This is to certify that the thesis entitled THE PREDICTION AND CORRELATION OF MODULI 0F POLYMER SOLUTIONS SUBJECTED TO LARGE AMPLITUDE SHEAR OSCILLATIONS presented by Ekong A. Ekong has been accepted towards fulfillment of the requirements for M.S. degree in Chemical Engineering L J Date February 23, 1979 0-7639 va FINE§z 25¢ per dey per it. RETUMIKE LIBRARY MTERIAL§z Place in book return to runove charge from circulation records THE PREDICTION AND CORRELATION OF MODULI OF POLYMER SOLUTIONS SUBJECTED TO LARGE AMPLITUDE SHEAR OSCILLATIONS By Ekong A. Ekong A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE Department of Chemical Engineering 1979 (+\ {0 ~‘. fig“ will ABSTRACT THE PREDICTION AND CORRELATION OF MODULI 0F POLYMER SOLUTIONS SUBJECTED TO LARGE AMPLITUDE SHEAR OSCILLATIONS By Ekong A. Ekong A non-linear constitutive equation of Acierno et al. (la) is tested on data of MacDonald et al. (lb), of stress generated in a 2% polyisobutylene solution in Primol 355, and subjected to a uniaxial oscillatory shear with finite amplitude. In fitting the data of MacDonald et al., the correct correlation of model results is strongly dependent on the adjustable parameter 'a'. The model quantitatively predicts a larger decrease (30%) of the dynamic storage modulus than of the dynamic viscosity (5%) up to a strain amplitude of 1.28 units, as observed by MacDonald. The model predicts only a slight dependence of both moduli on the frequency of oscillation in contrast to the Bird-Carreau model. At higher strain amplitudes, the undestroyed frac- tion of entanglement types corresponding to large relaxa- tion times tend to the same value as in steady shear with equivalent shear rate. A qualitative agreement results in model prediction and shear stress growth data at large shear rates. la) D. Acierno et al., J. NonNewtonian Fluid Mech., Ekong A. Ekong l» l25-l46 (l976). lb) I.F. MacDonald et al., Chem. Eng. Sci., 24, 1615-1625 (1969). To my parents ii ACKNOWLEDGMENTS The author would like to express his sincere appre- ciation to Dr. K. Jayaraman for his guidance, assistance and exposure into the field of rheology, without which this work would have been impossible. Also his thanks goes to Drs. D. Acierno, G. Marrucci, G. Rizzo and G. Titomanlio all of Istituto di Ingegneria Chimica, Univer- sita de Palermo , Palermo (Italy) for their model; and Drs. I.F. MacDonald, J.D. Huppler, L.A. Holmes, E. Ashare, B.D. Marsh of the University of Wisconsin, Madison for the use of their experimental data. TABLE OF CONTENTS LIST OF TABLES . LIST OF FIGURES. INTRODUCTION Background. . . Small Amplitude Experiments Large Amplitude Experiments Stress Growth Experiments . The Constitutive Equation . The Acierno Model The Bird- Carreau Model. . . Determination of the Relaxation Spectrum. Unsteady Shear Flow with Large Deformation. Nonlinear Effects without Harmonics in Stress Signal . . Determination of the adjustable parameter . . Spectral analysis . . The Structural Character in Oscillatory and Steady Shear. . . Shear Stress Growth Prediction. DISCUSSION CONCLUSION . NOMENCLATURE APPENDIX . REFERENCE. iv Page vi al.-l —-l N N ._e_e.—I._l N N mVNN OCDNO‘ “N #N #0) NRC \IU'IU'IU'IU'l OCDU‘le LIST OF TABLES Table Page 1 Contributions of different A0 regions to linear viscoelastic functions. . . . . . . . . . . . . 35 2 The effect of large amplitude on the different A region contributions to the nonlinear visco— e astic functions . . . . . . . . . . . . . . . 37 Figure la 3a 3b 4a 4b 10 LIST OF FIGURES Small angle cone-and-plate viscometer. The relaxation spectrum. Steady and dynamic data of 2% polyisobutylene in Primol 355. Large amplitude dynamic viscosity data at w = 0.188 sec-1 Large amplitude dynamic viscosity data at w = l.88 sec' Large amplitud storage modulus data m = 0.188 sec‘ . . . . . Large amplitude storage modulus data m = 1.88 sec’ Large amplitude dynamic viscosity data with Acierno and Bird-Careau model predictions at m = 0.188 and 1.88 sec-1 Large amplitude dynamic storage modulus data at at with Acierno and Bird—Carreau model predic- tions at w = 0.188 and 1.88 sec‘]. Spectral diagram of finite amplitude latory shear, 7° = 0.2, A0 = 1.2 sec Spectral diagram of finite amplitude oscil- oscil- latory shear, yo = 2.0, A0 = 1.2 sec . Spectral diagram of finite amplitude oscil- latory shear, yo = 2.66, A = 37 sec . O The oscillatory and steady structural parame- ter, Xi as a function of relaxation times. vi Page 20 21 28 29 30 3O 31 33 41 42 43 45 Figure Page 11 The oscillatory and steady structural parame- ter Xi as a function of relaxation times and shear rate. . . . . . . . . . . . . . . . . . . 46 12 Shear stress and normal stress growth data and prediction for 2% polysiobutylene . . . . . . . 48 vii INTRODUCTION A number of useful properties and behavior of visco- elastic materials have been borne out of the classic small amplitude oscillatory shear experiments conducted by Ferry (1). Recently the Weissenberg Rheogonometer fitted with small angle cone and plate geometry as shown below has further enhanced the study of this flow regime on various classes of polymeric liquids. First the polymeric solution is placed between the cone and plate. A sinusoidal motion of amplitude yo is imposed on the cone while the plate is held steady and the gap between the platen and cone maintained constant. The sinuso dal strain imput can be expressed as 70¢ = yo sin wt (1) The torque on the plate and the axial for constant gap between plate and cone are related to the tangential shear stress and first normal stress dif- ference respectively 3T 16 = ¢ ZWR3 (2) I¥ ____ V 5.1.1; (Vb—,l r35 V. Figure 1a. Small angle cone-and~plate viscometer I‘D N] = r¢¢ ' TOB = $32 (3) The determination of T¢¢ and r¢¢-TBB after the initial transients have died out through the equations above are obtained with the assumption that a linear velo- city profile prevails within the gap. This assumption has been shown to be valid for small amplitude and small fre- quency motions (1b,2). The output stresses, T9¢, 16¢ - T99 are observed to oscillate sinusoidally with frequency w and 