A "zero-parameter" constitutive relation for simple shear viscoelasticity. Key words: S_hear flow; _shear thinning; v_iscoelasticity; Cox- _Merz rule

Transcription

1 Rhelgica Aca Rhel Aca 29: (199) A "zer-parameer" cnsiuive relain fr simple shear viscelasiciy J.C. Dyre MFUFA, Rskilde Universiescener, Denmark Absrac." Based n he Cx-Merz rule and Eyring's expressin fr he nnlinear shear viscsiy, a Wagner-ype cnsiuive relain wih n nnrivial adjusable parameers is prpsed fr simple shear viscelasiciy. The predicins fr a number f nn-seady shear flws are wrked u analyically. is shwn ha ms feaures f shear viscelasiciy are reprduced by he mdel. Key wrds: S_hear flw; _shear hinning; v_iscelasiciy; Cx- _Merz rule 1. nrducin Afer several years f research a number f useful cnsiuive relains are nw available [1]. n rder reprduce experimens accuraely hese relains all cnain a number f fiing parameers. n his paper he fllwing quesin is asked: Wha is he simples pssible cnsiuive relain which sill reprduces impran feaures f viscelasiciy? T limi he discussin, nly simple shear viscelasiciy is cnsidered, and nrmal sresses are ignred all geher. Saring frm he Cx-Merz rule, a Wagner-ype cnsiuive relain wih n nnrivial adjusable parameers is arrived a. The nnlinear seady sae shear viscsiy is, by cnsrucin, clse ha prediced by Eyring's phenmenlgical hery f liquid flw [2]. Varius nn-seady shear flws are hen cnsidered and wrked u analyically. is shwn ha he cnsiuive relain reprduces ms qualiaive feaures f shear viscelasiciy, wih he nable excepin f he versh usually bserved in he shear sress grwh upn incepin f a seady shear flw. 2. The mdel The well-knwn Cx-Merz rule [3] saes ha ~(~) = ~(9)1 ~=~, (1) where ~/(~) is he nnlinear shear viscsiy as funcin f shear rae and r/~' (9) is he frequency-dependen viscsiy in he linear respnse regime. The Cx-Merz rule is a useful empiricism beyed by many plymeric liquids. The quaniy ~/~(9) is bained [1] frm he equain rl~(g) = ~ d'g(')e -i~', (2) where G(') is he shear relaxain mdulus. By definiin, G(') deermines he sress r in he linear limi frm he shear rae hisry by means f r() = ~ d'g(')~(- '). (3) Frm Eqs. (1) and (2) ne expecs he Cx-Merz rule be saisfied if rl(~) = ~ d'g(')e -~' ()~>). (4) A sraighfrward generalizain f Eq. (3) include Eq. (4) fr he sainary case is he fllwing cnsiuive relain r() = ~ d'g(')ip(-') "exp -!, li'(")ld" 1 (5)

2 146 Rhel~ca Aca, Vl. 29, N. 2 (199) Equain (5) is similar Wagner's cnsiuive relain [1, 4]. The difference is ha, in he "linear" par = x f he relain, y in Wagner's mdel is here replaced 2~ by Y. Als, he "damping funcin" is here exp {- ~ _ ' [ ~)[} insead f Wagner's exp {-,~-' ~[}. The presen chice f damping funcin is suggesed because his damping funcin sums ver all shear -1 displacemen aking place beween ime -' and, independen f he direcin f he displacemen. Nex, a Specific frm f G(') is chsen, namely G(')=El(' ) where El(') is he expnenial in- -2 egral [5] El(') ~ e-u = du. (6) ' b/ Fr cnvenience we here and hencefrh wrk wih dimensinless ime, sress, and