'drugen b'",ding OClw"e" the residues on eilh.. ,;de of a I'ro-G ly pair is though! 10 be imporlanl "il h regard to Ihe iJ-l urn$ in the U. ry mooel for Ihe deformalion of elaslin...·." It is illustraled in figur~ 15.4" . Figure 15.5 shows a Iypical o""ru m;]p, thu, il""cr.o lc>d. for r,i)"t:ine u"i ... In spite of Ihc facI Ihm long·range hyd rogen bonding is laken illlo account. Ihore is SC"<;n 10 b< a grea, deal of ,he eonf,gur.uion .pace a'-aila bk '0 ,hcsc ""idllCS, Thi, is shown by Ihe larb'" are", corresponding to reiOli..-l), low energics. tha, is. 10 I"", ,I,an 5 h a l m"l- ' Thi, conforma,ional versa'ilil}' io;. of COUI'S<, cunsistent wilh 'he tkxibili1,ing dTe"" of ,be glycinc unilS men,ioned carli~r, In the ;;«;ond Slage. ,rallsforma,ion mal1i= wore defined so ,hal Ihe skel.I.1 bonds. or lbe ,'irtua! bonds replacing ,henl. could all b< 1ran,(ortn.d into a sinr,ie coordinate ' )'$Icm for v"",or addi'ion. These ,ransf"rmation ma'n""" wcre Hohzmann aver, The gcnerd,or malrices are ~ucntiaily muhi· pli"" in Ihc order of the residues in , h. primary seqo.",,,, struclure of ,be polypep,ide being "",dek:d for ,lie limil "f lo,,~ ch"ins, Ihal is. for larg<: degree of polynlcriza'ion -'. T he "alue! or J;/f I"~." ,(11~""~.1 '!~.L ' I",nl 'p JO 'l"nOW~ ~"!'~~JJ"i ~I!·" .(II""!"OIO"OW '''''"'''''''P H pUP. 'u'I'q~ u~lloMS JOJ PU""J '! J.,"IUJ:><J""JI UU!I!'UUJI ""I~ JUQ AIUO '1'"'1'01 '(uo!l!wdwQ,' JO IU"I"""bpu! "O '!) '~"4tl 0"'1 '41 JO SIU"OW" ""lI" I'" '41 JO IU"I' .u:><J>pU! "'I PI""'" s:un I"J><.I"101 0 ·"1 ""'41 po" '(""""4tl 0."'1 ~41 ,oJ) .,J nI UJ><.I,u." UO!I!i IiUJI"UliI' O,"llJ!~Jtl PI"O'" UII'"P ,,~\O"" JOJ 'I'pow ",~~d.o.,,'.l ·, . S:U"I _UJodWOI UOlllS"V.J1 JO SlU~w"''''"'w ''''''lJOOW I ",Ow 'P"" ..SlUJw",n,",w "'IOJd :x>tl"""",onu sopnp"'! ",o!,np"o, '!41 JO IJood ns "'! :>'>""'!" I"U"!I!PPV ' u il'''IO "'110,'" JOJ IOpotl' "'" (ld,oMI AUu 'J'IIp.w IU'II JOJ 'pU" Ppow tlOJf>"I!O o~, 1!p>.J-"i!P pun ' ppow ~JO""I"" UIOp""' >41 uoodn. ' 1""lI",oo4' puc 1"IUOl,,!,>eJn ~,oq "1\"'" "llS"I>ow'O '41 41!" IUJw -""'~" 1>0011 AJJ.' Ii! "',." ""I"A '41 '~U!puoq u,~o'P.(4 ~UllJ_iI'uol u:>.,' JO "'~'" SIlo" luno" ," ·:.sno '!I'! IU" "OW '4' iii ·....'!I!">'>[ "'.,.. '",llJ"lnll", "",:moo. oJ,." 11'1 ,0J 1"'1"1"'1"' "" lnA '41 '''',",''04 ·AU,.. "41 "" w:><J.')J "!I'''I-' '41 J" ~"'!"!"''''' ''''!"~~ ''!JJW,(IOOOJ "41 ~"\IM 'f1'/pu" J.r/~{ r')ul P 41"'1 01 ....'!n,a""" "'!~ ",lIunqJ q:>n1 ·:.sn" ~U~ UI 'p~"JJJt'! S! JJU\\lJ;)du'~1 '4' S" 1....'!1'4 t> pndwoo :)')ow 0, ,SUC40 Plno" 4"!4·" J" ..,,"" '<>Jnprul' fi POPU'IU "'l PI"'" :»"IC,xiwOl JJ."OI "'II 1" uO!lnw'''J''O] '"I" S:u ~41 ·..'I'!ldod·'IOO =41 JO ,,"'" ""ts "'''I,'' ,"':M :).)"" =",,.,1>.,,,, PJ'm" ""1"" ·.. Hl "Jn~~ U, poquO<>p'. ' ""pnP'" u:>.x) .oq lIu'l'uoq u,llo'p,(q oiu",·jllol 4"q ·" "! "un ,u!"(I$. 'OJ """J,n. /<3""1> 10UO!"UUOJU0::l !OS I _31.,. OOC I DOC o>z (\\ 1"'-1 ' ee, ~I \0 0\1 \ 0 I S)lIl OM13N 3)l11!:l3ssn!:l m S31.1!:l3dO!:ld ON .... S3!:ln1:ln!:llS 9tZ BIOELASTO MERS 247 is 10 be e. peeled from lhe <me_l,hase rando,n ,,,,I work model and , in essence, a parallel silualion oc<:urs whene'", more plasticizer is added 10 a pol)mcr 10 m:.ke il "mort pliable." Ailltou!!h Ihe lwo.pl,"se m<Xlel fm s"..ollen elaslin is incorrect. Ihe "hydrophobic bondin!!" bel"""" lhc nonpolar elaslin side chains ~nd waler docs ma ke an i",pollam comribution 10 the cla.' lic force in cia>!;" in vi\'o'. This bonding ca use. an OIdering in the "aler S1fuclur. ncar Ihe nonpolar gro up". and thi. decreas< in e"tropy gives ris< to an "'"TeaS< in the fr(>l; entrlO' of the syslem. Part o f the r<:lrae li ve fnrc.: in st"'lched cla. lin is lherefore Ihe "lIempt of the system to d""",,,>< il' free energy hy min imirillg itS h)'drophobic bondi ng, It doc. Ihis by relraclill g. siroe Ihi. eau...,' a dcswclling thai decrea..,., lbe amount of waler available for inleracting WiTh Ihe nonpolar side chai n., 15.3.3 Resilin Thermoelasticily measurementS h;,,·c been carriM ouT on thili" lendons from a dragonfly'''' The'S(: wert heroic ' _,periment' because Ih. minute size of lhe sample.. rC<[uired Ihal Ihe enlire ""rics of mea,urtmenlS be carried out ulldcr" microscope! They were condurtoo in ;In 01''"'' ')"Iem. in swelling C<[uilibriulll wilh waler. Quam ilali", correclion. for changes in >wolli ng were nOI carrie'd out. lbe jU'lificalion being Ihat ,h,·r. was >ome ,,'idence 'h~t they wen: ,mall , Th. resuil. obtai noo by aM I),>i, or the d'l1a were only ""n,iqU:IIl,ila,i\'C, bul clearly ahowed Ihe .. me lrend, as elastin. More .peeific;,lIy. the cla.lic force " ''IS almo~1 entirely en\ropic, and the swollen Tesilin ,,-as wi'll rcpre> 15.3.4 Olber Proten, IOlastomcr$ Abducti" from a ,;caliop wa. al>o ' ludiM wilh "'!!'ltd 10 it< lberm""la,li" 1"01><' _ lics'·". Again. only approxima te dal" were obtainoo. bUI Ihey supported II .. primarily cntropic nalure of the force and a r~ndom nelwork slructure, OcIOpU' arlerial clastome, (OAIi) "'as ,wdied more 'Inanlilali,-.:Iy , wilh co.fCClions for changes in ,welling}."". The rorc.: in this CaS<: is .Is<> .Imost em irely cnlropic, but Ih.rt m•• y nOw be an important dilT.ren"" relative 10 tho other protdn bio!omcr'_ fk< of the large nun,her of polar amino acids in Ihis polymer. Ihere may nol be ,ignifK""H contributio", from hydrophobic bonding. The", arc no publishe'd I"", moela'tic resuil. on any or lhe spider,web , ilks", yel_ Any such re.ul l~ m~y hoc inhertnlly unrdiable. hO,,'C,-.:r, Ir 'ht$e maleri.l, are parlia lly Cl)'sla lline. lhen ."c lling. deform"lion. alld temperalure, could chang> Ihe nal"", and amounl of the crysla llinit y. Struclllr~1 changcs .uch as these. ","""urting during the Ihcnnod."ti.:ily mea'urem~nl'. could "'" ~ C ,he lc in Ihe prt"""1 "ontc,t. 2(8 STRUCTURES AND PROPERTIES OF RUBBERLI KE NETWORKS 15.4 Stress-Strain Behav ior 15.4.1 G"I'I<ml Rcsuh~ As is lrue for daSlomel"1. in general. Ihe m<>duli of Ihe elaSlomeric prOieins ,·ary wilh defonnalion. and SweIli Rg l•. The dependence on deformalion i. parlicularly slrong in Ihe casc of elongalion. which is almost alwa~s Ihe deformalion used wilh bioela'lOmers. Nonelheless. v.lues of Ihe elongalion modulus Ihal arc rel.lively conSlanl ha'·e been reponed for Ihese malen.ls. This would be e"['«ted in ,·iew of Ihe very . pecitic w.y in which Ihe cross·links are imrodll<:<"ll .•' ''''pl [>Ossibly in Ihe cryslallilc-<:ross·lin ked spidcr·"..cb silks. In any casc. lhesc ,·altlCs arc listed in III<:: flrsl ro"· of I~blc 15.3 fo r the prolcin elaSlomers being di<eussW. The valuC'l arc remarkably similar in magnitude. oonsidering Ihe diversily of Ihe struclures in which Ihesc proteins a ppear. As a rcsuh. lhe values of lhe ,nolceula< wei&hl bel,,·..,n cross·linxs is also '''ry similar. as can be seen from the socond row of table 15.3. T he)· arc a ll ""II within lhe "mge 10 ' ,II,,, 5--15gmol- ' which is chamekriSlic of oommerdal nonbi"logical cla'I,,",ers 1!.2">.". Som. 1}·pic.1 force eXlc,,,ion curves ~'" pn:senled in figure 15.6.• Iong ",i lh Ihe lim;ling Gaussian curve . Dal. were a ....dilabl. for all of lhe prole;n daslOmen under dis...ussio~. save for .bduclin (possibly bec".u"" of (he u"u,ual shape of sampkos of Ihi! malerial). All ,·.luc"8 of thc f"ltt ha'"C been "I>nnali;.ro by dividing Ihem by Ihe limiling modulus vl< T 10 facilil"IC comparisons. Since Ihc magniludes of!he mod uli a", nul ,·cry diITerenl r"r these maleria ls. as documenled previou.ly. Ihis permits focusing on Ihe shapes of Ibe isothenns. The most slriking fealure;s Ihe faci Ihal all fo ur of Ihe.$C da$IOmcrs sltow \"C ry signincan1 non·Gaussian uplums in Ihe force al high elongalion •. Also very n<>lable ,,'" Ihe ",Iali,·ely low val\lcos of Ihc "longalion"1 which Ihi, oc"urs. One way of inlcrpreling these dala i. in lerms of equim'ent fn:cly joimed chain," . In Ihis approach. a number n' of consecnl;'" . kclclal bond. are grouped imo • single virlual bond . T he "umber II ' i, made large enough W Ihal Ihe COm:lal;ons oc l",«n the r,rsl and lasl skdetal bond~ in ~uch a gr([uping bc<:omcs negligible. Thi. make. ooruttuli,"C ,·irlual bonds unoorrelaled try design. and thus Ihe rea l chain; , "'placed hy ils freely joinlc"1l <'quinlen!. T h;s is a scmiquanlilutiw way of ch. ",clcrizing non..(;aussia n behavior in lernlS of chain 5liffncs,. The il iffer Ihe chains. Ihe larger Ihe non-Gaussian cffcci al oon5lanl defo, mal;on and Ihc lorger Ihe '·alues " f n' required I" f"nn Ihe u""orrelated vin",,1 bonds. ·hblt IS.3 Some Nclwork Ch.mc' <,"",1 Prolcin 11'001'''0'01<''" i'T<>r<'I, f.1..1'" Ra.i ~ . e (MP.) n..l O.M 10'·", (1""" ' ) • ~"'r> r""" """""'"" '. , AM",,;. ., .. 0.10 " ...,,-. .".,"..... ""',,., OA~> U.• 7 '. , v,oad od_ .. 0 .)0 •• BIOELASTOME RS 2_9 • ... • • • • IS • • • -- "...... • • ••• ••• ••• • - -".. .... --- __ ... ' lUstln UTIIOU '.' Fi~~ .. 1~.6 S'rm-~ra in i_ocrm. in rlonSa'w. ro, biod,,,'om0 G,u ssian Ii.", being s,,-en b~ 'he d ..1>mer. and ,1>0 cum, ]. b Figure 15.6 S"]!ge", 1hal1he ,i,;oid silk i. tile lowest in . tilTness of ,he bioelaslomers shown, and the ""lOpu, ,'rtcrial elaSlomer is the highc>l. If c.ch of Ihe cur...,. is iitl<:r frc'ely joinlc-d "'Iuival,n1<. Ihen the number of amino acid ",rca, uni1S requi=l p"r virtual oond i, apprOXim"'c1y 7 r(>r I h~ .iscid ,ilk " rtd Ihe ",';Iin. 8 for Ihe claSlin. and 12- 14 for Ihe OA E'. Sin.. e the simple Hexible polymer polyethylene [ CH,CU, - 1 has approxim alely 6 r.peat uni" p"r vi rtual l">ond on Ihi$ approach. all four "r the bioe]a'tomers are """n 10 hal"< ,ignif"alll chain Sl,lTnoss. It is vcry interesting t ha1 1he 1i"'1 Ihree of the><". Wi lh "alue, of n' of approximalely 7. are Ihe o n.s ",·ilh relalively large "umbers oflhe amino acio nd is much highcr in Ihe case or Ihc OA E. T his i$ presumably d"e 10 il' relatively high conlcn1 of lars,. polar amino acids. a, described in la ble 15.2. ] 5 4.2 1, la,tin T he C<Juivalenl frttl y Joi nted i"lerprelaloon of an)· Jala o f Ihi' Iype ;$ se,-erd y limit<:<.l. h"w~,·er. b"."ouse ,I is nol vcr)· ,n,,\ocular" Th,u is. Ihere is no rcal direcl ."Onneetion belween the cxlent of Ihe "",,·Gau.';.n hchavi or and the detailed molecular Siructure o f 1he chains. Such a c01lneeli(>n has Ix-.: n mado in the Ca li< ,,( cla"i,,". Sires., '''·"dI n isot he r"" ",-ere F.ene,·aIL-.J u,ing"" approach ycry , imibr 10 1he Ihcoret",,,1 c~l.;ul"lion. de", nhc<1 pre,i"",I)· f"r I .II In Ihi, co,.;. ho",c.·cr. the ··deg"" of 1>olyn,eri'.alion·· .,. is very ""1''''1 ,,,11 . an,1 ,·alllc-< "'-er~ en"""" th~1 250 STRUCTURES AND PROPERTIES OF RUBBERLI KE NETWORKS were oompardbk 10 Ihose presenl between cross-li nks in oati'-e ela,lin . SpecifICally. these '-a l"", ""ere .' ~ Sand 10, AI",. alanine unils were placed al th. ends of the chains to model Ihe alanine sequence, fo und both befOl"e and afler the ly.inc"S usWto gi'nemte the eros.;-link$_ [n this approach. Mon tc C~rlo melh_ ods were applied to the ~onronnation"[ energy maps to gi'ncratc largi' numbers of ch.in, h,-i ng Iho d ..ired confonnational characlerislics. The end-Io-end distance r waS lhen calculaled for each of Ihese confonnalions. and the various ".Iucs of r placed inlO a hislogram. The histograms were subsequenlly , moothed using Ihe cubic .pline method. thus prod ucing a oo nl iml('u, distrihution , as described in chapter g_ Ocri' -a l;,-cs "-er. laken al variou, pOinl, along the distribulion, aod w.'" used in the final «.p of lhe calculation , Ihe genrration of Ihe Ihooretical slre' ... 'tr.li" i"'thenn. from Ihe sla ndard th "",-ch.in model. The cakuw lion. were •• ,,;ed out 0 " the clastin repeat sequences dcs.;ribed ea rl ier in Ihi. chapter. A typical distribution thus produced. a"d ii' Gau,,;"n approximation. are ,hown in figure 15,1", As expected. th. Ga ussian a pproximation is not '-ery g<X>d for chai", that JrO this short. Some of the calculated m ..... str.lin resu lts repr~sentcd as I~c nominal Slre!oS against the c1onll"lion itself are ~hown in figure 15.8. The rep",,,,matioll i. Ihe ",me a. th"l used for the experimental data in flgnre 15,6. The ",nle calculaled results. now in the Mooncy-Ri,'lin "'presentalion " ,29. are presented in figure I 5.9, At least some of the .. thc'Oretical irot herm. did show uplurns Ihat were qualilati,-ely simil.r 10 those ob",rwd in the experi,"cntal ~t...:li.s on clastin . The upturns in Ihe Iheoretical iSOl herms are due 1Q Ihe limi,oo eXkn,ibility of the elastin repeat "'q ue"""". M",I no,,,bly. thc)' "",urred , - C , o o 0.' 0.' 0.' 0.' r/ nl .'it;u" 15.7 Monl< Carlo ~DO", .. d nonnl tiz.cd radi.l di'lribuliO ni"c_'<TIllln.,C S( ,', (k~n:c of I"'lymeri'"H,oll x of J. and lhe Gau";.n di,!ribullon h.,ing lhe .. m..: vah", of lhe unpt"urM dimell' "01" . ,.. (, _,- -J" . • r •r r ',•" • rc, C- 0 T 0 0 0 - 0 O ,r 0 ., " -- ~#---- ... ... ... a , P H " 0 .., fi,u" 15.8 Thw",,","1 ""minal 01.",,, , • f>- , ,o C.- , t- o T ~ -, f- ,,.. o o P H " """ , ,. ,. r;gun: IS-' Thco ... ,,,,, 1 mooulu, ,h"wn ",. fundi"" of tin eh.i n, ..! n, figu", I S.8". i"\~"" , cl",,&.,IoOO\ fo' ,he d ",· 252 STRUCTURES A ND PRO PERTIeS OF RUBSERLIKE NETWORKS al appr<)xima\ely Ihe same elongalion. ca . I,g. al which Ih. e.perilnemal ISO· Ihcrms showed Ihi' boha,·ior. "I leaS! in Ihe most rehable case of ,ingle ela$tin fibor$ (whOch a"oid the problcm of multiple fibers being oul of regisl " in larger samples), Unfortunatel y. it is very diflieuh to extend Ihe .. simulations 10 higher elongations 10 charnClc,;'" lhe upturns m<)r<: thor<)ughly_ This is becaul\< of Ihe ra pid decre ... in Ih. number of gellCmled conformal ions "' high elongali<>ns. and dilT>cuhi., in cvalualing dcrivali,"eS of Ih. di'lribUlion funClion"_ The .. oon-Gau ..... an upturns in Ihe mooulus could be extremely uscful for lhe .urvival of the organism', In the ca"" of arlcries. the incre"se in modulus could decrease the likeli hood of balloonro-out a..", called ancuri~,. and Ihe hurs!i ng of Ihese aneurisms. or Ihe ruptur<: of Ihe a rlery in g<;ncral. Thi s would be in addition to Ihe usual reinforcement of Ihe"", blood '-.:sscls by ha"ing Ihe cl"'!in eX1ensi,,,ly interpenelrated by mongo partially crystalline fibors of collagen. IS4.3 Rcsihn As shown in figure 15.6. the Slre"-Slr~in i"'lhcrm, for Ihe re,ilin "ctwo,h arc significamly more 1\on--Ga~ssi" " Ihan Ihos.c for clastin _ It would Ihe ~fo", be of considerable imcrcst 10 allempt to "ndemand lhese differences through additional Mon te Drlo simulalion,. Such $Iud ies a",'ait the requircJ sequencing informalion on rcsilin . 15.4.4 Spider-Web Silk The "i""id silk fibers in the .pide r ",'eb ha,'e al~ady been discll5sed wilh ~gard to Ihcir Slre>s- .tr~in isotherm •. It i. interCSiing. howe~cr. 10 c<)mpaI'C some of mcchunkal properlies of viscid 'ilk wilh those of Ihe frame ,il k occurring in lhe ""00 web"'. The fr.me silk fibers ha,'. a modulu, apprm.imatcly 3000 limes higher lhan Ihal of lhe "iscilerclie. which mean, they degmd. mllCh of thi s absorbed energy quickly inlO hcat'"_Thi, helps pr<:"Cl1t the il1=1 from being shOt bad out of Il>c web by Ih. relracli.'. force of Ihe fibors, 15.4,5 O1her Protein Ela~tomcrs Thcre"r< also inle rc.;ling "niaxial and biaxial ,,««-strain Jala "'hieh have been rq>Ortro fo< "rterics from scvcmt cephalopods_ namely" >/pilius)". 'J here haw "I"" bctn .1u<Jics "f "'luid ma"tle w,,11 in i"Oalion. " deform"lion crol\ BIOELASTOMEAS p.,.,-aded with muscle t"'HC. yea lly c'Omplicating Ihe interpretation mechanical p"'pcrli", in m"lccular lerm •. 253 "r tlleir 15.5 Oy na m ic"Machanical Pr ope rties 15.5.1 Vi""ClCla stic Respt)nses in Geneml The bioclaSl0mcrs ,n vi,·" generally functi,," in cyclic d.f"rmalion recowry pr<JceSK'. an c, .mpic b1h Iher. modynamically opcn and thcrm<>dy namieall y cloied sY"cms have been in'-esli· galcd . Of course. lh. values of Ihese vuriable;, occurring phy.iQlogically are of grealesl relevance. As is Iypically thc casc. lhe erTee,i,.., fre<jue!\Cy range is i!\Creased by the melhod of rcd,.cOO variabl.s_ In Ihis 'echnique. dala laken al 0"'" I<mperalure ;"" shif!ed along ,he fre<jllency a.i, by ~n amount Qr (ha( yield, a smoolh joining with da,a pertaining I" anOlher le"'per-~turc . In Ih;s I>'ay. CU"'e, The Ibree vi'lOOelaSlic quantil;', of greale'l ;mport.1nce are Ihe 'Iorage mod · ulus 1:.". Ihe 10<1; modul us t;". and Ihe 10>< tangent Ian 6 (",hieh is .qual 10 E" f t:')" The loss (angenl i. of p,".,ieula. in'crC'1 sinc'C i, g<>es thtough • rna,' imum a( lhe glass (I1m,i,i"" leml":'.'"rc . I" Ih. nl<),1 relia ble in"Ntiga(i(>n •. Ihe ,hm faclOrs aT "'" delermined 5" a, 10 g;''C ,he 00.1 joining, for oolh Ihe £' vs, frequval of pla"ici,,;r) or with inc ..a", in frequency, A. men,ioned earlier. d""",.", in 'h" ~mb,rnl I<mpora,urc of an anima l'. body could be dilollSlrou, al co nSUlnl l.:'" in ' he CaSC of fi'h. amphi!>i"" •. and "'pl il «. since lheir b<>dy Icmpocrd'ure, Ca n drop bel"", OT. The erTeel i, bcco=, ~ 8 •"• 7 ~ 6 ~ ~ ~ ~ ~ ,. 5 . , 0 , , ., 0 ,, log f (Hz ) . at. """ 1'. "B.r..... (E') and I"", moduh IS. .20, 26.01."," J. j J7 . 42. .9 • 56 • 1\4.13. .·i~u .. 15.10 S'(};:r.,'~~ uli ..m~"lu"" 0, 10, 'g "' ~. (fr<>m mea'"lOp loool1on' l . - . I ~·C and" ,. •• • • • • , • , . log fa. • . tan 6 ' • , . •< ••• ••• • . • . >1 . w>t.r con. ·" ••r IS. II Ma.",. Cun 1»-",,1 oncta";n'·, ,he d.", The in ~I";.l~ i' nos sIoO"'ing 'Ia'~~n "" for 1'1>< bro~.n ,.... sh",",' d."in -1% "'ere • of 4J,] "'1% . .O'''tnt ' of 28.1 .. t.n' aU ~· •·,tor ,'S.U;, •• ~ - ., • ,• • , ., L, ~ ., temperature 00 1"01 F~a .. 1~ 1l Tho iop " thm ~f (lie l""' p. 10'0 • ,110>"-,, •• • f."n of'mI- • •• ~ I• ~ • '.' o· ., , ,. , . • , -. . • m 0 • • "'- • • 15. 1.1 (0) n", ,," ... g.: "".,h ,1t r", "' 12 ~ 'a"r "'Wen!>, corroclro ro, ,l>e horj,onlal _,h;n,. "",I (b}'h< ma< " .. 1<' '''"Ion,, '· Fi~"'" ,nnl>nl"UI' UI pm'''J OJ~ . ,.",W J "I""!S 0, is,'p":).",,, ""'" J" ~"() " '1 S"~"'" b"'J J,""" I '" w",!, U!'lS uIP 'u~'uo~ J~ll'." u! .."n",,,,,1' II""" "'I"'!'"'''' I'u,' 'UOII"od",o, 0, "",,,,,,,,, "".\ OJ~ '''''!I!I'"()~ In"'~{}I",,,Ji4o.! 'OpU" """"' ,"P"'"F""'''!.\ ,til. 'SO!, _Jx10Jd ~!JlW01sop " ~ullIuuG "'1"'04' "! ~n!'!I'"ld I"'n J"",,, "41 J" "'U<,,; l>"!~,,,U'" so,' I()AU ! U!lsnl' JO UO!W.JI,A'I>(J UOHUJp.\4>cI JO SPolJa t n l 'ApJOOn:>l" U! S,!u,.JoJd ""'41 ""~Id PI""·' wql """UCls"m",p AUR >q "I wOJS IOU 01' :».'111 (>lin 'J""'O""I" roo~ "Sl II ."""ni»JJ ' '01 Al"'!'"IOJ .'41 In ,.",u.>doJd "1'"1""""" '\ >11 ul U!""P 01 J"IIU"S .u'A Sl u'\.lnM";XJI 1"4' I! I"'"!",qo Illns", '41 WOJJ P'''' >q u~, IR41 II" Inoqv .J'),C." "'II qSnOJ41 JI><\I I,.JOlo.! 01 11'4< SlI """'I' puo IU.>dO 0.1011= "4' 4"!'l·" I" k"""i»JJ ~41 01 >sop h..!J.\ I! I! >S"4' s" ... bu,"IulJ "4.1 ',ZH f 'PI"UI!xOJo.!d" '"~u",,i»JJ "UO "luO I" J"'lpOl' u;»q '"4 uH~npqy -PJ"I",\U! IOU I! uo!,nwJopp ifu!!'", AIp!dlU • 'UOll"'!ldde i!4' UI '"~y ~4 1 0. ~~ns ·..""'U! SU!dw nf Aq p;o<" U"!UC4"'W Ilnd~,,,,, ~'I' JO (Jed i~ fX>S" OSI" '! II -'''''U!''~I' JO ,u8!< "'04' 0, uIlI"'I plno." UII'''''' '41 4~1'I1I\ :ISO'll "'''1'''1 ""':)U~nb"'J I" .{ Iquwns:uSO'll 01 J"I!W!' h..!,., :»~ '"IIJ.xloJd J!"ul""""!" II! 31""!PU ! IU~I U!l!S", uo RIUP JO Hl"OUm p:t1!1ll!1 U ~J" OJ~4 1 ' WJI'lwd " ~"'oooq I'Inoo "''I I '1"'4." ," k)u~n~J ~4' J~."Ol plno ... SU!Il'_"S,P "q J'I"." ~U!7.P!"~ld '41 JO 'wO! JO ssol 'UOIIOJS 'x,,, "41 U1 "4unJ !»'i"n,.lp "'I []!II\ '¥ 'I"!-'''~w 1"'lllJoPP ,(II~""I"(' ""U U! ",nl!"J 3Sn", AII"nl"'," 1~41 ,wq Pl"'-'" SUO!' -"-'q!~ .(:)U~nWJJ-4l1!'l "UO!I""JIsqO JO 3:m>S:>Jd "'II "'I ,U31 "GJlH 'P"UI ""oy poolq JO 3SR:> '41 U! ·J".. 3."o H 'w,!uriJ" ffufA!1 r. U! p"'0lUn""UJ AII"uu<>u ,'pu""b ""J ~41 3.'Qq" II""" In:no Plo04s ",ollUod,uJI pun uO!I'sodtuo, "41 I. UOIIIS""JI $SuiS ' 41 'r; UTII q '! U!IS"!' "'II 1"41 h!IU'!PU! ·"U"'!)"wr.Jp """,,,>Il! !lnpow '''''I P"" ohm). '1'04 .... p -u"nIulJ J3«814 I" ·J-:lA:l.'I\01 1 ':IlIJ"1 "'" <..",uall""' .. "'I') h..!"A"""-' "'IIUI" JO "'u"'~ -UP puu 'Uuw, "'" ":>OlJ' Su)dwup pun ,U"SUUI ""01 ''I' ·AlhIPUods.:wo:) ' '''1 ~I!nb.! .o l,,(>Oul ">01 SI! PU" 'hu~nIulJ JO Iuapuodapu! lsu~nIulJ IV 'or,8u!n8jJlu! :>.lOW U'M~ "'" 'lIu'l -IJ.'" JO »IS,p w~"Uo~ pUE :'IJn)tJ.xlwJI wm'uoo In . A)u""""-'J JO '):>O.u.' "4.L ''''nllUodw~1 p""E"':>OP ~Q' I~ "''''Ill ""1S~p 'U!"W'" U!IOWJ""II ~'II 'UOl\IPPU UI 'W>I'.(s u.>do ue .. ,u,"u1!'iIJ0 SU'AII E '" UI1'"I" 1"'11 ')f~ ''1) ,(q "'_" ",,04 ',>5lJO:> "P,"unlJ<)J ,n "'I' > S ~ 1I0M13 N 3)1 1l11388f"111 ~O S31l113dO~d aN'i S31101:l0US !I!ll BIOELASTO MERS 2l>7 experimenls in which Hry sl<>w calorimelric <>r dilalOmelfic measuremenlS ,lfe used 1<> dClermine Ihe "equilibrium" (Iong.lime) glass lransili<>n Icmperalure al con,laDi compo,ilion. The value. are appro.imalciy 2OO' C al 0.00 g waler per g claSlin. IW' C al O.OS g Walcr per g .Ia'iin. and 15' C al 0.30 g water per g cla<1in Thi s. of course. ,mmN ialel y raiSt! the qocSli<>ll of how >ome of the cla,lin in ,i,o might I"", some of ii' hydration. Th. fact Ihal elas!in is ,ery hydrophobic SUggc>1S Ihal the ,,·a ler ofhydrali(}n is n(}! very tightly held. and Ihis is conlirmed in th. absorption i...,therms it e,hibil,I". /\t 100'10 relative humidity. swollen claslin 'xm\ains appr<>ximatcly O.l>g w~'cr pcr g claSlin. bul wilh (\nly a 2 ~. decrease, 10 98 '10 . Ihere;s only approximately 0.3 g waler per g claSl in lelt Thi, , h"w$ 'he ca'" with which s",,,lkn ciastin can be dcil)"dratN. Thcre are molecules in Ihe extracellular malrix, such a, c11O",lrilin sulfale. Ihal are n(}w known 10 dehldrale swol1<:n etaslin .ignif",',"II)" al physiological CO""'n' lralions. Olher malerials Ihal miglll h"w simil"r effects are cholesl"",1 and cho· te.ler(}1 .Slers. This c'Quld bt: highly releva"l 10 cha nges in lhe func1ioning of elaSlin in di<ease stales ~"d a< a re.u ll or aging'~ 15.6 Som e 0 1her SioelaSlo m eric Gels IS.6.1 G<;ncr;l.l l'",pcrlies Ther. an: a wide ,-ariel}" of biol>olymers Ihal form gtls in which Ihe !l."',ork SlroclUre .-esull. from physical cros~, link., suc~ as cl")'st"II;les or helical ··junclion wnes'·'~~'. These gel. are I)·pically fo .. mtom" of Ihe chain segment' aggn-gale sufficienlly 10 f<:>m' a maer"",<>pic tel. Since Ihe crOSS·lin k poinlS m Ihese malerial. are ph)·s",a l aggregale, of one type or anolher. Ihey arc generally ·'thermon;wrsiblc."' Ihal is. Ihey rdiqudy "pon ,nerease ill ICm(leralure. There are anal<:>gies in Ih,· realm of non·biological malerial,"; for •• ample. Ihe poly' elh)"lene gel'" used in Ih' spinning of ul1ra. high.slrcnglh f,bers. Wilh regard 10 mechanical prO(lerl;e,. bolh low-defonnalion and high-nc 10 >om" ~xlcnl·j. Olher. more applied approaches incit>dc con' pression under cOnSI,ml 10adlO. and penetralion using slandard l"obeS. On. <>f Ihe mOSI ,mporlanl . Xp<"rimc"IS involves 'n(}niluring lbe buildup of I~ modu lu, '" lhe gelalion reaclion proceeds. Since Ihese gcis are quile weak. c,"en when Ihe gelati"n reaClion is complele, such ",c~.ure",enls arc .lmOSI .I,,·.y. carried o ul ~,ing "". "f lhe dynamic cycling lechniques. On"" l!;c modu lu. of Ihe S"I has ",ached a n apparenl pl"leau. ,u&<,'O.'ling Ihal 'he gela,iun reaclio" has noa red ""mplelion. t!;c fn.:qucn<:y of Ihe """illaliOll may bt: ,·ariN. 