1 J. Non-Newtonan Flud Mech., 72 (1997) Effect of a spectrum of relaxaton tmes on the capllary thnnng of a flament of elastc lqud V.M. Entov a, E.J. Hnch b, * a Laboratory of Appled Contnuum Mechancs, Insttute for Problems n Mechancs RAS, pr Vernadskogo 101, Moscow, Russa b Department of Appled Mathematcs and Theoretcal Physcs, Cambrdge Un ersty, Cambrdge CB3 9EW, UK Receved 8 October 1996; receved n revsed form 20 February 1997 Abstract The capllary thnnng of a flament of vscoelastc lqud, whch s the bass of a mcrorheometer, s analyzed n terms of a mult-mode FENE flud. After a short tme of vscous adjustment, the stress becomes domnated by the elastc contrbuton and the stran-rate takes on a value equal to two-thrds the rate at whch the stress would relax at fxed stran. Ths stran-rate decreases as the domnant mode changes. At late tmes, modes reach ther fnte extenson lmt. The flud then behaves lke a suspenson of rgd rods wth a large extensonal vscosty, and the lqud flament breaks. Predctons are compared wth the experments of Lang and Mackley (1994) Elsever Scence B.V. Keywords: FENE; Vscoelastc flow; Capllary thnnng; Mcrorheometry; Vscosty; Relaxaton tme 1. Introducton Whle real materals often have a wde spectrum of relaxaton tmes for the vscoelastc stress, most theoretcal and numercal calculatons gnore ths and only use a sngle relaxaton mode. Ths s not wthout good reason. On the theoretcal sde, we have only a partal understandng of how a materal wth relaxaton tme chooses to behave mdway between a purely elastc and a purely vscous materal. On the numercal sde t s stll not easy to compute wth flow tme-scales shorter than the relaxaton tme,.e. hgh Deborah number, and so wth a wde spectrum few of the modes can have ther Deborah numbers larger than unty. Usng many modes also has the danger of smearng out any nterestng behavour, and of ntroducng many adjustable parameters for fttng data. Fnally calculatons usng a spectrum have normally found after much effort that the results were domnated by a sngle mode, the slowest. * Correspondng author /97/$ Elsever Scence B.V. All rghts reserved. PII S (97)
2 32 V.M. Ento, E.J. Hnch / J. Non-Newtonan Flud Mech. 72 (1997) Ths paper examnes a flow n whch the spectrum plays a sgnfcant role. In desgnng a sutable problem, t s mportant to allow the materal to select ts own tme-scale and not to prescrbe one. Thus we must not mpose a velocty and a length, as n flow past a sphere, whch would set a tme-scale. Instead we mpose a force. We are also nterested n nonlnear dynamcs rather than a smple lnear vscoelastc response as n smple stress relaxaton. A problem whch s both nonlnear and allows the materal to select ts own tme-scale s the lqud flament rheometer (LFR), n whch capllary forces squeeze a thnnng lqud flament (Bazlevsky, Entov and Rozhkow  and Lang and Mackley ). After a complcated ntal formaton, whch we do not study, the problem smplfes consderably to the stretchng of a unform crcular cylnder, whch s homogeneous n space and vares only n tme. Before we start, t s necessary to dscuss the form of the consttutve equatons. In lnear vscoelastcty, a wde spectrum of relaxaton tmes can be represented by a dscrete or contnuous spectrum. We choose the former, wth N modes havng elastc modul g and relaxaton tmes. The nonlnear development requres a non-trval assumpton: n ths paper we assume that the modes are descrbed by uncoupled FENE dumbbells wth dfferent maxmum extensons L. Ths assumpton s possbly approprate to a dlute polydsperse mxture n whch each speces contrbutes ndependently to a separate mode. In cases n whch the spectrum comes from dfferent nternal modes of a sngle speces, one would expect consderable cross-couplng between the modes n the nonlnear regme. To our knowledge ths cross-couplng has not been studed. Gven the consderable uncertanty n the consttutve equatons, t s nterestng to compare our predctons wth observatons. In some recent detaled experments, Lang and Mackley  frst measured the lnear vscoelastc spectrum for the benchmark fluds S1 and the seres A20 A100. They then tested these fluds n a LFR. Now, for a Maxwell flud wth a sngle relaxaton tme, Bazlevsky, Entov and Rozhkow  have shown that n a LFR the radus decreases exponentally n tme wth a rate equal to one-thrd the stress relaxaton rate. The queston thus arses for a flud wth a wde spectrum whch of the many relaxaton tmes, f any, s measured by the LFR. Lang and Mackley  found n ther experments that the rate was approxmately g / g, except for the S1 flud. We seek to answer the queston. We also try to explan n terms of fnte extensblty why the flaments break after thnnng by an order of magntude. 2. Governng equatons The LFR has been descrbed n detal by Bazlevsky, Entov and Rozhkov . Consder a unform crcular cylnder of radus a(t) beng squeezed by surface tenson. Let the axal stran-rate of the axsymmetrc extensonal flow be e(t), so that the radus decreases accordng to a = 1 2ea. (1) For the rheology, we take a FENE flud wth N uncoupled modes. We use the Chlcott Rallson verson of the FENE equatons, although n the purely extensonal flow dfferences between dfferent versons are very mnor. Each FENE mode s descrbed by an axal deformaton A z (t) and radal deformaton A r (t), whch n the partcular stretchng flow satsfy