2m of the input strain rate respectively (3). Also a phase shift is observed in the response functions. The input strain rate function and the stress responses can be represented as ie¢ = Re i 1°e‘“t1 (4) o ‘ t 16¢ = Re I re¢e1m I (5) o . r¢¢ - 100 = Re i(d¢¢-d88)+(r¢¢-T88)e21“t} (6) Using the theory of linear viscoelasticity (l) we define the material functions, the complex viscosity, n*, the first normal stress difference displacement coefficient, ed, and the complex first normal stress-difference oscil— lation coefficient 6*, as 0 0 0 16¢ = - n*ie¢ = -(n'-ln")19¢ - 0 d¢¢ deg = _ ed|76¢|2 O 0 _ ”O 2 _ I ° N .0 2 T¢¢ ' Tee ' 6*(le¢) _ ' (9 ’ 19 )\9¢ These material functions often represent oscillatory flow properties of viscoelastic fluids and in the linear region are experimentally observed to be functions of the frequency of oscillation only. It has been experimentally observed that if the amplitude Y0 is increased, n*, 8d, 8* become functions of the frequency as well as amplitude and linear viscoelastic theory no longer applies (lb). We therefore proceed to redefine large amplitude material functions as follows 0 n*(w910) = $8 = n'(w,i0) (7) ed(8.y°) = - (d¢, - 8691/1101? (8) e*(...°> = - (13¢ -'Tge)/(+°)2 = e'(w.y°) - i8"(wsYo) (9) Several constitutive equations have been proposed by rheologists (4) to appropriately relate the stress tensor through material functions to large deformation rates. These models centrally recognize the presence of nonlinear effects in the equation of motion and introduce a number of parameters to correctly predict the nonlinear responses when large strain rates are imposed on polymeric fluids. Recently Acierno et al. (1a) proposed nonlinear constitutive equations based on the network molecular entanglement theory presented in detail by Lodge (5). This model was presented by the authors for polymeric melts and concentrated solutions with only a single adjus- table parameter. It has also been used on polymeric solutions by Graessley et al. (6) for start-up and relaxa- tion experiments. The objective of this work is to test the predictive capability of the Acierno model in obtaining the material functions n'(w,70), G"(w,y°), 8d(m,y°) and 9*(w,y°) that are to be determined experimentally by fellow workers (7) on the NBS fluid polyisobutylene in cetane. These experi- mental measurements are forthcoming and we turn to the literature data of McDonalds et al. (1b), who use 2% polyisobutylene in Primol 355. They collected G'(u,y°)/ G'(w) and n'(w,y%/h'(m) with 0.1Sy°$l.2 at w = 0.188, 1.88 and 18.8 sec". Also all data were taken at 25.0 i 0.1%. Our interest in this model stem from a) its successes in correctly predict various transient data reported by its authors (8) and Graessley et al. (6), b) the fewness of parameters and constants as compared with other models and c) the universality of the adjustable parameter in terms of flow regime and type of polymeric fluid as pro- claimed by its authors. Since the adjustable parameter 'a' is to be obtained through the guidance of dynamic shear data as opposed to steady shear as earlier suggested (8), we further evaluate 'a' by predicting the fluids start-up data of Huppler et al. (12). The Acierno model relates uniquely the relaxation times of the fluid to the existing entanglement density of network functions. By comparing the concentrations ob- tained through steady shear and averaged concentrations of the oscillatory data using the same 'a' value, we h0pe to ascertain some relationship between the shear rate 7 and ”Y0 that has pervaded much of the literature. Finally a brief comparison between the Bird-Carreau model and the Acierno model in their ability to predict n'(w,y°) and G'(w,yo) as a function of amplitude on the fluid under test will be presented. Background Many experimental rheologists have characterized the Non—Newtonian behavior of a number of polymeric fluids through steady shear experiments over two to three decades of shear rates. However the applicability of a constitu- tive equation to describe stress responses of Non—Newtonian fluids will depend on its ability to correlate other flow regimes as well. These flow regimes include start up and cessation of steady shear flow studied by Chen et al. (9) and Graessley (6), elongational flow experiments of Meissner (10), superposed steady and parallel/transverase oscillatory shearing of Schowalter (11) and oscillatory shearing of McDonalds et all (lb), and others (3,12). This list is by no means complete, each experimental set-up enabling the determination of the material function specific to the flow situation. In studying oscillatory shear flows it is necessary to review experimental and theoretical devel- opments of small and large amplitude shearing as well as start-up experiments. a. Small Amplitude Experiments In the introductory paragraph, it was shown that small amplitude experiments enable the use of the theory of linear viscoelasticity and thus the characterization of fluids in terms of the material functions n*, 8d and 8*. Furthermore the theory defines the linear viscoelastic functions listed above to be governed by the relaxation spectrum H(A) unique for each fluid (1) such that , _ H(A)Ad1nx n — (1+w2A2) (10) ‘R(A)uzx2dlnx G' = _‘ l+u2A2 (1]) Also for small amplitude shear flow 8*, and 8d can be obtained directly from the following relations (4). 