'hus obla ining E'. f ;". aof imerest. Thi, I)·pc of e~(lerimem is parlicularl)· ,",-,ful in identif);ng ~nl"nglc"'enl nctworks. in which only "an,icnl inlcr-<:".in cnlanrtcmclIi< hold the nelwork 'Iruc'ure 'ogc'!;c". Ob~i<>u,ly. in Ihi, """" Ihc modulo to 10 'em", the frequency of Ihe dcfurm"lion de"",,",,,,. 258 STRUCTURES AN D PROPERTIES OF RUBBERLIKE NETWOR KS There i, anolher ime","ing a.pect to the long. time boha"ior of gcls of Ihis type. 1'101 only is ther<: rdalively liuk pOlymer pl'\!!iCnt in these: highly .-wollen m.leri.ls. bu! wh.t pOlymer then: is. i. fr¢quemly susccp!ible to bac!erial auack"'. This type of ne!work dcgrada!ion ""n be prcven!ed by !h•• ddi!ion of sm.1I "mou,," of baclericide. Ihu. proving. nice parallel !o the use of an!iox_ ida nts and ."tiolOna"lS to prevent network breakdown in commercial. " ... "biological claMO""'''. There have bttn very few aucmpts to eharac1erize g<:l s such a s these with regard to their large-d 10 e~lcnd lhe effccli"e range of deforma!;on rales in f,... nl'" experiments on nonbiologkal ma terials. The concenl1""dtion d."""denCll of Ih. gd modulus is widely interpr.,eold into a nctwork "'octure by ,mall crystallites"" , Another rdated but more specific example would bo n.,\>Co rk, in ",'hi~h Ihe cross-linh arc !egmcn" of double or \riple helices. "';Ih the .«:<juen«. involve,\>;derali"" "f result' inlarMed "si~g " latl;,.., 'hl"<>ry. Account w(>uld Ii,,,'e 10 be ta ken of lhe pos BIOELASTOMEAS 259 and poly",er segmem. on the latt ic., a eircum'ta nce not anticif>lOtoo in the ,im' plesl versions of the bUitt theoric<_ Additional complication, occur when the nelwork chain, are sufficienlly rodlike to fonn anisotropic phases. This w"uk! occur only above a critical co,"",,,, tration, but for relalively long rods Ihi, co"""niralion is <juile 'mall, fr<:<juentiy amounlingto only are'" percenl poly,ne," ·". When Ihi, ph.. separalion occurs, two phases arc formed, the more dilute o ne n;maining isotrop;c. while Ihe more ooTl<.>:ntrated 0"" con,i,t< of an ani""tropic a,rangement of the tit. aligned rodlike chains. Tt..re have been calcu lation; on the str .... -sirain ""ha"ior of isotrop,c .y>tems of such rodlike "gment'''' If Ihcre arc no permanent cross·lin ks, Ihc claltic response comes from Ihe """ding of the rodlike segments . The ""'Is-Slrain isotherm in elongation is predicted to be S shal"-'c:r of these COnl"c! poin! •. 15.6,2 Some Specific Systems 15.6.).1 R"tuw""CI!il HiQPQlrm~" Gelatin. "'hich is probably rhe most important gel O)f this type, is " pro,ein d~,ived frO)m collagen by hydmlytic degrada'ion. It readily dissolves i" warm waler 1(, form a ~Iu li on O)fr"4ndomly coiled polymer. So I<)[,g as Ihe co"""ntrarion, are not .>lrcmcly low, for e.a mpic, less Ihan a few lenlhs of a ren:cnt. cooling loch a sol u'ion l'C,ult, in Ih~ formation of a transparent elasric gel. These gels ale, of ooursc, "'011 known in tt.c food induSlry_ The geta tion process involv", the pep!ide chains reforming SOme of lhe "",n-known ool1agen triple-helieal sequence, they had in the nalive stale. Rccau!<' of Ihe dilule nature of the $)'1tem. Ihcsc rcgion. are very li",ited in si7.c and num""rM. They are . ufficien!. how.v«. ,0 link thc chains into Ihe obse rved nelwork ",,><:tu,..,. The gelation proce", i\",lf has been studied hy ~ varie, y of tl'Chniqucs, inelud· ing n,casurement , of optical rotalion, m"'chanical properli"'!. NMR specl,a, dickctric rcia ""lion , and radial;on !lC"dl1cring. The studies arc gcnerolly £arried O)ot as a fmICtion of re>tclion lime, poly,"er con""nlrat;on. len'per-,,,ure. and. in the cas.: of rhe mochanical pro!,a.ic goal has """n to undcrsland the gelation process, parlicularly with rc~rd to ;t, mochani." " ne 1",0 mo,1 i",po"anl sleps a", ob"iously Ihe conformali,,""1 ch''''go> «" hdi<..". ami !h~ ",,-"'C'allon "f these hclk-a to form lhe 260 STRUCTUR ES AN D PROPERTIES Of RUBBERLI KE NETWOR KS physical cross-links. Severa l detai led mechani,m, have bc<:n propo&:d fo r Ihi . process. T here is "Iso a limiled amount of large-dcfonnalion data wh;';h ha s been used !O est imale «!uilibrium moduli and ultimate propert ies. mari"" p"lysac-.;haridc$ r<>nning hi""la'· One "f thc ,n(>11 intcresting ').re< IOmcric gels are the carragecmllls. der ived from seawcro. T hey arc pa rtia lly sulfated polysaccharides having an AR disaccharide repeat unit. where Ihe ./\ unit is derived fro m a I·J_linked p.o.gal"elo.., ,"""idue and tl'" II unil from a 1-4·lin ked 3.6-anhydre fonnat;on of elastomer;'; Sels. /\ va ,;"t y of carraseenan, u iS! . and differ from onc anolhor primarily in Ih. number .nd localion of the sulfale smups Ihcy cantain6S . Like gelatin. Il>ey dissolve a l dC""o'a'.x! temperalu",. and cooling of tl>e sol ution s thus Obtained ""ullS in thc formatiun uf thcnnom-ersi blc gels. In Ihi' case. how""" •. the cross· links arc thoughl 10 eonsi't of Ihreefold righl. handed helices ,"bili,.ed by hydrogen bondi1li. Another diffcretlCe i, lhe facl thai the s~lfate grou ps can give th" polymer considerable ion;'; charge. a nd thi, could certain ly aff"'l ils propert ;"s, The associaled c""umcrion. also haw to be taken inlO "',<" un l in detailed Ihrories for the gdatiun pr""cs.._ A related polysa<"(;haridc.•gar. is a <"oll1ple~ mi xt"", of conlpo"ent •• 1'0 ohlained from SC3.weed" 1lle most important component wilh regard 10 gela lion i, agaro",. whose repeat unil i, ,,,,olher disaccharide "'ith A and II unils of 1-3· linked P.()..ga iacto>c and 1·4_li nh-d J.6-anhydrO·O-L·ga!;'CIOliC. fcspc<;ti,-cly.... Agars arc quile ,i milar 10 lhe carrageenan,. ncepl for Ihe replacemenl of the !>-galaclose unit by ilS '. counterpart. and by Ihe near absence of ;onizahle unitl slIch as sulfale gro ups. Again. there are occasional dep" rt ur~" from pe. f""t Struc· tural regularit y thai ca n cause kinks in Ihe chains. wilh allerubnt eff...,,, on the gelation P"""-""_ There i$ $Ome cont,o>-ersy regard;"s Ihe e~ aet molccula,str<>eture a1 lhe cross-links. There ma y not only be double hcli~ fonnation. but also associa ti{)n "fthts<: helical ~uen= ;nlO microcrystalline region,. T he mcehan· ism for the gelation process may depend greatly on lhe rale at which the agarose 5Olution, are cooled. It is thoughl Ihal quenching gives risc 10 no nnucleated ,pinOJdal dccumpositiun. whereas slo,,'Cf co-oling gi'.'-'; nuclealed nelwork growth. AI"" """uri nl! in algae are lhe algina les. whi<:h are salts of alginic acid. These are apparent ly nol alternaling disaccharides. bUI aN made up of '''rying amOun" of 1-4-li nked p.u·mannllronk acid and 1-4-linhxt o·L-gul uronie acid· l . The ".taC1 compo,ition depends on lhe souree of the alginale. In .ny ea"". lhe chains consi" of homopolymer;'; blocb of Ihe fin;( spc<;ies mixed with homopolyrncric i:>lock$ of Ihe &¢COnd. and with block$ of ir~gular ~uences of Ihe tWO. The .Iginates gel upon the introduction of divalent c.lIions such"s Cal • . a .-.d the gel< thus formed are nOI thennure"crsihl'-"_ Jt i. thoughl thai .'ery specific ion binding occurs. with "ccompanyingco nformatio"" 1changes in thechoins, One modd fo< Ihe c ross·lin ~ ~gio", depic·" th,,", a. cryslalline aggregales in a twofold ribbon form. wilh the C.'+ ions eoopcraliwly bound among Ihe chain>. There h,,"C bc<:n a number of Studies on how the rigidity of the alginate Sci ,Iepend, on mO\oclllar ,,-eighl and concentral;on of the pol}'mcr and lhe concentr~tion and type "f tl>c di'·aknt or BIOELASTO ME RS 261 calion. Dynamic-vi ...,ocl.Slieily ""dies showctl """""Iastic effects to be small in these gel., which ~,oroU"'d!;C!; molecular thro,ies of rub~,hk~ dastK·i,y. V.lu .. of lhe O1olecula, weight betw"",n croSS, link. oblained i" tl,i. way """mcd 10 be in s.atisf"clory agreemcnl with values obluioed from swelling ~quilib,ium mca'"...,"'ent •. The ve'y compli. cated Slruet .. re of tile'SC gel., ho"",ve., ,ugge'" that the agreement may bc<:n fortuitous. Of the pla nt poly""",,ha,idcs, cell ulose is the "",st com",,"," H"d ("rtainly {he mOSt impon.nl"'. Since ""lIul"",. itself. is nol soluble in <...,,,,mOn SOlvent., il cannOl be gellw into swollen networks in Ihe way described for Ihe olher biopolynlC ri. It can, huw ...... be de,ivil;,wto este'" an,1 Clhcr~. and thes. are readily soluble and do f""n gels, presumably thcrnlorc"c,siblc on~s. H)"dro, ypr<>pyi cdlulose is an example of such " system . lke"use of (he derivati/.ation. these ma terials are no longer purely biopolymcric, howe,,,, Peelins are an e."mple of pol)"sao;harid.. lhat do fOTm gel' readily In Iheir nalUral . I:HC·'. The) <>OCu' in 11M: cell ,,"~lIs of cerlam plant. and arc import"nl in thc food induSll)·. They ~rc branched in their nali"e Slate, but the e>uactw rna",.ial is pr<:dOlnlnantly linear . The chain, con,iS( of 0-( 1-4"linkw u-gal.ctutonic acid r""iducs inlerru p!<...:i oct • • ionally by ' __ rh;,mosc unit. , and til. GoI'''on OC<.' u", uncler ""nclition' where inter_ molecu lar clCCUoSiatic repul,ions arc reduced, for cxample at low pH. Addition of dinlem ions, '''''h as Ca l + , fae,litales the process, a nd al leasl sonIC oflhe gels arc Ihcrmow•• "ibl<. T he cr.,.s-litlk poinll arc rather ill·defined aggreg:uc'S that arc non,tnichiomClr;C and arc thought 10 illv,,".., noncoy"l~nt a nd nonci<>;lroSiatic bonding. In the CaSC of mal"mal., Ihe m", t important poly"" ...churidc il unquo"ionably slorch, because of ,IS edibility. Naw"lly occuring Slareh occurs in grannies and consists ofl",'o fraclio"" the " '" i.li"oar and c"lled amylose and the second is branched and calied " m)"lop<:c''''·' ' Rolh are ba,.,d on giuco..,. with amylose being an n-( 1-4)-1inh,d ,>-!!!uco$<. while amy loP'Xlin ha, bolh no{ 1-4) and no{ 1-6) linkages. T he stud i, partially cryslt,lIinc, largely bc<:anse of Ihe amylopecli n, and Ihis i. "'~Ihcr w<prising bcc~us e of ' he h'anched na"",' of ,hi. componenl . Amyl"", is no,m"lI) the minor component, hUl can also undergo cry,la llilluion In lhe case of amyl""", quenching soluti"" , rapidl y p,oduccd gdati,," , with ronsiderable lurbidily ,,,,d a significa,,, ,hear lOo<1uh". CrySI"lliniIY. ho",·e~r. a. judged f"'Ill wi<1c-anglc ,-n'y diffraction measurements, ",'a, very .Iow 10 dc,.. lop. T his IUgj,'CSh lha l gelalion ,,-suit, f,mTl lhe r"rm.lion of .my lose double hel~. which lhen slowly agg"'lI',lc inlo cry,lollil<" c.pahlc of .. altering .·raj'S The turbidilY is though' to an.., from the scparation of Ihe s)'S{em into pol)'mcrrich and pol)'mer-dcfic~cnt ,..,gion •. Although th. gels are highly d"Slic, Ihd. complicaled struc,u,..,. loa'" discou,aged imCrpl"C'alion of 'heir pro!",rti"" in te,ms of molecular IiIrori,.. of ruhberlike da'tici{y Thc "m)lol"-'<:li" f"'clinn "f """,h ha, "I..., becn . ha,,, 262 STRU CTURES ANO PROPERTIES OF RUBBERLIKE NETWORKS In add;,ion to ,he alsal and plant p<>1}'&accharid~ mentioned 3OC,,"C. there are p<>lys:lccharides that occur outside the cell' of certain bitoltria and fungi. An example i, , ant h.n. a well· known bite1.r;.1 polysaccha,i!lcM , In some ""1"'<15. these are similar 10 the olher polysaccharides. ht lend 10 fonn loss stable gels. """'" "f " 'hich are lransicn t. cnlang1cmcnH)'pe nC1works. There are also an im.ll'Olysaccharides. most notahly th. prou:oglytan, (which con,iSl of a protein backbone wilh allachcd linear rolysacchari!lc. ,~ll cd i!JY"0, saminoglye,ms) and gl),wproteins (,,'h;eh consist of a central protein core sur· rounded hy very short laceharidc ch.in;; in a "bri;1Ic·brush" " ruel""'). These are u"."j in li"ing S)'Slem,. more for their viscodasticit y. for example as mn~us. rather Ihan for Iheir clastic properties, They are therefore not of direct intereSl in the present con text. Ther" is al«> a ,'ariel? of relalively "'lid. Sd~li k . malerials I h~1 a", only l,..~n sienl network structures , They exhibil thIS gel.ll ke structure be<:a"se "f the eXlen· sive entangling of the biop<>lymcr chain •. and dilution. e"en at const,,,,, temperalUre. eon"erts these malerials into ,imple sol utions , Guar galaeioman ""n is an example of a matrrial of this type"'. These malerials are al.., beyond the scope of thc present treatment. Ij.~. 2.} G/olm/a, and Rodlik" BiQpolJmCTJ Finally. there are globular biopolymcrs. particularly protein,. that ean aggregate into rodlih '''''el u.... which ulti_ malely aggrcgato into ~brl)us I)r branched network structures~'. For c~ ampk. actin. tubulin. hemoglobin -S. insulin. and fibrin form fibrous a'!<:mbliC'S; ,,"na· lUted proteim form branched networks; and ea",;n form, mictl lar agg«gate •. The unit. undergoing aggregation can them",I,'es bo rodli kc . as they are in the ease of myosin and col'ag~n. 'nt network "chains" in these syslCms are Ihought to bo References (I) Vincent. J. F, V.; C""ey. J, D" Ed •. Til f'r0{1<'f"'" 4 8W1QI'''~1 UniverSity Pm§: C.mbridgo. l'lSO; Vol ,].4, (2) O.,.t; .... J. M tn TIw' M "'hlmlNI P'<>p<"I<, a/ /J.,."kDI Mo",4II!I; J , F, v. Vince,,' and S. O. C"''''Y. 1'4•.: Camt>riJ~. IIn;ver';'y p..,..., C.mbridg<:, lyg(I; (J) (4) Vincen'. J. ~' . v , S'nM;'W"; 8 _ 'N;"Is. M:ocn,itl>.n: Lond~n. 1~82, W.inrighl. S. A,; lIiW. W. I),; Curo'O')'. S, I)" Gwioo. J, M, Mffm,,,kal IkJig~ ;" p. JJJ. O'g IS) O.,.lioo. J. M, Il.ub/x. C"'m, 1i-l. 1987. 60. 417. (6) ,.... ",.me,. 0 .. Ii B~krd ""J .\','",I•.,k Pol,.",,, "'n"'d .b. F"""ier: I..,OOon (7) Hu""n."l. W.: 1I.""".Mu'pny. S. H.. ]"h. l'h,'#",,1 NerM""*, , P'~"ffU''' "",I G~l, n",,,,er: t-"OO"" , 1'190 BIO ~l AS TO M E RS 263 (81 Lillie. M . 1\, : Go>hr><, I. M . In I'hy,ko" "..,., ..,.., Po/y""" aM G,b, W . Ilur (10) lilli<. M. A.: {;()$Iioc. J. M. JIj,>pOiy"'"" '990. 29. 1147. ('I) Cio>line. 1. M. In C""""p" oJ ~ffic",,<-)" in C""'I""~ri'" J'hy,islino. j , M /, E"p, BW 1\1215. 114. 2.W, (21) Sh,"wid. R. E.: Cio>li ne, I . M /, £'p, HWI 1985. 114, 2~ (22) Go,li"". I. M .. Shad wid. R. 10 Pa (23) Go.li ... I . M .. I)c.l.1 on'. M. E. Sci. Am. 19lI5. 252, 96, (24) G ooli"". J. M .. D. f:/a,"";"·. .lrd ed. ClorenJon P..... : O' fool. 1915. flO) w.i•. FotJo. 1', Ar.tk,,,,n. S O . .'(~'u" 1970,227.71 8, (1) Hoc .... C. A. I .; Flor~, I' I Hi..",""".·., 1974. 13,611, (l2) A"",.dy. A. L: Ma rk. J. E flj"poJP"N., 19110. 19, l!49, (33) A.",n. II. B.: (;osl; ... I . M. i/o'"poJy",,"" 1981. 20.1241. (14) Sage, E. II. : Gmy. W. R. In t ·/tm;" and f'-h.",;" 11." ..... t •. !I SoOObo'g. W. R. G ray,.OO C. Fr,",brau. Ed., Pk nu", t'r...· New Vork . 197b, (35) K ...1>",. II.: Viney. C .• K.pl"n, D.: Loml>io rdi. S. N.III'" 1991. 349. 5%. (36) Voillalh . F .'.,.;, A m, 1992. 21.(,13). 10. (37) K.l'bn, D, I.,: Ada",,- W. W.• ~·.rm«. II.: V'I>cluwJ<>xy Amcric;o n CI>cm"",1 Soci<.y: W•• hinglon. DC. 19'J.I, Vol '44, (l8) (;o, li"". J. M . I'{khol>. Co: 0""t<1I<. P.' Ch"nt- A. In J/i,,,,,u...'''I. 1-".,;,,, ,utd p'OCI!ISOtg of .lIm.,ial•• M. &i<:< ]'re;..; Wood bury. N\'. 199$; p. 2)1 (39) F",,'1,,,. C , 1" .... nW". E.• S.ndborg. L. !I. /. 8M . CI'm" 1974. 2411. 6 1 ~L (-!O) Lu.n. C.·H.: 1111)". D . W. J I'hl"'. Clwm. 19'!I. ')5, 18%. (4 1) Lo, o, C.·lt. : 1'.IIl ... 'T. M, : 1' ,-,,,,". «. u .. U"y. D. W. /Jiol"'ly""''' 1991. 1 1. 46S. /41) U"y. D. W.' """'" S. Q.: I'ark... T. M. Hh'I""!'''''''' 1992. 32. J7J (41) Sd<>tli,>o. r .: I'm ... ,!. «. IJ.: I ' try, D. \\'. (,,',"a. M. U. Hiol"'l ....",,, I99J. 31. 7'). (44) Uft)'. D. W. rn~, · . '. L'hrm. I", fA. I.-"~J 1'1'13. 12. BI9 (4S) Urt)'. n . W ,\"d. "m. 1'19). 11?( 1). 6! 264 ST RUCT U RES AND PROPERTIES OF R UBB ERLI KE NETWORKS c.; c.; (46) De Bolt . L. Mark, J. E. Poly"", 1981, 28. 4 16. (41) I:I< BoI <. L, Ma rk. J. E. J. Poly"', &1.. PoI)'m, Ph)'" F.J. 19~8. 26, 8(.5 (48) Flory. P. J. SlarUIiC, J. M. : Ro«nbioom . J. In t~T""<:f!lul(J, M a"ix 8,-""",mUlry. Ie A. Pie. ond A. It ReJdi, Ed •. 101......., N .... York. 1984; p. 191. (SO) Thiel, R. L.: Kunkel. D. I),: Yiney. C. BWpoIJ."IhJ 19'94. ).4. 1089, (51) AndcrSonia, Y. M"",omI)/. Sy'"p. 19%. 102. 159 (54) Simmon., A. H .: Michal. C. A .: J«Ilio'ki. I.. W. SCieMC' 19%. 27 1.84, (55) Bill meyer, F. W. r.xlhwi< o/P"'y_' SeieMr, )rd cd . Wiley: New York. 1984 (56) Kaplan. I). l.: F""""y. S.; Mollo, C. M ,: Areid;"""oo. S.: Soo"",,1. K,: Mulle •• W.; S«dwcll. S.• Ik< k ~ 'ill. R,; Viney. c.: Kcrkam. K . AI R .~ Hull, 1'192. O<:,ober. 41 (S1) W.'pod. w. B.. Snyl, R"btN. I"d. 1934. 10. 2).4, (58) Hoc"". C. A. J.; Flory, P. J. J . Am . CII.."I. ,Wt-. 1958 . 80.6523. (59) Flory. P J P.wipks of 1'01),,,1<" C/wtIIUIrY, Cornell Uni",,~\y I'r<-so: Ithoco.. NY, 195J (60) Ma.k. J. E. Riopel)'"",,, 19J6. IS. 18~3, (61) Hoe"o. C. A, J, : lloe,." M , D, J. A, Poi,..", &>~ . &;. 198(1,20,2'10. (62) Wci ... Fogh. T . J, .lloi. BWI. 1%1. J. S20. (6JI AOO",dy. A. l.. M. rl. J. E. YoI)'m . Bwll. 1991.27.217. 164) Forry. J. D. Vi>c""kw~' P'''I'<''';'' _/1'01),,,,,,,,, ] .d ro. Wiley' Now Vo,k 1980 (6S) (;\a.k. "'. II. : R"S$-~l l1rph~. S, n . ,,t. Ph/)'m . Sd 19117. Sl. S7. (66) R"...Murphy. S. B. Y~lp"I"' 1!I92. JJ. 2612. (61) ~ " ... Murphy. S. D. In Vlwv/""lcily 8iom",,,;'II," W."'ington. DC. 1!192; Yol. 4S9; p, 2621. (68) Ru»<>. P. S .. Ed . /tcw"ib/c l'oi, _,1< ~t. "",I Rcwled Sy"'''''' A"",rica" Chemical Sociely: W.'hington. DC. 1981 (69) Y.ng. Y.: Idc". M. L Karns>. ~' . E. JlocrlDiffy/" 1'186, 19. 1lIS6 (12) "'h.,,,oi. S. M .. fd. S.,.."",",/s, Cho",n"l:wotl , aJl{l T"""y Poi,."",ic N""'OI"k" aJoJCck »]onuto Press: New Vork. 19\11 at at 16 Multiphase System s One cia.. of mulrjpha>c cI"SlOmc~ an, thm;c """"bie of undergoing Mrdinind""ed cryslalli~alion •• , ,..a~ discu, ...~1 "'I"'rately in chapter 11. In this cas¢. I h ~ SttOnd phas< is m.de up of the cryslalli'e, thuS generaled. ",hieh provide considerable reinforceme nt. Su Sl raio-indu<.'-'''', <."" and the additiun or silica to sil .. rublle,,', In fact. the reinforcement of mnural rubber ami ",1"'00 mmerlal. i. one of lit., mOSI impor1am proccs.scs in .Ia., onler 1«hnology. h lead s 10 in mooulus al given " rain. " lid inlpro,·emonts of ,·a,ious technol"gic"]]) im l>orlanl pf\)p",lics. iuch as lea' and abrasion resistance. re,ilience. CXlc"ll,ibilily. and lenSlle "",ngth '·'·7.II.1< " There are al ,nc"",.,,, '" a 26(; STRUCTURES AND PROPERTIES OF RUBBE RLIKE NETWOR KS cales Ihal Ih ••"Ienl of the rcinfo"""menl depc1l,l$ strongly o n particle sil<', The m""immll reinfOfWmenl is oblained for particles with dian"'te" ranging frQm 10 10 100 nm. and larger panicles can ~Clually cause a delrimenlal clTect, The rein· forcemcnl phenom<non perlain. only to tbe rubbery .Ialc of Ibe elastomer. ""ilb Ihe $arne roly,ncr in Ihe gla<sy Sla le ffC<.Juenlly being essenlially unalTtXlcd_ The mo't important unsol,'Cd probkm in this arca is the nalure of the bonding between the filler partides and lh. po lymer chains"_ The network lhains may adsorb str011g1y onlO Ihe particle surfaces. ",hich wou ld increase Ihe effeclive degn..., of cros,.linking. Thi' .ffoct ""ill be esp<....,ially Sirong if panicle'S e"(O nlain .ome rcacti'-e su rf~oe group< thai ma y cross_lin\; (or end·hnk) polymer eh"i ns, Chemisorption. wilh permanenl chemical bonding bel"""",, filler particles and rolymer ~hains. C,tn be dominant, particubrly if Ihe filter i. precipitalc<1 into the elastomer durillg euring»-><''', Anolher type of adrorplion which can occur al a filler .urface is phy,isorplion. arising from long_r.. ng~ "an der Waals forces betwcen th. su rface and Ihe polymer. Contrary 10 eh~"';sorplion. Ihi. physical adsorplion doe. nol "" '~"'ly rt'Slri<;1 Ihe mOWmenl of pol),mer chain, rcl"ti ..~ to Ihe [,IIer surface when high siresses are appl i<'quare nanomeler of Wer surfaoe)" Excessi," coyalent bonding. lead ing to immobilizalion of Ihe polymer at Ihe fi ller surface. is highly undesirable_ A fI ller parti<;1e may thus be considered" ero ... \; nk of ''''y high functio"ality. bUI transienl in Ihal it can participale in moke"lar rc",,<~nl\<' menl, onder ' Irain _ There arc probably numerous other ways in which a mler changes Ihe mechan_ ica l properties of an elastomer, some of admilled ly minor COllSC'lU~nce". For ",ample. anolher faClor invol."" changes in lhe di<1ribution of . "d ·lo-<"d yee~o", of the Chains due 10 IIt~ volume laken up by Ihe f,lk, '· '''·... , This eff""l is ob"iou.ly closely relaled 10 Ihe adsorplion of pol)"mer chains onto filleT .urf.<xs. bUI Ih. ourfate also cffecli,,,ly segregales Ihe ",olocllk< in ios vicini.y "n<1 reduces enta nglemenl,. Another impo.tant aspecl of filler re;"fo=mc~t ari"" f.om the fact that Ihe v."tiC\cs inOllCncc nol only an ela'tomer's "atic propenies (such a, Ihe dislribulion of ils end'lo--end """Iors). bUI also ilS dy namic properties (s""h .s ncl"'-oTk Chain mobilily)_ Mo'" _'pecifically. the presence of fillers r<'cn:forc lI nli kely _T he effect of "."ilalion On lhe beh",-io' of filled rubbers i. probably of g",.t imponance" " , The elastic modulus of the filler is mw:h larger Ih"n Ib" modulus of Ihe surroundillg rubber. and Ihi . lead, 10 large strain gra· diems in the 'icinily of Ihe f,Tkr i"trlide_ Th i, effcel is '"ry diffICult 10 '100)" from the 1I10k<.·ular point of "kw. bUI ha' be<:n in\'eSliga !e~l usinS ,·,triou, eompn MUl TIPH ASE SYSTEMS 267 Ther. have also b<:cn numerous Studies ch~racleri,ing the p,"liclc, Ihemsd"es. The"" have employcSCopyw. ,·ra y ~nd nCUI",n i<:"uering".,. ...... ). and nuciear magnetic resonance spcc1roscopy" 'J. Non-spherica l particles are of growing ;n tere.t ,........ "'. as are particles having regula . ly spadng> within the elaslomcric matrix" ", As has !x-en mentIoned. ,h. Malu< oflh<" thoory rOt cia'tomer .einforccmonl i& rather unsati,factory at the present time ". Nonetheless. it will be u",ful to ha,'c 3n o"emew of wh,,, has \><en a(U)mpli,he 16.1 Some Theo",l;cal Approaches The first theorelical attempt 10 ""plain Ihe dependence of reinfnrce,ncm on Ihe conccm ration of fillc< was carried o"t by Guth "nd Gold " . These ""tho" mod· if1M th~ Ein'tc;n viscos il y equatio" for 'ph.ri",,1 paniel« in" v;"cous mooi"m by adding" 'luadratic teno '0 a"'oun1 for interactions 1:>ot ...",n particles. T he)" obtained the equation 'I =. 'Io( I + 254> ~ 14.1".') (16. 1) where ~ lind ~ arc the viscoll is ' he volume fraclion of r,lIe •. S",,,lh..·<><.