3 V.M. Ento, E.J. Hnch / J. Non-Newtonan Flud Mech. 72 (1997) A z =2eA z f (A z 1), (2) A r = ea r f (A r 1). (3) wth relaxaton tmes and FENE factors L 2 f =, (4) L 2 +3 A z 2A r wth dfferent fnte extenson lmts L for each mode. The boundary condton at the free surface sets the radal stress equal to the capllary pressure a = rr= p e+ g f A r. We assume that the axal stress vanshed, because n the LFT the flament s attached to large stagnant drops on statonary end plates 0= zz = p+2 e+ g f A z. In these expressons for the stress, s the solvent vscosty and g are the elastc modul of the modes. Elmnatng the pressure p, we have a =3 e+ g f (A z A r). (5) Ths equaton gves the stran-rate e n terms of the nstantaneous radus and deformatons, whch s then used n Eqs. (1) (3) to evolve the radus and deformaton. For ntal condtons, we take a(0)=a 0, and an undeformed materal A z(0)=1=a r(0). In Secton 6 we make detaled comparsons wth the experments of Lang and Mackley . Before then we wll llustrate our analyss wth a smple 8-model wth +1 =2 1, g +1 =2 g 1 and L +1 =2 /3 L 1. (6) Ths spectrum extends over two and a half decades n relaxaton tmes. Many of the spectra of Lang and Mackley  have the feature that g,.e. the contrbuton to the zero-shear-rate vscosty, s approxmately the same for each mode, except for a couple of very slow weak modes. The relaton between the fnte extenson lmts corresponds to polymers of the same type but dfferent molecular weghts M, for whch M 3/2 and L M 1/2. In our smple llustraton, we further choose a moderately strong surface tenson /a 0 g =1000/255 and a moderately small solvent vscosty / g =1/8 (and sometmes 1/16 and 1/4). Agan these values are not untypcal.
4 34 V.M. Ento, E.J. Hnch / J. Non-Newtonan Flud Mech. 72 (1997) Our analyss wll depend on the surface tenson beng large, /a 0 g. Because the total elastc modulus g s small, the elastc stress enters only when the deformaton s large. Thus, at early tmes, studed n secton 3, the stress wll be domnated by the solvent vscosty contrbuton. Snce the elastc stress s ncreasng rapdly when t becomes mportant and thus would soon exceed the drvng capllary pressure, we fnd n secton 4 that the stran-rate falls dramatcally. The elastc stress then domnates. A small stran-rate s requred n a second phase to stop the stress relaxng. Eventually fnte extensblty has an effect, whch s examned n secton Early vscous tmes The assumpton of strong surface tenson along wth the ntal condtons mean that we start wth no elastc stress. Thus the ntal stress s vscous, comng from the solvent vscosty. In ths secton we consder the early tmes before the elastc stress has had tme to buld up to a sgnfcant level. In these crcumstances, Eq. (5) reduces to a =3 e. Substtutng ths nto Eq. (1) for the decrease n the radus, and ntegratng gves a=a 0 t. (7) 6 Thus the radus of a flament of Newtonan flud decreases lnearly n tme, vanshng at the vscous breakup tme t vb =6 a 0 /. For a water flament of thckness 1 mm, ths s less than 1 ms. Fg. 1 shows the decrease n the radus n tme for our llustratve 8-mode model Eq. (6). The asymptotc predcton for early tmes Eq. (7) s gven by the dotted lnes. Ths analyss apples before the elastc stresses have had tme to buld up to a sgnfcant level. As after a tme N the fastest mode contrbutes a vscous stress 3g N N e, whch s comparable to the solvent vscous stress 3 e n our llustratve example, we requre that the shortest relaxaton tme n the spectrum be longer than the vscous breakup tme t vb N. In the three cases studed n Fg. 1, we have t vb / N =0.77, 1.54 and The agreement between the asymptotc predcton and the full numercal soluton becomes less good after t= N = for the last case. Clearly the spectrum of relaxaton tmes plays no role durng the early vscous tmes, except for the fastest relaxaton tme possbly delmtng the end of the phase. 4. Mddle elastc tmes At the end of the vscous phase, the elastc stress has grown so that t s comparable wth the capllary pressure and the vscous stress. The stran-rate must now drop, n order not to stretch the elastc stress beyond the capllary pressure. The vscous stress drops wth the stran-rate. Thus a new balance s establshed between just the elastc stress and the capllary pressure. Ths change s dramatc n the case of strong surface tenson. In the new elastc phase, the stran-rate
5 V.M. Ento, E.J. Hnch / J. Non-Newtonan Flud Mech. 72 (1997) becomes ndependent of the surface tenson, so long as t s large. It s also ndependent of the solvent vscosty. We assume that n the elastc phase the deformaton of the modes s large, A z 1 A r. Ths occurs f the surface tenson s large, /a 0 g, because a large elastc stress s needed,.e. large deformatons. We restrct attenton n ths secton to deformatons smaller than the fnte extenson lmt, A z L 2, so that the FENE factors are f =1. FENE effects wll be studed n the next secton. The deformaton Eq. (2) then reduces to A z =2eA z 1 A z. Ths can be ntegrated wth the Eq. (1) for the radus to gve A z= a 4 0 a 4 (t) e t/. (8) here we have used the ntal condtons, on the assumpton that the full relaxaton term (A z 1)/ has lttle effect durng the short ntal vscous phase. Neglectng the vscous stress and the radal term A r n the stress balance (5), we have Fg. 1. Early vscous tmes. The decrease n the radus a(t)/a 0 as a functon of tme t/ 1, wth the 8-mode model Eq. (6), /a 0 g 1 =10 3 and /g 1 1 =0.5, 1 and 2. The dotted lnes are the asymptotc result Eq. (7) for early vscous tmes. The dashed curve s the asymptotc result Eq. (9) for the mddle elastc tmes.
