Orientation in simple shear flow of semi-dilute fiber suspensions 2. Highly elastic fluids

ELSEVIER

J. Non-Newtonian Fluid Mech., 62 11996) 135 153

Orientation in simple shear flow of semi-dilute fiber suspensions 2. Highly elastic fluids Yoichi Iso ", Claude Cohen b, Donald L. Koch b'* "The Furakawa Electric Co., Ltd., Hiratsuka R&D Lahorator)'. 5-1-9 tligshi-Yawata, Hiratsuki 254. Japan bSchool ~?/ Chemical Engineering, Cornell Unieerisity, Ithaca, N Y 14853, USA Received 25 April 1995: in revised form 12 September 1995

Abstract The orientations of fibers suspended in strongly viscoelastic polymer solutions were observed. The suspending fluids consisted of a glycerine-polyethylene glycol mixture containing different amounts of dissolved polyacrylamide (PAAml with concentrations ranging from 200 to 2000 ppm. In the highly elastic limit of 1000 and 2000 ppm PAAm, a single fiber oriented in a fixed stable orientation. For the 2000 ppm fluid, the fixed orientation was along the flow direction, whereas for the 1000 ppm fluid, the fixed orientation was either along the flow direction or at a 20° angle from the flow direction in the flow-vorticity plane. The orientation distributions of semi-dilute fiber suspensions in these highly elastic fluids were insensitive to fiber concentration and exhibited sharp peaks at the stable orientations observed in the single fiber experiments. In contrast, the orientation distribution in the 200 and 500 ppm PAAm solutions were much broader. The distributions at the higher end of the semi-dilute fiber concentration regime in the 200 ppm solution resembled the distributions observed previously in suspensions with lower fiber concentrations in a weakly elastic fluid (100 ppm PAAm), although a single fiber in the 2(X) ppm fluid does not spiral towards the vorticity but aligns in a stable fixed orientation in the flow-, vorticity plane.

Keyword~': Highly elastic fluids: Orientation: Semi-dilute fiber suspensions: Simple shear flow

1. Introduction It was pointed out by Bartram et al. [1] that the spiraling motion of a fiber toward the vorticity axis in viscoelastic liquids subjected to a shear flow generally observed in weakly elastic fluids [2-4] and described by the theory [5,6] was not always seen in experiments. They found, * Corresponding author. 0377-0257,'96/$15.00 4'3 1996 .- Elsevier Science B.V. All rights reserved SSDI 0377-0257(95)01405-5

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at sufficiently high shear rates, that a fiber oriented initially along the gradient direction would rotate toward the flow direction and remain at this fixed orientation unless disturbed. Johnson et al. [7] were the first to look at the effect of different elasticity on the orientation. They observed the orientation of dilute suspensions of hardened prolate red blood cells in two Boger fluids with different molecular weights and concentrations of polyisobutylene (PIB) using small-angle light scattering (SALS). In the lower elasticity fluid at low shear rates, they observed an evolution away from alignment predominantly parallel to the flow direction toward alignment along the vorticity axis as the shear rate was increased. In the higher elasticity fluid, very strong orientations along the vorticity axis were found at the lowest shear rates achievable, but at higher shear rates preferred orientations away from vorticity were observed and interpreted as due to precession around the vorticity at a fixed angle. SALS cannot distinguish however between a precession around the vorticity and a fixed orientation away from vorticity. We also found in Part 1 [4] instances in which a fiber in a PIB Boger fluid did not spiral but rather aligned along a fixed orientation. We note that previous single fiber results are limited. Bartram et al. had limited observation time and Johnson et al. did not get complete information on orientation. Finally, information on orientation distributions in suspensions of fibers in liquids with highly elastic behavior is not available. In this paper, we present measurements of fiber orientations of single fibers and orientation distributions of semi-dilute suspensions (1 ~
2. Experimental system and procedures The Couette flow-visualization apparatus used here has been described in detail in our previous publications [4,8] (see Fig. 2 of Part 1). The increase of the PAAm concentration in the glycerine-polyethylene glycol-water mixture had negligible effects on the density and refractive index of the fluid so that the cellulose acetate propionate fibers used previously with aspect ratio r = 24 were still closely matched in density and refraction index with the fluids used here. The rheological properties of the polymer solutions were characterized using the Rheometrics RDA II Dynamic Analyzer. Our instrument was capable of measuring the first normal stress difference for the most concentrated (2000 ppm) PAAm solution when parallel plates of 50 mm in diameter were used. The sensitivity of the instrument was not adequate for the measurements of the normal stress difference of the lower PAAm concentrations. For these solutions we used a Couette device in order to maximize the torque signal in steady, oscillatory, and shear stress relaxation measurements. The Couette viscometer device had an inner radius of 16 mm and an outer radius of 18 mm. The height of the inner cylinder in the fluid was 33 mm [9]. The