>d" "s,,,,ned Iha, a ,jmil~r tqu.tion holds for changes in Young'l moduous: <J> f: .. I:Q{I + 2. :i4> + 14.1¢ ') {16.2} where f: is the modulu, rOJ tb< filled cl:o'to,"or, and f.;" for the unfilloo clastomer. Equation (16.1) """s gencrali,oo I~trr. hy Guth . to no'''pherical p;>rtick.": ( 16.3) where f is a shape f""l<)r used tn e"imale ,he ~,'cr.ge I'''rlid" "nisometry. Some au th ors'" ugll. that the ruhber ?d,or1:>od on Ihe tiller surface ("OOUM rubber"j'" " ,hould also be countoo uS " pari nf the fill .. p.arlicie whcn using tbe a oo,,,, equations in the ~n"l y,i, of .'peri,nc"t.1 data, 1"hcre arc num"",u, other ""Klel. f<)r rc;llforttmellt in polymer compos;te$"'..... >-.. , foc the t)·~ of intcr~cti"ns betW<."n f,lI". and cla'to""" ' >--0'. and the effect' of fill. ", on s",,,lhng..·9'. A review of many or Ihe .. thenri .. has ~n tly been given by Ahmed and Jones·. Most of them are not moleclll,,,. and ,he.e is a .. riou. lack of rigo'o,, ' extcn,ion or'he Mati ,t icallhcory of rllbber elasticilY 10 filled ela'tom.",". Seve •• 1 attempts have \><en made to apply ,och theM)" In e.perimen,.l data. h"" """., in an "ttcml't to ",tim~lc lhe ",,,,,ber of liller · rubber att.,,,hm,,n1S....... Bueche'oo mooilic -SUO! I:>'''' SU~"OU"J ~~I J" l"'~" '" J" "'''>!"'I"' "'II I! 1I·,"",,1tl" III" U! "11 .1. 'S'''~U!I -'"'''' J!>1I1 01 J<>!Jtl "'~W~loJ Ol"! "JJll~ 1"""'·>W"I~~ ~lp"'l JO Smpu31q wn,n ~41 U~41 '"III'" ""'~I~nJli ~JO'''I'U 01'" "'~IW ~4' ~U" "lldp -;uti ~q ~! '!'II )lU!Op JO ~""" :)0,00 '\iW~'>'IS f"11~ PP""u u" lIU!sn""J ~q J"'U!~lq" "'l l'IlI"U Jl""ll "'"""",,,-,d "'II J" U"""":>I)'ld"". puu p:lUlulq" IU=""'''J'''''' "'II J" ,,-""~U J'II JU UU!I~ppnp JW<>S -W~li~S J~Il ~-J~WU IS~P 'III JOJ .ppoW I~'! '''' '001/1 "IS!I"'" JO lIOIl;mJISUO:> ~1I1 '''''-'no~ JO 'S:>IE~" dlUO:> ""I~ "'IJ. '''"''' 'I1.d pue 'nJ""" '~W!I JO l~oP W:>JS ~ ,,-,!nb:>J ~!lu;mb'lJJ ' ''''!SJ>d,!p pool! IUIII '.>.JoJ"-'~1I1 'lI"",,,d,n, LOU "! II '"""p:lICJ~wojSl!" "11''''1 """"!d~1 "! 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JO IU~lUI""'1 ~1I1 '''ldlUUO JOJ :p>d0l',','p IIUI"'I ", e "'!J0341 1",,!I·< I,,"n '"1110 'fO' l"poW ~JO","u·olqnOp" '.11 ,n ·,o,SW ,...<S I'-'IIY 01 p:I!ldd" ~nnJ<S"""'" UO"'! OSI" ",00 SUII "~~10.""" pu" ""!I!)I J" PPO'" "I""M J~p ur.' "'ll. ·... 'W.(loo P-'ll!I JO S:>llJ:xlO,J In"u~II""w ""I uo <JU~IU'llIuonu~ JO 1""11' '41 popnpU! ~PII1S S! lIi ' lUSnnlUJoJ "J'ld" ~41 UO p3Stq S ~JO."l"" lOW·_tod P"ll~ JOJ ppolU 1""'1'''"1' ~ p>dol"""'P '01,11111", pun 4,,-,ul~H '~JO"" 1"""'" ~10lU u l ·1.1, ~ (/" + ''') 01 I"ub, 13S ':n IU"I'U"" '41 411."1 p:I'ldd. "'! pl""4S UQ!lnnbo U!I_'! l:I _·'ouo ,~ "'II '3S!"-"'IIO 'P'I""lb u "'I ~1'l!<S<.>JP IU'lU'l~umu~ "'II·" 'SJ~w"loo u~llo,,,s JOJ Slln~ po<.>~ ,",\!Ii N()OIII ' 4.1 (9 '91 ) .l¥(/" + ''')'' [.11 "'W"""'l' ~J "_"PU u'lI o_,, 1.11 SOlnpow ' 41 ''''''IS p>Jnp:IJ ~1I1 JO 'UIJ~I U! U'UU'" "'11" '(~'9 1) UOIWnb3 ·,.IS I"OIU 'P"U" .0 J>d "",n '~~J,,"'U '111" ~ pUU "lm)O 'U'"I O,\ I!"n J;xl ""'" """J,n, 0111 SI S "-'~~'" , ,. I A _ = I" {n Il :/, ~JO"'\3" 'Ill '" J"IIY JO uom"'J ~wnlOA 0111 01 1""UO'UodOJd", I" 1"41 p:lwnss" 11 -'''lIY '41 01 ~u, puoq "'II ~q p>JnpoJd SUI"q~ l"u0!l'pP" JO J"'llUnu 0111 ~! I" pu" '-""lqn, f"lIyun ~1I1 U! SU !C1(~ ~JO,,,pU ~'\ !P"ll~ JO J'>qwnu '>I.(l~! '" :u~~,,, "~... (nl) (._" - ").1~(/" S)ll:lOMl.3N 3)111 l:l388nl:l ~O + '~ ) .. {) S311 1:1 3dO l:ld CNV S31:1nnn l::llS 99~ MULTlPHASE SYSTE MS 269 16.2 In S itu Gene ra tion o f Fillers in Elasto m e rs t6.2 t General Comm~nh The pol)'mer wnich ha, been u ..:d mOSt .'Ien,iwly in . tOOi"" of this type is poiy(dimelhylsilo.. nc) (P OM S) [ - Si(CH,),O - ]•. II has I>c.."" chown in pan beeRU,", it is compallble with many of the OrS'l"omelallk m"lcrial, used 10 gencraie ceramic-type ph.,.,. and in 1",,1 b';'ca u",", il i. an ela",omer which requircs con,idrcing tille", in pol)',ne", i, iMsed on an imporlant adv.occ in Ihe cer.'ni", area. 11 invul",,, Ibe "'" of I}reparati~e tech· niqu<,< heaoH y based on chemical reaclions; for c.rganomclallic co",pound"·'·' '''. In thc ,'cmmics .rea. the adva ntage, are Ihe purity of the ""ulting produeu. the po .. ibility of .~oiding high 'empcr.,utcS (,hereby permitting ,urvival of organk gue,t IIlolocules). the control of ultra' tructu,e (atth. nanomet~r kvel), lbe relati~e ease of forming ceramic ··alloy~:' alrd Ihe po"il~lity of i'ICorporating a dillen"'t m.terial inlO thc ,....,ulting ceramic while it i. It ilt porous, prior 10 its dcn ,ificalion In the present "ppl~'I;OIl to ela,tomer •. the new approach ba, the advantages of "''Oiding tbe dillkllll, tirnc-colr'II,ning. a,J i",imatc bonding bet"""n the inorganic (cen,,,,ic) pna", and IJ><, organic (po!)"mer) pha"", m a way that i. mo. t likely to gi"" no"ci mechanic-al properties, The PDM S ela'tomers mentioned .b,,"e h.ve bee" rno~t s' udied in such org;"'ic~ inorg.nic hybrid ,'oml'O!ilcs. Althougb IJ><, reinrolUn,enl of ela,tomers by dispersed ",ramie phases" of prima ry inlerC5l in ,he prescnt context, Ihe C"dSC where the ceramic predominatc. a"d be"<:o,,,,,. , he contino"". ph .... is alro discussed. The elaston,cric pha.. lherefon: b..-come. di,persed in it. Yet a"olher "ppr""ch 10 the no,ocl r~info=- mcnl of elaSlomers i""ol~c,> mon"me'" sud, a. SI)rCIlC or me,hyl melh3cryl.lc. polymeri7.ed ;" situ to gi"c glassy pol)'meric dom~in'. and some related .ystem, in which there a'" .ilhcr magnelica ll y respo",i,'c p"rt"k< or crysta lline "<'olil('. Of primar)" ;nloresl in Ihis chap ... , i, ,I", rciufore.mcnl 1'H)"id",1 by Ihc'iC li lkrs. I, is 270 STRUCTURES AND PROPERT IES OF RU 8 BERLI KE NETWOR KS 'nalrix in which Ihe cerdmic m"lerial. are toeing gellerated, III Ihi. "ma trix ;",Ia· t;on" approach'''. x-rny and neutron ",," u ering techniques. for example. call be us.,;d to Qblain information that t!"~n&ttllds th<'SC parlicular system •. h should be useful in " ,'ariNy of areas. it>Cluding Ihe new "'I- gel ltthniquc for prq,aring unmodified o:::r~mics menlioned alxl''', 16,2,2 I'repar.tioll The mOSl imponanl reaclion of this type in,'ulvO$ lhe UL: Si (OC, H,). + 2H,O _ SiO, + 4C,H, OH (16.7) In the original $OI - gd \cehnique, Ihe gel thu, fom,ed i, liM dri<:t <>blaining cia>lomcr ",infofU:' mcnl . wnw of the <>tgano'l>I;la llie malerial i. a bs<>tI",d int<> the cro ... linked "ct· work. and I~C swolkn .. mple is placed into "'aler containing the catalyst (tyri<--~lIy a vol"tile h.s< ~"" h .. "nmo"ia or ethyl.mine), Hydrolysis to form lhe desired silica like p.1tticies proceeds rapidly at room t.mpera lure to yield in lhe order af 50 "'1 '1. fille' in less lhan an hou"''''·'''''', Alternal;wly. hydro~}'I"e r' mina'ffl chains are blended with enou!!,h TI;OS to OOlh cnd·lin k lhem and to pro"ide .ilic-~ by the hydr<>i)' Bolh Ir~nsmi!Sion and ..~nning ek"lrOn microscQPy have h«n used to chardeteriLe lh= " 0",1 composile materials',"'''','''''''. The information obtained in thi, way includes, (I ) the nalure of the precipitmed phase (p'H liculato or nOli ' particulate), (2) the a""rug<: pan ide ,i,e. if paniculate, (3) lhe di"rib~lion of panick ,izes. (4) lbe degrce 10 whieh llle particic'S are well defined (smootbn.., of the interfaces), and ( S) thc ckgrec of agglomeralion of Ihe partidc.. ~, A typical micrograph i. shown in figure 16,1"', Such resul" show lhat lhe parlidO'S fonnffl in this manner lypically ha'" a narrow di'lfibution of sizes, Wi l~ ",,,,t diameters lypicall y in the 200 2SO;" range''''''''. They arc g<:ncrJlly ",'oil di.pcTied and c<><:nli"lIy unagglo'neraled, ..-hieb ,uggc." tha, 'he rea~ti"n Inay in\'Oh'c simple h"mogcnco~, nudeation , T his is co",istenl wilh the fact 10" MULTIPHASE SVSrE MS 271 .lee""" flru .. 16.1 T .. n<mi»ion mk",,"ph fOI. poIy(dimclhyl ,.',% ~r1i"'ts lI,ovo;nl ;nd~nll y of one aooll><, a nd stp;lralfd by crou-linkfd nOl agglomerale unlc$$ . 'e ry high concenlralion, ,.~ ... n:achcd. The ... is lOme: e'-idc:nce Ihat. in at kUI IO~ cases. the po ... ,,>II: of II>< ""Iwo rk .;onS'",in, Ihe .... 0.'111 of III<-se p;lIIid cs. in Ihal Ihe a,~raie di~mete rs ob$w.·fd iftm 10 incr .... " 'ilh uh from ( ha ..~cte,iu'ion SIL>d>« of this type is Ih. conclu_ sion lhal basic calal)'su zcnerall y )'''Id plrtic1c:s Ihal are " -ell <' ··funy··''", Tltis conchloon is in ....... mcnl ...i lh ,""h. obt.aincd earl;"r in sol ~I ceramic.; in'"nliplions" '. poI)'~' " 'ould 16.2.4 S<-~n.rini Tr:chn;q1><S A number of ' _r~ and """tron ..~ucrini 'Iud,,,, ha", b«n carrifd ou l on Ih= ~lI.d d',lome,,' .11.1 ...... 1... '. S 212 STRU CTURES A ND PR OPERTIE S OF RU BBER LI KE NE TWORK S • a (1IA) f it. .... 16.1 Tho SA.XS an4 SA.NS ..,.ttering in'cn'i'ico 'hou&il' be be !>ironti"""." shown ••• func,ion of ,be 1'<""'" [Of a SiO,.. PDM S CQmpo.itc ",,,n.rin~ '-..:to' (in r..:iprocal AI'''. TIoc ,,~igll, of prdCn' i. indiao',-d in ,lIe tigure. and ,he "Umbe" 'lt8C1>cd '0 ,he ,wo c.,,'eO Jive ,."1",,, of 'he tern,i".1 ' lope (,,·h"'h arc .,.. inte'.., wLlh "'gan! '0 ch.mc'e,i,ing ,he rough ..... of tile inTeri"",,). ,h. phases may Ix: bioontin"ous (interP<'nl"ll"ating). are sl,,,wn in figure 16.2"" and arc di",ui;SCd furthe, in secti"" In .3. In general. ~uch data can pro'-id. cstimales of average partido size and possibly particle.,ize distribution. In addit;"n. lhe term· inal s lope~ give an indicate of the "aWn: of tn.: interfaces. with - 3 corresponding '0 rough interfaces a tld - 4 to smooth . Although s Nude~r Magnetic Reson.nce Another " -ay to char~ct"ri/c the i" ,ilu gcncr-~ Tcd panicl ..", is by nuckar l1\a~nc'ic resonance (N M It) imaging, ulilizing IH and lOSi magic-,m&lc spi,ming. with \".0di mensional fourier lransform 'pin.ttho t"" i>niqucil One spocific approach i. 1o study'H spin -.pin (r,) rela xalion ,im .... uf 'he prolun, in a PI)MS polymer ""' Ihey ,,'" hcing constr-~inCmc illuSlr:tli,,, results are gi,,,n io fig"", 16.3, \\hich .·I R imOb"> ..fa cloSiurncrie ' ample of I'D~' S ..~,"'ainin g in si", pr<"ipila'c<1 , ili.a, For .' igure 16.3 IH NMR images or a Si02 -PDMS clastomeric sample obtained with a two-dimensional spin-echo sequence having an echo time or 3.3 ms (A) and 22.7 ms (B)14) . The view is down the axis, and the resolution is 128>0: by 128x pixels or 21 1 and 236 mm, respectively (in A), and 211 mm in both axes (in B). Reprinted with permission rrom Garrido, l.. ct al. (199 1), Macromolecules, 24. Copyright 1997, American Chemical Society. "igure 16.4 1H N MR images of an POMS- aerogel composite obtained with a two-dimensional spinecho sequence having an echo time of 2.82 ms (a) and 122.6 ms (b) 145. MUlTIPHASE SYSTEMS 275 Its,ing purposes. tbis oon'posilc w," int~"'ion:olly mad~ !o be i")",mogen",,u,. with mllCb larger amounlS of silic;' being on tbe surra"", This W~< done by choos_ ing a sample with rda!i,'ely large dimen,ions. and ,'urrying out the plttipi!ation reaction for only" .ltvr! period ortim • . T he sample ,..".,ploycably indicalCS a rtJuccd mobility or the nelwork chains due 10 tM pr... nce of lIIe silica, I IS chang. in local ion w;tb lime can help Cb:lraCiCril<: the m" ' ,,menl of Ihe reaclio" from into Ihe sa mplc. Th is toch"iquc is "bvio", ly nonJcSl"'''Iive. hut if Ih. sampl" Clln be sacrifJCCJ. Ihen 'hots lohn rrom the <.liO(! in" gradien t column ""ilh regard 10 densily. by ciectron microscopy'''. or by x_r-.. y or 1>C~"on """"ering. A I'DMS el.""mer ",hich has t><.'Cn i"flaral".! into a silica ucrogel can "Iso he imagcll in Ihis way' ''. This is. sho,.-n in figure 16.4. and will be discussed f"nhcr in """lion 16.4, 16.2,6 Aging j'ermilling prttipil'llc~t sili~" parllcles 10 ",rna", in cont"ci "'ilh Ihcir "'1"COU~ c","lysl sol"Iion can permit Ihcm t~ "ago" or ··digesl .. ' ..·, ....... Eloclron microsc<>py results suUt't Ihal some reorganm";on is (IO."Curring.•..,tb the panicles becoming bener defined. more "nilorm in sile, and possibly e,'ell less aggrega ted , Thore sc<:m to be intor. 16.2.7 Densities Compariron, bet"...~" lhe ' -al ue> of wt"!. liller oblaincd rrotn dcnsily measurements and 1M va lue., oblainc>J di"•.;tly from ""cighl increases ~an give very useful informa(ion On tbe filler parlicle$_ I'M e.lorncrs''''·''' indi_ cales Ihot Ihcre are plOoobly either .'oids or "",e"ctcd organic gro ups in Ihe ~lIer panid"". and possibly both . in.:ompletc CO n",' rs,On of letr.lelho.ysilanc In ,i lica is a lso indicaled by ""Si NM II. me;lslI",,,,cn,,""'. 1 6 . 2,~ C~lor;melry I)ifferemial scanning calo"mctry mCasurements al low Icmp",.atu'C'l have becn carried oul "" POMS clam,me", oon'aiJling ,iii",. I'rccil'it;ltcd in .iw l... Tile prescn"" of the ,ilica was found 10 «>Jucc bolh Ihe •• Ienl of crY'lalli,,!;o" and Ib< ralc of cry>ialli/at,on when Ihe da>iomers "'ere in tile "nstrelehed stale. Thi, i, in inlCrC'ling co",,,,,t II) .imilar "udi,s or l' I)M$ ill 'he" retclle,l stale. wher<: 'he filler m"l' ratili,,,,,, ,he cry""lIi/A,I;on pr".:,,»'>U 276 STRUCTURES A ND PRO PER TIES OF RUBBERLI KE NETWOR KS ->,"- '" ~ " -" m 0 ~ 0 m a: ~ ," - - - ~ '" "" f i&u.. lo,S Compari"'" of'rOA "-==_.:.:::. . .::::...". . .::.:--'3..... , 300 , 400 500 600 700 BOO T ( ·C) tbormo~ .. "" fo, I' OM~ Tltt"'orb th., "'en: unfolloJ (U). Or co",.ined e;,hcr ,il"" pm:ipi!ai,u (A) Or oom L ru:rcial fume oi li "ilrO&<". 16.2,9 Thcrmogr~" imetr ic Ana lysi~ In a. least Wmc caS/:>. fillers pre\'.. n in figure 16.5"', The sampl"" in ",hieh Ihe Si D, "·a. introduced in ,il11..:em 1o have higher decomposilion lempcraturcs. A posSible mechanism for thi, imp"" 'emcm would be incrca.oo capabilily of the >ili.a produced in situ 10 Ii. up hydroxyl chain end. Ihat participale in lhe Mgrad,'i 16.2.10 M""hanical I'ropcrtic. and lOquilibrium S".lIing 16.).10.1 IIniaxial EXUflJWn T he reinforcing a bililY uf suc h particlco gene raled in si tu ha s been amply demonstrated ror a ,·.ricty uf dcforma,ions. includ ing unia xi.1 e,len~ion (~impk cI(>ng.~linn). bia~ilol oXlcnsion. , bcar, and torsion" " , In the ea $< of uniaxial cxl~n$i<)n. the modulu$ If' ) r""'lucnlly ine~a$C':l by mo .. than an ordcr of magnit ude. with the i"mherms generally showing the upturn, al high elongalion lhal an: lhc , ignatun: of good rcinforc.;mcnt"" ". Typical rc,ul1S are sbown in figure 16.6'''. AS i~ genera lly lb e case in filled elastomers. Ihere is ",e1\ to be cOII,jdcrablc irr~"ef>ibility in lhe isotherm'. "'hich i, th"ught H> he due I{) irn:<:o,w.blc ,Iidi ng or 'he cbain, owr lhe surface, of the filler p;rnides. Som" fill"'" mher tha" silie •. rur ",umpk tilani. (TiD,). d" ~i ,," "r.,...,;trai" iwthenn, MUlTI PHASE SYSTEM S 277 •• " , •• " z =• " ... ", ..., '.' •• • • ••• • •• .. . .. .. .. •• , .~' .· i ~ " .. 16.6 " R«lumJ ilre;; sIIo",".s" fu"",i"" of m:;iprocal elonga',,;>n f", PI)MS n,l· W01'" (rut ar< e;the, "Unned Of fIIk1l ,"'uk oil:m in ,,,. '" Filled .)'mOOls.n: fur ""'.)" obtai, ...) out of "'que""" to , .... , for f<wrnl>ih,y, ano! each cu,,~ j. I.bcl«l ~ il. ,h• .... igll! ('1<<<:""' offi'"" ,hoi Il thai arc ...,'-.rsible, indica1,ng ime""ling d,fTcl'en= in inrfa"" ch~miSlry, j) Til,,"i. ti liers, and Olher n(>nsilic~ filler<. '''" dj""u,sed r"nhc. in seclion 16.2 .13, 16.2.10.1 Adncsi<m &,.,-er" 1>lu«~,\ I, i. l'O',jblc 10 inlerp,.." C h,,,'. 278 STRUCT UR ES AN D PROPERTIES OF RUBBERLIKE NETWORKS -.., " \ 'A • ~. ,., ,., " , • • , , • •" • ,, .. , 0 , ,., OJ(I - t) Fit""" 16.7 Viol of ""Iume fra<1w," .. ,w, V..I Vif . c .,,'.Uilll! of"" unfilled I'DM S nel"-ork ,ebt;,~ \Q ,h.' of. filled rDMS netwm~ .•",i"" fill<, Io.Jing .,pn:...J.s voIum" r.,io offil." to ",bi:>e1 l1/(1 ~) (wbe", 6 i' lh~ vol",,"" frat.1Ooo of Type< of filler "~,,, . ilia. lilani. mi_,,-.I oxKks (0 ). ;;Iie. (0 ). and ,it"nia (,I.,,)''', <6) 16.1. 10_3 Hi,,-,i~1 f:_y,pn,,,',,,, A< .I,,,.<.l)' mc,nion"". Iher, ha ,'c bc<:n n"rn<:' O" , .'re'im.,," carrie"'s of deformalion are imponan! in Ihe characterizalion of cla<1ooncric male, ial" bul are nol much stu<.lieculi to impole Ihan simpic elongation. This is true even for cla"",nc", f'lloo in the usua l nlanncr. by blendinll of I",,(icics 1''';0' to cross-lin king '". II is obviously interest· ing a nd important 10 use a "ariel), of <.lororm'lion, 10 ch;"aclc,;';,IIo<>n'I.,,, '''''. TI,e .tn,i" can be oblainc" MU LT IPHAS£ SYSTEMS iIIll<" ... . ... ,'" ••• • -e .!z -- " 279 .. ••• •• • • ••• I ----r::r.===~~. ... !--=0123 ,la,''''''''.... 45 ",u, ~'II"" 16.8 S,,. .. -,'fli," iso,hcrrn. for I' lJMS ",t.:. ",mf"""d in in ckmgaljon (,.~"n to (he left of ,he ",rtk.1 d.,hOO line, willt" I " II aoo in bi •• ial <" ()''''_ The f,IIN """nu ,he dal. u..... (0 leol for rt".."; b,li,y, I'" rep''''''' oblain<;d for both elongation (,;:.ciproca ' dongalion ... _1 " I) and "ompression (<> ' > J) arc therefore depicted in f'lIure 16.g''". where the modulus i. 1'101\..-..1 again" " - ', J I i. 01>, ;<>". that '-ery s(wng rcinrorci,,& efT""is from the p"",ipilaled sil;c,. Oo:<; Uf ror bia< ial •• len.ion . a < well as r", dO"galion . Thi~ is evident both from Ihe large upward shirts of Ihe iSOlhcll"s a! a ""hole. and from Ihe pro"ouno...J optn"" oolh al high cionllation< (0 ' " I) and "( high c'nmpr",,;uns (bia~;al c",c",;on,) '" I > I). A, can al,o Ix >cen. S"" ~ ,lcs;rablc u!"urn s do nol occur for the rcfcrc,u.~' (unfilled) " ,ml'k._ 280 STR UCTURES AND PROPERTIES OF RU8BERLI KE NETWOR KS 11 is inlereSlins 10 nOle Ihallh ••• tent or Ihe Teinfo=",enl, as gauged by Ihe ",agI1itmlcs of Ihe upturns, i~ apprnx;malc1y lhc same ;n un;ui,,1 and b;,,,ial extensio n. The range of de(on"alinn OYcr which il """urs, however, ",-",m, 10 t.: larger in Ihe case nr lhe bia,ial exlension . The pronouncffi ma.,ima and minim" in Ih. iSOlherms for Ihe filled ctaSiomers in biaxial eXlension a re nOI underslood al present IS' , and rep",senl chaHenges 10 theories on the origin of Teinforeing effe<1S in claSlomc ... in gencnd. The cfTcct of filler conlent On the ruplure moduli l.r 'l. in elo ngation and in comprL.,.ion i, ,hown in figure 16.9. 1'h. derendcn("(S art; seen 10 be 'Iuit. similar, wi,h Ihe cu"'« r", uniaxial anJ bi",i,\! .,Icnsion being very Marly par-.. l1 d 10 one anolher. Figure 16.10 ,hows the effecls on lhe deformalion at rupture n, in clon&"lion and in rompre",ion. The 1"'0 behaviol'll are again quil. 'imilar, in thai increase in fil ler conlent ch"nges th"", ''''ue, of <>, in Ihe !I;"'''''lion of smaller dcfommtionJ (0 _ I). Similar ly, of the energy J-.~ requi«-d for rupture in both uniaxial and bi •• ial (""mpre,sion) eXlensioll c.n t.: oblainc-d by direct in legr-dlion of Jf'II". For bolh deformalion" lhe incr<:ase, in [F J, pr<:dorninatc IWot the decrease' in a" and, a, a result, 10, incre.ses ,ub.sl"ntially with increase in filler content " • . The Iwa curve, lic nearly par:,lId to one anolher, in a way ,imila r 10 thai ,ho""n in figure 16.9, ,'"1"", ,. ~ , • E ~ ~ ••• '" "0 •• £ •• Conl"nl f~ .. r of .mea " (wt%) 11'>.9 HfC<1 or lille' """'e'" on [he ,upi"'" "",dul". '" in"""r""'"," [r'L, ( 6 )'''', (S<... f'H"'" I~.~.) .Io"~"t;')" If' ,., (0 ).• ,," MU LTI PHASE SYSTEMS 281 • ' ~ - , , • 0 0 0 • 0 , _.._--------- "---_. _---_. -------------_. -_. Content 01 s il ica (wt%) f1~~ ... 16. 10 l:lfc<.1 <>f roll", (,"Qnt<:n, 0"' If>< in ""mp"";on ...,'><. IS« ~g"1"C> 16.S and 16.9,) /6, 1, 10.4 She", As desc, ibed in appendi, C. ru,c shear inv<>lves (\<'~",ion in Ih,tt pe,pe"d~" 'a, direelion. "'i,hoUl ""alion Qf ,"" principal • .<<$ or lhe ,'rain. II is ~on .'cnien l ly impo .....d by Slrelching " wide roclangular ,hoci in a ;n"oduccoth.rms obt"i,\Cd rur both unfliloo and in-'; IU-~Il oo rDMS ",,,works. represented in le. m, of the pure . hear modulu. G and Ihe rrirn;ipal eM.",;on ratio <>, "re shown in r,gul'(.' 16.1 L TIl<: r",,,11S in simplc . hear, in Icr",s of the , h.", " .. in 1. were csf filler shifts Ihe ;""he,m' upward '<) ,ignif,canlly highe' n1 u". "f 11,,, modulu,. In the ,,"!Ie of the filled nClwork!. ,h. re;s an ini,i.1 ,lccreasc in the m,> 282 STRUC TURES AND PROPERTIES OF RUBBER LIKE NE TWORKS ,.,,------------, ,. - Sn'w lU • E E Z ~ "., lU '·'11\.0 ' 0 ,., \;:;":: " '.,.. ,., LO • -- 12.' 1 •. ' " • . ... " ... • • • • • • • U " • • ,., a .l~ ... 16.1 1 s,,,,» ..... in i"'them" ror 1'[)MS-.n... tho da'" "« p""" . hear' ''. The filkd poi"" rep=-"' ror "'vil>ili'y. From lbe .. resulls. it "a n r~ ad ily be seen thai lite ~Ikd ""mples ha'" much higher shear modu li Ih~ " the corresponding un filled >ample. Also. al high defor. malion. lhere are pronolllOttJ uplurns in lhe rcJuecd slrcss or mod ulu. in the cas< or lhe f'lled ..'mpltS. which indicalC$ wry gOl:>d reinforcement. T he m"snilUdes of Ihe UplUmS are much less prono u~, how<\"Cr. Iha" Ihey wen: found 10 be in unia~ial exten,ion and compression . In ~ny ca;e. Ihc exte".ion m,io <.' or ,be .. main 'Y al wbich lhe upturn aprears is seen 10 de<;n:asc wilh inc"," .. in ""ighl ",,<"Cen1 or filk,. A, is ","all)' the c""," · ·'o. lhe maximum dcf<.>rmability d",,,,",,," as ",,11. C learly. the t>chavi<>, ohscr\'cJ here ;, very ,;mil,,, 10 ,ha, d"",ribc<1 p,...,-i"u MUlTIPHASE SYSTEMS 283 I\IS(> of inl~rtSI is 1he depend,,,,,., of I~e ul!;m31~ propertio;s on Ih. silica con· lent' ''". 1\ wal o""",,'ed thai o. and"/. dc"(;r~,,>c will, ill<:rea", in silica conlcnl. ""hil< the values of the mo"",. comcnt •• bUI appe'" [() I>c 1.,'\Oling oIT or c.-cn dcc.ca.ing '" highcr conlcnls. This is prcsumably due '(> de<:reasing extensibilily, The rurtur~ energy E, dcpend $ on .il"'a conlem in " ""ay vcry similar 10 lh~l show" by (i, and/,', 1b,l,JO.5 Torsi"" 'Ille "prar~IUS f"r Ihe lo~i,on meaSureme,," was ba~d (>n Gem', modificalion '''' of Ih. ori!!i'ml ""uipmcnt dcserihcd by Treloar", For Ihi' type of defo,malion. thc shear modulus can be calculaled from (i .. 2M /"1/.",'. when: .If i. the lors;on couple. ¢ i, Ih. 'or$ion in ,"dian, p"r unil Icn~lh of lhe 'lr~'n~-d .. i~ll.tlglh. "",I " ;, Ih. u"""3ined radius , Wide ra nges of "'1~lional d.fom'"lion CIOn be invesl;galoo . for uample. up 10 1500' tOj, Some i50lherm, obl~inOO on PDMS--SiO, ci ..wmers filk:d in .. inability. 10 dale, of reaching sufflciemly high lorsiona l dcfomlalion . .",- ",, ••• Z OH " 0 0 ~ 0 ·n 0 0 ~ •.•0,0;----+.----;';:----~ • ' .3 •• ,..a, radians •• ~lll"" 16.12 Shear ",cod"li f", PDMS ,il",,- cla,lo",,,,,, .•<' "oil ~"glh of 11>0 """roedul", ;"""'a"" COOI",,,,,u>iy "' ,he ","oun' or fillc. " ;0<",.>«1 ;n tho "0."",,0« ''''''''" \be rUPlnn: P>""" 0.0. I ).l. 1".' . 19.". "",I 2~ ,4 "'1%. Th< ,,,,,'",~I J"hru linoo 28' STRUCTURES AND PROPERTIES OF RUBBERLIKE NETWOR KS u . , _. ~ 0 • U 0.> ,. , ,., '" % S10 2 "" ,• • ," '" K '" T, '" T<1>II'<'.'"" ..!< pur_ P<"C!' 0( cl. rity. the cu"'", h,." bcefI >-hi~«I .',"If.,ily ~Ion¥ ,I><: O,~i"alc. f igu'" 16.13 16110.6 Drnum;" MI'chanirul PFOpalil's Mc~s"'Cmcnl or d)'n~mic mechankal losses can pl<ms'''. Of p'tltieular interest ill s",,1t c,pcrimen" ;, the ch"nge in fracli"na l lenglh or " &am ple "'ilh time. ,in", Ihi. characterizes Ihe amount of c~p 0' compres ..,,- - -- - - ----- - .c::. 