6 36 V.M. Ento, E.J. Hnch / J. Non-Newtonan Flud Mech. 72 (1997) Fg. 2. Elastc phase. The decrease n the radus a(t)/a 0 as a functon of tme t/ 1, wth the 8-mode model Eq. (6), /a 0 g 1 =10 3 and /g 1 1 =1. The dashed lne s the asymptotc result Eq. (9). x a = g A z. Substtutng our expresson Eq. (8) for A z, we predct the decrease n the radus n the elastc phase 0 G(t) 1/3 a(t)=a 0 a, (9) where G(t)= g e t/. (10) The functon G(t) s the lnear stress-relaxaton functon for the materal. If the materal were subjected nstantaneously to a sudden shear, then the shear stress would relax accordng to G(t). Eq. (10) s for dscrete spectra, wth an obvous generalsaton for a contnuous spectra. For our partcular spectrum Eq. (6) wth g constant, an asymptotc evaluaton of the sum for N t 1 fnds G 1.44g 1 1 /t. Fg. 1 compares the asymptotc result Eq. (9) for the decrease n radus n tme wth the full numercal soluton for three values of the solvent vscosty /g 1 1 =0.5, 1 and 2. We see that, after the ntal vscous phase whch does depend on the value of the solvent vscosty, all three cases approach the same asymptote whch does not depend on the vscosty. Fg. 2 contnues the
7 V.M. Ento, E.J. Hnch / J. Non-Newtonan Flud Mech. 72 (1997) plot to later tmes for the one case /g 1 1 =1. Fnte extenson effects have been suppressed by settng L 1 equal to a very large number. The asymptotc predcton Eq. (9) s found to be good, despte the large surface tenson parameter of /a 0 g =1000/255 not beng very large. The quas-exponental decrease of the radus n tme Eq. (9) means that the lqud flament does not break n ths elastc regme. Later, neglected fnte extensblty wll become mportant. We also observe that, f a flament starts wth an ntal radus varyng slowly along the flament, Eq. (9) means that dfferences n the radus ncrease by only the addtonal factor a 0 1/3, and thus do not produce a neckng nstablty. Ths stablty of the flaments follows from the stress beng saturated by the elastc contrbuton. We suggest that ths mechansm may explan the stablty of vscoelastc lquds n spnnng. Another test of our analyss for strong surface tenson s to compare the predctons for the contrbutons of the modes to the stress, g f (A z A r ). We have assumed n ths secton that A z A r and that f =1, so Eq. (9) for the deformaton yelds contrbutons g a 4 0 a 4 (t) e t/. These estmates are compared n Fg. 3 wth the contrbutons n the full numercal soluton. The agreement s good, agan despte the large parameter not beng partcularly large. Fg. 3. Elastc phase. The contrbutons to the stress g (A z A r )/g 1 from the 8 dfferent modes as a functon of tme t/ 1, for the model Eq. (6), /a 0 g 1 =10 3 and /g 1 1 =1. The vertcal lnes mark where t= for the mode. The dashed lnes are the asymptotc predcton g a 4 0e t/ /a 4 (t)g 1.
8 38 V.M. Ento, E.J. Hnch / J. Non-Newtonan Flud Mech. 72 (1997) Fg. 4. Elastc phase. The stran-rate e 1 as a functon of tme t/ 1, for the model Eq. (6). /a 0 g 1 =10 3 and /g 1 1 =1. The dashed curve s the approxmaton e 2/3t. Fg. 3 shows that the stress s domnated by the modes n successon: the modes wth hgh modul decay rapdly gvng way to weaker slower-decayng modes. From the ponts marked on the contrbutons at t=, we see that the mode whch domnates at tme t s the one wth the relaxaton tme nearest to t. Ths concluson apples to spectra smlar to our llustratve example wth g constant. The stran-rate n the elastc phase can be found by substtutng Eq. (9) wth Eq. (10) for the radus decreasng n tme nto Eq. (1): e t/ e= 2G 2 g 1 3G = 3 g e t/. (11) Thus the stran-rate at tme t s two-thrds the rate at whch stress relaxes at the same tme n a standard stress-relaxaton experment wth fxed stran appled at the ntal tme. Gven the concluson from Fg. 3 that at tme t the mode whch domnates G(t) s that wth the relaxaton rate nearest to t, we expect e 2/3t for N t 1. Beyond t= 1 when the slowest mode domnates, one would expect the stran-rate to be constant, e 1/3. In Fg. 4 we plot the stran-rate as a functon of tme. The above estmate s seen to be wthn a factor of 30% over the wde range of varaton.