1w * = ”*(w) - n* (28) (12) wed = n"(w) (l3) which has been derived by a variety of viscoelastic models including Oldroyd three constant model (13). Analogies have been presented both theoretically and experimentally between steady shear data and small amplitude oscillatory shear data and the most accepted is (14). ”o = ”(Y)ly+o = n (w)|w+o = H(A)AdlnA (14) 8 ' . _l%ll = §_éflll = H(A)x2dlnx (15) 27 7+0 w w+0 In determining non—linear material functions, the small amplitude experiments will be useful if the non-linear viscoelastic model employed demands the dynamic linear properties of the fluid. In the Acierno model a relaxa- tion spectrum is required and this can be obtained from the n'(w) or n"(w) vs w data in the linear region through an inversion procedure (1). b. Large Amplitude Experiments Harris and Bogie (15) carried out finite amplitude oscillatory experiments on the Neissenberg Rheogonometer with a clear check that harmonics in the input signal were minimal. They observed for their different solutions, third and fifth harmonics in the stress amplitude as a function of frequency. Their polymer solution showed only a third harmonic in stress amplitude slightly sensitive to frequency while the fundamental harmonic was predomi- nant. Philippoff (16) using a rotational viscometer (coni- cylindrical arrangement) studied the effects of large amplitude up to 700% shear amplitude units on the material functions n' and G' of polymeric solutions. He noted that the largest third harmonic recorded in the recording system within the range of strain amplitude was not more than 5% of the amplitude of first harmonic in stress. He further observed decrease of G' and n' at higher strain amplitudes, G' decreasing considerably. This experimental observations are consistent with linear viscoelasticity which recognizes a limiting ampli- tude beyond which the theory does not hold. So these data are in the realm of the non-linear models and along with other transient experiment can test the validity of the several proposed rheological equations of state. McDonalds et a1. (16) working on three polymeric solutions of different composition and a melt studied the effect of large amplitude oscillatory shear on n' and G'. They obtained results similar to Phillipoff's and furthermore noted the effects were slightly dependent on the fluid and frequency of oscil- lation. They chose frequencies well within the power-law region of their four fluids. McDonalds data lends itself to comparative analysis with predictions of nonlinear models since they maintained a linear velocity profile in the input deformation. Tee and Dealy (17) also found from large amplitude oscillatory shear in a small gap concentric lo cylinder rheometer that the extent of nonlinearity in the stress response is primarily a function of strain amplitude and independent of frequency. Strain amplitudes up to 10 and frequencies between 0.5 to 30 sec.1 were employed. In most of these studies inertial effects are neglected or compensated for. Dodge and Krieger (18) have argued that due to secondary flow, conventional analysis on oscillatory cone and plate geometry are not valid if fluid density is not considered. They thus elected to work with parallel plates or coaxial cylinders in their experiments (18). However Walters (22) noted that for relatively high visco- sity liquids (low frequencies) little error is incurred due to inertial effects in using any of the available geometries. c. Stress Growth Experiments This experiments more popularly known as start-up experiments involve a sudden initiation of simple shearing at a certain shear rate on a fluid at rest. It has been experimentally observed (2,6,12) that the response shear stress, 6K7,t) and the normal stress difference N](i,t) are functions of the input steady shear rate as well as time. At low steady shear rate, Q and N1 increase mono- tonically to their steady state value. In this limit 3 and N] are said to be governed by linear viscoelastic properties and are obtainable through the relaxation spec- trum for the fluid (6). As the shear rate is increased 11 the functions Gland N] overshoot their steady state value; then in some cases an undershoot follows. The magnitude and time of overshoot has also been shown experimentally to be a function of shear rate. Huppler et a1 (12) working on three nonlinear models, ONFS, NJFLMB and the Spriggs 4- constant models showed these models to give varying results on stress growth prediction while agreeing closely in pre- dicting steady shear and stress relaxation data. Graessley et al. (6) reviewed various models, the strain rate, rela- tive strain, averaged rate and structural dependent models on their ability to predict start-up as well as cessation after steady shear data collected with a modified (stif- fened) Neissenberg Rheogonometer R-17. They reported funda- mental inconsistencies between both the strain rate and relative strain models with their data. For the structural model (Acierno's) they obtained a reasonable fit with their data only when they adjusted the structural parameter 'a' at each shear rate where 'a' also fits fairly the steady shear and normal stress data. All these studies point to the importance of stress growth experiments in evaluating rheological models. We intend to use 'a' obtained from large-amplitude shear data to predict start-up data espe- cially at large shear rates. Such tests may reveal whether 'a' obtained through oscillatory shearing can portray a wide range of viscoelastic behavior. 12 The Constitutive Equation The non-linear constitutive equation proposed by Acierno et a1 (la) will be considered in this study. Acierno et al. report that their model adquately correlates data on tangential and normal stress growth in shear, stress growth in elongation and normal stresses in shear creep. The Bird-Carreau model (19) an integral nonlinear model studied by McDonald et al. (la) will be presented on a comparative basis in the study of large amplitude dynamic complex functions. The Bird-Carreau model has received wide attention and interest as it correctly predicts complex viscocity, non-Newtonian viscosity in steady shear primary normal stress difference and stress relaxation data (19,20). a. The Acierno Model The model is given by T = Ti (16) 1 (17) T1 9_ ' _ 13‘1— + A181; ((1%) ‘ 211'9- . 18 93:1 = (I-Xi) _. 