1 .0 1 I" ~ 0 ._ 0 - """-0 0.' 0 .7 0 40 20 60 80 Time, hours .'~"f~ 16.1~ Err«, ofcydi< ""'" on unfoll«! I'D MS anJ P D~I S ela"omcr ro lled wilh fum. , ,II< dtrcndc-nc< of (","&lh on limo: unfi ll "'p,...,,,!Cd., 1.1 1'0 r::::::::::::::::= r.... 0 .8 J- 0.7 10---02"0" --C4~0---'! 60""---;;80 Time, hours ""r,'k:.' r ill"" 16.15 f:rrOCI of ,)1<1", "res.< "', PD.\ t S and PIJ ~I S da'h>"' .... f,11ed " 'i'h >ilic. I'f',M<JJ. /lep., A." . :J)' coo'""") of M;.,...I I;IoU". 1",- 286 STRUCTURES AND PROPER TIES OF RUBBERliKE NETWOR KS this case a rc..:en 10 Show wry link compn:ssion SCt''', T his ' ''UesIS Ihallhc in .ir u si lica ""n retard Ihe chemical chang.'S causW by cyclic $1rcSS<"s. ~s wdl as those cau,"-' 16.1./0.8 11111"'" Rcsi,lI1$-modified SiD, a"d SiOd TlO, glasses willi I'DM S conlenl. ranging from 0 10 65 wI ~. , As "'peeled. ol1ly samples with relaliwly high ceramic conlenl. were suflkieml}' briule I(} be . ,,,di<>.! in Ihis manner, The inftue""" (}f 1'0.\ 15 ",nlenl and I'DMS m(}kcula r weighl <>n lhe ;"'",10' strength is shown in figures 16.16 and 16.11. respeetively. The larger Ihe amou'" of PlJMS usN. or Ihe high~r its mole<:ular weighl , thl: higher Ihe impaCI Strength For PDMS·modified SiD, gla.=, .Imetural analy,;, 'hows Ih;lI Ihi' h)'brid ",alerial has 50me degree of localized phase separ~lion of Ihe PDMS compon~nI • • ,-cn Ihough OK-t.rm inated I'DMS can be .uca:s>fu]ly ;m;<>rl'0mled inlO Ihe SiD, netw<)rk by chl:",ica i bolldillg. The »[)MS "ompone"t can beba'.... as an cl",tomcrit phasc because IhI: glass t ... ", il ion t""'reralu"" of PDM S is far below room '~mp"'alurc'''. Whcn Ihe m"leri,,1 i. ,ubjected 10 an impaci t... ,_ lhc f'DMS cXlmponent .-all kMorb a great deal of CIIergy by mOlioo. of the PD:l.1S chains, "" d can amelior"le Ihe growlh of cracks and fraclnre. Therefor<:, eonsider.ble loughening uf lhe glass c.1n be ""hie"ed by incre.1Sing Ihc a mount of "DM S intr(>(IUl-ed . TI>e elTecl ofincr~"st in I'DM S molecular "",ighl may be 10 incrca" Ihe pha.e "''''''al ion which lead. to the encrgy-aMorbing dom"in,'67 .- .t • -- .-- ,' • • • •, I •• • .. • WI .,. 01 f>OIolS • •• .. i M- ~j o. •j t3 n~ ~ .. 16.16 I >Cpc"dc<>,,, of I,,~ e,l<mole; of the imp_" ,"~ng'h on lhe "",O"nt of PlntS;n PDMS-",ud;Ii<.-.J SiO, gla,...,' ''. 'n .e imp'"' "",ng' h, "w" "h.. ,ntd frum ( 0 1 the Ch.'1'l' I> MlJLTlPHAS E SY ST EM S 287 ....,.---...,--------1 _ ...... we.ghl 01 PDMS Fa~;nu·w.;ghIlmpacI. I"Il _ lest) T ..., 16.1 1 Dtpdiliro SiO, Illas",." . M;erosropic ob,;c1"\·" ,ion of fmclure surf"c.."> can also pr(",ide uS"f,,1 informa· I;on ror ,imcrprc!ing impaci wcnglhs in lermS of Ihe physic"~1 proces..cs involved ' '''_ S~;I;call)". ,mOOlh f""~lure surfaces.rc .""",ialed wilh CQmpklti)' brittle f.ilure' . wilh littl< cfT""ti,'e ",-c,,'ro in Ihe ca,;c of $ltmpl., having relatively I",,· PI) MS cOnlemS. In conl rd" . -""mpics wi,h high PO MS conlcnl had fract ure surf.= showing some deG-'ee of "whilcnong" '" .hearing, This suggests • docli lc. energy.absorbing r"pon", 10 Ihe impacts. ,.'ilh increased resi.lance 10 crad pro· paS'lIion. 16,l _HJ_9 i.qlli/ibrillm S,,'dlinK In a ,welling e. pcriOl",'. the networ\; i. t}'pieally placed imo ;tn cxee", o f ""Ivenl. "'hi<:h it imbtbe. unlilthe dil~lional , 'relching or the ehaim p",.,."nt, fuu her absorption ... r solvent"·..·".. Gener.. 1 aspeclS or thi s non mechanical Iype of dcformalkm are described in chapter 6. Adsorption of ",me nelwork chain. "nlo the I~,,'icle! "f a reinforocd daSlomcr would. of COil"", reduce tbe «t~nt or swelling at c"tj ui li briutn and lead 10 in ~ otnogcneollS strain !iClds""_ Cha racteri7.at;on or soch equilibrium ,welling .. K furn:"on of ti lkr constitution. nile, Kmou"'. and Ihe method of gI,neraling and incorporal ing il inlo an elastomer a re the issues "f greatesl inter.", with regard to Ihis type of dcron"a,ion . An cxample or such ~n in"esti);"lio" was dc"SCribcd in figure 16.7, AlthOUgh this i, a potentially intcresting arca . ""ult, to dale are rather limitro. 16,2, II Compariso", "mung Va, ious Silicd-lla",d Fillers As i. now obvious. there are a variely or wa)s to gcncmlc silica-Iype fillers Il>cfll l ror reinforcing I' I)MS nCI"'<>rks "",l the ,·, te"t It> which ,ucb nilers pro"ide reinforocm"nl wa, ch''''lelcri"..1 III U study b)' Sun and Mar k t11. T he material. and techniques etnr1o)'ro we "" (Ii inco'potating" commercia' silie:! which has 288 STRUCTURES AND PROPERTIES OF RUBBERLI KE NETWOR KS I><:ell lreal"IY'.cd TEOS and a l>o oonlaining wme rOM S. The resulting ml"OS of (4) and (6). and the largesl value oflhe rupture energy for (4). A bener ",olecular understanding or liller reinforce",.nl C<)uld provide a dCloil"0 achic\'ed in Ihis polymer. suggesling th"1 such Ct)'Stallil,tlion is nol important for this Iypc of reinforcemenl"'. A large number of olhcr pol~mers hJ\" bc<:n lrealoo in Ihis way. as is de'iCribcd elsewhere ....... For exomple. Ihe ..,me lochniqucs ha,'e al'" b."en .hown to gi>'e good reinforcement in pOlyiwb ul)'lonc claSiomcro"', and in poly(el hyl act)'· lale)"~, The resull , for pOlyisobulylene are . hown in flgur. 16.18. In Ihe case of some polymers. il is useful to "h,><)sc R group, in Ihe alkoxy , i!;m" Si(OR)., so as 10 make Ihe molecule more hydrophobic Ihan TEaS. In Ihe case or Ihe poly(c,h),1 acrylate). il appeal'S Ihal Ihe silic. 1'n.'Cipita lion can be e"rric' Pr<>' ,·idc good re'nforcemrm. Onc in l ereslin~ diff.ren'..... ho""\,,r. is the ons.,,,·ation Ih .• , ,lie S'",!s-$I roi~ i""herms in thcsc cases f""l""",ly ha ve beller I'<:,'crsibilily' '''_ Thi! i, illusmlomer;" matrix. There is now inerea>.ing interest in inlroducing fillers oon';>Iing of more Ihan one "'idc ....... " •. ,,,, ... ,, ..,. OtiC of the hores is Ih"tlhcrc ",'ill be an ad \'3nlageo,", "delega lion or I'<:.p<>n.ibili,ic;" in Ihe .... mi_,cd ",id,,,. ", ith one 'YP" of ",u'eri,,] impro>'ing "nc SC' "f properli .. :lnd ""o,her improving a diff"r.,,1 SCI. "'''''h M ULT IPHA SE S YSH MS 289 ~ E z fi ftU'" 16.lg SIT"W-.tf.i. i"'l!>tnn. for pol~-i"'bulylc"" I!C\W",k< ",i.forom wilh 'ilie> l!"fI<",teJ i•• itu'" F.... 0",1<, "'p,,,,,,nl ",.uit. obtail\Cd doi"8 l"lract",,')"i'a ••• nd "i.~ .. «p<e>cnl "",ulu ",ing phenyltrie1ho.)"ila nc ("'hich ca n po",itoly introdu« o filler pa<1ickl), 16.3 Prepa ration of Bi contimlOUS Sv s1ems At SOme eotnpojiho", and undcr SOtoC ),y,lrolysi. cond ition,. bicontinuous phases can be oblainc-d (""il h the .i liea and 1)(}lymer phase< interpenelml,"g One "n"lhot) _Some nr II,. ""itl.nce rnr such "'ttrpenetralion i. from ,mall-.nfle . -ray sc"tlering (SAXS) and ,mall_angle neutron scattering (SANS) prof,lcs./oO· 'l. such as were illustrated in figure 16,2. The observoo peak suggests thallhe SiD, cannot consiSi of indi.idual pa,,"'Ic •. but "'Iher form. 3 ronli n"o"' nelwurk. Ol he, cvidcn<" i< fron! ,tress strain i$Othcrm, in elongation. Since Ihe two network. i"'«poncnate one ."mher. 'he n,echan"",1 propertic"S e. hibiloo hy the ,..IOrial can be ~ery peculiar. In the Ii.."t deformation "f tho ,·irgin ma'erial, the silica net"'·oTk cao giv.: a ""y high inilial modulus. but 0""" Ihi. Slruclurc is hroken . addi\ion,,1 ,Icforma\ioll cycle, "an ,tidicale ",,,,oj, lower ""Iue' "I" th~ mooulu<. The mt"hani,,,, I""r the g"' ~""ion ul" hin . 0<.""""'£ cuher herol. or ~fte. the I' 290 STRUCT UR ES AND PROPERTIES OF RUBBERLIKE NETWORKS ... '0 "E E 0' z 0, '" o. "- o. o. o. o. o. o. 0.' fi~... 16.19 Strc>O-"", in isotherm, for PJ)~I S ... ,,",. ..... nro~ "ith tilonl' pani • • "" gener.ted in >it" I". Each <:II"" i. ",bt« resul« "",.;,,<J out of oJCq""""" to ttit fo .... , 16.4 In Situ Generation of Elastomers in Ceramics If Ihe hydrolyses ;n organosilica le-poTymer syslem. are c~rrie "C ERAMER "'" '''''. or " 1'01. YCERAM"'I',m" ... ,,.,. It is obviously of considetable inler.,IIO dClermine ho .... Ihe pol)'meric phas<:. orten clallomen.;. modi· fie'S th. ceramic in which il i. disre rscd. One ill""""i", proreny of importance in .uelt h)'brid orga nic-inorganic com· """it~ i. hardn=. 11><: ca«: of Ihe PDM S-SiO, sy1lem is iliumalN in figure 16_2(1 "·_ T he hardness oould be ,".riM greally by cha ngi"g Ihc r~ l io of OIganic. l<>inor8'mic c~aracler_ as me~s"red by lite mola, ra l; O of organic R groupo (lIeo:. e ll , si de grou ps) 10 Si alom •. Low valllCS of Ih. R/Si ra lio yield a briulc cerdmic, and high val .... }'icld an elaslo mer rein forc.:d in sil\!. The most in leresl iug mn gc of value •. RI Si _ I. co n giW" hybrid material (ltat C'~n be viewed as" ceramic of rMuc'Cd brinlene.. or an cla",omer of inc,eased ~ardness. dcpcuding on one', poinl of vic ..... The imraCI rrlc" Su' emenI S already described ..1 a", als<> >'CI)' ">cflll for characleri,ing ma lerials of Ihi. lyre 11 i. ~Iw """,ible 10 u>c Ihe NM R lechnique described carlier 10 char-deleri"" 00",1'<"';("" in Ihi. calegory. 'IS wa. shown in ligure, I~.] and 16.4. Of p,,,(ieula, inlercS! ;, Ihe one re,Uining 10 ",,'oget.. which are obtained hy d.ying .i lica gci. ,u pereriltc.1l1y to ",ispy, f",un like .I',"elure" It i. possihle I" Vdc," um impreg""le MUlTIPHASE SYSTEMS ~ •• ,, ~ 'ioi -''-'i i, Tough . . - - - ~ ---- ------ 0 "• ,, I,, r c " .. • fl';11I. .-----r---------..---------------.. • ----+---,,, 1,, -,,-----~ r 2'9 ' , , , , , , , ! , • ,~-~ •• 1 ElUtom. ,", ., •• RISi .'ta"'" 1(>.:1(1 Deporu!c""" of lhe O·"",k> hard..,., of I'DMS oo"'p<»J",s "" lhe ,.,jo of al l yl V'Jopo 10 oil""''' .Iom, ,... The "I"'" dn:"-" ,"Q'f<'JN)OO IQ bjmodal I'D~S. and the filled dfClos 10 unjrnodall'DMS . • uch an acwgci wilh I' OM S cOlllalnmg MllIte . inyl gwups 10 facililalC SUDsequenl cross-lin king with ra,halion'''. Figu,"" 16.4 show' two NM R imoge> obtainw on such a malcri"l. T ho ,·", ialion of,;go .1 inl"mil y aCros, tho ,ample can be ~ sed 10 oblain i"formalion Ott Ihc di 16.5 In Siw Generat ion of Ca lalysls in Pol vme rs In Ihi' lyp< of orsanic-inQrg"n;c composite. a relali".ly unrca~li,'< substance i. lr.nsform<Xl imo " calalylically "~Iiv. one wilhi" a polymer which serws as a pr(}1«:tive organic ,·" .. ironmenl. An illustra l;on of lhe lechiquc invol,·e'j nickel fO, male dissol'-.:d inlo a solUlion of poly(elhylenc ".ilk) (PEO) in elhylene glycol. ~nd Ihen Ihermall y d...·<>n'posed in $il " to forn, findy di\'id<Xl (high ,urface area) nickel particles ..•. The dccclllposilio n reaclion is ( 168) When Ihe reacl<Xl "">lure i< ,ubMjuclItly dried . the nic~d parlic"" arc p'otecled by Iheir being on<·.ps"I.1<..:I in llIe PEO matrix . A "'~n nin g eleclron micrograph of weh a malerial i~ ,hown in fig"re 16.2 1. The nickel parlicb (al a loading of approxima lely 1.6 wI ~. ) a rc d".,ly in e"id"l>C<'; they seem 10 be ...,,11 di,per>ed. and appea r 10 haw di.melers of Ihe order of 0.4 ~m . The bonding belwee" Ih. particles and Ih. p<,lym<, matri. is se.:n 10 be relalivdy poor. but thi~ should nO! be a disadv,mtagc In Ihe pre.... nl ( nonmoch"nocal) application. Fn 292 STR UCTURES AND PR OPERTIES OF RUBBERLI KE NETW ORKS Flt«< 16.11 !leann,n, ~I .... eI..... "'" IIIJCfOp'OplI 01 a PEO lIIi umple (7.6% Nil a, on ori&i· Tho: Iml,h 01 the bot ~I '0 ~ jUIO. """""'ion 01 »JO~ '''. the Sltm< reSOlut ion .. ~gure 16.2 1. i. ~ho"-n in fogu", 16,22. n.. partieln a re ...,n to be: ,,"en di~pctscd. in agrttmen t wilh the elCCtron mk toscop~ resu lts The pr<) l ~tM cata lyst s.ample can lhen be ul ili~ by addins portion. of il10 a t.a~tion medium. which dinol."C$ lhe protocli." polymer and ",teascl Ihe nickel panklts r~ calalynns the ",aclion, For <.ample, the Ni. PEO sample "·a, .·ery dTt<:ti.'c in the hyd'0l"nation of an ~""lefin. ~nd should h••'. lrealer wn.enicntt and e.. tcndW .... If ~f•. The PEO matrix ...... chosen b«aUK of its solubility in a " 'ide variety of sol."nu. bolt il bas the disad.·.nlal" 01 high pcrmabi1itJ:.!0 various pses (.. hkh OCIuld panly inaai''&I< I"" ca..,lyst). Otl>cr poI)·men'''· . panicu· t.rly elastQtnCrs ,ItCh as poIY;$Ob.nyltnc""', ...it h ill unlllWllly low ~bilil y, ..·ould probably be bell.. luitM for tbi, unusWlII,.... 01 application. 16.6 In Situ Polymerizations of Glassy Po lymer' 16.6.1 150"",* Systems It i. ulso possible to obtain reinforttmcn t by polymeri~i n\: mono"",r. such a• .. yreno, 10 ylcld hard glu.y doma in, within Mn clallomcr ,:toll,. In PDMS. 10'" IXH~nl"'~lions of Ilyrcnc &i •• lo,,·-mol«:ular· .... iglll ~I ymtr that acts mOrc lik. a ""fl. pl"lIci~inl liquid IbMn ~. rcinforcin, I"'"k.., '. The .!>stncc of •• idcna: fOT pol)"ly"'M (PS) pankl", allo"..:r iI},"''''' oonccn"alion, is dCmOn.,"'dIM b}' the .k<:tron mkr"""phl 1" ...."'"<:<1 in ~Iure 16.2). The pla'ikirini ~fT~t ;s FI&~ ... 16.ll T h< EIJXA nick, 1 dlstribu1ion map'" of Ih<.om< .. mplo .ro. ,hm>'n in foguro 16,21. Ficu ... 11>. 2.1 Scann ins .lKtron milyr al 1M bollom COff<'l"'nd 10 2 ~m . 29. STRUCTURES AN D PROPERTIES OF RUBBERLI KE NETWOR KS O. ' $ O. '2 <:'e 0 .01 E Z .~ i... :.. 0.000 ' 7.5 ~ .,- 0 0.0 0.02 -----:; -- _ ~~ '~ D.3 W""" 0 10:000 0 - 0 c , '-T-;;'.--'--.t.L~ 0.2 0. . "., 0.& 0.8 1.0 .1~..... 16..u S,,,,.,._.,,,,in ,.... horm' for ,II< PDMS PS com"",i,.."". E:tcb c " .. igllt percen' of PS present in tbe <:ompositc, and Ib< d..bod lit><> Iocat. tbe ",I"i"ely li"".r """io., of Ih. cu".... cha raclori,,ro by II>c M""""y. Ri'lin ""nllan"". indic"'OO by the S1ress-slr~in r.suh. oblaincC<-'il "'inforcemen1. or panicular ;me,.."t is the ract thai in add it ion In the inc ",a... in modulu. a nd ullimate Sirength , thc", are inere.... in maximum eJongre .o(\c" ."d presumably 10'" Iheir rcinforeing capabi lilY. 1'0' thiS reawn. simi!;i< ,lUdi". I,,,,,, ,·."ied Oul minI'- poly{diphcny\sil<"",,,,) as the Ix"" MULTIPHASE SYSTEMS 2'95 n:inforeing phaSC,:.-oo, It has a rdati"ely high ~ of 49"C. i. cf)'sta llin", and has an ntraordinaroly hLgh mchmg point of 550' 0 .The parlid<:1 of gla,sy polymer generaled by such in Silu pol)"Rl<=.i1.ati<)ns aT<' generaUy appro,imalely spherical. and lhe system< aT<' ;"olropk T he following seclion shows how II,,,.., Iwo con$trainlS ,an be rela.ed. 16.6.2 AnisoLfopic SySlem. II is possible 10 COnverl Ihe "~"'iaU)' sphe'ical I'S don",in. described al><>", 10 rodlikc ell ipsoidal panic!es 109 "'. Firsl. lhe I'S- eb>lontcr composile i! raised to a temremtare w"n above Ihe T, of ilL<; 1'5, H i, then $!retched uniaxiaUy. and rool«l whik in lhe ,Irtlch«l ;tale. The parlicles art deformed into prolate ellipsoid,. and relain thi. shape wt.cn cooled, When Ihe deforming for<;/; is "",,(,..«1. Ille mer is obs,2S. They were found to have axial ralios of approxlmalci)' 2. and 10 h,,,lt Ihcir axes preferentially orienroo in lhe di""'lion of Ih. high_Iemp"rature strelching"". The reinfotttmenr rh~ y providc"d waS cha racrcrilt-' 01"",,· sis~I),, ·_ It i•• 1'0 p".,iblc 10 gonenOiC ohl'OIe dhp>oi'ls b~ 'tr.,chin g such a I'$--I'DMS elastomer biaxitlJIy; for example. by inft"t;on of a similarl y wl"tcn«l of Ihe m:JteriaI 2ll . .hee, 16.7 FillerS Respo nd ing to Magnetic Fields II i. po~ibk to I"el~'''' particles of molals or 'HC,"' oxides b}' rcial<-d I""hniq ae" Examples are tbe ill silu generation of hydra led iron oxide by the hydrolysis of rerr;c chloride, and iron and iron oxides by I"" Ihermol)",i. of iron carbonyl,. S""h panic",", can . of course. also be prepared ""I"lfalely and liI<;n hlcndcd in,o a polymer . They . .. n Ihe " be lock«l inlO position eilher by cooling Ihe polymer ;n lo Ihe cf)'slalline or glassy "ale. or by cross· !in king ir ill lhe ease of an elasomer' 2'.' '6.l' 7_ or partK:ular i"l~rcS' 'tl Ihis e"legor) are I~"riclc, Ih~1 re.pond 10 au CA,ern.1 magnetic field, For example. magnelic rerrile porticles di'~rsed in pOM S ca" be aligned in a magne,ic f,eld durinl\ cToss·!inklng. ]" Ihi, wa}'. aniSOlropic m",han· i<;al pfopcrtlC' eS<;lIliall}' ~r h"nc,, 1 parlides. The reinfottt",enl i, found '" he S'llmfica nll)' higher ,n the di"""I,on p<,ralld 1<) Ihe magnetic lin .. of f •• ,,'e" 21.2" . as is illu,lraled in f'gure 16. 27. (b) «j fiC- 1(;,25 Scanning ol«lroo miample. Ib) dnngOI;"" of 1.1. (e) M UlTIPH A.SE SYSTEMS ,- --- 297 --• ,,, • ." ,• ..,. ',c---Cu:c- --;:.--- ,,---;.,•...J · ·O$•• " 11K." 16.26 SIrc»-.lr.i" ;.."he,,,,, of ,he dr .... r.,io .r>«l ,cpr=. dala use<J 10 'o>t rOT ",,-ersibili,y. !I..printed ... lth permi .. ion from Wang. S. a nd ~ t afk J. E. (1990). ,I/Mromoiff.u,., 2.1. Copyhghl 1'197. America" Chern"'.1 So This techniq\IC ""uld be rombllled " itb Ihe in ,iw approach by l!"n.raling metal or ll\Clal oxide magnelK: p'orliclcs in a Plagnetie f,eid l "'''', for c~amrk by the t hermoJ}'.,. Of photolysis of a melal <.rbonyl. 16.8 Fillers of Controlled Crystalline Stru<;ture Commercially. lhc n;;nfortl.-nlonl 01 rOMS has eOl"cnlratcxl on In.; ","" of high. surface-are•.,mea. which is much u ....1 hccau ... "r 1he e~cdlcm ",info""mcnl ;1 provides. T hi. malefial i. am()fpho",. whelhef generaled "'p"ra..,I)" ,,, in Sil u. a~ ;S the earhon black uS«! in ,he rdnfor .....'1l1.c"l of o lher dasse. of ~lae "",,,fuIIO 'Iud)" cryStalline follers. prcferahly of known "nlclu 'e~. An .,ample <:>r a clas., "f such in<>rW",ic ~uh"a"ces "r~ the >.«>1;1«. ,,"hieh aro hydraled , il ie,..", of aluminllm "ott ,odium o f Ihe gene'a' fo'mul, Na, O . AI, O , .nSiO,·.<1l , 0, "rhoi. """,111= ha "0 lI<."n ~ x tcn,ivcl)' "Ullied . I.,gel)" 1I<.""u,", 'ho)" arc ,,,,,-d (,.-il h "n, '" "",ee ..'a'il ies). "nd Ihe)' hav", IInpo"~nl 298 STRUCTURES A ND PROPE RTIES OF RUBBE RLIKE NETWOR KS ,. ~ 60" ,E , 00, 11 0. 50 E Z • .w F~u", 16.27 Str<>S .. ", in isotherm' fo, I'DMS comp<>;itc.; containing ma~flctOcally ""p<m,ive I"Mieb''' . r"'ch cu've is iJenlifooJ by lhe ""ighl pe""IlI o(foIle, p,=nl.I~. >"eng, h or ,be maW'.. ie /kid (in pu») applied du,ing <,,,,,,unk,ng.'oo the oricnt .. ion (1",.11<10' pe'pendietll. ,) or 'he .. mpl. strip ",I"i,,, '0 ,he lin •• offoroo 0('1Ie m.g· netic fl<'ld. app1;cation. al ca1al~ts . ...,rp,ion agontS. ion o,~hangc rWns. and..., on,<»·m Two studi.. ~ "ve investisatc-d the use of zeolites for ",info"",mC"1 of PDMS elastOfrlCrs'""·'''''. Two types of ?colites, ha,iog cavity si>cos o( 4 "od 10 A. re.f't'C" ti,.. ly. were employed. and the reinfo,cement they provided was characteri7.ed by Mres~_ ~lr~;n measurements on the elaslom.", in e("ngotion. Some of lite results "t\O shown in figut\O t6.28. and demonstrate thc good re;nfo«;cment prO\idc-d by Ihese Iypes of f,llers. II W""d thai the :r;C<)litc wilh Ihc la'ger pore .i", would b<: botler (1(""'" lraled b)' Ihe polymer eh~ins, Ihus gi,iog .ignifocanlly beller reinfQrcemcnL Small. angle neulron sc~lt"ring prof,les for 'he I' l)MS el~'lomers filkJ "'ilh the IWO Iypes o( '.colile were nearly tho: smne "'. ho"'.",r. Thi. argues agai".t chain ;Tllcrpene1r.uion ",;<:urring in Ih, largc·pore z«>litc "nJ nr" in the sm3lH>o~ onc. Sintt. slati'ltitaUy. penetr.lion by I<x>ps would be much mo~ li kely Ihan by ,,,,II. il m h,,·c ,....,Ii,e p<>re diam"",> "'''''' Ik on 1wi", Ihe diameler of 'he potcnl,ally pene,r... ,ing chain . c"cn Ih,n. there rna)" he in~"ffident driving force for lbe ch"in. 1" ~o imo " n,1 Ihrough .uch ",,,·ilics. MULTIPHASE SYSTEMS 299 .,r-----------, ., " "~u •• 11>.18 St"""' .... t •• m iwtbe .... 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DC. lWO; yO)!, 224: p. 2Oi. ( 190) ~"'nnan, A. 1\" Wang. B.: R(Joj,ig"t •• I). ( I ~I) ( 192) (193) (1 94) (195) (1961 E.: W,I ~ ... G. l . }, I""". O't,,,,,,,....r. I'oi)'m , 1991. I. Ib7. Sobon, C. A .• 110»"0. n, K .. IIr<M44.49. l>Oyie. w. !C .; Fal:>os. B. D, : ll oot. !. C-: Simmon •. K. n .' Chiang. Y. M. : Uhl""' nn. l) R , In (JlftwfrlKlN'" I'",,,c.tt;',,, "f A ~ C""",ie" J. I). M. ckonzi<: o"d O. R , Ulricibm&h. 1''''. 1990: Vol. 18(); p. 1iJ. ~b ' k. J. E __ ~"n. C- .c' I'ot/)",. Hull 19~7. I~. 2$'1. 306 STRUCTURES AND PROPER T IES OF RUSBERLI KE NETWO RK S (197) Zh.o. M K ; Nin8. V. P.; Mark, J. E. In AdwsJt,<,IC"",p""u Mm.rja/" ~ 1. D. 5<0<1: •• Ed •. "merican Cer.mia Society: W"le"'illo. OIL 1993; p. 891 (1'ilI) Wonl- S.; Mark . j , E. PQ/J''''' Suil. 1992.29. Hl. (199) Mayer. A, B. R.c M3fI: . J. E. In N"""fII""unJ Ma",iaI,. G.· M. Chow and K. Ii. G<>n .. l,~ Ed •. American Ch<-mieal Socie1~: Wa,hington, DC. 1996; p. 137. (200) May.r. A, IJ. R, ; Mork. J, E. Pol),,,,, /lull, 1996.37.683. (201) Monon. 10.1 .. F.d,; 1("'*', Techno/Of". Van N"" r.nd Roni. G . 8, ; ~brk. J. E, J. App/, l'oI},m. So"i, 1987. J4. 2853, (219) Sur. G. S.: J. E. PoJ'-'~ ' 8"11. 1981. 18, :l69 (220) FI.nl:. W. fl.; Wh~t •• T. Ii.. Jr .• "" •. P".fW".... in Mokn.I", S"" S'#8"i"8 CawIX,i. Sil<•. 101"",,;,,<: Am' lorda,n. I99S (ll)) AI·Gh.n> 101.,., om Ilc''''''g<. Appendix es A. Network S tructural Parameters The con"entianal way of form'ng a ",,!wo'~ ,. by mean. of cross-linking a long "primary" polymer chain, at random p"-linl>. to neighboring chains''>. Thi, is referred to as random cr<>M.litl king, and tbe mOlt important e""mple i~ the vulcanj~"on of natuml rubber by sulrur. Some of the time-honon.'tl ""n<x-p's in 'his area are described in appcndi~ II. R",:<:ntl'rogrcss ill .ynthel .... polymer chcmi"ry, ho,,'ever, makes it possible to for'" claSlOmcric nel"'orh by more conlrolkd ",actions, for example, by end·linking functionally terminated cha in , as described in chaplers 10 and n. An<>tber way of forming such "mood" oct,,'ork< i. by linking chains together al well-defined poin", ,uch as reacti,,, side chains, along Ibcir Iongtbs, Tbcst more specific and more conlrolled methods of nct work for· mation have the ad,'ant.go of providing infor",,,tion on the structure of the r.suhing network, particularly the molecular ,,'eight bet,,'..,n cross·link& and thu, a mea.ur. of lhe degr.., of eross--linking. Describing the dilTe",nt "'a)'s of r~presenting such "ltu,,.urnl inforonation i~ !he ,nain object of !hi~ appendi • . To begin, a chai" extending hocl"'I."n t",o linkages or junctions is called a "n",,,,ot ~ eMi n." and the total "umber of chains and june,io", in a no,work i. denoted by " anol /', r'''pc<:!iwly 'The ,"olcc"Ia, "'eight of a network chain is d""o!c-o hy M" and, since ne'w"r~ chains g<;nerati}' c,hihit a dislribulion of molecular weight;, .