viscsiy, he laer quaniy nrmalized s ha p/~ (c = ) = 1. The final cnsiuive relain is r() = ~ d'e~(')~(-') "exp - -'ilf'(")ld" 1 (7) -A~. Av + Av.,~v X A V X +A X V x ~ -1 (3 1 2 Lg(x) Fig. 1. Lg-lg pl f varius quaniies characerizing he mdel. n his figure, and hrughu, dimensinless ime, sress, and viscsiy are used, he laer quaniy nrmalized s ha /~ (c = ) =. The figure shws: (1) The prediced nnlinear viscsiy as funcin f x= ))() [Eq. (1)], (2) Eyring's nnlinear viscsiy as funcin f x = ~ (A) [Eq. (8)], (3) ]r/~(a~ =x)] (V) [Eq. (11)], and (4) he real (+) and he imaginary ( ) par f?~(~ =x) [Eq. ( 2)]. A cmparisn f he and A pins shws ha Eyring's viscsiy, which is knwn give a gd fi many experimens, is reprduced reasnably well by he mdel. Cmparing he and he V pins shws ha he Cx- Merz rule is beyed, hugh n quie accuraely in he ransiin regin. The real and imaginary pars f ~/~ (~) lks much like in experimen 7 V A The use f E~(') as he relaxain mdulus is mivaed by he fac ha his chice leads a nnlinear viscsiy which is clse ha prediced by Eyring's phenmenlgical hery f liquid flw [2] which fis many experimens: sinh- 1 (p) n(p) - (8) T see his, ne ha he Laplace ransfrm f E 1 is [51 /~l (s) - n (1 + s), (9) s he nnlinear viscsiy is given by n (1 + p) n(~) (1) Frm he ideniy sinh-l(x)= ln(x+ 1]/~x 2) i fllws ha Eyring's nnlinear shear viscsiy fr large is clse ha prediced in Eq. (1). This is illusraed in Fig. 1. Ne ha he presen mdel crn- pared Eyring's has a less sharp ransiin nnlinear behavir. Figure 1 als shws ha he Cx- Merz rule, as expeced, is beyed apprximaely by he cnsiuive relain. This bservain is based n he fac ha he frequency-dependen linear viscsiy is given by n (1 + ig) rl~(g) = ~ d'el(')e -i ~'- i9 (11) which implies fr he real par and fr he negaive imaginary par ~/' (9) = Arcan(g)/9 r/"(g) = n [V + (.2]/). (12) We nw prceed calculae he ime-dependen nnlinear respnse in varius siuains (fllwing Chaper 3.4 in [1]). Cnsider firs he sress grwh upn incepin f a seady shear flw, i.e., he case when he shear rae is given by p()= fo, < (13) ( ~, >O.

3 Dyre, A "zer-parameer" cnsiuive relain fr simple shear viscelasiciy 147 Then Eq. (7) implies fr he sress r-: = (b) (d) r- () = i' ~ d'el(')e - ('-) (18) Equains (1), (15), and (18) imply v/+ (, ~) + e - ~ l/- (, ~) = n (1 + Y), (19) ~ Lg (f) Fig. 2. Sress grwh upn incepin f a seady shear flw wih shear rae ~. The quaniy 1/+ (,~), given by Eq. (16), is pled as a funcin f ime fr: (a) ~ =.1 (reflecing he linear limi), (b) 7 = 3, (c) ~ = 1, and (d) ~ = 3. Like in experimen, ~/+ (,~) fllws he linear q + () fr shr imes while i sabilizes fr large a he nnlinear viscsiy, a sabilizain which akes place sner he larger is ~. The versh f ~/+ (, ~) fen seen in experimen is n reprduced by he mdel n his case Eq. (7) implies fr he sress r +: r + () = Y ~ d'el(') e-i' ' (14) r, fr he quaniy ~/+ (, ~) = r + ()/~), ~/+ (,?) = 1 d'el(')e -~ ' (15) where r/- (, Y) = z - ()/~. By means f Eq. (16) we hus find l - (, ~) = {El ()-El [(1 + ~)]e~ }/~. (2) Figure 3 shws r/- fr