3X]. /_E_1_ ( ) dt )1]- Ai 81- . - _ 1.4 61 - Goixi xi - Aoixi (19) 13 and 5T1 _ drl eri - r-VvT The Goi and Aoi in equation (19) are the linear elastic moduli and relaxation times of the fluid respec- tively. These are obtained by the construction of the fluid relaxation spectrum H(A). The H(A)'s are calculated from experimental curves of one of the three functions G(t), G'(w) or G"(w) by an inversion procedure to be shown in section C. The H(A) vs A plot is plotted on a log-log scale, then the log A axis is subdivided into equal inter- vals A log A. The mid-point of each sub-interval represent a set of discrete relaxation times Aoi and the corresponding values of the ordinates H01 gives GOT = HoiAIUA (20) Equation (19) relates elastic module Bi and the relaxation times A| to the existing structure through the structural variable x1. Acierno et al. claim the set of xi represent "the degree of connectivity of the macromolecu- lar network with respect to that of equilibrium" (9). The rate of change of the variable xi is given in equation (18) and is equal to the rate reformation of type 1 functions due to thermal motion minus the rate of destruction due to the existing stress. The destructive term is conceptively formulated by considering steady shear flow. In the absence 14 of thermal motion, "i, the number of junctions of the ith type which on the average exist on the given molecule at any instant of time would be destroyed in a time of order l/y. Thus the rate of loss due to the imposed flow would be aniy, where 'a' represents the ratio of a proper average contact time (attributed to thermal motions, entanglement property and other intrinsic properties) and its rough estimate 1/7. Then the rate of junction loss relative to equilibrium junction concentration is given by axii In order to relate this term to the stress level Y is arbitrarily expressed in terms of the second invariant of the stress tensor during steady shear. This concept is further generalized to all flow histories. Equation (16) assumes that the total stress development of the fluid on a strain input is obtained by the superposition of all stresses induced by network junctions corresponding to relaxation times that contribute significantly at the time of consideration. The model predicts zero second normal stress dif- ference in shearing flows. If unsteady oscillatory shear flows is considered, equation (11) predicts that except for 112 and 1]] all other components of the extra stress remain zero. Suppose lwt II x (D .<. O (D D i 7 15 Then I 12 d 1 12 l 1 : lwt 61 + ‘1 a? ( a.) ‘iRelYO e I (21) 11 d 111 12 t T __ = .08 1w —GT Aldt T) 2A (_.g:_QRey{11 (22) 9.51 =(1'Xi) _ 6X1 /"in (23) dt A1 A1 261. By defining 6i = t/Aoi, ai = aAoiyO (yo assumed real) (23a) 12 = aTi 2 '1] : Ti 61 , N1 = a T; , 81 ”A01 (23b) eqns. (15) - (17) along with eqn. (13) become 1.1+ xi1°4 51.11 = aixi1'4COSB-ie1- (24) dOi 1.4 dN' _ 1.4 N1 + Xi 33%»- “iTixi COSBiOi (25) 1.4 dXi _ x. m -1 - x, - .1 /N1- (26) The stresses 012 and 011 are continuous functions of time and can be represented by the sum of odd and even harmonics respectively by a Fourier expansion. on 1.6. T- = [ . 1 5;; A 2n+1,icos(2n+l)wt .sin(2n+l)ut] 1 (27) + B|2n+1, 16 Till = cg”, , }'_;[D-2n+2 .cos(2n+2)wt + (28) n= ’1 E'2n+_-2’isin(2n+2)mt] Also ;$;E_= 2:%k2n+1’iCos(2n+l)wt + (29) B2n+1,1sin(2n+1)wt] :8] = Coi+ fizmzqcc’smmz)“ + (30) £2n+2 ,isin(2h+2)wt From eqns. (7), (8), and (9) - k n'(w,Y°) = 12:; A1,. (31) k u 0 _ I - n (“’87 ) " G (wsYo)/w ' 12311 (32) k 6d(way'o) = gcois 6'(N9YO) = $011, (33) i=1 and 8"(u,y°) = 12E“. (34) Solving the coupled equations (24), (25) and (26), Ti(t) Ni(t) for several values of t can be found. If 'a' has been predetermined, the sets (ljéé—fl ,t) and 11 (T Lil-n) -YO 17 can be used to obtain the constants A2n+1 , 32n+l , Co, 02n+2 and E2n+2. At this point the predicted values of the amplitude of the various harmonics as well as the large amplitude material functions can be established. b. The Bird-Carreau Model The model is given by (28) I? -LuLt-t', u(t‘)]{(1+§)£']+;£ ldt' (35) with the memory function u given by u[t-t',ll(t')] = j:% "n eXPL-(t-t'llAZn] (36) "‘ A2n2[1+1/2 u (t')A]n2] where = Tl A _ A' . rIn 1” 9 Aj" ' .__\1__.__.,. 3:1,2 (37) RIP (n+11a3 The terms no, A], A2, a] and 02 are the model parameters determined from small amplitude oscillatory shear and steady shear experiments. The empirical constant 8 allows for nonzero secondary normal stress difference. Further details of the model will not be presented here as they have been published elsewhere (lb,4,19,20). In oscillatory shear motion McDonald et al. have derived n*(w,y°) for this model as * w o = A l- iwAZn Bn 2m n ( d1 ifgfinnn n1+w2——-)-3— n [1+ "12:!an m2) :1‘ 18 2 2 - Z nnAn(1-2m Azn-BlwAZn) I 2111-1 Bn 2111-1] n= )(—?J ( m-l 37a (1+m2Agn)(l+4w2Agn) m= ( 1 1 A" = (1+1/211n11012) (37b) Bn = 1/2A§n|+°|2An (37c) This result assumes that no higher harmonics are observed in the stress response; n'(w,y°) and G'(w,y°) can be obtained from eqn. (378). c. Determination of the Relaxation Spectrum The elastic moduli Goi and relaxation times Aoi's encountered in the Acierno model can be determined through the construction of the relaxation spectrum H(A) which is obtained from the G"(w) data. For 2% polyisobutylene in Primol 355 we used the n'(G"/w) data of Huppler et a1 (28). An initial approximation of H(A) is made using Tschoegl's second approximation formula (1) 2 n ' w HO(A) = ; (e (m) - 4/3 d—Sjfil + (38) To obtain denim! d1nw and dZGu m d(1nw) 19 values, we fit the G"(w) vs ln w data with a cubic spline, using the subroutine ICSVKU of the IMSL collection. ICSVKU starts with a given set of knots and shifts them among data points one by one in order to determine the knot locations that minimize the least square error. It then calculates the derivatives at the data points. More points were generated in the main program LSQSF within data limits for better precision in the next stage of H(A) determination. Next, an iteration scheme was set up by comparing calculated values of G"(w) and its observed values. From linear viscoelasticity, G" m = °° __‘£A__ 11 ( ) [mHO(A)1+w2A2 dlnA ( ) Then H](A) = H@(A)[:G"(w) obs/ G"(w) calc]h=l/A (39) He failed to achieve convergence with the relationship w= /5/A as presented in equation (38). This relationship according to it's author (21) represents a shift factor in the time scale at some point of the spectrum and was not fully understood. However on using the more conven- tional m = l/A relationship, convergence was obtained after nine iterations. Then the ability of H1(A) to reproduce 20 1000 £:100 * 0 1 J J 0.1 1.0 10 100 A(sec) Figure 1. The relaxation spectrum calculated from G"(m) (data of Huppler et a1. (28)) for 2% polyisobutylene in Primol 355. 