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glycerine-polyethylene glycol-water mixture behaved as a Newtonian fluid with a viscosity of 6.6 P in the shear rate (frequency) range examined, i.e. from 0.1 to 10 s -~ (rad s-l). To characterize the elasticity of the PAAm solutions, we calculated the values of the ratio I of the elastic modulus G' to the loss modulus G" at a frequency of 0.5 rad s-1 and these values were reported in Table 1 of Part I. We have also performed stress relaxation measurements in which the fluids were sheared at a steady shear rate of 1.0 s-~ before the shear was instantaneously stopped. The relaxations of the normal stress for the 2000 ppm fluid and the shear stress of all four fluids were recorded and analyzed in terms of a Williams-Watts equation for the decay of the stress as the latter was clearly not a single exponential function [9]. The Williams-Watts equation [10] which represents a stretched exponential function is given by A = A,, e x p [ - ( t / r ) h]

(1)

and is frequently used to describe the spectrum of relaxations encountered in dielectric and dynamic light scattering measurements of polymeric systems [11]. In our measurements, A is the measured normal force or torque decaying with time t after the cessation of the flow. Eq. (1) fitted our data very well for the 200 and 500 ppm PAAm solutions and somewhat less adequately for the 1000 and 2000 ppm PAAm solutions [9]. Such fits allow us to obtain the parameters r and b and extract an average relaxation time ( r ) given by e x p [ - ( t / r ) h]dt = ~ F(l/b),

( r ) --

(2)

II

where F(l/b) is the gamma function evaluated at l/b. The values of r and b obtained for the four fluids and the corresponding average relaxtion time (~) are found in Table 1. The Deborah number defined as ( r ) ~ based on the average relaxation time for the stress and a shear rate ;:, of 0.5 s -I will then vary from De = 0.5 for the 200 ppm solution up to De = 3.2 for the 2000 ppm solution. As noted in Part 1 and elaborated further below, the orientational behaviour of fibers in viscoelastic fluids seems to be more sensitive to the magnitude of the elastic forces in the fluid as represented, for example, by I than the dynamic response of the fluid represented by the Deborah number. Using the Couette flow-visualization apparatus described in Part l, single fiber experiments were performed by placing a single, black, tracer fiber in each polymer solution. The fiber was placed in the center of the fluid in the Couette, away from the walls and the top and bottom

Table 1 Best fit values of the Williams-Watts parameters for the stress relaxation and the average relaxation time PAAm

Stress

r (s)

b

( r ) (s)

200 500 1000 2000 2000

torque torque torque torque normal force

0.333 0.449 i.35 1.76 15.7

0.417 0.370 0.387 0.387 0.705

0.99 1.87 4.91 6.39 15.7

ppm ppm ppm ppm ppm

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surfaces. The orientation of the tracer fiber under shear was then recorded continuously while counter-rotating the two cylinders in a synchronized manner so that the tracer fiber was kept in the stagnation plane and was always in the view of the camera. When the fiber started moving out of the stagnation plane due to some disturbance, the rotation rate of one of the cylinders was either increased or decreased to move the stagnation plane back to the fiber. While this was being done, the rotation of the other cylinder was automatically decreased or increased to keep the shear rate between the two cylinders constant. Two separate observations lasting 2 h were performed for each polymer solution. The experiments could go on for 2 h without any substantial drift of the fiber position, indicating that the secondary flow in the cell was very small. The absence of appreciable translation of the fiber also indicates that we were operating below the critical shear rate for the elastic instability described by Muller, et ai. [12-15]. After the recordings were performed, the video images were analyzed at intervals of 30 s so that the time evolutions of the projection angles ~ and ~b were obtained from the observations. The angle is defined as the angle that the projection of the fiber axis into the flow-vorticity plane makes with the vorticity whereas ~b is the angle that the projection of the fiber axis into the flow-gradient plane makes with the gradient direction (see Fig. 3 of Part 1). The time behavior of the azimuthal angle 7 between the fiber axis and the gradient direction can then be determined from ~ and ~b. In the fiber suspensions, two types of observations were made. First, a single black tracer fiber was placed in the middle of the fiber suspension in the Couette, and its orientation under shear was recorded continuously for 2 h while the fluid was sheared by counter-rotating the cylinders. The recorded images were digitized and analyzed every 30 s. This way, time sequential data for both projection angles z~ and ~b were obtained. If the fiber migrated in the cell as a result of hydrodynamic interactions, it was followed until it came close to the wall at which time the recording was stopped. In order to minimize the degradation of the polymer, single tracer experiments performed for all the fiber suspensions were limited to two or three observations of up to 2 h each. After observations of a single tracer fiber in a suspension were performed, the number of the tracer fibers was increased in order to obtain statistical distributions of the fiber orientation. In this case, the inner cylinder was kept stationary and the outer cylinder rotated at a constant rate. This way, all the tracer fibers would translate relative to the camera. The fibers were recorded as they passed though the camera view. In these multiple tracer experiments, only the distribution of the orientation angle ~ was recorded and analyzed. For the suspensions based on 200, 500 and 2000 ppm PAAm solutions, about thirty tracers were initially placed in the middle of the gap away from the side walls and top and bottom surfaces. Upon shearing, the tracer fibers were recorded once every hour for 5 h. This interval of 1 h for the recording was chosen because it was much longer than both the time required for the single fiber to come to a preferred orientation in the single fiber experiment and the time required for a suspension in a Newtonian fluid with a similar fiber concentration to become ergodic [8]: Thus, each hourly observation can in effect be considered as providing independent data. This 5 h shearing experiment was performed three times for each suspension, producing three data sets tbr each suspension. The three data sets were then combined to obtain the orientation distribution of the suspension. The three data sets for each suspension were similar to one another, indicating that the experiments were consistent.