9 V.M. Ento, E.J. Hnch / J. Non-Newtonan Flud Mech. 72 (1997) As explaned at the begnnng of ths secton, n the elastc phase the stress s elastc, equal n value to the capllary pressure. The only reason for a non-zero stran-rate s to stop the elastc stress relaxng. In an axsymmetrc extensonal flow, the axal stran-rate requred to stop the stress relaxng s one half the stress relaxaton rate, not the two thrds we have found. The addtonal one sxth s requred because the exstence of the stretchng flow means that the radus decreases, and so to balance the slowly ncreasng capllary pressure the elastc stress must also ncrease slghtly. Fnally, we comment that the way n whch the radus decreases n the elastc phase s crtcally dependent on the spectrum of relaxaton tmes, and could not be smply represented by a sngle mode. 5. Late tmes lmted by fnte extenson As the deformaton A z ncreases, the fnte extenson lmt L eventually has an effect and we enter a new phase. At the late tmes of fnte extenson effects, the numercal system of Eq. (5) n Eqs. (1) (3) s very stff. The stffness s partcularly acute for large fnte extenson lmts or low solvent vscostes (as n the A20 A100 seres of fluds). The problem s that, when the stress s domnated by the elastc part, the vscous part, whch s used to calculate the stran-rate e, sthe small dfference of large numbers. Usng the deas of the prevous secton, we can develop an alternatve expresson for the stran-rate to use n ths stuaton. Once the vscous part of the stress s small, Eq. (5) becomes a balance between capllary pressure and elastc stress a = g f (A z A r) In ths secton we do not assume that the radal deformaton s neglgble, because the fast modes can be hghly relaxed at the begnnng of the phase. Also, we do not assume that the FENE factors are unty. Dfferentatng the equaton above wth respect to tme, we have a 2 a = g A z A z f (A z A r)+a r A r f (A z A r) n. We now substtute Eqs. (1) (3) for the tme dervatves. Ths yelds an expresson whch s lnear n the stran-rate e and the relaxaton rates 1/. Solvng for e and usng Eq. (4) for f,wefnd g f 3 (A z A r)/ e= g f [ 3 2(A z+a r)+2f (A z A r) 2 /L 2 ] (12) Ths approxmaton for the stran-rate was used n the numercal calculatons once the vscous stress dropped below 1% of the total stress. The resultng numercal problem stll requres cauton as the fnte extenson lmt s approached. Reducng the tme-step so that the FENE factors ncreased by a small factor each step proved adequate.
10 40 V.M. Ento, E.J. Hnch / J. Non-Newtonan Flud Mech. 72 (1997) Eq. (12) for the stran-rate s the generalsaton of the earler two-thrds the stress-relaxaton rate. It s based on the same physcal balance, beng the stran-rate necessary to stop the FENE elastc stress relaxng from the value whch balances the slowly ncreasng capllary pressure. Fg. 5 gves the decrease of the radus n tme wth the fnte extenson lmt L 1 =50 (and so L 8 =7.9). Compared wth the result (9) n Secton 4, we see that ths large fnte extenson lmt has lttle effect before t= 1. On the other hand, the stress contrbutons gven n Fg. 6 show consderable change before t= Once the FENE factors exceed 1.3, the stress contrbutons relax faster. Thus the slowest mode comes to domnate earler, n our example by t= compared wth t=0.6 1 wthout FENE effects. Because the faster modes have relaxed early, the domnant slower modes must bear more stress than the nfntely extensble case. After t= 1, the radus decreases faster than the nfntely extensble case gven by Eq. (9), see Fg. 5. We see that the flament breaks after a fnte tme only when we take nto account fnte extensblty. The precse tme of breakup depends on the value of the fnte extenson lmt L 1. For L 1 =50, the tme of breakup s We return to the dependence of the breakup tme upon the fnte extenson lmt at the end of the secton. As the tme of breakup s approached, the slower modes are vrtually fully extended, A z L 2, whle the stran-rate e tends to nfnty. Under these condtons, the rate of ncrease n the deformaton A z n Eq. (2) becomes neglgble, as does the 1 n the relaxaton term. Hence Eq. (2) reduces to Fg. 5. FENE effects. The decrease n the radus a(t)/a 0 as a functon of tme t/ 1, wth the 8-mode model Eq. (6), /a 0 g 1 =10 3, /g 1 1 =1 and L 1 =50. The dashed curve s the asymptotc result Eq. (9) for nfntely extensble dumbbells. The dotted lnes are the predctons Eq. (15) usng an effectve vscosty *=1667g 1 1 (mode 1 only) for t , and *=2716g 1 1 (modes 1 and 2) for t