21 Am.o u m c Lowv — mcamwd soc» :zcuomnm cowumxmpmc ace» woumpau—mo soc» ouou.u.o.>l region giving the least contribution. In the G'(w) data the wAml region contributes significantly but the greatest contribution comes from the wA>>l region. In Table 2 the network con- centrations, xi, of the different spectral zones are pre- sented due to the imposition of the strain amplitude of Y0 = 0.41 on the fluid. We observe the maximum effect of this large strain amplitude at the large relaxation times as xis in the wA>>l region deviate more from unity than those of smaller relaxation times. In other words the model 35 a Pm. mee.o Am.F mA.N me.m am.m pe.m_ ~.mp m.m_ we.FP we.“ mm.e. mm.” mk.e eo.e ARV”; - so.w me.o_ _.mF me.mp no.5, a.eP mo.op Ne.m ee.~ mp.. em.o N~.c - - Auvwe _.om Ne.mp mN.__ em.e mm.e me.~ Ape.F mmm.o ~oo.o ekm.o e-.o amp.o emo.o mmo. umo. .043 cup o.mm o.oe c.2m o.m~ o.ep e.m ~.m ~.m o.~ N.F e~.o 88.0 m~.o up.o Aeemvre‘ FAAKB 9843 va<3 Nae\ee»e amp 1 A3V.a emcee oee u Aav.e P-uem mmF.o n a mcowuocae u_ummpmoumw> somew— on m=o_mmc o« acmcomm_u mo mcovuanwcucou .p opnoh 36 predicts that the effect of non-linearity is restricted to larger relaxation times in the oscillatory flow regime. Since n'(w) is principally susceptible to the changes up to mAml region, the model predicts that the effect of nonlinearity is oscillatory shear will be least apparent through this material function. 0n the other hand the model predicts a drastic effect on the G'(w) value since the zone wA>>l which gives the greatest contribution has been mostly affected. These two observations are correctly supported by McDonalds data. Furthermore, we notice that in Table 2 in the wA>>l region there is a percent increase in n;(w,Y°) Spectral contribution. This seems to be borne out of equation (49). If 'a' is chosen such that the nonlinear term 1 + aiosi/Nzoi is close to unity then the decrease of each spectral unit will be moderate. However if wA>>l then the denominator becomes .8 . 2 (l + a1°8i/”2013 +w2Aoi2=(oni) and "i (wsyo) ni'(“) 2 1 ' (55) But no significant increases was incurred as shown in Table 2. In sum the model verifies McDonald's assertion 37 ARVAP- A3v.wc m._+ A... e._+ e.P+ mo.F+ _F.o+ e._- _.m- e.m- o.m- _.N- e._- «.2- mm.- mm.- Ac».av._ev . . ARVAF- . . . . . . . . . . . 3 r... - w P. N m- e w- o a- F o_- m FF- P mp- N NP- 8 m- em 8. am e- me.~- - - .mimqmwnflmv O . 2mm. mmm. mmm. mmm. mmm. _em. sea. Fem. was. mam. com. com. com. mam. emm. _x m.mm 8.“, 8.02 em.e mo.e me.~ em.F em. New. can. eNN.o amp. ewe. mmo. umo. 2‘3 m.cm_ m.Nm m.em m.em ~._N ~.m_ N.m o.m _.m o.~ ~._ e~.o ee.o m~.o up.o Aeemku oe_ o.mm o.oe o.~m o.mN e. e.w N.m N.m o.~ N.F ee.o ee.o m~.o N_.o Aemmvwe‘ FAA&3 Pe<3 _.VV&3 emo.o n .. A3V.e\fio».3v.e Npo.o- n F- A3V.e\fio».3ve Ne8\ee»e o.me_ . Ac».ev.e emwea ~m.Fme u fie».ave Pe.o . o» 2-888 mm_.o . a mcopuuczw ovummpm -oumw> com:_pcoc on» ou mcowuanwcucou co_mmc o« acmemmwwu on» co mnzuwpaEm mucmp mo uummeo mg» .N epoch 38 that in nonlinear oscillatory shearing experiments, more emphasis should be given to G'(m.v°) data rather than the n'(u,y°) counterparts. In Acierno's paper (8) it was shown that there was a drastic truncation of the relaxation spectrum during steady shearing experiments when shear rate as low as i = 0.01 was imposed. Also an a = 0.4 gave the best fit of their viscosity and first normal stress coefficient results. In Table 2 truncation of the right part of the relaxation spectrum was less drastic as the greatest change in A01 to A. occured at A01 = 160 sec. to A1 = 151 sec. at Y0 = 0.41. This would have an obvious advantage in terms of computer time, since we will not need to reconstruct the relaxation spectrum at this moderate shear rates. In this approximated form of the model equal intervals of AlogA is assumed even though large Aoi's are affected by relatively large 7°. Thus serious error may be encountered for large 1° as constant AlogA will be inconsistent with the model formulation. The effect on Gi needs a little study. From the model G = HiAlnAi = HiGlnAi oi At a strain input yo X1. =1/(1+ oi /N201) 39 _ .1.4 HialnAi - H16ln(AoiX1 1 C) ll 01- G0,- -1.4Hi61nai/Nzoi (56) However this error term is found to be negligibly small at large Aoi where the problem is centralized. ii. The structural character in oscillatory and steady shear Having obtained an 'a' from the preceding section, an attempt is made here to at least study qualitatively the model's nonlinear responses of the stresses and more especially the structural parameter xi at higher shear amplitudes. It must be pointed out.also that this analysis suffers from the severe limitations we impose on the velo- city gradient Iland the assumption that the contrvariant convected derivative, 6/6t is equivalent to the total time derivative, d/dt. We resort to the study of spectral responses bearing in mind that the total response is the arithmetical combi- nation of all the spectral regions that contribute signifi- cantly. The IMSL subroutine [NOGER has been employed to solve the system of ordinary nonlinear first order dif- ferential equations of eqns. 24-26. Initial values of Ti = 0.0, Ni = 0.0 and xi = 1.0 and an allowable set error of 10‘4 along with a range of the step size serves as the 40 input and DVOGER selects a suitable stepsize to calculate the next step which meets the error criteria. An amplitude range of 0.2 - 10.67 with relaxation times of 0.17 to 160 sec. was successfully tested. At higher ranges of relaxa- tion times 60 sec. to 160 sec. and large amplitudes, 5.0 - 10.67, the error criteria was not met. The program is presented in the Appendix and sample computer plots are shown in Figures 7 - 11. In Figure 7 with an input shear amplitude, Xi devi- ates slightly from unity and remains steady confirming the model to degenerate in a proper manner to linear visco- elasticity. The dimensionless normal stress have the expected frequency, 2m and a pronounced displacement that has already been reported (3). In Figure 8 we notice that a lO-fold increase in the input amplitude, leads to little less than a lOO-fold increase in the normal stress dis- placement. Christiansen and Leppard reported a correspon- dence of the slope equal to 2 for their solutions between the normal force displacement and the input amplitude (on a log-log scale) working however within the linear range (7L? = l). The xi is seen to deviate pronoucely from unity and exhibit a sinusoidal-like motion with a frequency of 2m approximately. There is also a lag between x1. and Ni noting that "1 is directly the forcing function. The Spectral behavior of large relaxation times 8.9. A01 37 sec of Figure 9 is intriguing at best. The xi curve I“) («'1 20. 