varius values f ~. As in experimen, ne finds ha r/- (, ~) is a mnnusly decreasing funcin f ime fr all ~, and ha ~/- reaches zer faser he larger is Y. We nw urn he calculain f sress relaxain afer a sudden shearing displacemen Y. The shear rae is given by ~()= y5(). Subsiued in Eq. (7), his gives r() = (1 - e - Y )E i (), (21) which is easily shwn by rewriing Eq. (7) as [ d ] ~-r~(")d" (22) r()= d'el(') --~, e- Afer a parial inegrain Eq. (15) reduces /+ (, ~) = [E~ [(1 + ~) ] - E l() e - % + n (1 + 7 )}/)), (16) 1. where use has been made f he fac ha El () varies as - n () fr ~. n Fig. 2 r/+ (, Y) is pled in a lgarihmic pl fr differen values f Y. The figure shws ha r/+ is always mnnusly increasing. This is n quie like in experimen where here is usually a characerisic "versh" f /+ as funcin f ime befre he seady sae value is reached [1]. Cnsider nw sress relaxain afer cessain f a seady shear flw, i.e., when ~7, < (17)?() = (.., >O. O f Fig. 3. Sress relaxain afer cessain f a seady shear flw wih shear rae P. The figure shws he quaniy /- (, y)/~/(~) as a funcin f ime, where ~/- is given by Eq. (2), fr: (a) Y =.1 (reflecing he linear limi), (b) ~ = 3, and (c) ~ = 3. As in experimen, ~/-(, Y ) decreases zer as ~ c faser he larger is ~

4 148 Rhelgica Aca, Vl. 29, N. 2 (199) which is valid whenever p>. Fr he relaxain mdulus G(, y)= "c()/y, ne hus finds 1 - e -7 G(, Y) = E1 () - - (23) Y Fr Y ~ O, G(,?) reduces he linear shear relaxain mdulus El(). Equain (23) shws ha G(, Y) facrizes in a funcin f ime muliplied by a funcin f Y, as expeced fr a Wagner ype mdel [1, 4]. Nex we cnsider he calculain f he nnlinear creep cmpliance J(,'c), defined as y()/r, where y() is he al shear displacemen in ime when a cnsan sress "c is applied a =. The calculain f J frm a cnsiuive relain is usually cmplicaed by he fac ha y() is nly given indirecly. Fr he presen cnsiuive relain, hwever, y() may be fund analyically in he fllwing way. Firs, Eq. (7) is rewrien fr he case under cnsiderain as "c er() = ~ d'e~(')~(- ')e y(-c) (>O). Equain (24) is linear in he variable C()= exp [y()]: "c C() -- d'e1 (') C(- '). Afer sandard manipulains ne hus finds C() = "c + P)e ~ + "c ~ due - u 1 [ln (u - 1 ) - "c1 z + 7~ 2 r finally, by inegrain wih respec ime, l+9(e~o 1) e J('rO)rO = 1 + T - - Y (29) + r ]~ du 1-e-U~ 1 (3) 1 u [n (u- 1)- ]2+ n 2 " n he linear limi Eq. (3) reduces J=+ ~ du le-u~ 1 u n 2 (u - 1 ) + zc 2 " (31) (24) The creep cmplicance J(, r) f Eq. (3) is pled in Fig. 4 in a lg-lg pl fr differen values f?)- As a final example f he use f Eq. (7) cnsider he cnsrained recil afer a seady shear flw is inerruped a = by suddenly remving he shear sress. We wish calculae he s-called recverable shear (25) y~. Wriing This equain is nw Laplace ransfrmed in "cc(s) =/~1 (s) ~(s) (26)?)() = ~)~, < (32) (-f(), >, r A ~(s) =. r (27) n (1 + s) - "c Here, use has been made f Eq. (9) and he ideniy C(s) = sc(s)- C(O) = sc(s)- 1. C(s) has a branch cu n he negaive real axis frm s = - 1 s = - and a ple a s = )), where cn )) = e~ - 1 (28) is he seady sae shear rae [Eq. (1)]. The Laplace inversin f Eq. (27) is perfrmed by defrming he inegrain cnur run frm - slighly belw he negaive real axis, runding he ple a s = )~, and reurning -c~ abve he negaive real axis Lg( ) Fig. 4. Creep cmpliance J- y()/r, where J is given by Eq. (3), pled as funcin f ime fr: (a) ~) = 1, (b)?) = 1, (c))) = 1, and (d))) = 1, where ~ is relaed r by Eq. (28)