11 V.M. Ento, E.J. Hnch / J. Non-Newtonan Flud Mech. 72 (1997) Fg. 6. FENE effects. The contrbutons to the stress g f (A z A r )/g 1 from the 8 dfferent modes as a functon of tme t/ 1, for the model Eq. (6), /a 0 g 1 =10 3, /g 1 1 =1 and L 1 =50. The dashed lnes are the asymptotc predcton g a 4 0e t/ /a 4 (t)g 1 for nfntely extensble dumbbells. The vertcal lnes are where f =1.3 for that mode. 0 2eA z f A z. Thus the FENE factors are gven by f 2e. (13) Hence the contrbuton to the stress for the modes becomes 2g L 2 e. To the extent that ths contrbuton to the stress s proportonal to the nstantaneous stran-rate, and that the dumbbells are nearly locked rgdly at ther maxmum extenson, we can say that the FENE flud s now behavng lke a suspenson of rgd rods, wth an effectve vscosty *= 2 3 g L 2. (14) The sum here s to be taken over the hghly stretched modes. As the FENE flud n now behavng as a vscous flud, we can apply the analyss of secton 3 wth the radus decreasng lnearly n tme at a rate nversely proportonal to the effectve vscosty. The approach to breakup at tme t=t b s thus predcted to be
12 42 V.M. Ento, E.J. Hnch / J. Non-Newtonan Flud Mech. 72 (1997) a= 6 * (t b t). (15) Fg. 5 shows ths pecewse lnear decrease, usng the effectve vscosty correspondng to one hghly stretched state up to t= and two modes thereafter. It would perhaps be approprate to use three modes after t=1.86 1, and further modes a lttle later, but they have lttle effect n the lnear extrapolaton to breakup. Note that the lnear extrapolaton wth just one mode n Fg. 5 s not poor. Assocated wth the lnearly decreasng radus Eq. (15), secton 3 predcts the stran-rate ncreasng rapdly towards breakup e= 2 t b t. Combnng ths wth the earler expresson Eq. (13) for the FENE factors, we fnd the growth of the deformatons of the modes towards breakup t A z=l 2 b t 1. (16) 4 Ths predcton of a lnear approach to full extenson at tme of breakup s compared n Fg. 7 wth the full numercal soluton. Fg. 7. FENE effects. The deformatons A z of the modes as functons of tme t/ 1, for the model Eq. (6), /a 0 g 1 =10 3, /g 1 1 =1 and L 1 =50. The dashed lnes are the asymptotc predcton Eq. (16).
13 V.M. Ento, E.J. Hnch / J. Non-Newtonan Flud Mech. 72 (1997) Fg. 8. FENE effects. The decrease n radus a(t)/a 0 as a functon of tme t/ 1 for dfferent fnte extenson lmts from left to rght L 1 =20, 30, 40, 50, 60, 80 and 100, for the model Eq. (6), /a 0 g 1 =10 3 and /g 1 1 =1. The dashed curve s the asymptotc result Eq. (9) for nfntely extensble dumbbells. The dependence of the breakup tme on the fnte extenson lmt s studed n Fgs. 8 and 9. Fg. 8 gves the decrease n tme of the radus of the flament for varous values of the fnte extenson lmt L 1. Note that the longer dumbbells need to be stretched for a longer tme before they begn to devate from the nfntely extensble result Eq. (9). The tmes for breakup are plotted n Fg. 9 as a functon of the fnte extenson lmt. The tme to breakup can be estmated by patchng together our two asymptotc approxmatons Eqs. (9) and (15), the frst from the mddle elastc tmes n whch all the FENE factors are set equal to unty, and the second from the late tmes n whch all the FENE factors are very large. We choose to patch these approxmatons for the radus of the flament at a cross-over tme t c, defned to be where ther slopes are equal. Of course, at ths cross-over tme, the FENE factors are nether close to unty nor very large, and so nether asymptotc expresson s strctly applcable. Our cross-over tme s therefore but a crude estmate. Settng equal the two expressons for the tme dervatves of the radus of the flament, we fnd t c from G (t c ) a 0 G(t c ) a 0 1/3=. (17) 3G(t c ) 4 g L 2 The sum for the effectve vscosty s taken over all modes whch relax slower than the cross-over tme, t c. The breakup tme then follows by the lnear extrapolaton Eq. (15),
14 44 V.M. Ento, E.J. Hnch / J. Non-Newtonan Flud Mech. 72 (1997) t b t c 3G(t c) G (t c ). (18) Ths estmate s compared wth the full numercal soluton n Fg. 9. The step n the estmate between L 1 =40 and L 1 =50 reflects a swtch from 2 modes to 1 contrbutng to the sum for the effectve vscosty. For short FENE dumbbells, L 1 ( /a 0 g 1 ) 2/3, the cross-over occurs n the mddle of the spectrum of relaxaton tmes. Here we may use the approxmaton G(t) 1.44g 1 1 /t (for our partcular test spectrum, Eq. (6)), whch leads to t b =4t c and t b a 0 g 1 t b /4 g L 2 g 1 1 3/4. Ths estmate s good for L 1 40 n Fg. 9. For long FENE dumbbells, L 1 ( /a 0 g 1 ) 2/3, the cross-over occurs where the lowest mode domnates, G(t) g 1 e t/ 1, whch leads to t b =t c +3 1 and t b 1 3ln 4 3 L2 1 4ln a 0 g Ths expresson produces the same estmates for Fg. 9 when L 1 150, although these estmates are about 2 1 too large. Such an error s to be expected gven the crudeness of patchng the asymptotc approxmatons, and s probably due to the fnte extensblty makng the radus Fg. 9. FENE effects. The breakup tme t b / 1 as a functon of the fnte extenson lmt L 1, for the model Eq. (6), /a 0 g 1 =10 3 and /g 1 1 =1. The dashed curve gves the estmate Eq. (18) wth Eq. (17).