15. 10. -15. -1O -20. -25. 41 I 0b 0 1 iT(sec) L l .6 9.2 13.8 18.4 23.0 Y Increment 8 Plot 1.60 E-04 A p101 4.00 [-037 c Plot 4.00 E--2 Figure 7. Spectral diagram of finite amplitude oscillatory shear. (A) Dimensionless shear stress, Ti (8) Dimensionless normal stress, Ni (C) Structural parameter, x- relaxation time, 101 = 1.2 sec w = 1.8 sec” , 1° = 0.2 25. 15. 10. ~10. -15. -20. -25. 42 l V V U V l J J l 3. O 6.0 9.0 12.0 15.0 T(sec) Y Increment 8 Plot 1.600 E-02 A Plot 4.00 E-02 C Plot 4.000 E-02 o'l Figure 8. Spectral diagram of finite amplitude oscillatory shear (A) Dimensionless shear stress, Ti (8) Dimensionless normal stress, Ni (C) Structural parameter, xi relaxation time, 10, = 1.2 sec. w = 1.88 sec-l, 10 = 2.0 43 0.0N1 oo.~ o» ._-umm mm.p n 3 .owm mm n _04 _x .qumsmcmq Pasauoacum Auv wz .mmmcum _oscoc mmm—eopmcmswo Amy we .mmmcam cmmgm mmmpcowmcme_o A m.m o.m m.~ o.u m.F o._ m.o d u q q d d - Aommvh \. / . 0-0/J A o.mp1 o.opl o.m1 o.o o.op o.m~ o.o~ o.mN 44 rapidly drops during the transient period and dwells much longer at this low point except for sudden peaks regularly interspersed. The stress functions also lose their sinu- soidal nature and only one broad peak is present in the normal stress function. These curves can be approximated by means of a Fourier analysis recognizing higher order harmonics in the shear stress and normal stress. The presence of these harmonics odd harmonics for torsion and even ones for normal force have been shown experimentally (15) and predicted by other models (e.g. the NBC & Carreau-B models) as well. This further portrays the importance of spectral studies since correlation of harmonics with experimental data may aid in choosing the correct defor- mation rate invariant and thus help ascertain the physical meaning of the adjustment factor, 'a'. In Figures 10 and 11 we compare the structural para- meter subjected to steady shear with that subjected to oscillatory shear. At an equivalent strain input Figure 10 shows that at the larger relaxation times more destruction of entanglements is achieved with steady shear than oscil- latory shear. This is when we judge the destruction in oscillatory shear in terms of the averaged structural con- centration occuring during a period of revolution. However when Figure 9 is considered it is found that the xi function of oscillatory shear at large relaxation times behaves as though it was subjected to steady shear. We next superimpose .mmsww cowamxmpmc 45 co coppocse m we .wx .mcwwmsmcma chzuozepm xummum use zeo»m__wumo one .o_ mcamwd wOKG FOP cop F-0F , m1o_ o_ d 1 N NI mapm> mmmcm>m n o _-umm m n + m:_m> smack“ u > cmmcm mzpm> “mace n < , o .383 11 £3. . 8.... omm m n >3 omm mm.~ n 3 F1 0 Fl L o_ x s —.I .L. a t . . v Q Q . Q. Q Q 1 < P 1 OF 46 .mumc cmmnm ucm woe?» cowumxmpme we :owuocse m we .wx .mcmpmamcma ~cczuoacum Anemom vcm acoumppmomo och .pp acumen mCKm . 1H (a N1 _ omm mm.~ n 3 ex xcmmum . wx acoumFPmomo 1 1 1 1 1 1 X I PIC—.5. 1 F o_ 47 the averaged xi functions of oscillatory shear on their steady counterpart subjected to the same degree of strain rates as shown in Figure 11. We notice a correspondence between steady x1 and an averaged oscillatory xi subjected to twice the strain rate of the latter. If we recognize the statement earlier made on averaged xi at large relaxa- tion time this correspondence is viewed with some skepticism, on the other hand such correspondence to some degree may infer that the truncated relaxation spectra of the two modes of flow are interchangable. This further add credence to the study of large amplitude shear oscillations and it's ability to generate 'a' that can describe non-linear behavior. iii. Shear stress growth prediction The generalized curves of Acierno et al. (la) for dimensionless tangential and normal stress growth in shear were utilized to obtain the shear stress and normal stress growth functions using an 'a' obtained in the preceding analysis. Since the relaxation times of polyisobutylene were used much interpolation was done to obtain the suitable parametric aikoi curve. However because of the regularity and smoothness of these parametric curves we feel that the associated error due to interpolation was minimal. Then 61(t) and N(t) were obtained by simply adding up all the contributions of each relaxation time at each time increment. The model results are presented 48 l V 104 1- /\ ul ~ \ / \\ / ‘5 / N],y= 1 67 N: / g [I . O],Y=16.7 0.1 e / 5‘ 3 / // /’ “<7‘ V10 1- / / / A _ -1 r— - --I\——e ’0],Y - ‘P ’ 1 67 'U \ . C It! 2 1 10 ‘ 10'1 100 . 101 102 t(sec) Figure 12. Shear stress and normal stress growth for 2% polyisobutylene. Data of Huppler et a1. (27) Data — - — - - - - Acierno model. 