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During the course of these experiments, the tracer fibers initially placed in the middle of the suspensions were found to migrate very little, meaning that the fibers observed were not experiencing wall effects throughout the entire experiment. Based on these results, the number of the tracers was further increased for fiber suspensions in the 1000 ppm PAAm solution so that the density of the black tracer fibers in this case was n,L~= 0.05, where n, is the number density of tracer fibers. In this case, it was impossible to keep track of all the tracer fibers. Therefore, the tracer fibers were recorded continuously as they came into the camera sight for the duration of one rotation period of the outer cylinder. The focus of the camera was set at the center of the gap so that the tracer fibers close to the wall were not recorded. The suspensions prepared in the 1000 ppm PAAm solution were sheared for 5 h, and the recording was performed after 1 h and 5 h. The 5 h experiment was performed twice producing four sets of distribution data, which were then combined. The four sets of data for each fiber suspension were very similar, indicating that the orientation distribution had equilibrated after I h of shearing, and that the experiments were consistent and reproducible. The suspension with n L 3 = 10 in the 1000 ppm PAAm solution was produced by mixing the suspensions with nL ~ = 1 and 20 after shearing experiments had been performed on the two suspensions, This meant that the suspending polymer solution for this fiber suspension had experienced somewhat more shear than the rest; however, there was no apparent effect on the orientation results (see below). In each experiment, the suspension was mixed manually in order to randomize the initial distribution. During the course of the experiment, the rheology of the fiber suspension was monitored frequently to ensure that no substantial degradation of the suspending fluid had taken place.

3. Orientation in highly elastic fluids 3. 1. The 2000 p p m P A A m f l u i d

A single fiber placed in the 2000 ppm PAAm polymer solution quickly aligned along the flow axis when a shear rate of 0.5 s-~ was imposed. The fiber then remained in this orientation for the 2 h duration of the experiment. Two separate experiments yielded indentical results. The time evolution of ~ during the two experiments is shown in Fig. l(a). The time evolution of ~b shows a similar behavior with a constant ~b at 90 ° indicating that the fiber is in the flow-vorticity plane. The time evolution of ~ in Fig. l(a) shows that for both observations the fiber remains aligned with the flow direction. The fluctations in the lines are due to discretization errors. The angular distributions for the fiber suspensions with n L 3 = 5, 10 and 20 in the 2000 ppm PAAm solution are shown in Fig. l(b), where the population of :~ is plotted vs. the absolute value of ~ accumulated at 5° intervals. The three lines in the figures corresponds to the three different fiber concentrations. The population is plotted in units of percent per degree, so that the sum of the values for each line multiplied by the size of the interval, in this case 5, is 100. The actual total number of data points for each line varies from 511 to 344 as shown in Table 2. All of these distributions have a sharp peak at levi= 90 °. This is the angle of alignment of a fiber in the single fiber experiments, and it can be seen that the orientation distribution is directly

140

Y. lso et al../ J. Non-Newtonian Fluid Mech. 62 (1996) 135-153 90 ¸ ~ - -

60. 30. O. -30-60. -90" 0

a0~0

~o time

6oo0

8000

(sec)

15

cO

o

¢= O.

10

c tO CO

-5

O. 0

0.