15 V.M. Ento, E.J. Hnch / J. Non-Newtonan Flud Mech. 72 (1997) decrease more rapdly than Eq. (9) at the cross-over tme. An alternatve to patchng the radus of the flament would be to patch the stretches Eqs. (8) and (16). Ths alternatve suffers the dffculty of havng to decde whether an ntermedate mode wll relax on beng affected by fnte extenson or whether t wll domnate the weaker slower modes n the stress. There are two ways to vew how fnte extensblty leads to the breakup of the flament. In the elastc phase of Secton 4, the stran-rate s two-thrds the current relaxaton rate. An effect of fnte extensblty has been seen n Fg. 6 to be to ncrease the relaxaton rates by strengthenng the sprngs of the dumbbells. The relaxaton rates thus ncrease wthout lmt, leadng to a faster than exponental decrease n the radus of the flament. Alternatvely, at large deformatons the elastc stress of the dumbbells becomes proportonal to the stran-rate,.e. the stress appears to be vscous. We have seen n Secton 3 that a vscous response produces a stran-rate proportonal to 1/a, and so the radus decreases lnearly n tme. The spectrum of relaxaton tmes does not play a central role when fnte extenson effects act, although they do nfluence when those effects come nto play. A flud wth a spectrum could be adequately represented by a sngle mode wth g L 2 set equal to g L 2. Wth many spectra, a sngle mode does n fact domnate ths sum over the modes. 6. Comparson wth experments Our theoretcal calculatons above are based on an assumed form of the consttutve equatons, a system of uncoupled FENE dumbbells. There can be no certanty that ths form apples to any real materal. It s therefore nterestng to compare our predctons wth experments. Recently Lang and Mackley  have nvestgated the benchmark fluds S1 and the seres A20 A100. These are solutons of a commercal grade of polysobutylene, wth a wde range of molecular weghts whch wll produce a wde spectrum of relaxaton tmes. The S1 flud s a 2.5% soluton n 47.5% decalne and 50% polybutene ol, and the A20 A100 fluds are 1 5% solutons n decalne. Lang and Mackley measured the storage and loss modul G ( ) and G ( ) n oscllatng shear n the lnear regme. Software n the rheometer then ftted a dscrete spectrum g, wth 11 modes spread evenly over 3 decades on a logarthmc scale, see Table 1. Nonlnear step-stran experments were then conducted and nterpreted n terms of a Wagner exponental dampng functon n an ntegral consttutve equaton. Ths gave good predctons for the measured stress growth and measured steady vscometrc functons. All these tests were made n smple shear flow. Now we have used a FENE consttutve equaton rather than an ntegral equaton wth a Wagner dampng functon, and t s not possble to nterpret the dampng coeffcent k as a fnte extenson lmt L. The dampng functon produces stran thnnng, whereas the FENE sprng produces stran hardenng. The FENE modfcaton of the lnear Hookean dumbbells was ntroduced to control extensonal flow effects, and s not good for descrbng nonlnear shear effects. As noted above, the good ft of the Wagner consttutve equaton was only tested n smple shear. Hence, we must dscard the expermental data for the dampng coeffcent and treat our fnte extenson lmt L as an unknown adjustable parameter. We do however keep n full the measured lnear spectrum.