5 Dyre, A "zer-parameer" cnsiuive relain fr simple shear viscelasiciy 149 where f()>, Eq. (7) implies fr > r, = - j d'e 1 (')f(- ')e -j,_,f()dr + ~ d'el(')pe-~ ('-)-J; f(r')d'' (33) - ',,. d,el(,)f(_,)el f( )a = yo e~ ~ d'e~(')e -~ ' (34) Defining F() = exp f(")d", Eq. (34) becmes d'el(')f(- ') = ~ e~ ~ d'el(')e -~ c (35) The Laplace ransfrm f Eq. (35) is r ffq(s)[sff(s)-l]= ~ [/~l(s)_/~1))] (36) ~) -- S 3. Discussin n his paper i has been argued ha a simple cnsiuive relain exiss which has n adjusable parameers (excep he verall scaling f ime and viscsiy) and which gives a qualiaively crrec picure f shear viscelasiciy. The relain Eq. (7) was arrived a by requiring he Cx-Merz rule be saisfied and ha Eyring's nnlinear viscsiy Eq. (8) is be reprduced apprximaely. This ensures a nnlinear viscsiy and a frequency-dependen linear viscsiy which are bh clse hse bserved in many experimens. Figure 5 shws he nnlinear seady sae viscsiy f he mdel cmpared experimens n fur plymeric liquids. n Fig. 6 he abslue value f he cmplex frequency-dependen viscsiy f he mdel is cmpared experimens n hree f he sysems f Fig. 5. n bh figures here is a qualiaive agreemen beween mdel and experimen. Frm sudies f he lieraure i is esimaed ha f he published rhelgical daa n plymeric sysems may be fied similarly by he mdel. A quaniaively saisfacry fi is nly pssible fr few sysems, hwever. T bain his, ne r mre fiing parameers mus be inrduced in he mdel, which will n be aemped here. The chice f he linear relaxain mdulus be E~(') may be jusified frm he bx mdel, i.e., he ~(S) 1 [1_[_ )) gl(s)-gl(~))] " s )) - s E 1 (s) (37) ' The recverable shear is deermined frm e~== lim F(). This limi is given by he residue f he --~ ple a s = f Eq. (37), and ne finds Y~ = n [2-/~1(~)], r Y = n [2-r/(~)]. (38) n he w limis ne has ~1. ~,.~ 1 = TYO, y (n2,?>>1. (39) Y~ (~) is mnnusly increasing which is als he case in experimen. Als like in experimen, y~ sabilizes n a recverable shear f rder ne a high Y. 2 3 Lg('~) Fig. 5. Nnlinear viscsiy f he mdel [full curve, Eq. (1)] cmpared experimens n fur differen plymeric liquids. As hrughu his wrk, bh he viscsiy and he shear rae are repred in dimensinless unis, he scaling parameers being, respecively, he linear shear viscsiy and 1/T where T is a characerisic ime. The figure shws daa fr (a) linear, mndisperse plysyrene in 1-CN (O, Fig. 15 f [6]), (b) Ply-l-lefins (e, Fig. 1 f [7] based n daa frm [8]), (c) ply (mehyl mehacrylae) (x, Fig. 15a f [9]), and (d).75% plyacrylamide (, Fig. 3 f [1] based n daa frm [11])