0

30

60

90

lal Fig. 1. (a) Two observations of the orientation angle c~ of a single fiber in 2000 ppm P A A m solutions. (b) Population distribution of [ctI for fiber suspensions in 2000 ppm PAAm solution: nL 3 = 5, O; nL 3 = 10, ©; nL 3 = 20, -

governed by the behavior of the single fiber. Furthermore, the distributions for the different fiber concentrations are remarkably similar indicating that the effect o f fiber concentration is negligible. This is not surprising in view o f the fact that a slender fiber aligned with the flow direction in a Newtonian fluid is known to give rise to a fluid velocity disturbance that is small and of O(1/r). The similarity of the distributions for different fiber concentrations is also evident in the second moments of the distributions and the values o f the order parameters (P = 2 < cos 2 ct > - 1) that are shown in Table 2. The value of P gives then an indication of the width o f the peak in the flow direction. This value remains around - 0 . 8 for all three fiber concentrations despite a four-fold increase in fiber concentration.

3.2. The 1000 ppm PAAm fluid As before, two experiments were run with a single fiber to determine the fiber dynamics at a shear rate of 0.5 s - ' . In one experiment, the fiber aligned along the flow axis at the inception of flow as in the case of the 2000 ppm P A A m solution, but, in the other, the fiber ended in a stationary orientation at ~ = - 7 0 ° (Fig. 2(a)). The projection angle ~ remained constant at 90 °

Y. lso et al. : J. Non-Newtonian Fluid Mech. 62 (1996) 135- 153

141

except when the fiber was drifting from ~t = 90 ° to ~ = - 7 0 ° in the second experiment. In both experiments, the fiber ended in the flow-vorticity plane although in one case it was aligned along the flow direction and in the other it was oriented 20 ° away from that direction. The distributions of the angle ~ for the suspensions with nL~ = 1, 5, 10 and 20 are shown in Fig. 2(b). All these distributions have two distinct peaks, one in the flow direction at I~1 = 90 ° and a smaller one at I~1 = 70 °. These peaks are found at the same angles as the two stable orientations observed in the single fiber experiments. Also, the distributions are very similar for all four fiber concentrations. The m o m e n t s of these distributions are reported in Table 2. In the case of nLa= 10, transient distributions were recorded and the data for several transients were analyzed using multiple tracer fibers [9]. The results show that the early distributions had only one peak in the flow direction and the second peak at I~1 = 70 ° observed at steady state was absent. These distributions were obtained after the suspension was subjected to a total strain of O(1) allowing the fibers to rotate from their initial orientation to the flow-vorticity plane. These early distributions are consistent with the behavior one would expect if the initial rotation into the flow-vorticity plane followed a fraction of a Jeffery orbit. To achieve steady-state conditions, some fibers must then change their initial orientation, moving away from the flow direction and into an orientation around I~1 = 70 °. In the highly elastic regime considered in this section, the orientation distributions of the fiber suspensions are strongly correlated with the behavior of individual fibers in response to the imposed shear flow. This leads to to orientation distributions with fairly sharp peaks at the preferred orientation of the single fiber. The effect of hydrodynamic interactions between fibers on the orientation distribution is negligible as the features of the distribution remain essentially Table 2 M o m e n t s of fiber o r i e n t a t i o n distribution

nL ~ m

D a t a points

P t = 2 < cos-' ~ > - 1 )

_

2000 ppm P A A m solutions 5 10 20

511 428 344

-0.78 - (1.82 - 0.78

1000 ppm P A A m solutions 1 5 10 20

1711 I 150 1670 1110

-0.70 - 0.78 -0.70 -I).88

500 ppm P A A m solutions 5 10

327 337

0.44 0.24

200 ppm P A A m solutions 5 10 20

234 301 375

0.52 0.36 0.32

142

Y. lso et al. ,' J. N o n - N e w t o n i a n Fluid Mech. 62 (1996) 1 3 5 - 1 5 3 906030 0 -30 -60 -900

20'00

4o'00

60'00

8000

time (sec)

15

o c o o

10 ctO

0 Q. P.

30

60

90

Fig. 2. (a) Two observations of the orientation angle c~ of a single fiber in 1000 p p m P A A m solutions• (b) Population distribution of I~1 for fiber suspensions in 1000 p p m P A A m solution: nL 3 = I, ,: :; n L ~ = 5, ~ : n L ~ = 10, ~ : n L ~ = 20, •'^',.

unchanged over a fairly wide range of fiber concentrations. Observations of tracer fibers in suspensions made with these highly elastic fluids show that the fiber does indeed remain in the flow-vorticity plane. Although the tracer fiber remained aligned at a fixed orientation, occasional excursions away from this fixed orientation were observed at the highest n L 3 [9]. However, the number of these trace observations was too small to be statistically significant and we suspect that the excursions away from the fixed orientation are rare at all n L 3 considered here since they do not appreciably affect the orientation distribution.