\I, repr ..... n" the av~r~te of lhis distribution, The number of ch ain~ m..,!ing at a Junction i. called 'he "functiona!il)'" if> of thai junction. "network ha"ing different fu"ct;onalili .. Jl diff"rem junctions may be characlcri,.d by an avcr-~ge fu",clionality, "- ch ain connected to. ju"~tio,, at o"Iy one end i, called a "dangling chain," and one that i$ attached!o the same junction a' both of its ends is called" "1001'," "- Jangling chain docs not comribute to the clastic free energy ofa network at equilihrium: neith~r does a loop that is ""I pen'ITa'M by another nctwor~ chain !hat i, ,tself eia &t .... atiy clTc'CI;v~. A network with no dangling chain, '" loops, and no junction, hav,"! a f"n"lionalit )' Ie .. than Ihree, is callc'tl" "perfect 'I '"' 308 APPEND IXES On<:s. if prescn') w f(mn a mok<'ule in lhe form of a "Ire"''' (" 'hu,h b~ di nts. an: indica1<:1:1 with filled cin:k>. SOme or wh ich m"y con lain reaeli,·c groups Iha! cou ld fuuhc r re~ct with one anolher 10 form the final'lCl work .,"nelure . T here are v + 1 '" v junetion. in ,ueh a lree. In lhe second Slep. (pairs or Ih"'" junctions arc eombinc"i n1\ '0 v - ( ! I "'" - ~. and int roduces c. actly ~ indopendcnI cyclic palhs. Th;, '1Uamily {i, referred 10 as Ihe "cycle rank"" or the nelwork. and is Ihe number of chains lhal ,nUSI be cui 10 rNuce the nelwork ~It' k 10 " lree, In phamom like lheories of rubber ciaslicity. described in chaptcr 2. the c)-clc milk is lhe unly .,r""lural para"",.er re<]uircd fur dC$Cribing Ihe daSlu,ily of a nelwork. irrespecti'-c or whether the notwork is perfecl or impcrf""I ·-~. There arc Ihus Ii>.., parJmClc .... used {o charaClori>.c nClwo,k $(ruc' ",.; specifi . cally. M, . v. I'. <1>. and For a perfect 1I.",.,ork. only tWO of lhese are independent. T he remaining lhree paramN~rs rna)' be oblainc"m lh~ equal ion' e. 1''' 2,,/<1> ~= ( I - 2/<» " (A . I ) ~/ ~ '" ( I - 2/ ¢»pNAI M , whore ~ is It.. v()h""" of the network in 'he $ta lC of fo nna!io". p is the cnrrc· sponding density , and N" i, Avogadro', numbor, A derivatio n of the rela t;on, gi,"" by eq _ (A. I) can lit fo und ~I"'wh~re'. An addilional e~pres'ion which is u~f,,1 in interpcti ng e,per;men {al data nn swolkn nelworks i$ (A.2) "'hen; II i< the number of bonds in the e>pre"ion (;). ,. ("<>. nP. x, is the number or "'gm."I' (, he I>onio" oflhe ella;" whose \'olum" i, '" NE"rWORK STRUCTURAL PARAMETERS 3()9 pol)',ner. and M, is il< molecular wei&!ot 1'<' bond. In te.m, of these v•• iabll'S. the third pari of cq . (A I) becomes {/v'l = (I - 2/¢).~.~'~/x,v, (A.3) ,1 where is the volUlne fraction of polyme r during f"m,ation of the "etwork. For impe,fect netwo.k •• til.. exl'fC'Ssion. give" by cq , (/\ .1) req ui •• modifica· tion •. The prescnce of dangling ch "in, . I""p~. Or bifu""l;onal j unctio", requi.e, the idcntillc.tion of til."", chains and jll"",i",,, that are "aet;'-o" or "ellCetive," An a~li"" junclicm i. defmed by Scanla,,' alld C.",· as on~ joine where ". and I'~ arc lh. numhe .. or aClive cham~ and active junction,. respectiwly, a< defmed by Sc.1nla,,· and Casc·. Flo.y" late. showed lhat this cq "ation hOlds for ne twork, of ~ny ru nctionali ty. and that the ne'worKs d" no, have '0 be random. Flo.y a lso . ;,o,,'cd " tha t the ""f,nilion of "<,i,'c junetions and ~hain\ by Sca nlan" and Cas<" dq><:nd< on the loea l tol"'logy of a net,..",k and loence leads 10 ambiguitio:s. In~tead. Flory dcf"",tI tht 'c,m '"e!Tee,i"e" cba ins to be thOloe tbat cffeelively oontribute 10 the ela_" idt y of the net w"rk . and relatN their number "..to the cyclc rank by ( A5) (icncr-dl "xpressions rda!in g ,he clTceli," nU",ber "f cha ins 1<) (>Ih" n~twork param"t"rs in impe.fect netwo.ks are not ~vai l "blr at tbe prcscn! time. For an impe.fect tetr.fulICtional netwo.k. how<·vcr. the number of effect;ve cha im i. gi,'e n 1»<,,12 (1\,6) where "" is the ,o' al number ofcha; n~ in 'he n<:t""'k_ The quant it y ii. i'the a\"t'rnge moleculM ,,'eight of the prinla, y molecule, (the prjm".y o. pn'Curoor chains being e.oss·hnked into the net "'{Jfk , truelme). E~prcssions relating the cydc ran k and the number of ac' i"e Junetio'" to M , h",,(: bc<:n gi"en by Qucslel a nd M"k " , Fo r impe,fecl n<,wo.h. the "altlCS of Me may be de tcrmined f'om gel eu""" aad sol f.action measurements, making OlSC of the I'ea rson· (iraes~ky '· o r Charlcsby" .qua tion., The number of .keklal b-ond, in typ,cal IIet",,,,k n, '" "ge, bet"""" 100 a n~ 1000. which conform$ to the naussi"n pict"", .. f !he si oglc chain", Netwo.k< consisting of chain, having much fewer Ih"n 100 hands arc more like ha rd ,hermosets and do not show lhe t}'plCalla,&< e" cn,ibili ty of a " da""meric ""t,,'o,k, Tho .. h.,-ing chai,,, wilh mol\: than 1000 bonds typically tak" "ery long times to ","eh '''-I"ilibcium u,-der load. ;ond the ~cI, .h,'y to"" "'ho,, ,,,"ollen by ",I""", ,or" •.~l"'n\cl)' f.~~Ic. In" IY ri,,,' rlC,w",k d nc "Io,.. 310 APPE NDIX ES 80 A. A ;.rb~ric"1 domain or Ih;s rad;u~ cOn1ains aooul 4O-SO e,o",-I;nks and Inc;, pcndent chain". Thul. a ne,,,,ork chain sharcs;1s sp."" with many other network cha;ns.. whi<;h lOpololPe;,lIy may be: eithe, close or distant . This high deg,ee or ;n"fpcnetratio" of netwo,k chains ,"",ull$ in cntanglement s tl,.t are permanently tr.lprcd inlO the ne!work structure as a result or c,oss--!;nl: ing. Such enlang!emenlS are di>Cu ....' Referenc(!s (I) 11ory. P. S. PriMipJ,. <1' Poly ,,""' CMmi$TrY. Cornott t Jnil' (2) C"'a". 1\. V. In Sd,.ttNiDId ·IIThnoio/l.)" <1' R"""",,,. I . E. Mark. Jl. form.n. and F. R. Eirich . Ed •. I\""d<mic Pr",,: San l>iego. CA. 199-1: p. 339. 0) Flory. P. J. rroc R"J. Soc. /","don. A 1976.351 . 3SI. (4) I) uis<'. J. 1\ .: S.. ,'<mt.n. A, J. In P~Y''''''of /V''''''y,MU", SnlM•• J. A. Prin •• Ed. r-;""h Ilollaoo: I\m>l.rd.m. 1965. (5) (]mc"lm<>Ie(Vu-, 1975. 8. 865. (1) M.,k . S. E.• Erman. D. RWJbe.I,k~ FJ""kiIJ' A MoIfful", i'rim< •. Wiley_ (o/p", !iti. 1960. 4J. SOl . C.... I.. C. J . Pmym. Sci. 1960.45.397. (10) Pea rson. I). S.: Gr.=ky. W. W. Mm.'ronrokc"i., 1918. 11.528. (II) Flory. P. J. •1/~ rromWr"/" 1'182. 15. '19 ( 11 ) !'1oC)'. P. S. C~",. He.. I'J.I.I. 35. 51. Ill) Queb~ . 1\.: I~n""r. S, II . Pr«. lI."y, So< . .*,~, A 1959,249. >67. m B. Definitions in the Area of Rubber Technology The Icchnol"llY of lubber prim"Tily inyo!>", ,''' mo", apl,lied aspecl. of Ihe preparation of cros,·lin ked. filkr.reinfo!"CW elastomeric mat.rial~. In many cascs. there is not much scienlific "ndcrstalKl ing for tn any of the prO<"edurcs uSC<.! in the industry. ,in •.., the}' have app"",ntly bttn developel' of ,hese n>~'.rial •. This s..:tion gives SOme <"If the mOre impOrt"nt defini.ion. in ' he arta , M or~ detailed definitions and discuss,onl of the,. and rel,,,c<'epts arc ",adily available in .h< lilcraturt " ' , ""me B.l Basic Definitiun s A ~ ... _ : A .ub"allCC Ih., ."",,10""" 'he runnJ; or an ela""""'r in ,he c... of ,u lfur ,,,le>n;',,'ion, Mony o[ an: , ulfur.(on'.ining COI"po"'''''' inc,.",,,,, ,he d f",,,,,,,,y of an .=lcMUf. The", an: >OfDC pa ... Uel' be", -..;,h " and ""'al" )'''' In .. ".I1 ·"",lccule ch< dcvadalion or an elastome'. " 'hkh ')'pieall)' 00:"'"' from ...",,"on> w;,h oxyg 1"'* Arti,.,,,,, A."b"...." ,h., ,,,1><,,,,,,,, A..l1""~.nl : An pro",:" "~diti'·c 'ital an 01'''01''0' f""" o. """ive ".~ ... ~"hon unde, .... condit;""., S""h ",o'.""~dliOll. ['" <>aO>l'ie. Ihe d.. "" el ..'ome .... ')'f". AoI;"..,." ." An .ddi.ive th., P""""'" an cI.""me., fre~ ..",ly "f ,ho: un""uMo:<\ [",m dcgr.-td .. i"< ,u t1ion wi,h "'.0"" during ",,""ie<. Some aa:ompli'h ,hi. by miV.,ina ' 0 tI,,, 'n,r""" .,0.1 f"nninl' b.am 3'2 APPENDIX ES 1Iond1"!l . ~t , Sub$,."..., u>«l to improve ,110 adhesion or bonding be,,,,..,, a n """.oJ" f,ber, fih .. , f.brie. """.ltic , urface • • oJ ro 0<1. ,ha, "'.., 0..\ 1. I'IIbbe,., EI.\ ,omcrS p",duccd fro m bu,adic". ( Hu) (>01)'m:blio.: Mode of f.ilu", ~ lie" • wilh.1I th , .. principal 01 ...... O<jo.l. ClIIori • • lt4 '" .... _1,....1..:1 ,oIj·.lhj·"" O: EI~"omcr fron, lhi. normally t"',mopl. "ic pol)'l"c, by . ul><;' itution of a "'-'ge CnOU&h numbe' of eMori"" "on" or colo,o\ ulfon. ,c ¥roup> fOf h)"dr<>gb«: SI)'",.-....butadicu""i., ; 1""".--"0,,,,00" of addili,.,. into "" c,,,,om..-. 'ypically hy . mi lhug o r otho, bkndinll tccl1nOqU< CompresWon ...1: 00(0"".1"'" ,h .. ",m.i", .nCS!l of"" 01''''0'''''. E I· I).~ ' , 1]'''0'''''' "'$«Ion <'h),i'''''. proP)''''''''. d>en< comonomer ('0 fadli,.,. "'''''' 'inki''I). h ."""",.: Materi.1s added during oompound;ng 10 de<....... ,lie <0" of an <"'"on ..". r oliS'" lif. : !""m ller of b1ur>.1 ruhl"" ffOm ,0< go"y"Io .h,ub. G II~": SynoD)'"' (or S RI! ("y"'n<-bnlodi<no '"OI>or), HOld rubbe,: Rubber"" be.vily lomo, . H .. d..... , 1/";>1."", '0 indent..;on. Hr ... "'''''''" N.. ur~1 r"loI:>difyinS oo.>ion of in ~'''y f"ih,.'i"g n'e of an <01 .s a "u'f)' in • lat", "'hile ,be (>01)'mllear a"d by f 'h, cia>lom '0 ,,,\>bor .oor.. """,ined '0 '0 ,«, "'<,..Ul"< ",> ""w DEFIN ITION S IN l HE AREA OF RUBBEA TEC HN OLOGY 3 13 ,ft., M"I~ .. . ffKt; L\n'«.", in mooulu. of an 01""0'".' . .. pcoi.lly ,,'hen filkd. d.f",· m'I"'" (particularly .fte. 'he r. ..., defoflnalion '~r",;tion eyek), I"""";nford"ll fill«c M'lerial. 1)·pO;'. lly i..o,gan;.c. .""h •• day. lak-llm . f,nely divided mcl;>l •• nd ro on. odded 10 a n ob"omt, ,,, k,w ,hcnn,,1 or dc<.1ric. 1 "'nd u",i,';'y, O; I .., ~ ,Io. , Addi,;on of an oil. ,)""",Uy " hydrocarbon. ,,, ""it\c.olly ,11< P'" '' by ",,,ish' "f a , ub:ll.JICC pt, 100 ""'" of cia.","",.,-, l'ta!.Iicil." l o,...onokcula, ·wrigh' ma«,.;.1 .dded \" . pol)'mer. ,)'pi<~ l ly ' 0 ";oflcn" i, ("","", il> $1 • ., """';Iion """1"'''''"'''1. I'"""", .. , S)'no"ym f.". "",i.at". , I............ ,'.ltanLt.,c: C'o><- linKod b,,, u",illc"t imporl.nl moch.nleal Ph,p.:'t; . .. I" "iocubrly 10 c.. mpk. ar< ca,bon block a nd .. Ilea. RtoIl!on«, Ra.i" "f "'gy ,.."urned o<"-io nkin,. which ;n'of....... "'i.h ,1>0 .uh: of 'ho clt"om~ S«;o,ct. lIm<= Amo"'" or time 0 ,"bbe, m;,IU"" ","0 be ",,,,ked before i, bcc<>""", ""pro' i"",.... ,h. ,h..... "I"'" .h., .ff"", =~ 01""",".", Sil>ro.t.: M;.nome' fo r si.",,,... I."".n, 11<", "'''h rc~rJ ' 0 poIY""' 311< SInok... ~t: N.turAI rubh,.(;n k'''1; k>w . "<>ush '0 ,he high e>lcn,ibili'y n",,,,. lI y .~""" .. d ,,'i.h 01 ••1"",.", ,h .. i" ""II belo\\' ,h.. u« prod""e h.nI 'ubbc" Sohdi ... Sli M: S'y,',oc-buladicne rubber I"OO"'"""<J by ."ionoc l'Olyoneri""ion ;n solu· tion •• prionor;ly 10 g!'" higj",' "oo\e<:" I.. weish" .nd n.m' .... ' mo\e<:u l.. ",,,,gil' J;"ribu';on., Stobili",,, A" ,nlidog ... ,I"n, Iha, pm'""t." 1",lym... p.an",u l..ly ber",. >nd duti"&lbe .",... I;n ';ng "'1', Si.ioQn&: O;'I«>I,,,.,,,i,," of on .100Io",e. ,urf...", I) pically due ,,, ""J.,i"" of " IJ;li ...... SIOVOM-. ;001.. : M. ",u", of U", c," enl <>f ,,,ioJ,,.r,";,," '" en.,nlite .,,,,,,,u,,, or lile particic<;n a f,lk,. Soolf... _ : An o'lI""ic poly,uirOt" roa'b, f o"" 'e.Juired '" "'"",a!C 'I''' """ured ~y"ltome"I, I" , >On: .pee,r", ",.. " ,,, i>cavily "",,,.I ;,,ked (.ueh ., h"rJ gi.-." 'h" ,,,,te, ,.1< '" 31( APPENDI XES ,ubbe" "p,"y ""in., and pho1lol fonnalOo.l prim.rily 10 mold ritid objec1, ~J,h<, Ihon oIouo"",,,,, ones, \ ·MI<1t.I,OI"': C"',,'Hi,, ~ inH an oI."ome, by healing. m;"ure of poIymor and .Iomen· ,. 1 ,ulfur, f""lu and ..,,;,'oIOrs, AI"" used loosoly for "uy IH'" of ",""'linking pr<><:ess, Refe rences (1 ) Mo""". M .. FA. RubI",,1'ffh.wl"1!Y· Va" No.00I< of Pol!"."" Sci;on«, 3rd cd. Wiicy: N.-w Yo,k . 19114, (4) Rodrigue" F. P.iM/pI,,> of /'01)""," F.~&wning, 2nd cd. McGra ...·ltili: Now Y",I;, 1982. (5) Alge'. M. S. M. Poly",,,. &im< ... Di(Ii&!. lta" .. r: New York. 1'192, (~) M Ot. , J. E,; I::,n"n, II .. Eirich, 1'. M" 1:'''- .o;,.~"",< """ Tt<"""I",}· of 1/,,,,,,,,.: 100 ed. Aclodemi<: New yo,k, 1m. (2) I:irkh. t·. R .. Ed. C, Deformalion and $ treliS Th •• Ia'. of dcform" li<>n and .,"'." in an ciastic body may be described in v.. i<>us ways. but ,hi. "ppcndi ~ , umma';/cs ,Il.: mm' c(>1um<>nly "d<>l'tOO dcfinili<>l" in the area of mbbc' elaSlicily_ OllH:r definilions of . lress and ddormalion in nOn · linear elastk bodies may be found in ",wral mo", general l",al m~n ts in lhe literature '-<, C.l De formation eLI G.""f1)) Aspocts The deformation of" body may bc des<:,;!x<1 by the di splacenlen( &ra,lient (enW' A. defined as ,1" '- '" "" ih', ilX, IlX, iJ.<, ih ) aX, 8X~ OX, "" ii,, ) i).r , /)"1. iJX, (Ix, (I X, (e l) where _', and Xi a rc tl,e .-o o,di""tc, of a I."",t ,n Ihe dcfnm",d and the undc· f<>m.c'd ,la te, of Ihe body. n:sP'-'TlCS of r"hher da,(;"'il)' "'" o Il"" pn..,.,rr IOO in lc'm~ of hO"'''8''n<-o,,' principa l deformat'on,. in ",hich case "'1_ (c. I) SLlnplilN" ,n ,,,,, '" A PPE N DI XES X, X,. h " X, 0 X. 0 " , 0 ~ - , X, 0 X, 0 0 0 X, 0 (e2l X. The s,a te of deforma lion rcp",""nled b)' '' (ell Here , "1(' is the ,'olnmc fraelKm of polymer P"''''"1 during II .. fonn"tion of Ihe nClw "r o o 0, 0 0 (C.4) 0, <>, is (he ratio of lhe knglh along lhe ith axis to the ~ng(h in the initial state (in which the polymer may be swollen with sol,,,nt). Defined in thi, manner. a represents 'he diswnion of the ""mplc. and i. referred to as the "distortion tensor:' T his definition co",,,niently undi,(ort~ " /' ¥ " H~u" ' /' . ,~ " ,~ /' I " C I An ""deformed .nd. defurm<"\! ='""gut" rri,m, CJ' DEFORMATION AND STRESS 3\7 cross·linkc-d in 100 iWlm pic slaic For "elwor~' c"' .... linked in Ihe .nisotropic , laiC. furl her care nlLlSI be 13 ke" in dc-scribing Ihe ' laic of Ihc deformalion ."d it, relalion,hip I" SIres>. T re"ln,enl, " I' sl>ch ,,,,,w,,,ks may Ix r<>und elsewhere· $ C.1.2 ISOlrop;'; Swelling The 00"'1'''''''''10 o f Ih<: le n'lOr n 111;'1 ",ea '- (0" ,." , I "~, ) "o o .. ("u/ . , )1/1E (U ) (o,.-/v, )'''' C.I ,3 Simple Ten'ion l 'he eh;mge in ,'olu,,,,, uron m e'ehing d nel""',,,k is of occond order ootnp;lrcd wilh changes in Ihc line'" dimcn,io'li. and is (herefo re generally I"ken (0 Ix ~.crO. Taking lhc of ,(reIch ol,,"g Inc x, " "S. ond IClling ~ I .. ), leads I" , 1 ~ , 0 (c'6) where V/ ~ " ">C/o, Lelling '" -' ft . . . , , , n ,,z {l I" a nd uS"'g C o " (0 ) CIA Biaxial b len, ion When lhc tlimens;",,, of" rubbo:l "hcc( an: changed ;ndcpo:ndcmly by lhe faclor " I along lh. _<, ax;, ""d by co, a long the x, a. i, . Ihen Ihe lenso, l ;. &i"on by , ~ ", , , " 0 - 0 , ( ..:,, ( 0.. .-1., )1/)", , , / ,~ )I ~ , ~,) , I" '"I oJ) 'f) " " , , , (", / _, )1,"(" ,,, ,) (C,S ) :(ZI':») -1>.:1 00 SO"!I _.,3d" ~I!I U 'I" ~ U''''''IIOJ ~41 ~""ulOJ',",1 JOUn p:lUlolqo .. ' ( VVp ) "I " pun '" '" J" OO!lnp' ~41. " ,Jnl1lJ:x!w" '""1'00' "'J o~\lU"" " '! '" 4"""'''''! (1: I":») .t>" "' p:I[>I1pu, '! 'poe[ ~I J" ""'lI"'''P .4' s~ .n olI"U4~ ;)\Un]"" "U'"~. ~""I' ~,oN' -w'I,d ' 41 JO s:I[I!l ' 41 ul "SU"'P "' lp,x h:)J "'II IU:>S:)Jd", " IP puc -" IP ".11' ",~~,,, H""" ', p'j+ "1P'I + '7/''1 '' ~( I-'Vp) - (<:1';») MP - ""41. '~8" u, ""'J "1(}4WP1~ '41 "' .>8""4' '~I "I I! ~UII"n!>:o u~4' pun' 1':) :)JnS~ U! ""'''4S Ul'I,d ''11 J" " '''U '41 "0 ~U !'''' V pu n ''I- 'f =-'0) '41 '~Jl<'n :l1P - ~'o'" I"U'IS:>IIU~U! "41 lIU!I!'''' '4 'r v .oJ"u" ""-'J (",ISUp '''''1' l JO ",u,uo:1 )0 SO!11lJ UO""'I~' ""''II "'II 'Ill," p:l1"P"'''" OJ" (] I ':)) ' '''')0 "u,uoomoo""",,, IOO!'ulJd ""'41 '4 ' 'sn4,L - 1"~llu'I'! OJ" UI"J I< puc ..,,,, J" luoIP:"!1' l""I'-"'IJd a'll JO) ~1104Ul IJII '41 u, :.l!"u4' ' 41 mO'J p:I"'"lqo "'I " 0 0 o 0 " o I, ( 11':») ~.w o l~ 1 Aq u'AfS .. I '<> $lU!OO 1I~ I" ~lU'" ~41 II "SOliS ~41 J' p.:>U1~lq" " ""... " JO aWl< , noouolImu04 V ·~ "~I I'"la J:>q q~J JO ~P" 's ~4' "! J]JPIUUl,S , c U~ ~ U, XI '" U;I"~A UO_' 'UUl J<»U~l IS3J1S '4L 'SUI"~ '! =oJ "'ll 4"4" uo ~"Uld "'II JO luUl'oU ~I )0 UO"""!P "'II .,~""oP' ~'PU! pun.'»S "'II PU" =oJ ~q ' )0 "l"·'!PU! "pul 1$J~ "'II :')J~4 - " " .I 'I,l (01 ';») "J. 'OSUOI "'II ~q I',! 1;1 11,1. 11.1. ".1.1 "" 1. '" ",,",!S SIIUIOO E I~ ..... ")0 "UlS ~41 ' I mO""~" 1 SS8JIS " U ,(!.<jJl. ) «(>J) o o o "'("'1".. ) o "o Z'J I"' r "f/' ( <- ;''',, ) ~I 110'" ~4L -(~'J) -t>" "! I ,. '0 I)UU " .. Iu !!UI I1'>S ,(q S~ X IO N 3ddV STJ 8(£ DEFO RMATION ANO STR ESS I, ,dl 611 T = L, L J 1.,L,dL, .. .. !,L,L,dL, f2 _,I i.,i; L,I,,,..L, i 319 d f) L ,L, L,I-1 t, + I,L,L Jdl-1 + IJL,L,dl., L,o L, _ I, -, 1., 1.,1.,,1 .. ., 1. '0 1., t1 -~ I., 1., I. ,I., d L1 L, -I.,. +', -1"10 I.,L,L,JY " 10 (CU) He!e, 'he ~on>p<>nen' of Slr= I, IS de~ n c-d a, 'he fo""" div;,ted by ,,,,, deformed a",a on which il acl" The last equali, y of cq. (C.13) i, wn11en a. , I':[>,.\, 'd>', (d6 A h = The d~f,ni,i<m of . ness 'hen foUo .... fron> ,' (C.1 4) ,-, CQ_ rC I.I) as ."("'-') -"('.') Vm. rl'fu'r.~ 2.V\; -= 2..; (D.') --OJ.; V = (C. IS) (U6 A) (h, T r/' where the "",:ond linc o f "'I. (C.I S) is oblll i/)cd by a trivia l algebraic Ir.nsfOfma, ,ion. Ta king Ihe dilTerCNIial of "'1 _ (C A) and ."bsti 'uting into CQ. (C.14) leads '0 (dl!.A), - (,L' ) j I, ,IV + I' , \ L, t'("i 'doi) (C.16) hi The lerm in the fi " t pa re","""" on Ihe ngh' ,hand-side of "'I. (CI6) "",=o's 'he hydro""lie p","su re. which k"ds 10 work again" Ihe volume change . The St<X>nd lerm repr=nl' the work done in d;'to'ling Ihe claslic body_ Th~ change in 'h" Hdmhoh7. (OT d .. lIc) fr"" eneTgy i. gi""n' a. lhe . "m of '''"0 tenns: (et ?) where 6A '( "f. V) i,:. funclion of lempcr-.L1ure T and mlum. V. and dcnoles 'he contribution frnn, inlcnnoleculor f""", snell as lhose in si mple liquid., The q""nlilY aA ....( T .,\) ;s the el. " i" free .nergy an.ing from Ihe ela'ticity of the not ... ork chain , Su bslillning "'I. (eI 7) imo cq (C. I S) lead. to Ihe f01l0,,·ing .xpressioll for the . t"'<$: I,~I' +" ("61\,,) -00' • 2<>; , (C.IS) T. O· ""hen: p' .. ~ (1l6A 'Ia l' )r is Ihe conl,;b",i01l 10 the pressure from i1\l~rmolec .. la r interaclions such tho", ",..;: urring in si n' pk liq"ids. One may clinlinutc p' rron, C'q. (C.1 4) by ,aking the difr"",no;: ""'W,,,,,, ,'.h,,,. of Ii along ,\\"o ,"rce,io ,,<. ThIS lead, 10 II", Tre lo", re1:'lion,', I' 320 APPENDIXES " - I . .. ~ [n;(°tf (C I9) The above d,,,,,,riplion may be ge neruli,ed 10 nOn principal $t",S"'" und (C20) when: Ihe superscrip' T d onOI •• Ihe lran!pose. Lcning A be a fu nelion of A Iran.forms cq. (CD) ;n lO [8.' I (C2 1) (JIl.A ) r = T, 8 A ,IA Here, T, dellole! Ihe Irae<: operalor. ,hal is. Iil<: .um of Ihc diago",,1 clemenl. of lJoe I""""r gi"'I> il> {lie >quare brade{s, The definilion of Ihe Slre.lS (enoo r T foliow$, by lhe generalization of Ihe expression given in "'I. (C IS). as T ~ ~ .I(lJ.tlA ) V iJA r .IT (Cn) References (I) Mu'"~£ha". t '. D. '-;"/1" D York. 1951. (2) Truc>/J 1'11<",;"•. SprinScr,V60: Vol, m i l. (3) Eringen, A, C. ,V""/WdT 1'Mor), ofConlimwus AI",/io. McGraw_Hill: N~ \'ork, 1%2. (4) Ogden. R. W. In .If",/umiu of SoIi,ls. II. G . I lorlO", "od M J. Sc""II. ~d. I"'rpmo" r ..... : Oxford. 19l!2. (5) Trdoor. L. R. G. Tk I'hy,k, <1' RuM", lilwlkif,. 3ftl a. .. 1975, (6) Flory. P. J, no",. Famm.}· S<>c. I%(). >6. 722, (71 I:rm.n. Il. In NeI~'(H"b, 86. 81h ,vel.·",h G""'I' "''''#''8. Elon;'r' [I';nofC, L>rk. 1988, r. 497. (S) H.• """", W.; H"iJ,. S.: K",""" . 0., F"" WI. 1... J. In N".",h, U. 81h /'..,)''''" N",~"OIb G,o"" M""inX. EI,,,,,,,,,, "binor<. l) Pol,.,,,,,, 1,,,,,,. D. Summar y of Thermo dynamics and Statistical Mechanics Con,ider a 5)'litem of N Ololecules:11 fi~c-d tCOlp L, e.p(- Io",/k'l") ( D.I ) ,,"'here the ,umlna("", i, carried OChanics and (hcrmod) nanlics, as oullintl! in leXlbooks of .talisti""l me-chanic. (..,.,. e.g .. Uill'. '" Ch"ndlcr' ). The probabililY 1'1 of the .)"Stem to be in the slate where its cnergy is ,,'; ",ill be f'i ~ . ' p(- £./k1") £o>p( - ;;:;/kT ) ( b .2) , T he CQr rc,pondtl'lCC bel""",n 'la\i~tH:al mtth."ics and thermodynamics '" ""tat.. lished through the Helmholtz (r"" energy A(N. V, 'I") by the .. Iot;on A (N , V . r) '"' _ tTln Z ( N, V , '/") The iofi"ite~ilnal work "II' d"'1< (0 _3) On (he sySl cn. is till' ... - ",IV + f dt. (1M) ",hom,. .nd / aTe the pre!«un: ,,,,,I for"" .cllng 322 A PPENDIXES dE = dQ + ,10 '" T A _ F. _ TS on. ol>win. Ihe •• pression for dA in term, of +flit dA .. _ s Comparing t<j, (1M) (0.6) with the dilTcrcntial form. (") dA " , , , uT V,I. (") <11" + " uV T.! (") ,11' + " u l. T. ~ ( D .7) and ysing "'I, (D,3). leads to thc the entropy (.\1. pressure (PI, (orce (fl. and energy (f,) of Inc system: ' __ ( " ) in V ,I "' kT(iill1~) {Jl + klnZ V ,I. , --("') ' '''('''';) {Jv m T.£ f - (") __u('~"Z) {JI. ry T,~ TY ,)1. "("'iT) ".L_,,' ("" ') liT 01' (D.II) V,£ The Ihird of the " I)(we expressions is Ihe one YsW for ol>taini "g the maeroswpic force required 10 keep a network al ""lnSlan l length L_ The single-ch. ;n configura lion partition function may I>e ,,'rillen down in analogy with the pa rt ition function for Ihc macroscopic sys lem deiiCribed "00'''''', Replacing Ihe nwni>t.i" •• xpres,;"". eqyivalcnt 10 "'-1'- (D .3) .nd (0_8)' A(. ) .. A(T) - kTln Z, , _("') _kT(iilnZ,) _ klnZ {J r , iJT , ' f- (") __ u(""Z,) _ "(""') r "(",,,,,) ,J, Ii __ ()r T -01' _ , ( D.9) I -01', whcre A(T) is" (unel;on of lCl1lpcr~lurc. The singlc-chain ""<>Illigura,;"n 1'",;1;<", fUI)Clion Z, al fixeJ lemperalu," is defined as the sym o,'er the energie, of all available isomeric Males of the ch;,;n al fixed end_to_end '""lor " SU MMARY OF TH lRMOOY NAMI CS AND STATISTICAL MECHANICS %, -" >=, e.p(- E;jkT) 323 ( D.IO) Fo, cominuously ch"~l!.ing ,otation,,1 ""Sl .... , .11. ( 0, 10) may be ,,·, inen ill integ,al form ,.. /.,<1, .. (S"':) '1'"I exp( E/ kT)J{ I ) (D . Il ) where lhe fronl faclor ( S,..' ) , nonnali= the partitio" funclion " 'ilh respecl 10 lhe 'palial o,iolllatio~. ()f th. chain a•• ';gid body, T he illleg'~tion . horefo", is carried <>ul ovor "II inlemal confi~uratio"s {I ) or lh. chain SUbjecl to the .on· strain. lh". th. cnd·l"· W{. ) '- 1- ( D.12 ) Th is e<1;, F, Finally. s"ho;ti'Uling eq, ( D.12) int" C<j . (D ,9) rt'Suhs in the c'p",,-,;on< ro, A. S. r. ~nd E in tern" or the distrihution [unction. Ahema, i~e derivalion. or the , la'i ,tical m«hanical di ". ibution law. have heen dc>cribcd in the li te"'tu", '. References (I) H ill, T . l . A n l~rroJ"O, pp, t8" 19 (2) Ch.ndlc:t. D . Im,,,,,"caon '" MoJ.>rn SWI;"kot/ Mcc/oank. , O.fo.J IJnivt"i'Y ~ , New YO'~, 1987, (3) Flory, I'. J. SID,,,,;,",,I .If (5) Wall , F. T. Pro;: Nal , ACM. M i. 1971. 611,1 710, E. Fluctua tions in Phantom Networks Fluctuatio ns of junClion pOSitions an d chain dimensions in a phantom networx may be deduced once lhe r , matrix given in chaplcr 2 is known. In Ihi' appendix We challlclcriv: Ihi, n,atrix and iI, in,-crs<: . and deri'" impo,I,,,,t ,elalions ;m'oJvinS IIl1clUalions. For add ilional info'm"tiol'. Ihe reade, i, ... ferred 1o lhe ""pe ' by Kloczkow, ki el a1.'. which i•• conei.. presentation in motrix fonn of p,ev;ous wor k by Eichinger'. GraessleY"·. Hory'. and I'earson". E.1 The M atrix T- ""d its I"verse All qua nt ities in this appendi x are related cxdu$ive)y to fr"" junction, des • , . ",. ,," E. t A """like "r..,:t .. ", ron,i"ing or ¢ wi' ~ J>Crmi .. ion " 1o<'ko~'S'i. A.. M "k. 1. 1' .. ~'m .n. 1\ r'om (l9l1'1~ M(U"""",~", · .k.<. " COP)'rig~l " Cl>cmical So.:i '" 21. 1'197. Ameril.