16 46 V.M. Ento, E.J. Hnch / J. Non-Newtonan Flud Mech. 72 (1997) Table 1 The modul g (n Pa) for the dfferent relaxaton tmes (n seconds) for the benchmark fluds S1 A20 A40 A60 A80 A e2 6.89e e1 1.28e2 2.65e e e1 5.12e e e e1 4.63e e1 1.62e e1 5.44e e1 2.68e e e1 2.88e e e e e e e e e e e e e e e e e e e e e e e e e e e e 2 For our comparsons, we also need values for the surface tenson and the solvent vscosty. We take = Nm 1 and =3 Pa s for S1 and = Pa s for A20 A100. The asymptotc theory tells us that the results are reasonably nsenstve to the values selected for these parameters. After charactersng the rheology of the benchmark fluds, Lang and Mackley then placed a sample of the fluds n the Lqud Flament Rheometer. They measured the capllary thnnng of a lqud flament (see ther Fg. 13). The ntal dameters of the flaments were 2.44 mm (S1), 0.45 mm (A20) and 0.9 mm (A40 A100). The dameters decreased to 0.02 mm over a tme of s dependng on the flud. The expermental results show no early vscous phase. Wth the parameters of the experments, ths phase s expected ether to be very short (t vb = s or less) for the A20 A100 fluds, or to be unnecessary because of low surface tenson ( /a 0 g =0.15 for S1 and A100, and 0.36 for A80). In the former case, the phase wll occur n tmes shorter than those resolved by the experments, and most probably durng the settng up of the ntal flament. Ths s a problem wth the Lqud Flament Rheometer, that the process of makng the ntal flament s not yet well charactersed. Thus we do not know how much the materal s deformed durng the process. For the fluds S1, A80 and A100 we choose to assume that the materal s not deformed,.e. the ntal condtons A z (0)=1=A r (0) are satsfed. For the other fluds t s necessary to assume that the materal s deformed before measurements start. For these cases we assume a pre-stretch P, settng A z(0)=p L 2 L 2 +P and A r(0)= 1 P. 1/2 Ths corresponds to the length of the flament beng stretched by a factor of P 2 by the combnaton of the settng up and the rapd vscous phase. The arbtrary factor L 2 /(L 2 +P) s necessary to stop modes beng stretched beyond ther fnte extenson lmt. The value of the pre-stretch was adjusted so that the radus changed least at the begnnng of the numercal calculatons. One would antcpate ths value to be P=a 0 g /, n order that the elastc stresses start n balance wth the capllary pressure.
17 V.M. Ento, E.J. Hnch / J. Non-Newtonan Flud Mech. 72 (1997) Frst we consder the S1 flud. Ths flud has a solvent vscosty whch s not small, but whch s 21% of the total soluton vscosty (at zero shear-rate). Also, as noted above, the capllary pressure s not large compared wth the total elastc modulus, /a 0 g =0.15, although after the fastest mode relaxes at 0.01 s the rato ncreases to 0.63, and after the further 0.3 s t ncreases to 2.2. Thus the capllary thnnng of the S1 flud s not entrely wthn the asymptotc analyss of Secton 4. Numercal soluton can however be made of the governng equatons. Fg. 10 gves the decrease n dameter as a functon of tme. The ponts are the expermental observatons of Lang and Mackley. The three contnuous curves are the numercal solutons usng the expermental spectrum wth three choces of the fnte extenson lmt, L 1 =17, 20 and 22. No pre-stretch was appled. The best ft of the adjustable parameter s L 1 =20. We note that Eqs. (17) and (18) predct L 1 =47 usng the observed breakup tme of 16 s. Also plotted n Fg. 10 s the predcton Eq. (9) for the mddle elastc tmes. Ths has the correct magntude untl the fnte extenson lmt s felt. Dscrepances at early tmes may be due to the fastest modes ( =O(0.01 s)) havng relaxed before the frst expermental observaton. Also, the vscous breakup tme t vb s 0.73 s, due to the large solvent vscosty, and ths s not small compared wth the fastest relaxaton tmes. Fg. 11 shows the deformaton of the varous modes for L 1 =20. Note the lnear approach to the maxmum extenson, as predcted by Eq. (16). Fg. 12 shows for the contrbutons of the dfferent modes to the stress for L 1 =20. Note that the slowest mode never domnates, because ts modulus s so low. It s the second slowest mode whch domnates for most of the tme. Fg. 10. S1 flud. The decrease n the dameter 2a(t) (n m) as a functon of tme (n s). The damonds are the expermental observatons of Lang and Mackley. The three contnuous curves are, from the left, predctons for L 1 =17, 20 and 22. The dashed curve s the asymptotc result 2a 0 (a 0 G(t)/ ) 1/3.
18 48 V.M. Ento, E.J. Hnch / J. Non-Newtonan Flud Mech. 72 (1997) Fg. 11. S1 flud. The ncrease n the deformatons A z (t) as a functon of tme (n s) for the case L 1 =20. Fg. 13 shows the decrease n dameter for the flud A20. Ths flud has strong surface tenson, /a 0 g =45, and so one would expect a rapd vscous phase to produce a pre-stretch of about ths value. In fact the expermental data are best ftted (wth no jump near t=0) wth just ths pre-stretch P = 45. Wth ths optmal value of the pre-stretch, the full numercal soluton and a modfed asymptotc result for mddle elastc tmes, a 0 (Pa 0 G(t)/ ) 1/3, whch takes nto account the pre-stretch, both follow the expermental data farly well untl the fnte extenson lmt s felt. The breakup tme of t=0.3 s s best ftted wth a fnte extenson lmt L 1 =77. The approxmate theory of Eq. (18) wth Eq. (17), modfed to ncorporate the pre-stretch, gves the poor estmate L 1 =38. Fg. 14 shows the decrease n dameter for the fluds A40 A100. For A40, the ntal capllary pressure /a 0 s 3.19 tmes the total elastc modulus g, but ths doubles after the fastest mode relaxes at t=0.01 s, and doubles agan when the next mode decays at t=0.02 s. A pre-stretch of 5 was found necessary to avod an early jump n the dameter. The best ft of the breakup at t=0.7 s was obtaned wth a fnte extenson lmt of L 1 =77. The approxmate theory of Eq. (18) wth Eq. (17) modfed to ncorporate the pre-stretch gves the estmate L 1 =89. For A60, wth an ntal capllary pressure of 0.76 tmes the total elastc modulus, a pre-stretch of 3 and a fnte extenson of L 1 =24 (cf. 15 from Eq. (18) wth Eq. (17)) ftted the expermental data. For A80 and A100, wth ntal capllary pressures of 0.36 and 0.15 tmes the total elastc modulus, the expermental data was best ftted wth L 1 =4.5 and 0.84 (cf. 6.5 and 3.2 from Eq. (18) wth Eq. (17)). These values for L 1 are of course absurd f one nterprets them as the fnte extenson lmt of a polymer molecule. The results for the fttng parameters are summarsed n Table 2.