4. Orientation in moderately elastic fluids

4.1. The 500 ppm P A A m fluid Single fiber dynamics in the 500 ppm PAAm solutions were monitored in two experiments and the results for ~ are shown in Fig. 3(a). The behavior here was different from that in the highly

Y. Ls'o et al. ,: J. N o n - N e w t o n i a n H u i d Mech. 62 (1996) 135

143

153

elastic fluids with higher PAAm concentrations. In one experiment, the fiber moved into

a

s t a t i o n a r y o r i e n t a t i o n at :~ = - 4 0 ° a n d 7 = 90 ° i n d i c a t i n g t h a t t h e f i b e r h a d m o v e d i n t o t h e

flow --vorticity plane and was oriented at an angle 40 ° away from the vorticity. The fiber remained in this orientation till the end of the 2 h experiment. In the second experiment, the fiber did not stay in a stationary alignment but moved gradually toward an oscillatory motion about the vorticity axis. Although the fiber's orientation oscillated with time, the period of oscillation of about 350 s was much longer than the period (176 s) of the Jeffery orbits in a Newtonian fluid. Furthermore, the fiber's orientation remained confined with the flow-vorticity plane as may be seen from the plot of the azimuthal angle ;' in Fig. 3(b). Both of these observations are in sharp contrast to the alignment along the flow direction observed in the highly elastic 2000 ppm PAAm solution.

9060-

300-30-60-

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2obo

4o~

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I~1 (c) Fig. 3. (a) Two observations of the orientation angle ~ of a single fiber in 500 ppm PAArn solutions. (b) Orientation angle ;, of a single fiber in 500 ppm PAAm solution. (c) Population distribution of I:~Ifor fiber suspensions in 500 ppm PAAm solution: n L ~= 5, • : n L ~ = 1 0 . ~>:nL ~ = 20, : '

144

Y. lso et a l . / J. N o n - N e w t o n i a n Fluid M e c h . 62 (1996) 1 3 5 - 1 5 3 90-

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8000

time (sec)

2.5

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2

i 1.5

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0.5

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60

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Fig. 4. (a) Two observations of the orientation angle ~ of a single fiber in 200 ppm PAAm solutions. (b) Population distribution of letI for fiber suspensions in 200 ppm PAAm solution: n L 3 = 5, ",.): n L 3 = 10, ~ ; n L ~ = 20, - .

The orientation distributions in the fiber suspensions with n L 3 = 5, 10 and 20 in this fluid are shown in Fig. 3(c). They are quite different from those in the highly elastic fluids. The population distributions have been compiled in this case in angular intervals of 10 ° instead of the 5 ° intervals used previously in order to decrease the noise in these broad distributions. The distribution for n L 3 5 has a broad and nearly flat peak near the vorticity direction with most of the fibers confined between ]~1 = 0° and I~l = 60 °. As n L 3 is increased to 10 and 20, the peak shifts away from the vorticity direction and moves to larger values of ]~], thus creating a population depletion of fibers oriented along the vorticity. The effect of hydrodynamic interactions between fibers on the distribution is clearly noticeable in this case and as n L ~ increases it pulls fibers away from the vorticity direction. =

4.2. The 200 p p m P A A m f l u i d

Observations from two single fiber experiments in this case are presented in Fig. 4(a). Both led to a stationary alignment of the fiber with ~ = - 4 0 ° and 7 = 90 °. The fiber lies again in the flow-vorticity plane and is oriented at a 40 ° angle away from the vorticity as was observed in one of the two single fiber experiments in the 500 p p m P A A m fluid.