-. " FLUC1UATIONS IN f>HAN10M NETWORKS 325 g"'ws symmetrically frum lhe U'IlH-a\ c\,aill bell'''''" JUllClio"s I and 2, and ,he"" Iwo junclion. con"ilulo lhe ~"'l lier of jll""l;"" ', 11.c second lier is m,dc "I' of the , i" jun,'lio"s numhen,d.J.o 8 in Ihe fig"re. The nClwMk is ";'"m,'(I 10 g'ow in thi. m,nntr wilhout f, "ming any . hon.., i",ui'C<.! c}'chc «:>nn""'ion" The matrix . ' for {hi, {clrarU"CI;on~ ' >lrueWrc i, gi>1:" hy c_ [ 4_11=-1-_1_1 _I 4 -, - \ -, _1_1 I_I --, 1 r ,-,-,-\ 14 _1_I I·'., • • ." •• ., ., I .,. , • • •• ,., •• , ., ., I ., • ,., - 1- \ -\ , , _1_ , 1 I . I I I , J • ., ., -,, - ,. , .,, . . •• (E, I ) •• •• ,, , • •• •• The in>'crs<: of r is required for eakulali"s Ihe "'", .ge nUClUalionl on lhe phan, tom network, 1\. , • (10.2 ) ..- / " /;,--." '<:' I 326 APPENDIXES where all olT.diagonal clements in Ihe upper righl pMI of lh~ malti.' are I.. ro, oxc and the ",IUlioll of C<j , (E.J) is I(~ - I)"+' - I) a'-I(4) 1)' _ 1) (1':.4) In lhe lionil where lhe number of liers goe.' 10 infinil Y. a, is independen ' of n: (LS) lim". "' ~- I .~ In Ihe following, an infini te network i. con.idered, The nll"tu.lio ns in "hain dimension. arc obtained from lion, by lhe r~ 1 3lion nucluation~ {(U' ,)') = « u R, - u R,j') ~ ((l!.R,)') + « IlR;)') - 2(l!. R; ' 6 R;) '" ( J/2 l1( I ' ~ ) . + (r; ').. - 2(r ; ').1 of june- (E.6) where ' ii is the magnillldc of t!le "o<:lor joining t ,",'0 junclions i and j. The laller arc not n"""ssaril}' at the end, of. ,i ngle chain. and may be SCp:lratcd by "" 'eral chain' of the network E.! Expres si ons for Various Fluctu ation s £.2. t T wo Junctions Juined by a Single Cha in T he first four clements of r - ' follow from "'IS. (E.S) and (2,26): {(l!. R,)I) [ (tl. R, ,tl.R, ) (E.7) Substituting f"'!11 cq. (E.7) into cq . (E,6) leads 10 lhe nUCIuations in the dimension! "f a phan tom nClwork chain being gi"cn by FLUC1UATIONS IN PI'IANTOM NETWORKS 1'.2,2 Two Junction~ Scpara",d by Sewr~1 321 Chains When tWO junction> '" and n arC ",par~I~J by d other junctions, their awragc fluctuations arc gi~c" by thc matrix l ({t.R..)') (t. R.· t. R~) ( E.9) T he deri.a ti"n of toc off..!iagonal lerms of the mat", in C"'l. (E.9) i. ""mewh"t lengthy and ;s gi.en in ilS moSt gener.1 fo"" in the "pre""i, of Ihe parer by KloclJowslr.i et "I.' The mean-,,!u",e l\uctuations in the Jimcn,io". of Ihe palh octwcen junctions no and" is ohwined hy ~u h.{ i'u,ion from c'-!. (E .9) inlO «I. (IU') . T he result is «ar- , » 2 = - {o;> . - 1)" ' - 1 - ---,- . ~" - 2)(d+ I ) (9 - 1 )~ 2 =~¢ - i) (,;.- I { ' - I (';' _. I)J (r: - )0 (1'.10) , ('ill. ,,"'here the scronJ li ne follow, by the us<: of (,l_l. (rille' In toc limit d ~ 00. "'I, ( E.I O) reduces to " (J/2Tj{J + I) ~ (d r I) (E. I I) Equation (E. \ I) shows ,h"t 'he (". lio of lhe """,n_square flWl""t;Ons of the dimensions of .n infinite path in a netwo,k 10 those of the single network chain i. constant. E.2 .3 Points on a Network CII.in The nel"'ork cha in i. a,;smned 10 co". i" "f" Kuhn or free ly join tM seglllents ~nJ 10"" an aehcd to t",o ";'funetion,,' ju oc!iom at it ' n! rCnlilie' . A pOint on loc chain i. dcC,nM ai !he j unction of tWO Kuhn "'gille,,,, , The fme!'Omll diilancc, ( and 0 of the ith anJ .!'th po;na. resrcctivcly. "re ,lctinc-d .s ( .. i - I , --- 0« <1 (E. 12) () < f/ S " 328 APPEN DIXES The nuetua! io~ , of!1I<: two poi~lS i "~dj o~ the ch3i~ ha'" been derived ' !o be (~R;. t.II;I) l ~~x «t. H,il 21', 9- 1 .p(,. - 2) «J - ()(Q - 2) + ¢ {4> - 2)/min{(,O) - Q1) -" [ Q- I ¢(¢ 2)~ ,p 4> - 1 .p('; 2) + (~ 2»min{(. O) - (0) - ,; 4> - 1 ,;(4) - 2) til O{I - Oj(4) - 2) + ,; (10.13) Here. (1:. 14) (10 .15) where (rl)~ j, !~ me.n-"I ,,,,re .nd-'o-c",1 diSl"nCC of a ,ing)c ""glllen\ ~uch 'ha, (r'). = " (ri)o. The n1in «.0) ("n",ion in ""I. (E.I J) may be wrillen.s" min«,O)~(+O I( 111112 ( LI6) The nuclua,ions in 'he dimen,ion. of a chain joining 'he poill" i and j a ... oblaincd by u,ing C , J[ "-2 11] , «t.Ty) ) '" -h 'I - , (E. 17) (,2.4 I'oin" on NCI\.·ork Chains lhal " ... Sep"rdlN by Sever.. l JU'lClions The IhIC1U;"ion rnaui, for 'he c~sc where points i and j Me on ,wo cha ins SCP.l· ra ,,,,1 by d junc,ioo, is given by' FLUCTUATIONS IN PHANTOM NETWORKS 329 {( t..N,)l) [ (t.. R,· t.. R;) ¢ - I P t «,p-2)][(tI - I) - 0(<> --- +<>< .... _ 2) _ 2) .;.(41 2) + 41 "" _1 "'] (E. 18) where (,~d 0 arc mo".ured reMi." to Ih,· junCI'o". 10 the loft of l"'inL< i andj. resp'-"Clivcly. Til 3/2(41 - 1)[ 'I «t..'<j» " 2.., \ ¢l4I _ 2) l - i i_ I)J -t'LJ [« I _ () -t O( 1 tI O) _ (+(_-O -20<1 If ~ (.r" -·"l If (E.19) wllere Ihe diSl31'lCC belween i and j " 'I '" I' + 0 (if poinl i is on lhe kft l ide of po;nl j along Ihe conto ur. Otherwise . 'I = o. l' luel ualion'. eo"dalions. and 'mall.an gie O,,,!lron scanering from end-linked Om,,,,ian chains in n:g"l~r bimod,,1 networb kave also been deri,·cd alon8 the lin.., of Ihe above pre* "of ( - References (I) Io;lo<"d ow.ki. A.; Mark . J. E.; E,m.". II. M(I{'OI~"kc"k., 1\11\9. n. 1423 . M",,,,,,m,hTu/,, Iq 1Z.~. 4%. (J) C",«>l/n:ule, 1~77. 10.6'16. (7) Io;loc,kow,ki, A, Mark. J. 1'. : Er man. II. Mu. ·,,""'>i«~ I•., 1991.14. J26t> (.1) I;ichinget. R. F. Distributions of the Chain End-to -End Vector F.l Examples of Distribution Fun<;tions F. l.1 The F"",ly Joink"del is th~ f"",ly jointed Chain. in which the oonds a~ con ne<:led by uni.-e1Sa1 joint~ . uch that . wo consecuti,,, oond. adopt all directions reiati,·c 10 cach olher with equal proMbiiity. Thus, bond "ngle.. "re not prt$Crvcd and the orient", ion, of ditTo",n, bonds are uncorrdated in this h}·""thetical model. The distribu.io" fUl>Ction or lhe probabi lit y density W(r) for having o ne end of" chain at 'he origin of a coordinale system. a,m the other end ,11 a distance r from th~ origin. serves as a """ful quanti,y fo r de" '''''Y 1'.1 . 1.1 1-".. "':mct ExprusiQl' For" chain of n bonds each of length I. the exact expression for W (r) is .1 W(r) ~ (8 ~rr) where T , {-- (- I)' [n -cr/ll - 2Ij"-' "(n - I) S;r'(n _ I)! 2 (F. l) is Ihe integer in 'he r",,1I<' sp<.'Cilic>d hy I(n - r/lJ/2) - I :::: r < (n - r/ ll/ 2 ( F.2) Equalion (1'. 1) " 'as ob"ined by T reloar'. bu' tbe form gi"cn above i, fnon' Flory!. For a chain with a ,utTlCicnt ly large num ber of hood,. a n appro,imat. e: ,,(~ iI 1" In Sil~' (I) ( F.3 ) DISTFU8UTIONS OF THE CHAIN END· TO-END VECTOR 331 where i1 is Ihe ;n,'e~ L~ngcvin fU""'ton L - t{' l nl) obl" int~1 fr"m II>c mlut;on of Ihe c~prc$Sion ,/nl = cu,h i! - ( 1/iJ ) ~ 1.(8) (1',4) The fUlIClion L(P) is lnow" as 1h" Longe>";" f""c,;on, F././ .2 Th' (;UII<>U'" A/'proxim"U"" The ...,ric" eXI'",sion or the righi-hand 'ide ofeq . ( F, 3).;os done hy Kuhn and Kuhn". k"J. 10 {'(')' In W(,} >= cumt"nl - n ;" 1 "I - + .. .) '( ')' +-350 '" (')' "I ~ .- . 2(1 nl (1',5) When 11>c cnd-iO...,nd ,li,lance of Ih,' Chain '$ much ,mallcr ,ha" ,he fully I.retchcd lenglh (i,e .... hen ,/ nl ¢; I)_ Ihe higher o,d.. Itrm, in '' ( F.b) Equalion ( F.b) obey. Ille normali,... lion wndi,;on r W(,)4",'d, (n ) The mcan-> (r') ~ I'"' , ' W (,)4u' d, ~ liP (1'.8) " A more w;dely " ...'<1 quanlily in rubber ela>licil)'. and in chain slati.tics in general. i. Ihe charaele.iSl;, rdl,o C, oblainoo by di viding Ihe n> (I'.9) From eq, (F.B). Ihc ch"ra"kri stic ,atio for lhe frtcly join""d c oo n""l""""'" of the j,,,,al ,hcmKc,,1 ' 1rUClurc of Ihc ~ hain. A m"", real",.., <:.>pc"ioo for ' he cn,I-lo..,",1 cha in'lenglh diSlribution i. Ihe Hermile ""rie>. e~I'ft'S"'~! in 'ern" of 1m' ""0" momenl. (r" ) of Ihe end-Io-end 332 APPE NDIXES vC'C1N_ ll>c r"'al io~~ 1 isomeric . ,ale (I( IS) ""hem<: a llows for easy cakulmion of lhe c,'c t> momo nl , for any givcn chain of known chemical mu<;lurc _The dis1ribu · lion Function i.', W (r) .. (,'") r'' ( 'I) e~p -~ ·3.. ' [(! +). 5~, + ). 5 ·1~, + 3· 5·1,9g,+···) - (JOg. + 3 [5g, _ (9g, , I + )780)(, + 51975g,. + --)(;lI. ), + [89g, + )4()2g, - (27 8. +- (F.l O) " ) (r')l + 97281 + ... )7>,) " +. (r'). I " ·he,,, ~,,'" r'{ - 2'(pu... - 2)1 + 3'(pA. - 3)1 ~ (-I )' p' , ,.) ( F. II ) ,, 0 0'(12) Wilh a.,, _, - [ J 5 7 Y (2p+I) I''(?)! » F'/'l.! 1'11" I..<:mg·ch,,;n Appr<>.,-;""I/;()II: TM G",/SSum Chain Calc"[,,tion, show lhal for suflkicn 11y long chain •. the r.lli os (""')~/{""): in "'I , (1'.12) become negligi bly . 01;011 comJXlred wilh unily. Thus. allll>e f :, tern,. ' -ani.h in~ . (f. t I) "nJ 1he dimibut ion funClion reduces 10 1he ,,'cll·kno"-n Ga ussian form, (1'. 13) )kT , A (r) .. ,1 (1") + =-r.- ". 2{"' )~ ( F. (4) In lhe case of 1he frwly joimcd ,-hai n, "'I. (F.ll) reduces 10 "'I, ( F.6) by virlUe of "'I . (F.9)_ Sho'tu chains ....'<[uire lh" nSC "fcq . (I-'. tO), in which lhe number of required 1cn" . co ntaining highcr o rder momen1, inc",~scs as ll>e length dC'Crcases_ F. I.l,j 1"/", F;xman·A llH"u m . /r;butitm'· A complete .. rics e. pression for lhe d;\I,;hu,i,," (u ,," I;,," bec"me< pr<,h;hi1i",ly '''11lpl;<'a''' DISTilIBUTIONS Of T HE CHAIN END ·TO· END VECTOR I' " 333 { LIS) W(. ) '" c.p - :; - - • {r'). Thc quanlilie. hand be adjU~",d 10 gi"c lhe besl fil 10 aClual di stribution funclion; fo r short cha,nI. ror e."",pic. those given by Monle Carlo ,im ula li on., F.2 Transformation of Distribu ti on Flmctions under Deformation Let lhc dcforn,"lio" he rcprcsc:nlClobabilil y dlSlribution of r i, ,lcoo lCc dcfonnC d, , de,(1),I, o (F, 16) whcr. det is Ihe dele"";,,,,", of Ihe defonn"""" gr.dient I"",,,,. ",Ill i$ equal 10 Ihe ralio of Ihe ,'olum. elem.nl in Ihe dcfomlc-d . that in Ihe undefo""ed ".Ic. The prob.,bililY di , 1I'0( ,.) ( P. 17) T he d" .. ibblion fmlCtion in Ihe fi,m] .Iale may be; wril1cn in lerm, of Ihe di,... .. ibulion fUIlClion in the ,wdefnrnlC-.j ,t"'" as W (. )d. -'-' (deI A) '11'"(1 ' r )d. ( F 18) wherel - ' is Ihe i,,,,,,,. of A, In lerms of the original ,,,,,rdinates. Ihe fi~31 fom, of the di!1rlbUlion fun"lion ii W (. )d , .• (dct1) W{1ru)d' o (f.19) References (I) Volkon,,licr,1 M«iPm,in "0",,, In1 Mok",,,,.. ,~ . (3) T,ek> Ph";,·,,.[ /(,,,*,,,, HQ,"cil)·. l,d Odord,19 JS. (4) M ark. J. E.• I:rm"n. II. cd, CI.",ndon 1'....: l!uhl...,lik~ 8M!i";',. ,j .lloIff.I", !',iff",,". Wiky. In,c,.,ci,,,,["(,, New York, WAA, p. 1\\Ii (S) lIo),d, R. II.. I'hillips. P. J. n.,. &irM.' bcrg. ,.. . Y., Khnkhl<,... ,.., II SWI',,"c,,1 I'h}'., j,·, "f \f~. Tol""I,,'.I<s, "',ncr;';." "f I'hy'",,' "'<J<>d bu'Y. N Y, 1<J94, 0) Kuhn, W.. (;";,,. I' K"I~.id- /. t<j.12. I(\t. ~-M I."",,," 334 APPENDI XES (S) Kuh n. w.: Kuhn. It. 1/./,. Ch;rn. MI" 194J. 26. lJ94. (9) M.nioc<. w. t .; Suoc,. U. w. Confo"'/aU"""/ TIoNwy of IA'" Inl< .....-irk. 1994. (10) I ~.ma n . M.; Alb .I{"""'!e,. G. Fortran Program for Monte Carlo Ca lculations G.1 Program (Calculation of Persistence Lengths and Mean-Square End-To·End DistanceSI """",m """'''' , ..Iv di""",,,", d"",,,,,,,,, <(J.J).<~ lI .f~lOO)~'<1lOO.J, r~200).11 ZOO.l.J).p«>d(200.l.!) d,,.,..,""" 11(100,).]) dim<"""" <1<1<1)1 c···································· ················· ................................ . , ~OTi\ TIO~ f~ OM;··~Oll N l> "TI O~ S c ANl> GENElV. L MET ItOOS FOR , FIoo-y.!·. J. """""""........, ,,,. I. )A' . , < , < , < """""'",.on 0( OF ~ O"I" nO) OEN r ~ "T ' NG CO) ... ,;.~"" ""....om "umb<' 0( 1M,nd, ,1.,0, ""') b< " ,.,In.) """ """m coWl): ""Iumn "«I<~ , how,n, ,11< •• Y , ,,,, , 001"1""""" ..... til< """"_or < on ,II< 10001 ,)"cm < r~lOO) : ,,,,,",., 10",1< for ,b<_, < ,,,(100.)): \11< ...... \0 " ... _ " " b,"~"''' ,I>< ,,'\d "om , "" ,II< ,..,in. ~i " L in";',> ~ ,II< " "'" f"l>< ""'"""""'< "'''''h .,_ .".,Ix,. .....,.", C , < , < _h ron1f'U'>"'" "1.flU ): to"" """"','" "'" '<W<> .... , I"_b(~, ... of ,II< i~,,,, r> ,(200.).-'1: '.-.nilo.o .. ",," ,,,,n •• "" ",".flo odm,ifo<...eh b.>nd "lfJ.k): ,."00",, of ,,, .,,",,,,,,,,,,,,, "'''''''''- ,O!- ;·,h _ mo, numl><, ,0( ,.n, o (dum, , ,~" «"",,km<.1l'''~ p(.): «Iu'I,""." M"".,C."" "'Y »O~".'c ""i<" r"' .... [" """".n1l< "'_''''oa ;' ' ....... n "",nhe, .... ~, 336 APPENDIXES o o < < o 0.,00, .,.,'h of bond .: """,bef ofbond.lodd n"mba in orck, '0 dof'n< ,10< middk bond) ,II< <"-in "om '00 ;!i' ... 0( ,,.." ~ p""h<+ .00 .. """, . ... ... < ... (~ eoo '''rI' ,..;," ",,"" i'" 0"" 0···..........······•·•·•·· ..··.... ····•···•············ .. ··•·•........................ co"",,, ,"" fi~ .. n>«! MC.da, '" «ad ,II< "'I"iM:I ....ri>_ "I'<" (I,rol< ·MC..f'I"' '''-III c 0"' _ " ' ,he me n.n>ed Me '0 ... ~ ,he t "I'<" (l.fil< ,. '"", Me. '"",,1 .. ~" (1. ' )"' bond - (ok 1)" 1 , <1(2)" 2 ' rl(l)' "I)" ,j " ';1£(1,') ',urn",", of 00..." ~ ·.n ..... 1£(1.. ) 'bo>od anll< - ""0' '.,h "'n,«l") 'bond ""I'h - '._ ·..... 'ion. of i......,,,,· on,",,,,,, _ '.ro I. fol. fiJ ~'riI£(2. " J ~T ;l£(l") ', .... pro"!,;I;'y ,. '.pI ..nl£(l,,) ',.",11< t proo.t,;1ily - '.PSI' ~ri' '1'''''''' pooo."'b,y 'f'I"' .......F' . ... ".nonnn'< 00,10 ,-h>mo' p; _ J,1(1)9MJ.I ,..... _'"".·,,;,180 .If. m 1><1. - ~n(,hlV! '"mp' 0, """" - O. ... ,my _ O. "",,,,,- 0. '''! Q. d<>4J11 ijk _ (,_ dolni _ U doll.lOj _ U 11.10 10 ,",'.>- ' f( ll 0, "m - pi • I'll' FORTRAN PAOGRAM fOA MONTE CARLO CALCULATIONS 337 do;l(l; w1 .• ·1 ..., ;11" •.'"'''' r~;) _ r.1 ;11"•., .. p<--.l . "".".,~,,) f>(i) w m , 'be " .. «>I" <» ,.......•.••.••...•.•... , , r<"<'".''''' of ,I>< ",p ....,...,. ,........,.,.,. , ........,.......,........,...... ... . I""""""",",,, ...,,; , I", ...-h Ix>OO <107)(1, - 1•• · 1 n; - f~i) 1(o,1.1) • • lr. \>d. 1(;.1 .1) '(i,I.J) - O. ~i.1.I) - bt" ' ooo("i) 1(1.2.1) - -, II. ' «>«f,;) ,(o.2.l) - ~ n(foi) 1(0.1.1) - t.. .. ... n(f,;) ~i,).l) - ••If.·~n(r,; ) 1(i.J.J) - «('i) 1lO «>aUn'" ...,. ,.,....,.............................,...,................................., ,.... , ,..."i, < < fon",nl , 1>< ,-..00"''' p,OO(ij.k) " ,I>< f"od"" or ,.. ..r""",,'U,,, ""',,"'" ," "'I 14 < up '" i',h ''''''' do~ k m l.l do~I - 1.l SOOO proo.I(l.k,I) - I(I.k.l) ; _ 1 .... 1 k _ I.J dolOlO do lO.:I> do SOZO I U 'I'I - ~. m_ U '1'1 ~ Q do SOlO SOlO SO.:l> !OW p,-OO(i. k , I ) ~ 'I'I ,..................................................................................... . < < ",lLub';nl tlH: p<>Sit . .. '""",' oF...,h hoo.-kI""... lI_" " ,I>< bt"m'<><j' r..«d ,"'lid;""" .ffixeo.! to ,0. ",,,-,,,< to f"" I.,,,,, '»'''''' ",(I.I) ~ .~I) ""l.l) : . ~ 11 "'(U) - '~l) 00 !040 i - 1 .• ",,~. I.J ,"(i.k) - "",·I.kl ' ;046 "",.1) pm<~' · l ~ . I ),,~ I) , "";.~I t ",'''''' · I.\.. JI·'·~. ) 1~'~l\i l . k .2 '·<~l) 338 APPENDIXES , ••••••.•••.•.•..•...••......•.•.•.•..••......•.....•... ..•......•..•...........•...... , <>10: ... ,; •• tI,. m>;.jlo: _ , '«to< in 't<"" 01 ,I>< """'.' ' Y ft. "", ,j.,9:1Jj - '.l 9W d<Jr(j) - nf(midj}rutm;.J· j J1 ,..............•....••.. ................ ............................................... < <>k.Ia,;nl ,he (t2 co.·2 H) ... 1><<< F! i. ,1><."$1«_n ,I>< .. diS. bond and ,hed.o; • • nd·~ ...,"', ,.", _ 0. doJJ.OJj - I.l JJ.tl .... d , -.um . . , ....,.. $.Um.,.. ' ",m " I/h<>nd " : ,.m""_"''",,,I''''· ,...................................................................................... < ",k.ld;n, ,he '. Y , ,,. , comp<>n<"" of ,10< .,.,,;,t« < ,he ..... n ", ... ", ...... _lo- yt(''''' 'nO num, " ".m. "' ''<1n. l) ""m, < "<1n,2) 'lUrt'lr ,,"""_,,"all!' "no.J) '''' - "'" , "nn.I), · 2 • ,,<\n.l)·· 2 + " nn.»)" 1 .1 - ,,"m""~ ,2 · ,,"mYiZl"~ ""'- .... = .... -""1 cr - roq:n!bond"l .) ...................................................................................... , , ",k.latona ,he r,on, ",,"' 00 ror ori d(I - (J '....., .../"",-1_ WI 0 "-,,,.(1,') ' . "'''kl2.· ) " ~f'O"",' of tlo< _',,1«2.') ")-"""",(>00".' of ,he '''''"",,,,,,,,n' r<";",,,,,, ,-«'''' · '.. 1 p'"'''''''''' """Of · '..1 ,,-.. '«2.' ) of ,he ,,,,,,,,,,"'" ,,-,,1«2.' ) '",,'n ",..." md _K>- ""',0< . ,nIO(2.· ) .< .......,,"",,"' <>';' " -,,,,,(2,' ) ',,«.~ " Ti,«2,' ) '00 ""22 f""""r( MII.S) clo.>o< (l) ,,~ ~ of .. 2 "",'21 FI) · ·. dO ',oJ _ '."'1 FORTRAN PR OG RAM r OR MONTE CARLO CALCULATIO NS 'ubro" "" .. nZO, i 1.lM.i< - 1101<39.,,,, • 11.\1 d, .......,. i«911 <10" ilf,iI! 'Qodu"'.I,_O.",jff."lO) tM. ,If ~ 1 idum ~ """";.,_ .. un',ml do 11 j - '.91 id"", ~ """"i.·odu", • ic.m) ir{o) "",om " ""","n"" modf..·i idum ;".tol) iy _ Ml um <"M J- I+(91 " Y)im 'J\j,I', .1.0< .j.h . llI'« ' .... iy - 'r{o) ",n - ,y',m i ...'U," idum G.2 Form of the Datil Sot ,...,r.,,,, • _d "Ombt, rOf .........' "uml», ,"mbt, "W<-' ,,,,,,,,,,,,,I.".... r", ''''.'. " ",,,", ' ."" poc"" probobili, ... r", ' .. n>, .."" ,. ' .Dll .. ,,,,,,", "a'" 5J W 51 70.\) , O. 120. G,3 Output of t he Program "Um""," of bond. - SI _ '''';' - 10_53 bond "'"I,h • 1.000 """' '''''' 0( ~ m .... "" _ _(I(l0l) )2(1." -' :'0.0 "'" probo bili'1 - .<SOIl ",oo.bi~', - .Z~'" go, ...... Pf0babilH, .11'1< rcrfo><m 11"""". .""O¥>.r< ,,,,,,,,,,,,,,,,,", ,__ rom""",,", ",<"" . L '''''''' ,,~W< 1) '" .... n 3J9 J~O APPENDIXES .............. I".O --...... ..... _ IUl l.J~ ,_or,"l_"lIfl) - ;».11 DO _ "s.JIISlAQ H. Some Historical A spects The r""t material e.'er u>ed (or its ela'tomer", properties was natural rubber. obtained from t~ latex of the tree 11''''0 "rusilimsjJ'~, Thi' tree is indigenou'l to Soulh AmeriC"~. particu larly Ihe Amuon valley in Brazil. Ik<:ause of its .u... ceptibility to plant disea.. endemit to tlois region. conlmercial production is ""m<_ what limiled. However •• imilar dimatN; condilions occur in other tropkal I'<:gions, particularly in SoUlh,Easl Asia . whell' mo,t of I~ world's supply of nalUral rubber is no'" oblained, The rubber particles arc of micron siv: and are dispersed in an aquC<1u, medium , The materi"l obta ined directly frOnl Ih. lalex contains a nUnlber of romanlinants. including carbohydrales. protein s. mineral saill. and fan y acids. some of which stabilize 'he rubber .... pe'''ion'. The.. are removed in the "arious purifiauion >il.", carried oul aft.r the ruhber is coagulated. or pre· cipitated. from Ihe lalex alier it is ha,,'csled , This i. typically done by tilt: addition of acid to the lalex, The remainder of thi' appendix gi''eS a hri<:f overv,"w of som< of the hislorical milestones in the discovery and utiliza,ion of nalural rubber. and of 5t)me of the synthetic elastom.,...' dc"eloped to repla"" i, H.l The Earliest History The earliest known ,«o.d of nalural rubb« being used becau.. of ils ela.,om<,-i. properties pertains to ga mes played during tJ>. Maya" ci"ilization. possibly back " $ fa. a. the 11th ""nt,,!,),!" ) Such gam'" were o r.s.:rved by ,he crew of Columbu,' voya~ to the New World in 1496. specif"",lIy 00 th. i.lands .,f Trinidad and Hispaniola . In what must ""nainly have bt",n one of i" very fint application •. it waS u5ed by the A>:lC<' indians !<> f<:o"n ruhbe. ball' tha t w~,.. 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Jo 'n4:mu, pJ[ 1P.' !:).,!IUU "'II II:>!'!·" "Ul"II"'!JP JO s~ldU'1IS IP..<;IAOS ~'''II III~nOJII pOl! '=1 O""II 'III 00 UOllJ!lU'OJUI '111""'1''"000 """'"1110 1'"11 .(''ll '._,'\'l1nA uo,",wy Otjl U! ",!'J.\O>S!p J!''11 uO JO ,\w'P" ,y '!J"d ''11 OII"'UOO:>.l ,u,wepuoJ"1 ~P"J puB ""'U'''':! ':t """'RU ",W'l'U:>J.,j 0"'1 'SN 1 ul -SJc,'\ IXl ~IJn,U JOJ 'lu'wdol'A'P I"P-"wwo, 00 AII"!IUJ$S' ""A> "U~I puo "1\'SO"""~ pOI~:>J1 "'".. '(n!-"I~lU =111 ,:'~U>Ao,,!p .. .\IJ~J ~1I1 JO 'I!d. ul .,,!-''''' OS"" ,,,til" '"'I ,,"""'tI .""","'!> 81elS P8~U!l ·S SOJ:l -un 81.[1 U! J8Qqnij leJmll N ~"H ""''''''>A IU UO IIII)()l; ,!,ql SUI 'o.qll"' I.,nluu JO ""U:>J~'.~ ~A!'U"I'" '111 OlllU!P"~1 Ilu"", '111 In'll ",.>1 punoJV '>JU """ '\1"" S 3X ION3dd.... nt SOME HISTO RICAL ASPECTS :U3 part of;1< ability to cOlltract. until ;t was warmed. Both observation. are now known 10 he duc 10 Ihe ~r)'Slall; '.alion induced by the SIr<:(chi ng process. whi<:h ~ <ussed in detail. in the lighl of mooem Ihcrmoo)'namics. by I'lor)"o, Approximately SO yc"nla lcr. Lord Kd,,;n sh",,~l lb'l . • ",ording 10 thermodynamics. the result. of either of Gough's f",t and second ser;"" of CJlpcrimenls preordain.:d the n:.u lt, of Ihe other , Gou~h·. CJlperim~nlS were re pealed in 1859 b)I J. P. Joule. who was able 10 u:;c Ihetnw,,<:>uples 11, ",casure th" Icmpe""u r. changes mon: a<x:ur~tcly,",II .IJ, Since Ihis was after the ;n"ent;on of vulcaniza tion. d<scribed Ixlow. he was also able 10 use c,o,""li nk.:d ""mples. which save more reproducibk: rcsulu, Thi s work was carried oul juS( .. f.w yea ,... after enlropy was imrooocro a. a Ihcrmodyna,nic ,ari able. Considerably lalcr. mort comprehensive stress - l eml"''''l~re Sludie< were carried out on n, tu",1 rubber by K. U. Me)'.r a nd e. Ferri. in 19)5""'. and by R. L Anlhony, R. H. CaSlon.and E. G uth. in 1942""". In ' pile of Ihc Jc:nglh of lime th"l has elap.. d, such "Ihcnnoelaslic" expe,i,,,en,, are still being carried OUt, Ihe current inl".'l being chara~lcri""lio" of the thcr",,,,lynamic nonidealilY of claslOme,...'o.",". A less .. riou. vicw of ",,,ne of lhese early c 3" APPENDIXES 0.0... 2).1 .... ' , _ .... _ ......,N_ ....... ..... _ ~ I'll .... II .' "n.. For Side" b~ • '~ Gary L.....,n i, ,cpri nlcd by pcnni .. ion of Ch, onick 1'¢lO'Ure<. San I'",ncioco. CA . All ri&h" "",,'Vtxl. gas 10 a rubberlike subs.ance ' ·..·... A few years later. W. A. Tilden in Engi."d produ«d isoprene by the destructive distilla tion of rubber. Upon storage. some of his monOmer also oo nvened into a hi gh-molecular-""ig.ht material with SOme el~stic prope:r1ies'· ... Th .... were the ea rl~st beginnings of still SOME HISTOR ICAl ASPECTS 3(5 lion mechanism'"''''> '. T hey proposed that thc elastic foro: was due 10 Ihe decreased entropy of Ihc oriented and slrc1Ched chains. thereby explaining Ihe Ihenn<>: should be proponional 10 .he aMlute teml" were in at Ie.st approximate agrcement with much of the e ,peri~nlul data a_a ilable al Ibm lime, AI..:>. quite remarkably. they had wme features Ihat "'Cr_ very similar to lhe equaliono obtaintd laler using milch more sophisticated approachesl"", " , H, 3 Cross-linked Natural Rubber In 1939. C. Goodyear in Ihc Untied Sta lcs ."ul!lll ways 10 improve Ihe rropcnie' of natural rubber, particulJrly ilS stickiness and liquid like behavior " In his sludi ... he "CI,"idently found that hcating a mi"ure of natural rubber. sulfur. a~d lead carbonate coI"ened it into an Insoluhle mass . It is now known that 'he su lfur alOmS bonded or "cross-linked" the rubber cha ins imo a TlCtwork ,truclure . T he resulting material had properti.. thai made il" vast impro\'Cn>.ploiling tl,l. new tech nology, Goodyear began In publicize .11d creale ;ntere'l in hi' invention b)' pUlling on rubber ··.hows.