6 15 Rhelgica Aca, Vl. 29, N. 2 (199) _A -1-2 X X X (~ 2 ½ Lg(w) Fig. 6. Mdulus f he cmplex linear frequency-dependen viscsiy in he mdel [full curve, based n Eq. (12)] cmpared experimens n hree differen plymeric liquids qued in Fig. 5. Fr each se f daa he dimensinless viscsiy is shwn as a funcin f he dimensinless frequency defined by he same characerisic ime as used in Fig. 5. The figure shws daa fr (a) linear, mndisperse plysyrene in 1-CN ( ), (b) ply-l-lefins ( ), and (c) ply (mehyl mehacrylae) ( ) psulae f a unifrm disribuin f acivain energies fr micrscpic min. Cnsider he min f a freign micrscpic paricle in he liquid. Suppse he paricle feels a spaially randmly varying penial energy, and ha i mves by hermally acivaed hpping beween he varius penial energy minima. Then he linear mbiliy f he paricle (he velciy divided by an exernal frce acing n he paricle), is a gd apprximain given by [12] i) p *(~) = p () (4) ln(1 +i ) Assuming he Skes law is valid fr he paricle, ne has/~ * ~ l/r/* which shws ha he linear shear relaxain mdulus f he liquid is E~ (') in his apprximain. Because he Cx-Merz rule is beyed by he mdel i is n surprising ha he Gleissele mirrr relain [1] is als saisfied: The linear limi f r/+ frm Eq. (16) is lim /+ (, ~) = El () - e Y (4 ) Gleissele's mirrr relain saes ha /(7) is equal his limi evaluaed a = 1/y, hus 1, ~ 1 r/(~))= (1-C+ln~)/~, ~>1, (42) where C = is Euler's cnsan. A cmparisn f Eqs. (1) and (42) shws ha he mirrr relain is indeed saisfied a gd apprximain. The cnsiuive relain Eq. (7) reprduces ms qualiaive feaures f shear viscelasiciy. (An excepin is he versh usually bserved in /+ as a funcin f ime, where he mdel predics /+ increase mnnusly he seady sae value.) The fac ha qualiaive feaures f experimen are generally reprduced is n surprising, given he similariy beween he presen mdel and he Wagner mdel, which is well-knwn give a saisfacry descripin f experimen. Hwever, i shuld be ned ha he presen mdel, despie he similariy Wagner's mdel in he use f an expnenial damping funcin, des n belng he class f single inegral cnsiuive relains f he Blzmann-superpsiin-ype invlving a nnlinear srain measure. This is because, in Eq. (7), y appears insead f y. As shwn by Bij e al. [13], fr he frmer ype f mdels he Cx-Merz rule may be accuraely reprduced nly if ne uses a specific nn-mnnus srain measure. This prblem is avided here because he analysis f Bij e al. des n apply his mdel; hwever, i shuld be emphasized ha he Cx-Merz rule is afer all beyed nly apprximaely in he presen mdel (Fig. 1). The use f an expnenial damping funcin in he presen mdel is inspired by Wagner's wrk [4]. This damping funcin, in effec, cus-ff relaxain prcesses wih raes less han he shearing rae, an idea discussed by several auhrs [14-18]. An impran difference frm Wagner's mdel is ha his damping funcin is exp [-~-'71], whereas we here use exp [ - ~_ r 12] ]. Fr a mnnusly increasing r decreasing y() his des n make any difference. n mre general flws here may be cnsiderable differences beween he w appraches. Fr insance, if he ne shear displacemen beween ime - ' and is zer here is n damping a all in Wagner's apprach. n