Y. Iso eta/..; J. Non-Newtonian Fluid Mech. 62 (/996) /35-153

145

The distributions for fiber concentrations of n L 3 = 5 , 10 and 20 in this fluid are shown in Fig. 4(b). The distributions all have a broad flat peak at the vorticity direction with most o f the fibers again confined between [~1 = 0° and = 60°. it is seen from the figure and the values of the moments in Table 2 that the distribution becomes broader as the concentration increases from n L ~ = 5 to n L 3 = 20. The effect of hydrodynamic interactions on the distribution in this case is very similar to that observed in the weakly elastic case (100 ppm P A A m solution in Part I) which we are now approaching. We have monitored the effect of fiber-fiber interactions in these moderately elastic fluids by continuous observation of a single tracer fiber in the suspension. The results obtained in this case are quite different from those in the highly elastic fluids. The time evolution of :~ for a tracer fiber is shown in Fig. 5 for the 200 ppm PAAm solution with fiber concentrations of n L 3 = 5, 10 and 20. The time evolution of the azimuthal angle 1' for n L ~ = 5 and 20 in the 200 ppm fluid are shown in Fig. 6(a) and 6(b) respectively. In the case of the fiber suspension of n L ~ = 5, the fiber is seen to be frequently disturbed out of the preferred orientation at :~ = - 4 0 °, causing excursions of the angle ~ from - 4 0 ° to 0 ° or beyond. Thus the motion of :~ is not symmetric with respect to the vorticity axis, and the fiber is seen to spend most of its time between c~= - 4 0 ° and • - - 0 °. Due to the symmetry of the flow, we expect that a fiber would also fluctuate between ~ = 0 ° and .~ = 40 °. For n L 3 = 2 0 , the amplitude of oscillations of the angle :t becomes large so that the fiber is now oscillating symmetrically around the vorticity axis with an

(a)

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time (sec) Fig. 5. T w o o b s e r v a t i o n s o f the o r i e n t a t i o n angle ct o f a single tracer fiber in fiber s u s p e n s i o n s in 2000 p p m P A A m solution: (a) n L ~ = 5, (b) nL 3 = 10: (c) nL ~ = 20.

Y. Lvo et al. / J. Non-Newtonian Fluid Mech. 62 (1996) 135- 153

146

90

90

80

80

70

70

60

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50

40

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8O 7O 6O 50

40 3O 20

10 i

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2000

4000

l~rne

(c)

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J

8000

(see)

Fig. 6. Orientation angle ;' of a single tracer fiber in a fiber suspension of nL ~ = 5 in 200 ppm PAAm solution. (b) nL ~ = 20 in 200 ppm PAAm solutions. (c) nL ~ = 20 in 500 ppm PAAm solution.

amplitude o f about 60 °. In such cases, the fiber frequently moves out of the flow-vorticity plane as it flips and crosses the gradient-vorticity plane. Fig. 6(b) for ;, in this case shows that excursions out o f the flow-vorticity plane are n o w more numerous and much larger. For the intermediate concentration o f n L ~ = 10, the fiber is seen to approach the stable orientation of :( = 40 °, but again fiber-fiber interactions move the fiber out of its alignment (Fig. 5(b)). An interesting change of the orientation is seen in the first observation after 3000 s, when the fiber moves away from the alignment at ~ = 40 ° and fluctuates around the vorticity axis. The results for ~ for the 500 ppm P A A m solution are shown in Fig. 7 and are in many ways similar to those for the 200 ppm solution with some expected differences. We highlight the differences here by comparing the behaviour o f ~}, for n L 3 20 for the 200 ppm and the 500 ppm solutions in Fig. 6(b) and 6(c) respectively. Because of the higher elasticity of the 500 ppm solution, the excursions away from the flow-vorticity plane are much smaller in magnitude. Although the amplitude o f the oscillations in ~ for n L 3 20 suspensions with 200 and 500 ppm P A A m are similar, the fiber moves out of the flow-vorticity plane much more often in the 200 ppm fluid (see Fig. 6). Consistent with this observation, we find that the period of oscillation in the 200 ppm fluid is much shorter than in the 500 ppm fluid and close to that observed in a Newtonian fluid or predicted by Jeffery (see Table 3). ---

=

147

Y. /so et al. / J. N o n - N e w t o n i a n Fluid M e c h . 62 (1996) 1 3 5 - 1 5 3

(a)

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Orientation angle x of a single tracer fiber in a fiber suspension in 500 ppm PAAm solution: (a) n L ~ = 5: ( b )

n L ~ = 10; (c) n L ~ = 20.

5. Discussion and conclusions

5.1. Dynamics of a single fiber Although the number of experiments performed on isolated fibers in fluids with varying elasticity was necessarily small because of the long time needed to perform each experiment, a consistent picture emerges regarding the effect of elasticity on the final orientation of a single fiber in an elastic fluid. In all our experimental results, summarized in Table 4, we note that a fiber becomes confined to the flow-vorticity plane. The orientation of the fiber within this plane Table 3 Approximate period of fluctuation (s) PAAm

100 ppm 200 ppm 500 ppm

nL ~ 5

10

20

200 260 370

160 150 360

160 150 340

148

Y. lso et a l . / J. N o n - N e w t o n i a n Fluid M e c h . 62 (1996) 1 3 5 - 1 5 3

Table 4 S u m m a r y of orientation b e h a v i o r Fluid

Single fiber

Tracer fiber

Orientation distribution nL 3 = 5

100 p p m PAAm 200 p p m PAAm

Spirals toward vorticity (~ = 0 °) Confined to flow-.vorticity plane. Fixed point at ~ = 4 0 °