·· in London in 1851 and in I'ari, in 1855. He a lso accumulmcd Mer 200 rubber paten" in his name. but had 10 'Ilend much of his time and energies deal;ng with i"fringomenlS On ,hem. In one of the mo.c famous of ~ hese. he was "'presenled by the legenda ry D. Webs'er. who w". the U.S , Secretary of Siale at Ihc lime, In 'pilc of IhcsecfTorts. he died heavily in debt. in 1860', Ih""ock and Macintosh, on the oth,-r hand. con tinued finding successful applicalion, f"r thi, new _ukani~':llion tochni,!".' For <.ample. Ihcy turned out rubber fIre hose. which ""a, firsl used in the gr"," fire of London in 1827. In one c"",,,,,,iy importan' evenl. in 1846, Hancock buih wlid rubber lire. for 'he carriage of Queen Yicloria' , These were clearly the predecessors of 'he pneumatic tires thai are lhe basis of the enlire automobile indu.;lid that the Costs "'Cf< one hum~" life for e"c",' 4 kg "f rubber. "-'nich is approximately Inc 346 APPEND IXES amoun! tbat goes imo an automobile tire! Many rubl>cr Ir""s were also 10.1 in th<;S(: region. from improper tapping t«hni'lucs" . H.4 The Plan tat ion Movement East There had long be<:n interest in planting and cultivating the hevca ruhl>cr lree in 1'1= 011>.,. than South America .nd Africa, The 1\",1 attempls look place in IS73. when a Lend"n curator wa, able to bring some hevea =<1, from II ranl back to England. A few did produce plantS, but all died I>cfore th")' could I>c planted "I",,,·here. In 1876. however. an English colTce pl"nter named H. W;':kham waS able to tran'pOrt ten. of thousands of Ihc «:cds to Kcw Garde n$ in Wndo". w"ere many were sUCCC$SfuUy 8crminated into viable rubl>cr plants"'. They were sent to C<),lon . ",·h".-.: thcy wetc u'<;d u, c,,,. bli.h Ihe first rubber plantalions in the Far Ea.!. Th .... w"re then expanded to Malaya. SingapOre. Indonesia. India. la";!, and other countries. Thi. permanen tly hTOkc Ihc Soulh American and African monopOly. By 1912. the South·EaSl Asian plantalions malched tho", of Dra,jl for the first time. and 2 Yta f1 latcr c,,<:«< point where t~ could produce the desired latex efficiently fo. ""ernl dc·cades' . H.S Some Additional Scientific Developments In 1888. a !klfa'i velennari." b)' the "arne of J. Dunlop made lhe first pneumat ic tin'S. for biC)'cles ' ·•. II is hard to overestimate the impOrtance of thi ' de,'e iopmenl. ,ince the biqde wa, Ihcn, and still is. th. most common mean. of tran'pOrtation in the world. The new Iyp" of tires made the bicycle c,'cn more att"'c'!iv•. and thus encouraged vast impro,""",ents in the road 'ystems everywhere. Even more important is Ihe fact that automobiles Were undet rapid de,'elopmcnt arou nd the". juS! in time 10 take advantage of the irnpro'-ed road syStems, Not su rpri singly. therefore, Ihe pnwm.lic tire became of central impOrtallCC 10 the development of Ihc automobile industry. as well as Ihe rubber industry. By 1888. the highly e.. perimental motorized bicycle. and IricYtics h.d evolved into four·..-h""led automobile... which were now on ."Ie, The first of thest to I>c fitted with pneumatic tires was a I'tugeot enlem:! in a race in frar>e<: hy Ihe Mich.lin b, oth ..... Allhouglt many of these e~rly d"vclopmcm, of pneumatic lires too~ place in Europ<. leadcrship in this field «>on moyed to the United State>. part;':ularly to Akron. Oh io. In 1906. an Akron chemist n~med G . Otnsager di scovered the fi f11 of a long ",r~ o f COlnpOurld, that ",,-'demte the "uk"ni~ati(-", prOCNS', ATOund this time. im portant addi\io nal wor k waS being carried Out on the SOME HISTORICAL ASPECTS 3(7 rubber, For example. in 1'Xl1. !. Kond"k()~ ;n Russia waS able 10 prod"", " rubher lik< mal~rial by healing 2.3..Jimc th ylbuladienc in the presen<'" of polash ,a. Hi, eompalriot S. V. Lcbedev had sonw ,uece," ill polymerizing bUladic~e in 1910. A Iso in 1910. U ,,;vcrsity of K,d chemi'try prof~'SSOr C. D, Harries prepared SOlne .yn,hetie rubber by polym'. Tho m,,di~cation prei",red bycQp<>lymer' izing isobutyle,,,, "';liI a comonom~r providing ""Sllturaled groups for cr<>S.<-link· ing ",as ca lled "butyl I'I1bbcs in tires. and i, now uS«! a. one of \he la}'ers in tPock" tires. Two other clastomci"l should be mcm 'II)X'"rb thermal stahil it/"·l'. 1'ht <>ther lype ;, Ihc ethylene- propylene lerpnlymcrs. which oonshl primarily of elhylene a nd propylene w;lh just cnough of a Ihird <;umono",el 10 p",,;de 5OIl1C un .. turalOO , ite, 10 facilitate cros~Jin k i ng"'"';>O , During the 1'I4Os. !here WetC aho impOrtant new developmentS with regard to the Iheor;es of r~bb thmugh 0"" a"olhe<- Soc" thereafler. the .lternati ve ""fline neN""'" Iheory was propo.oo by P. J. Flory. J. Renner. and F, T. Wall. llot~ theories arc di",,~sscd in detail in chapter 2. Alle"'plS 10 "",>nelle and 'ignifi~""lIy nlend !h"'" thc~"ics wen: nol "cry ;uc.,,",ful "nl;1 Ihe 1970.. The description. or the,,", 11, ,,,1<-r,, Ihenries "," ke up a "cry significa nt pan of Ihi" hook. l~ APPENDI XES H.6 The Effeds of W orld War II By 1940, war already Sttrned ;mm;ntnl and II>< Un;led Sial.. was 'lOckp;l;ng natural rubber. e(f«li,·el}· doubling iu normal imponed amOunl of around a halfmillion lon, a )"<:.r'" In 1941. Japan occ upied Soulh· East Asia. oulling otT ,ur>plies of nalural rubber 10 Ihe Uniled Siaies. In ils first response 10 Ihis supply crisis. Ihe U.S. go,'ernmcnl ordered Ihe planling of len, of Ihousands of aOres of guayule. This shrub. which Ihrives in lhe w.slern parts oflhe Uniled Siale, ~"d in Me_.ico. also conlains rubl:>cr lalex. II has the disadvanlage of yickling Ihe rubl:>cr only wilh dimcully: Ihe planl muSI l:>c ground up and e> Imcioo. Ihus requiring a conSlant s\lpply of new plan 1._ In ii' second response. II>< governmenl orchestra led II>< gearing up of U.S, induSl ry 10 produce various wnlheli" rubber $UbSlil uICS .. '·l'_The impelus for Ihis colossa l projec1. came from B. Daruch. J. D. Conanl . • nd K. Com pion. ",·ho convinced President Roose,·ell of Ihe desperate natme of Ihe ,iluat;on , In one of Ihe mO>t re"marbblc Success slor;", in modem induSII)'. a capacity of I,~ billion pounds a year was reach<:(! in 1945! H.7 The Postwa r Period AI Ihis point. Ihe South-Ea'i Asi .n rubber plantaLions were recovered from Ihe Jopanese'_ Ahhough con!.idcrahle rch"hililalion work was ~uired> the planl._ tions wcre e,"c'lIua lly producing at their nOJmat le,'cl •. dominating the world marhl in nalural rubber, One addilional scienlifIC dc,'clop"",nt waS Ibe disco,..,ry <>r Ihe extenl 1<> whieh ciaSI<>meric malerials can l:>c degr-.. dcd by reaction wilh subslances such as <>lOne_ Thi, led direclly 10 th~ dc'''lopm~nt of "rganic anli · dcgradanls. panicutar the mon",,.,,,, unilS add 10 th~ eltain ;" ex.ell y Ihe same way. Specifically. bolh double bonds in Ihe monomer ha ,.., 1(> be ol"'ned (""1,4.addilio,,··). and Ihc new double bond Ihus fnrmed in II>< "'p"31 unil has In h.,.., the configuralion'_ Nalural rubl>lyiwprene. and. bec.use of this stmctur.1 regula'ily. il can nndergo stra in·induced crysl.lli""'lion. Thi' is a '"Cry imponant advantage with regard 10 mechanical properties. since the crystallites Ihus produced greally reinr<>n;c the elaslomer'O- ' >"", as dCSl;rib<;d in ch.pler 12_ Polyi,oprenC$ Ih.1 l\a,.., ,nOnOmer addi_ tions othcr than the 1.4 ~arielY. or n.,,, mi>tures of Irons and ~is placemenl •. do nOi ha'"e Ihis cap"bilily ror self·reinforcement. One of t"" t,iumph. of modern synlhelic polymer el><mi'll)' i. lhe capabilily. dem<>nSlraled relali ,,,ly recently. or being able to produce diene polyme .. having muclu"," ri'"aling lho"" of n,'ural rubber in Iheir regU la rity. Shorlly .ftcr the "'"4r. the threal to nalur.. 1 rubber posed by the &)-nth.,;'; "ilhlnmers quid I)· became more pronoun.,-,,!'. Some or Ihc .. ,ymi>etic m"ler;"I, ha,'. no,,,1 and "~ry allraeli,,, propert;c,. For example. Iriblod copol)''''''rs "''Cr" dewt<>pcd in which the ~nd bloch aggregaled inlo hard glassy domain. Ihal conld ··d,·· SOME HISTOR ICAl ASPECTS 349 fU!>Clion as lemporary cross-lin ks for I~c cla,'olhCri. ccnlr~1 block "''' These "Ihermoplasli. elasromers" have Ihe Ircmenduus advanlage of rcprOCC'S$Creased. particularly by h"rricultural bl"ft/Jing techniques. There has . '..", been developmenl of arlifi· cial !rimulanl' 10 enhance lhe How "f lalex during lilt lapping pr()Cedure L. Rapid suide, are also being made in understanding Ihe onlymalic biosynthesi, of namral ru b~r in the H~V(!~ b'~zilic",is Ir..." and in Ihe applicalion of modern I..:hniq Cle! of genelic engineeri1\g". AnOlher im riguing exporimenl is Ihe scpar~ljon "f enzymC$ from Ihe nalural rub~r latex. aod using Ihem in Ihe in vitr" syn lhesis of nalural rub~r". F"r" •• riely or "'"sons. , ""h as Ihese. nalural rubbe, m~de up. in 1990. 37 0; ' of lhe world produClinn of ela,IO"'C~ of alllyp"". Oflhi. 10lal J'",duclion (25 hillion pou nds), the Unitc-d Stales consumes 6.4 bill ion pounds. 27Vo "f which is nalur~1 rubber. Much hinges on Ihe fortunes of lhe na lural rub~, indnsuy"'··J-J'. It bolh provides. critically i!nporlanl raw ",alerwl ro Ih. indu>1rially advaruxd countries and rep .... nl. a major elemenl in lhe economic. of a numbe. of developing ""unlries. H"w ml>Ch of irs markol "" ill be perrn.nonlly losl I" syn lhclic malerial. is nol obviou! .1 Ihis poim. and ",mains a quelli"" <>f considerable imporlance. REFERENCES (I) fl urley. P. E. J Iot_ _. S<-i .. C.t.-.... 19~1. AI S. 1279. (1) MoynihAn, l. 100<>Y ', 19lI9, Oc'oo.r, 14, (l) K.utTm.n. G . II.; 5<)m"" •. R , II. J . C_ . t:J. 1m, ~7, 422. CI"""", (4) S,inson. S, CMm. &X' N....., 19'10, May 11. 4S, (5) Semegen. S, T In R«NHt T«luoolvty. M . MorroR. ",d. Va" Nosrrand R';nhokl: New YOfk . 1973; p. IS1. (6) S'yd..,,,. l . A . Rubht. CII<",""Y' Ap\~iod 5<;'""", London. 1975, (7) La!. l . In f:m-,.tlvpfifia 0/ 1'01_, S<"~(r """ £ng;",,,,~. Wile)': New York. 1\ll!9; p. 106, (8) Reek. 'e M .. E;ohinger. II. E. ('11< .... t:..~ . N.w.. 19f1J. r-...:embe, I~. 7~, (9) Gough. l . Pro<". Li,. Philo.!. So.-.. JIa.I<M>I ... 1nJ!in. 18<15. I, 181L (10) I'lory, P. J. Pt"'rlpk~ of PoI,,,- C"'" ,i",y. C"".. II Un;"" ..;,y P ..... ' Ithac>. NY . 195J. (II) M."ddkcrn. I.. p~, CiI<m. 1«luool. 19'13,66, (".6 1. (12) Poge. C. G . A",. J, Sci. r ~4 7 , j4, 34 1 (IJ) louie, J. I', T,ans. RoJ'. Soc. ( (,,'''''''', 18>11. AI ~ 9, ~I (14) Me,..r. K, H, ; Fetri, C. 1M•. Chi,.,. "rl~ 1935, IS. S70. 350 APPENDI XES (I~) T",loar. L. R O. 1.". Phy.k. of Ruhb<, E/dcs,k/,y. 3rd aL a.",ndon I ' ,..· O.f"rd. 1975. (16) Anthony. R. 1...: Ca""n. R. H.: Guth. f . J. Ph}'" CIor",. 19 ~2. 46. 82(;. (17) Mork, J. K : Erman, H. R_.IiJ< ~ £1""i<;11' " MO/IToJ", Pr~ . Wiley_ I" «..,.;ene<: N~ York . 1988. (18) Anderson, K. / . MRS 8,,11. 19\IO,/ u1'I<, 7 1. (19) KauITman . G. D.: Seymour. R. D. J . CIw!m. f:J. 19'11. 68. 211 (20) Ka ,,- I . R. Na'k""i"""""'f"~ 1925. 13. \100. (21) Mora"~ .. , H. PoI}''''''''- n.. O.(fj,os aNi Gro~"h of a &w..c~. Wiky_ln'< .... i W....pilO;f Wwd. Au"ralian Droadca .. ins Commi>Oion' Syd .. y. 191'J. (23) Lo W . R. t .: Hou •• P. In lI.uhbt. r~n, &I. Von N ()S( ... nd R¢inho\ing'on. OC. 19'90. (2S1 W.. ricl. E. L. fOH'y y,,,,. of FiW,. 1M hroJlff,;o,u ofa Do", Cor r",h~oIotJ1. M. M"rlOo. &1. Van N",' tand Roinhold' No", York. 1913: p. 210. (27) Mark. I. K , ~rm.n. R.. &I •. k P"'y"". N""""ks. i"r. R£JNrch P'qg."",. University o>f !'¢nn'ylva"i. £>reu: Philadrrss: l'o:<w Y"rk. 19 ~~ . (11) Seymour, R. B.; K.uITman, G. D. J. ChnrI. fA. 19'12, I. ChnrI. 1989. 264. 18618. (33) Grtek. D. F. C"'~,. £~, . ,II' .'J \'191. May 13. J7. (.\4) R<:i>IIini,on. OC. 1994. f:Ja,,,,,,,... £/tn,,,,,,,,,,. " Selected General Bibli ography The follo""in~ '." ",I""o.k<, ch'o""lo~. 1 [j" "f "'l!",din~ 'ubberlike (I) Flo,y, l' " 1,.irr;:ipJ., 0/1'01,'"," Chfflrj",y , Co",<11 Uniy<"';ly Pr=!: !In",,", NY 195 J. One of Ill< m",1 "ulhoritative Ie", on 1"'1)",,,,,,,,, in gen.ral. ""ilh ,,".o,i,. <"'.. rage of rubbe,li ke oIo"i..:;ly. A ctassie: "h"'" i. ""... dcmllho biblt of I"'l yme, ""ie""" b~ 1M"y, and i. still in poinl ~i .. ilS .g<. R'iorou.•• wil. Jeri>'. ""n,"-.nd much ph~.ical in.lighl, (2) M....... P. I'ol .....w.: S"""lUr't tHId Hulk I'."f"tni<-" V,n No" .. nd: N.,,' York 1965, Good "",.. rage <.If WUC1ure-p.o[>Orly re'a'i"",lIips of 1"'1)"'.... in Ill< bulk st.'e, &c.p'ion.lly ,oOO fo< me<""nieal p"'p"r1i ... inch,din8 ruhbe,lik< . . ...ell a, ,,1 .. =n. '0- IS) Smilh, K. J .. Jr. In PoJ}"",,, &k~"'. A. D. k nkm•• F--Ie' on ,ubl:>crlike cl'>lieil)' Ihal " "\wi",,,",y 00' of dale. but "ill 'cry "sef,,1. (6) Monoo, M __ Ed.~, T.dmoiov. V.n N ... ".oo R,~nhold : Now Yorl . 1973 Sever. 1 1I"00'ai chaP'/x" fila,·IId,,". 3,J ed. Ct.",ndon ,',,-,,,: O,fo,d . 1 ~ 7S. 0 "" of lhe ,'oml.rd """ • • on ",trI>...,hkc t;c"y fOf!he I." "'''''ral dccad .., Eo,h<, 1I1llJ 'nfom'""'~ 1 <no cry".lIi·" "io" in .I."on",,,. 352 SelE CTE D G EN E ~AL BIB LIO G~APHY (10) Labana, S. S.• &1. Clttmurry dJId f'rol"'''w, '" C"",lu.Jwi l'<JIy"'u,. Academie Pren: New York. 1977. Pro«<:dinp of.n ACS Synrpooium, . imilar 10 U.. book ciled in "um ..... 4. ( II) Eirich, F. R" Ed , &it-"", and T.~y 01 RuiJb, 10 lhe m.n"f""'U~ of ,if'<>, ( 12) Bl}'eml$lry. Appli<1l Scl0 book ciled in number II. b", II.u much more informalion "" specifIC t I>Slomcf'$, (B) N.sh. L. K. J, elwin. Ed. 1979. 56. 36). Very del. ikd deriva lion of'D .I>>lic "'Iualion of ""0, u"n, II>< oimplo>l possible 'PI>I'QlII:h. (14) MaIk. J. E. J. C_. Ed. 1981. 898. Scmiqu.anli'01ivc disc" .. ion of bo.ic IOpic>, plu,..,,,,,, mol'< od.a"""" ,hoI wcre ollha, lime. (1 S) M.rk. J. E.: Lak J.. t:.t•. E/aJI()n~rs and «>nlinuu,n mccbana u>cd 10 COvcr the .ub;:.:t "The 11H:ory of High EIl",i.i l y." Very usrfuJ drocriplion. o f "0",1"00 problems. ponieularly in ,II< . "'> of 'I>eory, (/1) uNna. S. S.: Dickie. R, A .. Eds. CIoar""... I:aliIM .1 11i¥l:Jy C.<w·l.inktd Poly"",,,, Ame<>«ling> lhal are. ""l""ll0 lbe book cil«i in numbe, 10. (18) Furukaw. , J. Mak",,,,oI. CIk .... Snppl. 1985. 14. l. R •• of in' .rp"'l.. i,~ work c.med oul u"II$ ' "potUdo croM-b nk " model, (19) Kilian. Ii. G.: Enderl<. II ~.: Un.dd. K. C<JII. l'oI;'m .•'>d. 1986. 264. S66 I(e'liew of in'",pm>li.. work coni«! oul u,ing' .an der Waals model. (20) ul. J.: Mark. J. E.. Ed., AdwlllC<. in t·/aJ,,,,,,,, .. dJId RflI1bt-. Elas,ld,y, P1vengo of 0 wide .. rit'y of ,op1co.. including some ,,~-n., melI.eh Conferc~ ... hich emph •• izcd of ,II< more appii«l ,opH:< invol· "in, rubbe,li);t .Ia'lici')', (23) Queolcl, J. P.. Mark, J. E. In 1:JK)'led re ...... of 111< 'ube mOO.1 of rubbo,Hkc .Iao, j cil~. (26) Krarner. 0 .• Ed, BWIogicai e,k,vn..·" ,b. Elsevier. lotIdon. 1988. Proceedi"" of lhe Ne"wo.-k. 86 Me«in, held in Ewno, •• I)(:nm.r ~ , ",i, h ""me <mpha';. on biol""""l n<:!work. and on .""lIini. (21) Mork. J. E.: Erman. 8. R~hJor,I;* r UWI Ie;ly A Molt'Cuio, I'r;""" . Wik:)'. Inler lrk. 1988. InlrOO""I",), l<"boo ~ . gi"ing 0 brood ovcrvH:w ond <mph.",ing p~yo.ic>l . r"... """ ,.w ,,,., "'m< 1'''''-''''' SELECTED GE N ER AL BIBLIOGRAPHY 353 (28) Heinrich, G .; S'",~be. E,; Helm", G. Ad.. Pol,..... Sci. 1988, 8S, 33. Re.iev.' a nd anal)";. of ,II< ,III," of III< Slalhl"",1 nt«hanko of rubbe t li~e olastici'y in 1988. (29) bm.n , fl, ; Mark. J, E, Ann , II~,. Ph!". O"ffI. 1m, 4<), 351 J\.pecU; of .IaOIicil1 li kely '0 be of in'e",' to phy>ical oh<mi." de$iring a hrier o,'."'~, (30) Queslit,ol properties of"",work h wfiuen in a ronci .. , e""ydopoI M""ntk. 191>9; p, 481, Mcdw " "", I .nd ..,1""nl.,,,'olli "1 melh· ods fOf i)'mer>, (H) Mott , J, E. [n f·,,,,,,;"'s <>I M",.,.omo/,,<:.I,,, Sc;;nrr. T. s,egu .... T . Hig • • him"" . • 00 J\, Abe, Edt. IIlacLwdl Sci::hnif I~p;c. rove"'" .1 an intO.nali"".1 ......ina., (lj) ILIum",.,,,,, •. i\, ; f'\eot. C. Ii Spr~" Pt'()t'"dl"ll' in P~J'lia. Sprin~r Berlin. 1989; Vol, 42. P.oce«1inp of. ",or"hop held in 1"lich, Germ.ny. "hich "'", •• "ngOO 10 p.ovKl< for di><;u"io ... bt,lwocn throrc,ic,. n$ .nd "pcIin> of pol)',ne. ""t"'o,h (J6) lIurclut' d. W" )toM-Murphy, S II " 100.1 •. I'~,.;«II /I".wh. I'Q/y",c;s ""II G<>I• . Elsevier: London. I~!. Prooxeding, of II>< r-:ol w<>tks 88 Moc:(ong Itthl in l'tcib""8. Germa " y. "ilh e<msidetabk: """"tag<: or tcmp<>.a.y. rc"."ibl< ~ i >, (31) Q...... I, J, P.; Mark. J, E, In f.--""yclo/><JitJ ~f P~F'i, ',,1 .'Uci£",.. (HId r""""",V'. 2nd ed. R. i\, Mey"r>, Ed. J\cad<mic P'<>o: N<w y o.k, I9'JI, Upd .. ing of'he .nicl< cited in "u mbe, B. (38) Mork , J. E.; E""",". II .. Ws. f.'/QJ,,,,,,,,,ic PM),,,,,,, N",">rh. Pronl"'" Hall: Englewood Oiffs, NJ . !9'J2. IIook dodie8,cd '" til< m<mory or EU¥<"" Gu"', a nd 'hu. "'ilh . n omph.,;, on con,ribu ,ion. r",m many of h .. coikaSU< conlri\>utod , (39) Matk . J, E. C'''''f'1'f, 1'0/)""" Sd. ]9'11, <, I J~ Su,,'oy of applica'io "' of mol.,. cui •• "Iodolinl 10 prOOle",. '" II>< .re" of ol. >lomc.ic pol)'",",' .... "'0 •• •. (4<)) Mork, I, E, Ang~ ",. Mal",»!",1 Clt<mk 19'J2. :1021:103. I. Roocn ' advances in bolh ,he ,hro"'t""'l.nd <xp". imc"tal.,de> of .ubberlike < heJinning engin"". ,he prin, (43) Ma.k, I , E.: bscnbcrt, A, : G, ....Iq, W. W,; .0.1 .00<1 . 0"', L.: S,m"'.ki. E, T .: Koenig, J. 1.. ; Wi&n.II, G, D. Ph,-,;,:"I 1'''''l''mi" of l'oiY'tk''', 2nd «I , AIDn. S, J .• Ma.l . J, E , In ...."o.mne S I CI.""n and J. A. Soml)'cn. I'd., I' ",,, ,icly>ilo,"n'" '" . ,heone. C"""",. "'"'rIC< 1'01,..,,,,,,, 35' SELECTED GE NERAL BIB LIOGRAPHY (45) Mark, J. E.; Erman. II.: Eirich. F. R.• EdL &i(>I!dio of AJ-.=mJ Mdl~';"Is. I) . HI""r. Ed. Pcrga""",: O~ford. 1994: p. 17l'l. D<1''''. E"," li.lly a n of lbe rev .... ciled in nu m"'" 29. (48) Mo, • . J. E.: Ii,man. 8. In Pnfoml>pee" of rubberlike eI""icily. "pd.,. Author Index A ",k<:I<:s1 dir«:lIy i,wnlvcd ,,·,Ih da"o""D and rubbcrli l c .""'icily Ab<. ,\ . 2N. JOoI. l53 Ab<. Y. 127. 12~ . In. 1J) Ah.rMi. S. M, 86. 16S. 18S, 259. 164. 353 A"loni .. j, ), 265. 301 A.say. I. A. 263. 265. 295. 300. lOt> Al.mo. R. G . 85. 264 270.303 Alben. R. 100. 105. 1'19. 218. 332. 334 Alfrey. T, 179. 181 Alloo BolobLO, L.. 1.14. 1~8 . 153. 157 161, )64. 189.217 R. D, 265. 266. Bolo<:. F. (.2. N. 22.1. 211 Ln. 2.14 lko)·~'. R. F, 108. iJl Urennan. A. II. 265. 267. 269. 270. 288, 190. JOO, 302. 303. 30~ Rrinke<. '- J. 265, 21>8. 270. 190. 300, 303- 305 U'ocha,d. F. 61. 11 Ur",'ow. W, l54 Rro1>Jn." . R. W . 61. 70 IIru'->ml<. M . 187 IIrrd""",), A. 341. .149, )52 1I""l m _,n,( ..., Ilahor. I. 58 -6(1. 70. 1"4. IQ5. 134. 14~. 153. 154-1 51. 159.16(1, 164. 166. 161<. 113, 11S - I1S, IS6. I ~) . 21). 211 Ball. R. C. 3J..-3S, 31. 42. 61. 70. 224. 230. 001'001. l'.lJ~. 238. 240. 263. 265. 290. 300. Ru tidc:. J. 62. 70. 223. 231 - 2J.1 lJ.:.uon&"Orl"...-, A. 234, 353 llel"u"" M , 21. 223. 2n. 2.14 Beooi" 11 , 8, 21, 1.14. 146. 149.220,223. 224. 233 Ri'nc'1. 300. 301. M !fiU$< W. D. 2t>2 !fioo.r. 1( . 104. 106 EIoc:ri<>. F J 1.14.147. 189. I'J!!. 211 Oond.u. S. 62. 70. 161. 164. 2Jl OondLo, F. 116. 120. 112 CalC. L. C 309. llO 00_ ... Ii F, 6l!, n Cbomp1T. A. J. ~7. 302. lSI or"'" .... 108.110. 118,I:!O. 121.123, 124. 131 lJl .187 CIa,,,,.,. S . J. ()!\. 70. 72. 1.l4. 142. 147. 149, 189. W5. 211, 217. 21 ~ . 267. 269. 275. 28~, >03. JO.I •.1(15, J5l m 301. I(JS '" m. 356 AUTHOR INDEX Cohon, C. ~1. 134. 137, loIS Colb~, R. 11 . 38, 42, 134. 137. 148. 268. J02 Co,"n, fJ,., Y. 307.1 1(1 COllon, 1, p, ~. 21. 220. 2D. 2JJ CUffO. 1. G. 5~. 70. 1M. 95. 96. 91, lOll, 102. lOS. 200. 218 1', G. 55. 62. 70.141,149.166. 186,223,2H D"om. R. T. 8.16.19,21. 34, 35, J7. 42, 224.2)0( 1)01;101<, I.. C. 1>6"ti!. 94, lOS. 116. 12'1. 133.240,244 246. HI. 2S2. 2M, 264 0< 10<1.0. U. lS7, 161. 163. 164 d« Cloizoau, g, 21, 221, 2JJ Dickie, R, A. 94, lOS. 134. 145, 149, 352 Di M a~lio, E. fJ,., 11>6, 186 Doi. M. B -3S. 17. 42, 85 [)()nald. A. M . SIi. 219 Dondo<. 1\, 134, 146. 149 Donn,,!. l . 6. U.S. 267, 301. J02 Dubault , A, 157, 161. 163. 164 l)uiO<1", 1. 21, m.310 I)uplcs/.ix, R. 62, 70, 22(1. 224. 233 I)lLI de Gcn".., n m. F..J",,,d,,S. F. 8. 10, 16. 18.21. 23. 32_35. 31- 42,6 1. JO, 137. 148 , 1M, 185. 224. 234, n2 Eiehinsor, U. E. 12. 16. 21. 23, J2, ~I, 1>8, 70. 12. 85, 134. 1J7. 14, )24. 329, 342. 349.352 EiriCh. F. R. U.S. 299, 307, 3 10. 311. 314 , 352, 3~ lOi.."bad!. C. D . 147 ~i"'nb8. 173,175 In, 182, 186-189. I'll. 193. 195. 199,200.205.213,216. 2 1S. 223. 224.228. 2)0 _232.234. UQ 243. 241 I;,.'on, n, 223, 234 hl forry, J. D. 38,42, Ill. 148 , 149. 216, 219. 253. US. 264. 284, J()4 F.U.... L . J. J~. 42, 311, 320 Finkelma n". H , 135, 148. 16S, 166.185, 186 Fil.g.:roW. J, J. 207. 210. 213, 2S4- lS6. J()4 hu""n, M 100, \05. 19\1. 2\8.332. Hoi Flory. I'. J. 7. 8. 10, I I, 16. IS. 20, 2l. 24, 29.31,39.42,44, 4, 4~ . SO. 52- 55, ~7, S8. 61, 62, 61, 10. 72 14.85.81.89,92, 9l. 9S. %, 104. 10~. 107- 110, 116, Ill. 12l. 125- 129, 131-133, 134. 139. 146. 147.149. lSI. 152. 154, ISS, 157. 160 164.1<>6 168. 1IlO, 186 - 188. 19\1.203. 216.218. 22J, 233, 2J6. 242. 241, N7, 26J. 264. J07_310. 311- 120. 322. Jl1. 314. J29. JlO, l31. 3lJ. 33S. 342 - >45. l48. 349. F~nl. w. T, 9\1, lOS. 261. J(ll. J(l2 ~'=-J. K . 10,21. )7. 42 F'oin:, l . " 101. lOS, 19\1. 218 f'risch, H, L, 29. 66. 70, 12. 3()6 FUTuka.."•• 1. lS2 m G.lial$all». V. 68, 12. 134. 142. 146. 148. 149,189. 19S. m, 211. 21 1. 218 Oao, J, 104, lOS G arrido, L. 63-66. 7O-7l. 195. 21 1. 212, J()4 Ooyloi8. J(l2 OO$li"". J, M. 2]S_244. 247 249. 2S2. 2S3. 256. 162- 26-1 Gottlieb. M 134,147.189, 216 Gough. 1. 342, 349 Cro«. 21. ]3. 42. 43. M, 71. lOS. 134. 1l7. 141. 14J, loiS. Wi. l oIS, 308. J IO. lH. 129. JS1 o,<s<, G. S. 104. 105. 106 GroMki. W. 134. 147 Grun. F. 16l. 167, 169, 170, 186.330, 331. m 0""".,. J. M. S~ . U Gu'h. E. 7, 10, IS, 20, 267. 302, 343, 350 AUTHOR INDEX lI.id", 8 . IJ~. 147.211 H.1i1ollu. T . 1C4. 105 llan. C. C. 8. 21. 2B. 233. 2J.l Hedrick. I. L. IJ.l. 147. 217 !lei"rich. O. 4. 147. 268. 302. 3S3 lI4. 42. 353 Hm, I. E. 21. 134. 147. 157. 159- 161. 163. 164. 231. 233. 2:>4 lI iW"" I . S. 21. 220. 221. 2lJ. 211, 2J4 H iw. 1'. G. 224. 230. 234 Ilill. T. L. 59. 70. 321 . III H"",-.. C. A. I . S. 2 1. lOll-I 10. 131. 146. 149. 236. 242. 146. 263. 264 Ilofrm.". A. S. &4 1101<1<". G. >49. l~ lIothy. F. 59, 60. 62. '10. 7 1. 73. 8). 267. ~, lIu . Y. &4. 265. 27<J. 301. 301 lI ,idt. S. n 8J. 317. J20 1I... h M. 74. &4. 85 I"'o K.'pl,n. I). L. 238. 24.,. 261. 264 Kawal>a ... S . 204. 218 K 311. )20. 3S2 m. 302 K",u,. G. 2b5. 267. K rrm 357 K';gl>aurn. W. R. 163 Ku.n, T 11. Ill. 131. 146, 149. 218. 283. ~ Kuhn. W. 7. 21~. I~. 20. 167. 169. 170, Is;,. 2(19, ))0. lll. In Ulban •. S. S. W. 105. 352 Lal. I . 2 1. 147. 189. 216, :loll. 347. 348. 349. m u.rI 1-"11&0. N. II . 349. I~ Leibk,. L. 12]. 2]1 . 23]. 2.l4 Lern"",. P. J. 8S. s;, Uonmlo. M. A. 101. lOS. 134. 137. 141 _ 147,1 49.182.18).137. 1M. 192. 193. 195.197, 19').200.216,217.218 Loho<. D. J. 38.42.220. 2JI Mattonnachi. A. 220. 2)3. 234 M >49, 351 M.n", S. 235. 263. 265. 290. 301. 305 M.,k. H. 1'. 7. 20. 179. 181 Mark . I. Eo 1. 13, 2(1, 21, 23. 32, 40. 42. b] 8S. 86. 81. 90. 94. 95. 96. 97. 100.101 . 100.11)4. lOS. 101 137, 119- 149.166. 168.1 73- 178.180--184. 