49 along with experimental data of Huppler et al. (12). The model does a good job in predicting the magnitude of the shear stress magnitude and the time at which it occurs at a shear rate of 1.67. However at i = 16.7 the performance of the model is at best qualitative. For the shear stress growth function it predicts a much faster overshoot time and a larger magnitude of overshoot. It is also unable to predict the shear stress undershoot, a problem common to many acclaimed rheological models. For the normal stress growth there is a qualitative agreement between model results and data as the model tends to predict a larger overshoot than the data shows. However no con- clusive judgement can be arrived at for the normal stress growth as the reliability of the data has been questioned. The reason as to why the model is unable to do a good job at high shear rates is subject to speculation, certainly we cannot blame it on the 'a' factors as the authors as well as Graessley have reported this failure even when using 'a' obtained from steady shear. DISCUSSION Acierno et al. evaluated the adjustable parameter 'a', from steady shear and normal stress data. With the single value of a, they obtained a good fit of data over a shear 4 to 1x103 sec'1 on low density poly- rate range of 1x10— ethylene melts. 0n the other hand Graessley et al. working on 8% and 12% polystyrene solutions could predict steady shear stress data only by choosing several values of 'a'. For the 8% solution the values required ranged from 0.2 at 1 = 0.1 sec'1 to 0.5 at 4 = 100 sec-1. For the 12% solution from 0.4 to 0.9 in same range of shear rates. They were then able to predict stress growth data by choosing an a which gave the best fit with steady shear data at the particular shear rate. From Figure 2 we found that 'a' obtained from finite amplitude oscillatory shear data successfully fit data with a shear rate range of 1 0.001 sec'1 to 0.3 sec' Also with this 'a' quantitative prediction of stress growth data at y = 1.67 see”1 was obtained. This perhaps indicate the usefulness of non- linear oscillatory data as far as the determination of 'a' is concerned. Another severe test of the Acierno model is met when predicting stress growth data at large shear rates. In 50 51 comparing the model predictions with the Carreau-B model (22) predictions on the same fluid, there is a striking resemblance between both model results at a shear rate of 16.7 for shear stress growth. Both quantitatively disagree with data predicting too early a time for overshoot, much larger magnitude and a complete absence of undershoot. The Carreau-B model, a modification of the Bird-Carreau model is similarly based on network theory however differs strategically from Acierno's as it relates the destruction of network junctions to the second invariant of the rate of strain. In the Acierno model it is noticed that at very small times after the onset of shear, the shear stress contributions of much of the spectral regions (except larger relaxation zones) arise close to the overshoot of the stress curve associated to this unit. This conceptu- ally appears as if destruction of the network junctions on each spectral zone is instantaneous. By choosing a dif- ferent 'a' say, these spectral curves are only shifted vertically and even though the total result may correctly predict the maximum observable overshoot it will still incorrectly determine overshoot time. We thus speculate that the invariant used in the generalization of i in the destruction term of the struc- tural variable equation is much too severe and suggest a combination of the invariant of the extra stress and of the strain rate be used. This similar conclusion has been 52 arrived at by Carreau (22). In this work the predictive capability of Acierno model with the parameter obtained from oscillatory shear ot the normal stress functions is only seen in the normal stress growth prediction. The fact that same qualitative agreement is met adds credence to the 'a' evaluation. Claims made in this study for 'a' is restricted to 2% polyisobutylene solution in Primol 355. CONCLUSION It has been shown in the preceding pages that the adjustment parameter 'a' of the Acierno rheological equa- tion of state can be obtained reliably and uniquely from large amplitude shearing experiments. The computed results further show in agreement with McDonald's observation that G'(w.yo) should be the choice correlation function as it expresses more nonlinear characteristics of the viscoelastic fluid in oscillatory shearing. However the 'a' obtained from G'(w,y°) correlation unsuccessfully predicted n'(u,y°) between yo = 0.1 to 1.28. For the n'(w.y°) prediction an a' of 0.1 and 0.2 were required to fit data at w = 0.188 and 1.88 sec-1 respectively. A method of numerical solution of the Acierno model in large amplitude oscillatory shearing where harmonics becomes significant has been outlined. Due to the lack of data in this region, no numerical predictions have been given, however we noted a similar behavior between junc- tion concentrations of the large relaxation zones to similar concentrations subjected to steady shear. With a single 'a' value the model is quantitatively unable to predict shear growth data at high shear rates. For polyisobutylene solution at high shear rates the model predicts an earlier 53 54 and larger overshoot and no undershoot in the stress growth function. In view of these failings much more tests at larger shear rates and amplitudes and different flow regimes with different polymeric fluids will be needed to establish the singularity of 'a'. NOMENCLATURE A11 A1, An,1 Bi’ Bn,1 Bn E -1 C03, Con,1 Di’ Dn,1 dii Ei’ En,i E F GI Gil , mn gmn’g H NOMENCLATURE Def. by eqn. 38a Adjustment (Acierno model) parameter Fourier coefficients def. by eqn. 27 8 29 Fourier coefficients def. by eqn. 27 & 29 def. by eqn. 38b Cauchy Tensor = oxmox1n 1 _1' —° gmn(x ) 5x 5xJ . _ _6__)Si ij 11111 X' F1nger Tensor - 6x1m6x10 9 ( 1 Fourier coeffs. def. by eqns. 28 8 30 Fourier coeff. def. by eqns. 28 & 30 Primary normal stress displacement, dynes/cm2 Fourier coeff. def. by eqns. 28 & 30 Elastic energy, (erg/cm3) Total axial force, dynes Storage modulus. dynes/cm2 Loss modulus, dynes/cm2 Components of the metric tensor Spectrum density, dynes/cm2 /-l Dimensionless normal stress, eqn. 23b Primary normal stress component following shear growth. dynes/cm2 55 56 R Radius of cone and plate, cm Re Real partcfi'a complex number t Current time t' Past time T Torgue transmitted through the fluid, dynes/cm 3 Velocity tensor, cm/sec x Structural parameter (Acierno model) xi,xj Coordinates of a fluid particle at time t xlm,xln Coordinates of a fluid particle at time t' Greek symbols “i