500 p p m PAAm 1000 p p m PAAm

Fixed point at ~ = 40 ° Two fixed points at :~ = 70 ° and 90 °

2000 p p m PAAm

One fixed point at c~ = 90 ° (flow direction)

Orientation distribution n L 3 = 20

Peaks at c~ = 70 ° a n d 90 °

Broad peak at flow (similar to Newtonian) Broad peak at vorticity (similar to n L 3 = 5 in 100 p p m fluid) Strong peak at = 40 ° Peaks at = 70 ° and 90 °

Sharp peak at ~ = 90 ° (flow direction)

Sharp peak at = 90 ° (flow direction)

Oscillates with Jeffery period

Broad peak at vorticity

Jeffery period at n L ~ = 20. Longer period at n L 3 = 5

Peak at x = 3 0 °

Oscillates with longer period Occasional excursions from fixed points Occasional excursions from fixed point

Peak at ~ = 30 °

relative to the vorticity direction was a function of the fluid elasticity as measured by the ratio of the elastic modulus to the loss modulus of the fluid, I = G'/G". The fiber spiraled slowly towards the vorticity axis in the weakly elastic fluids with I = 0.02-0.07 studied in Part 1. In contrast, the fiber rapidly aligned with the flow direction in highly elastic 2000 ppm P A A m solution, which had I = 0.55. (See Table 1 of Part I for a summary of the I values for the polymer solutions.) At intermediate values of the fluid elasticity, the fiber orientation was intermediate between the flow and vorticity directions. In the 1000 ppm solution with I = 0.46, one experiment led to an orientation along the flow direction while another led to a stable orientation 20 ° away from the flow direction. In the moderately elastic fluids of 200 ppm and 500 ppm P A A m solutions with I = 0.18 and 0.26 respectively, the stable orientation was at 50 ° away from the flow direction, i.e. 40 ° from the vorticity direction. Putting these results in perspective, we found that Bartram et al. [1] were the first to report an alignment along the flow axis in an elastic fluid. They used a polyethylene fiber in a 2.5% P A A m aqueous solution. The fluid was sheared in a Couette device at different shear rates. For shear rates lower than 5 s ', fibers with aspect ratios r = 3.6 and 9.0 were seen to spiral toward the vorticity axis. However, at higher shear rates (exact values not reported), single fibers with aspect ratio 9.0 were seen to align along the flow direction if the fibers were initially placed in orbit close the the flow-gradient plane. The length of time the fiber was observed to be aligned along the flow direction was not indicated. Fibers starting from other orbits spiraled toward the vorticity. Based on the relaxation time ~ l / 2 r / w h e r e ~ is the first normal stress coefficient and r/is the shear viscosity, the Deborah number (De) of this fluid at shear rates 1 s -1 and 10 s

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149

would be 1.13 and 1.6 respectively [1]. These values are comparable to the Deborah numbers in the P I B - P B (polybutene) fluid experiments of Part 1 and would place this fluid in between our highly and our moderately elastic fluids. The dynamics of a single fiber reported for this fluid are more akin to those observed in the weakly elastic regime or just slightly above it, as in the P I B - P B fluid. As noted in Part 1, the orientation behavior of single fibers is not well correlated with De but is better correlated with I = G'/G" which unfortunatley was not reported in Ref. [1]. There is some precedent for intermediate fiber orientations such as the fixed point 40 ° away from the vorticity observed in our moderately elastic fluid in the light scattering study of Johnson et al. [7]. Their experiments were conducted using a dilute suspension of hardened prolate blood cells with aspect ratios between 2.66 and 5.82 using both Couette and parallel plate cells. The suspending fluid was a Boger fluid consisting of a polybutene solution with 1000 ppm PIB. The concentrations of the blood cells were intentionally kept low so as to minimize particle-particle interactions. Therefore, their results are indicative of non-interacting particles and should be compared to our single fiber results. At low shear rates, the cells were aligned with the vorticity axis. However, at the high shear rate of 27.5 s - 1 the light scattering pattern indicated that the orientation distribution had a peak in between the flow and vorticity directions. They attributed the peak in the vorticity direction at low shear rates to the spiraling motion c o m m o n in weakly elastic fluids. The exact nature of the orientation distribution at high shear rates was not clarified due to limitations of the measurement method [16]. In view of the present experiments, it is likely that Johnson's particles under high shear rate were aligned in a fashion similar to the alignment of ~ = + 40 ° observed in our 200 and 500 ppm PAAm solutions as well as in the PIB solutions of Part I, which is very similar in composition to Johnson's fluid. Theories based on the assumption that the polymer stress is small compared with the stress associated with the Newtonian solvent [5,6] predict that a fiber will be confined to the flow-vorticity plane above a critical value of the polymer stress. This occurs when the rotation of the fiber out of the plane predicted by Jeffery's solution for Newtonian fluids is balanced by an extra rotation caused by the polymeric stress. These predictions are consistent with our experimental observation that a single fiber is confined to the flow-vorticity plane when the polymer concentration is 200 ppm or larger, but rotates out of the plane in 100 ppm PAAm solutions. However, the weak elasticity theories do not predict the motion of the fiber within the flow vorticity plane that is observed in our experiments. The theories indicate that the extra rotation within the flow-vorticity plane due to the polymer stress is always less than the Jeffery rotation in that plane, when the polymer concentration is sufficiently small to validate the theories. Thus, for weakly elastic fluids, the vorticity axis is predicted to be the only fixed orientation. It is interesting to note, however, that the small extra rotation predicted by Harlen and Koch [6] is toward the flow direction; thus, one might speculate that at higher polymer concentrations a fixed point at the flow direction could be obtained. 5.2. Orientation distributions in .fiber suspensions