cnras, all shear displacemen aking place beween ime -' and cnribues he damping in he mdel f Eq. (7). Thereby an irreversibiliy relaed he newrk rupure hyphesis f Tanner [14, 18] is incrpraed in he mdel. The mdel may be regarded as expressing a cninuus versin f Tanner's idea ha enanglemens are ls irreversibly in he prcess f defrmain as sn as a limiing srain is exceeded; here enanglemens are ls cninuusly during any defrmain. n passing we ne ha Wagner's mdel has been exended incrprae ireversibiliy using a raher cmplicaed funcinal f he srain hisry in he memry funcin [19]. This

7 Dyre, A "zer-parameer" cnsiuive relain fr simple shear viscelasiciy 151 gives beer agreemen wih experimen han he riginal Wagner mdel. A pssible bjecin he kind f damping erm used here is ha, fr a peridic shear ), = Y sin (c ), ne migh expec ha he nnlineariy ses in a high frequencies, even a very small ampliudes (because he damping apparenly is a funcin f ya), and n f Y), in cnradicin experimen. This, hwever, is n crrec: Suppse he wrs pssible case f he nn-lineariy, i.e., pu he damping funcin equal exp (-cy' ) in Eq. (7). Then he respnse is z() = y~ ~ d'el(')cs [~(-')le -~ ~ ' where = Y9 [cs (r) Reg- sin ())mg], (43) g = ~ d'el(')e -(ic +y ~)' n (1 + x) - -, x= ic+yco. (44) A a fixed m he nse f nnlineariy may be esimaed frm [ g(x._)) x (45) Y ~ = g'(x) :in, which is he crierin fr he firs rder erm being equal he zer-h rder erm in he Taylr expansin f g as funcin f Y. Equain (45) leads Acknwledgemen The auhr wishes hank O. Hassager fr several helpful cmmens n an early draf f his manuscrip. The wrk was suppred by he Danish Naural Science Research Cuncil. References 1. Bird RB, Armsrng RC, Hassager O (1987) Dynamics f Plymeric Liquids, Secnd Ediin. Wiley, New Yrk 2. Kincaid JF, Eyring H, Searn AE (1941) Chem Rev 28: Cx WP, Merz EH (1958) J Plym Sci 28: Wagner MH (1979) Rhel Aca 18: Abramwiz M, Segun A (eds) (1972) Handbk f Mahemaical Funcins. Dver Publicains, New Yrk 6. Yasuda K, Armsrng RC, Chen RE (1981) Rhel Aca 2: Kulicke WM, Prer RS (198) Rhel Aca 19: Wang JS, Knx JR, Prer RS (1978) J Plym Sci, Plym Phys Ed 16: Marinez CB, Williams MC (198) J Rhel 24: Wagner MH (1977) Rhel Aca 16: Marsh BD (1967) (as cied by PJ Carreau, F Macdnald, RB Bird (1968) Chem Eng Sci 23:91) 12. Dyre JC (1988) J Appl Phys 64: Bij HC, Leblans P, Palmen J, Tiemersma-Thne G (1983) J Plym Sci, Plym Phys Ed 21: Tanner R, Simmns JM (1967) Chem Eng Sci 22: Chen -J, Bgue DC (1972) Trans Sc Rhel 16: Bersed BH (1976) J Appl Plym Sci 2: Thursn GB (1981) J Nn-Newnian Fluid Mech 9: Tanner R (1969) AChE J 15: Wagner MH, Sephensn SE (1979) J Rhel 23: ym=c ~-~:~-~,~ ~-r-lln(1 + i9) i~) " (46) i9/(1 +,,~.j-..,1 + (Received Sepember 12, 1989; in revised frm December 1, 1989) is nw easy see ha whenever c _ 1 he nse f nnlineariy akes place fr Y f rder ne. Fr c,~1, hwever, he nse f nnlineariy is a Y = c -1, crrespnding a maximum shear rae f rder ne in he peridic variain. Auhrs' address: Dr. Jeppe C. Dyre MFUFA Rskilde Universiescener P.O. Bx 26 4-Rskilde, Denmark