Def. by eqn. 23a aj Bird-Carreau model parameter Bi Dimensionless frequency a Bird-Carreau model parameter y9¢ 9¢-component of finite strain tensor 73¢ e¢-component of the strain rate tensor, sec-1 Y° Strain amplitude to Strain rate amplitude, sec-1 1 Shear rate, sec"1 6i Phase shift between strain rate and shear stress, rad. o Aoi Equilibrium relaxation time, sec A. Bird Carreau and Acierno model relaxation times, sec A1" Bird-Carreau model parameters. sec Errata G(t) 57 Bird-Carreau model memory function Viscosity, poise Dynamic viscosity, real part of complex viscosity Imaginary part of complex viscosity, poise Zero shear rate limiting value of viscosity, poise Angle of cone (<40), degrees Complex primary normal-stress-difference coeffi- cient dynes secZ/cm2 Real part of 8*. dynes secz/cm2 Imaginary part of 8*. dynes sec2/cm2 Primary-normal-stress-difference-displacement coefficient, dynes sec2/cm2 Dimensionless time, def. by eqn. 23a Shear stress tensor, dynes/cm2 Component of shear stress tensor, dynes/cm2 Frequency of strain oscillation, rad/sec Second flow invariant = (1:1) Steady elastic modulus, dynes/cm2 Dynamic nonlinear elastic modulus, dynes/cm2 Dynamic equilibrium elastic modulus, dynes/cm2 Steady first normal stress, dynes/cm2 Shear stress component.following shear growth, dynes/cm2 APPENDIX 58 AC 9 9 ... C (Rh. 0 o C .l .. e! 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C is an NXK-l by 3 matrix First and 2nd derivative respectively of the spline function at the data points First and 2nd derivative respectively of the spline function at the inserted points. The dynamic loss modulus (or function values) 6‘ = wn' The dynamic viscosity, n'(Input) The unrefined relaxation density at data points The unrefined relaxation density at inserted points .Row dimension of matrix C in the calling program (input) i,j,k, counters Number of data points Number of knot locations = l5/w, equilibrium time constants at data points Spline function at data points Knot location The frequency of oscillation (data for inserted points), abscissa NC HK 60 = log x1 Vector of length NXK-i (output) Frequency of osciilation (data for inserted points), abscissa Nork area Spline function at inserted points 61 T»: “F ”LT hLflPLTlCtL VQLVE‘ F". fi'iu) 01 I“ t) uvc1puw nata sir“ ET SVTS 5 Ti“ 8 I p p U - 1: RH. 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O 66 PROGRAM LAMVIS List of Principal Variables ALO ANO ANNO DIFF EPL EPN GO GPL GPN I,J.K.L REPN RGN RGNO The adjustment parameter The equilibrium relaxation times The linear dimensionless normal stress function Nonlinear dimensionless normal stress function Dimensionless frequency Percent difference Linear dynamic viscosity Nonlinear dynamic viscosity Equilibrium elastic modulus Linear dynamic storage modulus Nonlinear dynamic storage modulus i.J.k,l. counters Normalized nonlinear dynamic viscosity Normalized nonlinear storage modulus Normalized observed nonlinear storage modulus The frequency of oscillation 67 l’RCka-n SCV'IL'FQIKFUTQ’JUTPUQI C 1'”. ofiuiiflhF SUIOWL I'VDLUYS‘ 7hr PEL P'JIFTIV': ".VGCFR TU ”ELEPKT' ( jan rfiutLUENl_VALUEC Jr T. N Shh X “F [HE BCIFRXO HOFEL HY C S’LV1hf 173 You": noaLlnzav onbxxasv DIFF. EQUATION.THF In“)? C 2521;“:196 IR!- SI-aa'tls LVPLITUUEQ PRLGUCVCY 61'“) 101E ‘DJUST'HNT c a 4' 5. PX'CR'DL 0’“ (‘7"C'SIJ'. VI“. 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F 68 PROGRAM SONODE List of Principal Variables ALO B DFUN DVOGER DY EPS ERROR HMAX HMIN I J JSTART MAXDER M MTH WK Ratio of strain amplitude supplied by input The relaxation time The dimensionless frequency External supplied subroutine required by DVOGER IMSL subroutine, first order differential equation solver The differential function for input system of differential equation required by DFUN Specification of the maximum error criterion Contains the estimated one step error in each component on output 0n input, suggests the step size to be attempted on the next step On input, the largest step size allowable in this integration On input, the smallest step size allowable in this integration Counter for A Number of time steps in DVOGER Initializes the integration Maximum order to be used in the approximation Number of ordinary differential equations Method indicator Nork area YMAX 69 On input contains the initial time, on output contains the updated value of time using H as the increment Y is two dimensional array (8 by M) containing the dependent variables Set in input is the suggested maximum absolute value of each component of the dependent variable calculated so far REFERENCES la. lb. 10. 11. 12. 13. 14. 15. REFERENCES D. Acierno et al., Journal of Non-Newtonian Fluid Mechanics, 1. 125-146 (1976). I.F. MacDonalds et al., Chem. Engineering Science, g1. l615-l625 (1969). J.D. Ferry, Viscoelastic Properties of Polymers, J. Wiley and Sons, New York, 1970. D.C-H. Cheng, British Journal of Applied Physics, 11. 253 (1966). E.B. Christiansen et al.. Trans. Soc. Rheol., lg, 65-86 (1974). T.N. Spriggs et al., Trans. Soc. Rheol., 1Q. 191-213 (1966). A.S. Lodge, Rheol. Acta, 1, 379 (1968). w.w. Graessley et al., lg, 291-301 (1977). D. Cross, K. Jayaraman, J. Pryzybla, Work yet published. D. Acierno et al., J. Non-Newtonian Fluid Mech., l, 147 (1976). I-Jen Chen et al., Trans. Soc. Rheol.. 16, 59-78 (l972). 'J. Meissner, Rheol. Acta. 19. 230 (1970). Ch. Goldstein et al., Rheol. Acta, lg, 253-262 (1973). D. Huppler et al., Trans. Soc. Rheol., 11, 181-204 967). . Halters, Rheometry, Chapman & Hall (LondOn), 1975, J. (1 J.H. Oldroyd, Proc. Roy. Soc. (London) A245, 278 (1958). K p l20. J . Harris et al., Rheol. Acta., g, 3 (1967). 70 16. 17. 18. 19. 20. 21. 22. w. Philippoff, Trans. 71 Soc. Tee and Dealy, Trans. Soc. Rheol., 33, 595-615 (1975) J.S. Dodge et al., Rheol. Acta, g. 480 (1969). R.B. Bird and P.J. 901 (1968). E: Ashare, Trans. Soc. Carreau, Chem. Engng. Sci., 2 , Rheo1., 12, 535 (1968). N.w. Tschoegl, Rheol. Acta, 19. 582-594 (1971). P.J. Carreau, Trans. Soc. Rheo1., 16, 99-127 (1972. A : .q .. . . s . uIE... a. 1.65 a .. .... twat. ...: pa 9 54.....- -a. 0.1.2:! 1 iffbré}. .. .1. 3,