181. 1l!8. 216 218. 219. 228. 230. lJ4- 2J(i. 240- 248. 250. 253. 257. 263. 265-308, I II IW, 332. Ill. 343. 345. 341. :I-I~, )SI). 151. 153, 354 M",ucci. G. 42. 41 Malt;';". W . L. 67. n 81. 89. W . IC4. 107.1011,110.127.128.12'1. IJ I. 332. n '" McCrum. N. G. 126. 113 McGill . W. I . 209. 21 ~ M«lr"h. J . E. 210. 30) ~kK .nna. O. B. n. 83. 8S. 209. 218 M<""",. P. lSI Modal ••. A. L US. m Menduin.,C. 101, 105. 19\1.218 Me,nll. E. W . 134. 1l7. 141. 189. 216 Meyer. K. fl. 143. J49 M"chell. G . II. 1~5, 166. IHS Molle, M 134.1 48.186 358 AUTHO R INOEX 2~. Jl. JJ. 18. ~. ~2. SI. S2. 13-1. 148. IS1, 154. 1$7 161. 164. 166. m. 1tl6.. I~. U1 M""""JI. M . 114. MOJa ....u. It .l45. J!O M"""". M.26S. 292, 2'99.)(l6, lII.ll • • Mun...no. l , m l4'I. HO. lSI MU I1I",h.on. 1'. 1), .$.1, 45, 52. 1 15. MUlhukum ....\ 1. 85 no N-.i. K. 146. I~. ISO. ISS. 161 NatajimI.. A. 114. II S. I U. Ill. III Nishl. T. U Nodlo. I. 1$7. 163 N",,1. C. S2 N"",ura. S. 8S. 13-1. 148 N",-ak. B . ~ . U.S.)O I Hi.."",. Il. 209. 21 S Richard .. R, W , 220. 2JJ Richl ... n. 121. 234 fI:i&t>;. 7_ 10, 72. US. U.1. 295. m. )OZ. J06 MiBpIorff. II. US. 148. 1116 Mi.fin. R S. -". 5(l..S2. I I • • Ill, 21».211 Ro<, R. J. 117. 104 Itor.nd, C. M. 134. 147. 189.112. 217 Ro,"",. G. 23. 32. 166. 167. 186. IS7. 189 Ro. Mu,p/ly. S. II. as. 2lS. 257. 153. 262. 26.).3SJ R...... r. s. 2.S1_m. ~ Saam. J. C. u.s. WI. 282. 299 Sac,u",. T. 265. 270. :100, 304 Sai .. ~, 129. lB. 209. 211 SakW'Odl. l. 11 4. III Samuloki. to. T 65.11. IOj. 164. 210. )OJ. Obo .... Y. 201. 21' Obor. C. K. ItS CM>cr. R. 8. 21. 2)) Obtrth. 1\ , E. 2M. ltl6. o...r. R. 234 Oaden. R. W. 108. UI. lIS.HO ObmOlo. T. UI. 219 Oppn .... nn. W. 13-1. IH. 1018. 1-" ~. A. 117. 112 Orof,...,. T . 1\ . I I B. 121. Il2 SC.... /f... J, '1..u ScIom;dl . II . K. u.s, 266. 269. 271. 275. 276. ,',t. It 49. Sl. 271. )(M Schv..,,,. J. 55. 62. 71. m f'Io.orSon. D. S. D. 21. 2U 2Y>. .lOll. 110. 324. 328. J29 na. 2.l4. PKI>oIcI. W. 1M. 166. liS. 186 /'nnIin. E. 209.218 r;.;.". J. 8. 2 1. 62. 70. 221. 111. 2n. no. 234. JSJ P<>Iman,=. K. K 42. 2M. M . 282. 299 f'ra,..ru,.. J. M . 61. 10. $4 Price. C 21. so 51. IO? 109. II ~ 1111. 121).. l n. Ill. 112 I'Tin.. w. 7J. 8). I 11. 1l2. lS I l'rud ·hom ...... R. K. 62. 71 Qu It.ha,.. 0, ~. 11. 117, 1l2. 18S M.knor. J. SJ. n. 10 R...""". P. R ~",, ~ ... (,6. 72. 134. '.7. 2l.l O. II , 267 • .101 '" Sandor•. C. WI. lOJ Sarikoya. M. 263. 2M. 295. 300. J06 Sa""de ... I), w. 4~, SO-52. 205. 218 s.::.".. ~. J J. lO'J. 310 Scba 300. )01. lO4 21ll1 • .!'IO. 300. JOI. JOJ.104.)OS 117. Ill. 1(,6. 186 s......F". $, T . 341. 349 ScnIIyoto. J. A. 104. 10. 12. 21'. WI, )OJ. l5) Sha",t. M. A. 99. 100. 101 . 102. 10l. lOS. 1.l4. D'. 1l7. 144. " S. I". 143. 1019. ISO. I~I. 187. 11/9, 21'. 2IS.117. 219. 2M. 266. 281. 3O(I• .IO I. )()4 S...... M. lOll. 116. 118. 120. 121. Ill. III SO"' ..... II , W. 1116 S"JCIn, M. e. JS! Smith. K LJr, 1011. 11"'1 16, IN..21. III IJl. l ao. 1117. lSI Smith. T . t . 114. 1 ~7. 189. 206lQ11, 2 17 Sperlinl, 1_ I L 210. 21l Sp;cs.. II. W. 1.l4. 1'7 Stad .... R. Ill. IH SL:nYnlUlo. A . J. 21. 10\1. 1l1..lO8. J IO Stco •• R. S. 134. Ill. 146. 147_ 149. ISII. 11>1. 1119. 217.11~. J.4~.)$O S«,.,"'. R 1'. T, 147. 15~ S,,""b<. E, 21, 2l, l2. .... , ~ 1. J$l AUTH OR INoex . , 359 Sulli... " . J. L. 131. 1}4 117. l.u. I'S. 147, Vi,• .-e n', J F. II. 235. 262. 26) v i",-1', C 26) . 264 SUod.",,,,jall. I' , R 1J(l. 13l Su, • •• U. W . 61. n. A1. 89, 9-1. 1\14. 101. lot. 11 0. 121 129, Il l . U). ))1, Jj) ""'hl"I.,,'. M 8. 21 .89. lOS. 322, 323 . T. ... ka . T. TOlota. Y . 7~ .U ~. U . 1116 70 Ttfmoni>. y, 217. 240, 264 Thirion. 1'. 14S. I~ T.,....I;.. [). A. 6S. 11 Tond ~ . A. 151, 161 Trdc>a" L R. O . 1. S. It, 2O, .w. ' S. SI , S2. 5).70. 1(10, l OS, 107. 1011 . 111 II ) . 11 7, 119. 1 ~ 1. 1) 1. 1)2, U 4. 141. 141, IS1 . 160 163.200 :!(14, .106. 216. 2)6. :!A I :!A3. 1'7, 248 , 250, 263. 26J. 2111, 211 1.2111.2111. 2'1-1. )00, JI6, n3. }43 }4S,)48. J.oI9. HO."I C . +I, 45. n. ] 15. ) 20 no. no. """"""11. Ublm:lnn. n . 11. . 265. 21>9. no. 290 • .'00. ]O}.-]OS UUmolR, R. no, 215. :Bl. 2:14 Uloid •• D. R. 26'. ~. 290. JOO. J02 lOS Uny. D . W. 240. 24S. 263 Voi... I'.,th. T. 116, 2H . 261 IIiIP. T. II . n.32· 31. 42. 41. 101 . l OS. 11lII. Il l. IJl . 143, 18{). 19\1. 11 8. 231. l J4. 2611 . .!OZ. H2 330.13J Wa,ner. M, H. 42. U w.n. r. l' 1. 13. 111.1 67. 136, lll. l Si W."•. S. 211, 279--2~3. 2S1. 290 292, 29' ~)1. ]OS, J06 Wa llie• . M, j j JS. J1, 42. 16', 166. 185. 186, !29. 2)1. U4 Warrick. F_ L 2M, 269. 282. m, loll. 350 W ............. ~_ 1'0. 12 w ie ....,. J, It I().I. l OS W'gnall. 0, I). 65, 1 1, 10'.220 223, 2]3 , )SJ Wilkes. G, L 265, m. 11S. lA, 290. ]00, .I01.)OJ, JOS W~ I"". T A. 264, 268. JOl..IO) Wood. L. A I U , 113 .(0), Xu, P 2\8. 265. 2b6. 270. 215, 218_283. Z9S. JOI • .104 . .lOIi m. z..,'I Subject Index "'xh> 2~ Abra.ion f¢>i>lanoc: 26S Abwrban", 158 ,o.= .... lor3 11 .. t4. 276, 278. 279. 282.2'15 ..111 lIiron"nuo"" ,y"cm. 289 lIifo""''''''.1 213 lIimoJal di.tribu,ion, \ 17. 142. I?O. \95, 1%. 241 '),01<"" AOIio 262 A"ivalor 311 Ael;.joy of ",h'~"' SS . ~. 61 , 16 Addih,ily orfrec 10 Aerogel, 214, 215. 2'lO AffI""dcfonn.,;on 8. 10. 1St.. 167. 171. ."",!Ue< 179,192.213.224 Allin< limit 140. 182 Mr, .... ncl,",'ot k ) , 7. I~. 19. 23. 29, 32. ) 1. 44, 4(,.41j,~. 77. US, 151. t ~. ISS. 159.180,181.223 22S 232 AII",260 ASO""" 14. 260 Agi"$ 27S AI.n'no HI>. 238. B9. 244. 250. HI Algin., .. 260 Alum;". !S8 Am in" acid comp<>lilion 138. 245. 24" Amllop«:,i.261 Amylo", 261 ",,,,,",ism 252 '~ .. <m J6 A., ;dc"adM' 111 Anti",i<;I.nt 258. 311 Anlio",,""",lll ARocaled 8 i"~'.I II.Cleri.1 .... ck 2Sg Ili •.".1 ""'"P'"'>;"" U\O 1~2. 11\8, lIimoda l network> S, 1&9, 19(1. 192_1 16. 224 Jlimoo.lity. ·269 BiOln;"",,;'· m.,.ri.l. 263. JO\ lIirefri ngo_. " rain -I(). 81. 124. 146. I ~~. 151.209.218 Sircfringonc.... 'cmpcr.'yrc ",lallon. 195. '" 11100.""'"1 111 !londinH .~.. m J\2 !lounJ rubber 261 IInl<1. fail",. lS7 lIul' compressibility 51 lIu~ . . . \>1><",312 ~"I!.rfly .~.pes l2J ~u' y l tubbe. 3 12 c· Ihcorcm 62 Calurlll1 26. 21 W"'""", 362 SU8JECT 'NDEX ,. C"-'~n ti'4! , 16. It'. 188. 190. 2JS. 2J6. C.... m ... WJ Own fIo.ubi~lr 2J.II 0..;. fold>", 180 Ch.in inl«fl<'lC, ,,ui,,,, 1J'J. 1<10 C~ .jn-Ju,l(lion Cnl.nllinll 145 Choin ... ",I k ill",ri"",""'. IU. 190. 201, 207. 21 t . 268 bimodol 117. 142. Ill. IA.. 190. I'IS. 190\. de, .... of 116. 111. 119. IJ9. 140. 145 ...... N'y H . YI'. 60. 171 efT....,,,, dcfm: of 266 on ....u,_ '45. US. 161 in~u'19' "' li",1 195 ""imoo.1 IU. 190. 1'12 o..in ori ........ loW ChonclC ..... '" d~ m Cho""",,,",,, r>.tioo 96. 1)11 Cha.rpy pmdulum (im",,'I) , ..I 2&\ C.... mal flOI C........ I;n~. 34. .n. 57. 13. 80. 11. 1.11>.239. 2'1.!4l. 24S.1!O.!SS 260.294. .107 111 helal J"nction 257 I""m .... nt 25\1 of 001,"'"1 4. 5J.. 'So 56. 59. 75. 16 ...JU«d 7~ a..m~"", YJ6 pII)"'OC".1 79. 2S1. 260 ''''''''''''''I")' 10 Chi piI"'.I<'er 7S. 7b. III ChIOf; .. tod JlOIy<,hyle:"" JU ChIo< ..... lfOfl3,«I po>Ij"elhylcn< l l! Cbo>ocallhcon<s J. 1. 22 ~ 'j"''''''' 5l. 53. li S. 1)6. 242. 2H C""iul.I,,,n )12 C""ffic.icnl of dil.,,,,,, ~ C......... , ..... ,.. ri'" 212 C..d .. _Ie: ,,,,,,.11;"" , . Cold ",I>bct ) 12 Colla"," 19. lJ,i>. 252. H9. 262 Com""" .... '"1 JI2 C_.,.....bility 53 C""""' ...•"', J. !l.... "6. *.I. 1(18.. 10'1. li Z. liS. 1l-4. 201.!S1 sc, 265. 2lI4. liZ CODdilMmaI l"obobi~lieo Z8 'S. ""ni'_ ........ "'" 10. It. 11 C>noor.,n«!..chai. _ 2t . JJ. » .... 2. .... COGfilu,,"'jon *.1.51. 52. $I. '1. 152. ' 80.!]O Con.tr.,n«!.junc,ion n\Od n. .w. $0. $I • .1'1. 61. 62. 76. n. 100. '010. UI. IS!. ISS. 160. 161. 162, 11O.181.199.:m. !12 CQ" ..... ,". _ .. 137 Con" "";"1< 1)9 C""Iomt*: in,er.tetiom ' 46 en.:~ by ""' ........ 6S <.......i..I")' 2010 C.,. ... lh' .... 79.10. U. 141. 16S. 171. 110. 181.181. lAO. H7. 2S8. 261. 265 Crro.. lli'~liQ" 80. 99. 101. 116. 129. 145. 149. 195. 209. 212. 236. 241. 2" ;""'1"<'" 179 ~ .. in.,ndUC "'"'I"""'''''' "'''potft' 180. 211 CU ... 312 "n.. ) 12 Cy<1e: "'''~ 17.29. J7. J07 310 Cydioc def",,,,,,,;,,,,., S. 207. 209. lB. 184 Cydi< ....... 1&4. 2t.S o. na"", 141 I4S. lJO. 310 Ocom Erl ..... nl. tho<>rnp<*t_ ocm"....,.'u ... 116 Drlonna'''''' liS 120 ]07 I C_'" ..'B. l.lO-2J2. !5!. 2M. 278 llC. 295 Cn:p.: n,h"" 112 ,J. 54. <1<,...., "ni • • ial (' J> (:, ;u.... 1 phc"""1 CrySI>IUHI. fibers 252 C.,. ... lIin< ma,.ri,l. ISO. 29S Cryolallini'y. of 115. 207. 2-17. 252 C."talli,. "uc .... 179 7J 8~ Il of ,"","""me., 1! SUBJECT INOEX o.&r«: Min',,,,,,,,,,,,,,,,,,, 2t .!4 Ikh)<.!rau,,", 2S6 0,,10 ydrolyoino-nor.I<""",,, 141 Drndri......,. 6' I......."..... 240. 2.1 IJonu"""ro do . .... 12. Ukh ,oio;: M io ISS I>ic:lrroism. ",.in 81 IM1",,,,o oon''""t. "",,' m o.ll"o=ot..1 _lIIiott ~...""""".. ~I 1>011".....:1 «H"'''''"'~ ll. 40-41 Dim .. i"" ooefI"",,"., ~) , M, 6t.r l)iu.ion 4l 1>r1a.00> l>Ii8"rneri< 61 ""I~n"'rir: ~l "'IIIOIi,,- 63 1>01......... cyclic j1. 6l. 66 Drpolo """"""'.. 2 11 OIsacr
ricIn 260 1>O"nl''''I'<'';'''' 0( ""'''W~ ch.rr" t>2 Or..,';'nl .. ion i""t. 161. 16Il 1>i.pI"""""nl ,,,,,Iit., ' l>o .. ribu.ion 6. 10. I I. 17 JO. )'). 100. 101. ,~ p:.i,Udt "'" 102. 2n [)jOl,ibu,ioru brrn.OO.198 md·' no HI ~2 DNA ,.b; 14 I).."",,;ns 0( ,,""""''''' lO ll<>"bir:· ... ' ..wk ",,,,,,,I 268 D.u, ,de.... 65 1""',10 boll>_ 217 u.....,..,., .. 112 U)'''''mic moclu. .....1 rrop '" ''',,>lit EI;r .. ", f,.". <""'U 7. 8. 9. 13. I~ 19. n. 29 n. 17. l'l.40. 46. 47. 51. lJ. l-4, SI. 81. 104.1 74.20).1 18 EIa"", """'" 107 Ela>1""'"~ riTer:ti"" "",", ... I'l E....l. 112. 1:!6. 2ll. 2.16. 2}8 lS1 EIr:c1,. .. fidd 74 EIoc,roI)'" lSI [ Ioc"Oft mk,r»ropy 19. 267. 270. 272. 275 !~ , O""P'ror> 4. 22. 23. 0I0 • .,g. 100. 101. 101. 101 111'1.112. 114. 111 120, 122. 1.16. 143. 24~ 1;"'UI",," 1",1)·n.. ';",ioo ~ End I'''",," 11.2 F.nd-blldtd poIymo, 12. End.~.k'nI4. 16. frJ. 64. 66. 69. Il-f. 1.16. '43. 1U--190. 19S. 212. 21S. 190I in "",~I""';.>OS I'nd ·,o..,"" di ...... ""'" (_It,.) 9. 10. II. 11. 'N 100.124.12'1.139. 141. 1$1. ,99. HO. 16b. 28\. 1:l() 3:>1 F........ ic «In,rib :roo. Ionl ".'''''''' "<e» 46 Enl>r.p;.l .hoins frJ linta"""m"nl mul,irno Ela .. ", dofl"' ....qy 2-42 F.la>toc «I"";"" of ...... 9. '''6 r<>n.~ S6. ,ot. 110. 146. .!4'. Nl. !SI EIcrtt ..... $QI,,";q ~J Diw>rtion WI' II J4l! HUlk ""n,,,b",ion to ''''<11'''1 00 """" "In, 210. 291 2'!1. :!Ill, 296 tran, m, • .;.", 270, 21 l. :!9S bn,, ...·u 0( W",kl ~rr 3&l l)yna",,,, """,han .... 1 t' .... Zl7 l)yn,,,,,,,,,. "h.in 110 <",",,,.ju,,,,tion 206 """ 09. 1;1(1. 160 ~""'" ,., ..... """"""" I)OJ E,".. ntJemcn .. 1. I. 20. 22 l). 38. 104. 116. 1.16. Ill. 140. 14<1. Il2. 195. 251. 266- '" Ionrh/aJ 1010 ""mla""" UII "app«l Oil. I~' FJlthatp;c "'''''I''>''''t' of ,... (""~ 241. 262 l;"(~.Jpy 121 ':1\'",1';" ",'nl1ih",i"n, M 4. ", 107. lOS. 24 '. J6.l SUBJECT INDEX Ef,.,o>pio r""", 56. 247 t""n, foc;tDl' 7. l(I [d.""'lI ............ 20'7 r"""n F.n1rof>Y 01 1110 EPD.\, 312 Eq .... ,;"" of .'a'~ 9. 107. '01>. 14 5 Ippt<>ach j j Equillia..., .... ~ 2C4. :ns. 21$ E.b)iMo he ...... 1 I~ 79. 110. 11 E.hykn<>-propylt ... ropoly ....... 117. 126. 'N E~It.-"..p. 147 f.........., ",",u"", 89. '19. I~. 121, :!63 Exlrndm J I2 Ex.en.ibili,y. finite (limited) 87. %. 101. 11 1-14).193.199 Exl ............. 'Ios ( S, ,1. H. 54. $7. 1' 1. I~. u" U9. 171. 11'. 114. m. 231 . ., n. oat.ct..... 260 Gllonml ""'iation 10M. 14' (lao liqWd penn<>''''' Gd Gdatin 1~. 79. ZOW, 260 O !;,'O". 01 ""clion 6.! ""~ioII ll. 61 -70 poinl 79 G./o • . 6l. 66. 1 1. 7J 36. 23 S, H 7. 2SB.. .i.... 64 2 60-~ 6) f lilure <.. ,..Iope 2$8 Falli"ll ... Fille. cool 26s-.-m F .... onItr , ......... tra"'''1011 7. Flory ~.mbtr 24. 1l9, 140 FIo>ry-- Rehr>« ....... lion'7 FlucI"",1on1 4.8. 10. 12. 13. 18. B. 24. 26-:10. )8. 39. 42, 48. U. lSI , I S2. 160. Illil. 222. lll. 22'. 226. 233, 124 129 ~_~"' ........... 146 I' "r<>:. mlUC<>ll 46. 47. " . n. 11) F~<".nlion cu".. 248 F~ ...... p F\oIlCl;.,.w F""I'I IU """",,,, ... I,ly 01 ju"",iono (<1"""'~ftko) l. 7. I ' . 16, 19.20. 24.2'.29.40.42.57,6 1. 63. 69. llol. 1M, In. 189. 19',223. 240, .JIl7 ) 10 ..ca,ioI\< 1(111. 110. III . 119. 20'2 bioIop::ol n poI,..ltctrolll< 62. 74. 7'J ''''"PCr,,, ,,,,, ..,,"t;ve 77. 78 '~bIc 7•. J9.4) G,b'" (no< miDOSlYcaM 262 G ...... """'JI~ JI ! G Il ·S ,ubber 112 G .... , ",t.c1on,.nn.n 262 G .... y"1o rubbo, liZ GUll. pctdI.a 126 Ff'OCUl aavqo ... MB 1'..... ,i"".1 p"",i"'I~'ion 79 I'.......,:r. y 176. 177 irneaot N 7 of ...w~1 '1. 61 . 74. IN. 169 FrrnY-JO II.", rubbo, )1 2 lI.rd ..... .!91. 31 2 Hea. 01 dofonnalion 111 11 ... , 01 r..,;"" 1110 1I00i. ~II"'''';'''''' l~ Ildn'~ul". fra: SUBJECT INDEX Uigb-tempc:ratu~ "~tobi", 29~ loni'.able network, 74. 76 loniud g.1s 74 lsodcsn,o,i"" 24(), 241 b ''' ' opt¢-ncnu.tic phase: ".n"tio,," Ili"orical infomlot"", J40..l50 II<>ok.·, I."" 46 11 )'brid ino,,,, n",",,,,,,nic """,,,,,,,ile> 290. I""tropic IIOlution$ 176 ''''tropic ."'~ lI i"J 317 It.vea robber 312 !IcXllPCP!id< Ui,h 2~1 .Iec""". 1<14 "' Il yd'ogon bonding 24S lI)'d'ol)',i. 2#. 210 I')'d'or~obic bondinf ~47. 2~7 11)'d rophobic- hydrophilic inlcroClio", 242 Il)'droxypropy( mlulMe 261 U),,,<1'C>i. 26~. 312 Il)'Sl.retic efT""" 252 Ideal mixin, I .... So( Impact rai$l3ncc 202, 204. 286 Im!>""1 "","'gth :!S6. 2 ~ 1 Impc:.foc, n..",'ork. 19 In ,itu. fillers 26$-2'1\1 Incohc,tnt ..,.ll.ring 222. 2B. 233 Incomp""',ibili'y 19 Infrarro dichroism. Fou,;', ""nsCorm Ul. IS7, 159 Inf.ored. polari .... ('unric, t,o ... ,fo"" (FfIR) IS7, IS9. 160. 161. 162 Infran: '" "".Clure Intorpc""tralin& n.tlolork 195. 196 in'<rpting phases 2~9 i",nomok<:ula, int 365 In -178 Jan",.- Gulh tl>wabili,;" 28 JU""tion fu""'io .... Ii'~ 14. 24.40. IJS, 136. 155. 160. J07- 310 Kappo poI.meter 29. 32. 39. 47. n. 100. ,~ KittholT adjattncy matrix II K noll)' I<.,i"t! 312 KI1I,ky pIo' lli. 232 Kuhn Icn"h (of > ,,. Kuhn model 160 Lam8. 170. rn lhoo'y >I. 166. 2S8, 2S~ ugendre poIl'rlO'ni.I....:ODd 114 unnard_Jo .... po'sooPY 161 Ligl., "",,,<,;nl &polan,ation 124 dy namic 212 . pe<:trooropy 74 Limi ..d chain O>l.n,ibility 23, 119. IS2. 184.188.195.198,1:00.209.250 Line.. ' p,in, 1..... 25. 27 Liq"i~ cryst.llin •. ""'in-ellain 165 I.iquid cry;"W"" ""'"",k. 5. Il~ ,id<..,h:a'n 165 Local ch. in •• i. 157 lo» ta"$Cnt 210. 253. 25(;, 284 l.<».cnge shape> 223 L.obri<:an. 312 Ly,,"" 236. 239. 240. 250 M,,,,,osropic ~ h~""t'" ~ Ii 151. 15~ 366 SUBJ ECT INDE X Map "=- M uluu .. 1 SI Mulumodal cIta,n ...... 'h H. m Mgl" .........1 "",...."to 18S ;0<>1.,,,,,, apl'fO" """hod< I.,. M.,,;, ",ulilpljo.o'ion 67 M, .. m~1II .>...,..;bt~,y 6). 144. 211 M.'~""I"'" ...,j·I<>flII ....I'h 6,' Mdli", pOIn .. ISO, 139.211-1.195 Mcth)'1 """h""rylalc 26\1 M icron M"""""".. d ..-pu.o'"", 1'9 I~ I , 1~2. no Moo;lIlY oIu;" 2J8. 266. 217 oqrnon",1 87 -.,.. bulk ~. 59 d ...., l • • , 8, 22. 1.1.~. "'IU;~brium 79 litl«! ....."""" 21>1 ~J. 66. 11)4. llJ 1_ 1'.1. 2S.. 2$6 .7. 2001. 231.161. 301310 boIOOUft '""~P ulI-<;mos-Ii"k unori<: ntru 116 N' ..JiothYIaa) lam"'" 74 N ..... "" 62. 16S. 116. 117 OhOIR. J07-JtO nnp ;"h","""""",,,, 211 "noeIu'" 6. I I. 14. ,:14. 101 110 14}. 146 unf""';nl 223 N 1001. [Sot. 1.16. 11111. 2!10. 2$2. 268. JJj I~'.;:(I(I. ..... 11 ansi< (SANS) 220-2.W. 212 Neu,""' ...... ""'" <.,pc""",nl>).I, 22J, lJ.I. 212-214 N!'i""'''''o to 192. 1\111 N"BafJI ........ IJ6. 180 N""",.,.. ...lli,.. blc ... ,,.,orh 142 100, 1001. 111. 142. 161, 180, 1[1.1. Mooney-II .. ,;" equ,,,ioft Sf. 100. tOI, 114. Na, ...1 ru ........ . 9 $2, 61, 101, 101. It l, 11 2, 114. liS. 11 1. 118, 110- 122, 116, 141. 142, IS3. 19S. 2OS, 26S /'11""<"........... d i>tributions 6., 67.,., \III. 1.l9 ......"'k """ ... 1.16. Ull numbe, .,'."... 13$ Monodi_ di .. ribuuon, 18~ M""", Carlo.> ..... _ 6. 14. (>S, 9 1. 9J- 100. M"tIi~> Myooin261 ">PoIosY 17, ""'.'''""",1 I Sol. 159 ph,nlom 1J9 pLo''''u Ill. 1;!Q .a.", 016. 101. 137.261.1111. :?t2.lI1 ...,..... Z3J.-2$$ u",u,n 100, 101, 141. IH Molocula, IJ..I, 143, 159. 139, 2b1i,2'J,4 21~ M ul'iph.o ...1....". 26'· 106 ""-, t.,. MotIoI oIHIomen $. ,)4 f.f9 '\'Iodol ... ''''''''k. 61. IJ..I. 1.16. 141. 1014, 139 .\ [OO"'u. 80. 82. 101, Il'. 1.16 1>10. t1ll. 182. ,92, t9J. l 116, 141, 145, '901. tas. 101. lSO. 118 200. 209. 2". "",,,·ork. 2~. 252 NOph.on'omlik< ","'(lrh 161 NoDpoia,.pobr inlCnctioru 242 NonrooIor _ ...,_ bioda"""""" ZJlt NOIII'OInforcin. off.... JIJ N""Ic'r m.~n .. i<: mon, ,,,,,, (NM II) ~S. 19. 161.2-10,261. 21! 214.290 .x..' ,n SU BJ EC T IND EX OcIOP"' ."eri.l e""o",", Z3S. U l 24" Oil-drop model, Olymp;,; ....,""o,k. (,'J Oven sy,,<m. liS. 242. 247. 2Sl OpI"",1 ani..,,,ovy 124, 1 7~ Optical proporlies 146 Optical ,olali"" 2)\1 Or9 Ore-nosil""',.,.,..,,)',",,' .y"em. 290 Orien,.,ion-dcfonn31ion ",lotion 17S Orien,a,io" rune.i"" W . 1$4, I SS. I 17•• 177 O,ien •• ,ionol di'lribu'io" 169. 172 Osmotk comr<<>Sibilil ~ H 60 O>nIOli<; pressu", S'1. 75 o,."",ie .",'elli"i and .Rowelling 62 sa. ,n. Pai, corrcl.lIon f.",,1,on •. inlramolecula, '" Particle ,il< 17(l. 272 Partil;"n fu""'ion 10. II. 12. 34.167. 1M!. 174, 21S1s. 321 - 323 £\>elwork, 19. 1-15. 301 31 1 !'tome.,iM ch<"""">rp.pby t,(, Pcro,j 367 l'I."iei:l:o-r 3D 1' la"led li)!h' mic,<»o'ICII¢-I) 126. 128 l'<>I)'(d irnelhyl'i ime1hy!cno:) 126. UO p"ly(dlrnelhyl,ilo .. t>Oj (I' DMS) 61. 64. 6S. 66.6!l. '" "rit"teo.! 116 "noriented 116 Poly«,hy",n< o.ulr) 204. 211 . NI Poly(i""bulylo",,) 75, 114, l IS. 120. 121. 124.126.1.29. I)G. 288. 289. 292. 294 PoIyi",po<"" 162. 230-232 ,,,, , .4 i17.1l7. lUI """,· 1.4 118.121. 1211 Polm><' modilicd cc",mi<:s 290 Polm" t--"",I",,,, in)C'''lio" n. 109. 121, ,~ PO)I)'(rnelhyl hyd,ogcn .ilo,.ne) 136. 21S Pol),{rnelhyl rnethoco)'I.1<) 79. 117 . 126 Pol)'{me1hylphenyl"IoMr>e) 288 P"Iy(~.penle,* I) "'"",ic 126. 128 o 368 SUBJECT INDEX PoIy(oxy""'lhy"n<) n . 99 polys.ac 294-297 dculo",ltd 223 poly(IClrahydrolu",") 202. 212 P<>Iy(lctr:o.methy"" oxide) 126. 128 P<>Iy(lrirnolhy\e"" o.ide) 126, 130 PoIYl'",'h."" 194, 209. 21 1 polyNnyl """ate) 59. 60 poly("';nyl .!cobol) 11 4. It5 I'oly(vinyl dJlorldc) I:u.. t29. 240 1'0,.., ..... 6) Pool"'" period:l4l! I'rn$UfC pu~n8 79 Principal • • i, of link in8 16 Ra .....,m "",wofk modocl . 242. 243, 246.247 Random pIl... appro. imalion 55 Rale of ."""»'ion 63 Rtd...:<:d ehem;':'1 pol ..",i.1 55 Rcd...:<:d fOfces 46-52. l B. 1S9 Rtd\>Ctd ",""ulu, 101 Red"""" orienlation 15~. 161 R.f.",,,,,,, ;nteratl;"n sile model 55 R.r.",ncc: ... te of. nct.'Of); 26 R.infon:emc:nl 5. 265 R"nforcinJ ~l "f :u.5, III Reinfomns pa""''' size 261i R.".,ol units 56 R",lica a ppro>eh 36 1I",lica fo,.".,.l i"" )5, 61. 229. 268 RopI.,in~ chain. 145 IIcsil,en<e 2}{>, 21i5. 313 lI",ilin 215. 238- 242. 247- 249. 252. 256 cI."i. R.tank,313 R."",..;o" 313 Rinll-ehain cycliwion 124 Ring opc:nini pol)TIltri ... lion 189 Ro.alion.l i",.,.ric ".te Ibrory 4, 6. 67. 89 . 93, 94. 95, 96. 104. 108. I 10, 125, 126. 131.154.156.157.199.240.244 lI,ot.tionol i~ljon 87 Rotolionol polen,;al, M lI,otalioool £laic 107 Rou .. mood 223 Rul'lu", 1~3 . 190. 193. 194, 200. 204. 205. 253. 281 0"'f81 193. 28J. 288 mooulu. 28O SDR 313 &onning "" .. rimetry 79 &onnin& lun".1 m",r<>:ICO!>Y 2~1 Scott«;n, cr""'_lion 222. 224 S Scor 16Ii, 168. 178. WO. 2lJ S<mille.i l>k choin< 5. 165-178 Serni-open S)"SI .... 44. 53. 2J<, Semirigid cb.;n, 5. 176 Serine 236. 238 Shea, 4. 5 I, 118. 203. 205, 276. 278. 281 modul us 016. 115 P""" ) 18 Silica 6. 2tJ5_299 Silicoo·boscd polymer> 149 Silicones )13 Simpk . rompu"f 87- 106. 14lI. 212. 21 J Shl'""linb 3. 22. ll·lS. 17. 42. 44. 4lI. 49, 54. n IJ7 Smoked oh«l 3IJ Son ,ubber III Solution SIIR 313 Sorplion 53. r,J.-7(I S1'"I;al nciglloo.. 25 SpecifIC ",I' '" SUBJECT INOEX Spinodal d poi., III S,.ining III Sla r 6. 58.6l. n. 101, 10l. Ill . 1:14. 174.l1$ )20 • 1 rup'o" 283 ""mina l 193. 250. lSI ""'~ nominal (0100 010') 181. 1~2. 199.268.277 '.'''Olio" 145 o"iuial 46, ~7 Slreso-Offio,h.nn. 23, 40, 92, 1(1)..102. 109. III. lB, 141. 145. 182. IS3. I~I. 19) 195. 198 200. 203. 205. 206. 249, Slreu ~ .o"""os d.,. 250.252,259.216. 179. 2SI. 282, 2119. 290. m. 291. 298 . m me.",,,,me",, 142. 295 'If>. p'opc:rlic> in elongalion 145 .da';onslii", 3. 4, 42 53 . 119 Slre .. - •• mp"ralor< .d.llioo, 11)8, 110. III . 112, 19S SIr ... 1U", i"".. JIJ S" ... IUre-proptrlY .dui,"" 5 Slyrene 269 Sulfu. dono" J 13 Sopc.ab$o.btnl pol)'me .... ~3 S~.. lling 4. n. 44. 46. 47. SO. 51. 53- 12. 73-86.109. III . 112. 114. 115, II~, 119.121. 131. Il~. 141. 142, 145. 146. 155.161. 162, 182. 195. 212. 22J. BO, 212. 2H. 242·244, 2#. 247. 253. 2~ hiin T.d "",nBln JIl Tadil",,, 313 Tea. 5. 206 <"orV 206--21)8 ""''',n,,.e 265 Temperalure "",Ilk",.", 125, 2}6 369 .nlrinlie .i=>Sily 122 T<ml"""'o'O ,hin fac10rs 2SS rrn,ile load I~ ren,J1e slreng'" 20\1, 213. 265 T~",lO" .imrle 119, 180 oni .. i.19. 51. 51 T C1me'hox>~ ila"" (TEOS) 270. 2S8, 289 ' d"",n'yl ..rlho,il'.< ... 1).4, 189 TOImfU'><1iOl\al ""I~'o.h 7. 19. ZO. 24. 25 • 29, 57. 61. ll5, 144, 153. (66. 167. 210. 224,229 TOIr~p l he.rn<> .. ,'crsible gel. 2n, 258, UO. 26 1. ". Tho.mose" 202. 31l Tl>ormolropi< inlcnC'N>" :?Ii ·rin\C--,.mp.ralU.. supt'1""ih,," 25~ Tit;ni. 269, 216-278. 188. 291,), 295 Topol",ic.1 r.cighbo .. 2S Torsion 4, III. (12 , 113. 117. II~. In 20). 206. 276. 278, 283 T Otlghoni"g 141 '[ o"gh,.... S, 19J Tm",ibon mom~n" 157 ·fr.ppcJ enl. "siemcn!< n Troppm5 ("",or 140 of c)<:I",,,, 53, 66-68 Tm: 17 T.d"". «Ialion. 45 1'.ifoo<><1 ::06 1',... 1 ~ 9. 2\0, 21) Tube m",,,,' 42 T" bulin ;>62 T w" pi",,,, model , ,w"Ucn 010,1", 1')f<»ioc 241 ~4 J 370 SUBJECT INDEX _,ties Ul,imol< $. 141 . 141. 14.1 . 11!4. 1'$. 1111. 189. 193.2001. 213. m.~. . ~, , Ultinl.l. """'$1h 63. 1'1 . 144. 1'$. 191. Unimodal "",,,,,,,10 14. 19. 117. 1111. 1119. 190 19:. 202-21 I . 21 4. 22' . 2..'9. ~I Unp '" Unptrl"'bod !1Idi... of Ufl&tiotI 67·~ Umptrlu'bed .... 1< S. 124. US. 09 Uplum •• In .. ~ ... ;n ",,, .... rnt. 141. 141. 1<\1. 181. IS1. 191. I~J. 19 ~_'97. 199. lOS. 2"'. 250. 1$1. 276. 279. l1l UV .Mia,"" ,.... Vocuok f<)(nlalOon 266 V.1;1I< 2)6. 238. 2.l9. 244 Von ...... W;oat. ,.....,.,. of rubbn ..u"oci,y I<)OJ. ~ V'p<>< J'l!'""Su.. ""'............. " 12l Vin)I.lem'''''''cd cho;". 1-14 Vin)V';n)'lkIone ""I)'''''''' 19 v iOCQC\utic efT..,.. 131. 207 V""""lu lie Pf"I'O'.lco. 148. 256 vitooelbli< rnporne> 1$3 vio '1;"""""1)'711. 124 m, 12l Vi...... ly boIlk 268 ...._,y-,."'I'on"u~ ~ 12l V""",",,~pl>rtftl('ft" 122. IB Volume rluo .. , ..... " ,,,,... n. 74 V"Ieo"""""'" 6. In. J 14 W.M 1'101)' model S. 19. n W... proP'g;l'~lft ,"CC:I,,, 220. 221 w•• ~"" lin k 'hror)' IllS. 190 192 Wonnlik. _in 166 Xan'"" !to! X.,."y J,IT=.;on 1111. 261 X.... y 62. IlA. 221. 211. 267. no. 271.215 .....11.""" (SA XS) 711. 272. 289 ,.,..,I<,i... Zeoli, .. 26\1. 298. Z;,,_ia 2IJ.8 m

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