19 V.M. Ento, E.J. Hnch / J. Non-Newtonan Flud Mech. 72 (1997) Usng the two adjustable parameters, the pre-stretch P whch s adjusted to avod an ntal jump n the dameter and the fnte extenson lmt L 1 whch s adjusted to ft the breakup tme, the expermental observatons of Lang and Mackley of the capllary thnnng have been ftted reasonably well usng ther measured spectra wth the wde range of relaxaton tmes. There are some dscrepances at early tmes, whch are probably due to the fastest modes havng relaxed before the frst observaton. The poor charactersaton of the ntal flament s a problem wth the Lqud Flament Rheometer whch needs to be addressed n the future. Our asymptotc theory Eq. (9) for the mddle elastc tmes also predcts the experments reasonably well untl the fnte extenson lmt s felt, despte n most cases the ntal capllary pressure not beng much greater than the total elastc modulus. It would be possble to develop an alternatve theory for weak capllary pressure, but ths would n effect be lnear vscoelastcty producng the vscous estmates of Secton 3 wth a tme-dependent vscosty *(t)= + g, where the sum s taken over the modes whch have had tme to relax, t. In the LFR experments however, as soon as a few of the fastest modes (wth the hghest modul) have decayed, the capllary pressure does become larger than a sum over the remanng modul, and so our theory Eq. (9) becomes applcable. The predctons for the fnte extenson lmt by the approxmate theory Eq. (18) wth Eq. (17) are not good compared wth the full numercal solutons, partcularly for the cases nvolvng a pre-stretch. Fg. 12. S1 flud. The stress contrbutons g f (A z a r ) (n Pa) as a functon of tme (n s) for the case L 1 =20. At t=10 s the modes are, from the top, =2, 1, 3, 4, 5, 6, 10, 7, 11, 8, 9, where 1 =10 s and 11 =0.01 s.
20 50 V.M. Ento, E.J. Hnch / J. Non-Newtonan Flud Mech. 72 (1997) Fg. 13. A20. The decrease n the dameter 2a(t) (n m) as a functon of tme (n seconds). The damonds are the expermental observatons of Lang and Mackley. The contnuous curve s for L 1 =77 and a pre-stretch P=45. The dashed curve s the asymptotc result 2a 0 (Pa 0 G(t)/ ) 1/3. We can now return to the queston at the end of the ntroducton, of what relaxaton tme s measured by the LFR for a flud wth a wde spectrum of relaxaton tmes. It s nstructve to compare the two fluds S1 and A80. These have vrtually the same spectrum, see Table 1 and also Fg. 5 n Lang and Mackley for the storage and loss modul G ( ) and G ( ). The fluds dffer substantally n ther solvent vscostes, by a factor of 10 3, but the solvent contrbutes only 20% to the soluton vscosty n the larger case. Lang and Mackley further show n ther Fg. 9 that the steady shear vscosty ( ) s the same for the two fluds. In the LFR, however, the capllary thnnng s very dfferent, S1 breakng after 16 s whle A80 breaks after 1.8 s. Thus the smple shear flow charactersatons fal to catch all of the rheology. The expermental observatons for each of the sx fluds can be dvded nto two roughly equal perods. The frst half follows, more or less, the asymptotc result Eq. (9) for the mddle elastc tmes, the dashed curves n Fgs. 10, 13 and 14. Ths asymptotc result does not depend on the solvent vscosty, so long as t makes a small contrbuton to the total vscosty, g. The result does depend strongly on the spectrum. As the spectra of S1 and A80 are vrtually the same, the dashed curves n Fgs. 10 and 14(c) are the same. Hence, the dfference between the two fluds s the tme at whch they devate from ths asymptotc curve, t=8 s for S1 and t=0.9 s for A80. The fluds start to devate from the asymptotc result, and so enter the second perod of the expermental observatons, when the fnte extenson lmt comes nto play. Ths lmt would seem to be qute dfferent for the two fluds, L 1 =20 for S1 and L 1 =4.5 for A80.