Fibers in a suspension will impose a hydrodynamic disturbance to the flow field that will affect other fibers. Such fiber-fiber interactions, together with the stress acting on the fiber due to the average flow field, govern the orientation distribution in a semi-dilute fiber suspension. For the fiber suspensions in viscoelastic fluids studied in this paper, the exact form of the

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fiber-fiber interactions is not known. Nevertheless, fiber-fiber interactions, together with the Newtonian and non-Newtonian stresses, will govern the orientation distribution of the suspension. The continuous observations of single tracer fibers, although limited by necessity to a few experiments in each case, do provide important clues as to the reasons for the major features of the orientation distributions. In the highly elastic fluids of 2000 and 1000 ppm of PAAm, there was a direct correspondence between the orientation of a single fiber in the polymer solution and the orientation distribution in the suspension. The location of the peaks of the orientation distribution in the suspension matched the fixed orientations of the single fiber. Since all the fibers are in the flow-vorticity plane, they should cause a minimal hydrodynamic disturbance of the imposed flow field, meaning that little fiber-fiber interaction is expected. Hence, little effect of fiber concentration on the orientation distribution will ensue as is indeed observed. The few continuous observations of the individual tracer fibers in these suspensions show that the fibers appear to be more susceptible to disturbance from their stable fixed orientation at the highest fiber concentration examined [9]. The orientation distributions of the suspensions show, however, that these disturbances must be too infrequent to modify the features of the distribution (Figs. l(b) and 2(b)). In these highly elastic fluids, the viscoelastic stresses dominate over all other factors in governing the orientation of the single particle as well as the orientation distributions of the suspensions. The orientation distributions for the fiber suspensions in the 200 and 500 ppm PAAm solution exhibited more complex features. Each of the distributions for nL3 = 10 and 20 for the 500 ppm PAAm solutions had a peak between the vorticity and the flow direction reminiscent of the preferred orientation of I~] = 40 ° seen in the single fiber experiments in that fluid. The other suspensions, n L 3 = 5 in the 500 ppm solution and n L 3 = 5 , 10 and 20 in the 200 ppm solution had broad flat peaks near the vorticity. We will consider these two sets of distributions separately and present possible mechanisms that bring about these distributions on the basis of the motion of tracer fibers (Figs. 5 and 7) in these suspensions. The four fiber suspension distributions that have a broad peak in the vorticity direction are compared in Fig. 8. It is readily noted that the distribution for n L 3 = 5 in 200 ppm solution and n L 3 = 5 in 500 ppm solution are very similar with a more pronounced peak near the vorticity axis than the other distributions. The motion of the tracer for these two suspensions is very similar: the fiber is fluctuating within a region which is roughly bounded by one of the preferred orientations, ~ = _+40°, and the vorticity direction (Figs. 5(a) and 7(a)). Fluctuations of this type will create a distribution with a peak between the preferred orientation and the vorticity direction as shown schematically in Fig. 9. Due to symmetry, fibers will fluctuate with equal probability on either side of the vorticity axis. Superposition of these two curves will create a distribution with a single plateau around the vorticity direction with a slight depression at the vorticity direction. This distribution is similar to those observed for the two suspensions with n L ~ = 5 as may be seen by comparing Figs. 8 and 9. The fluctations of the fiber out of the stable orientation at ]~] = 40 ° may be attributed to fiber--fiber interactions. For the fiber suspensions with n L 3 = 10 and 20 in the 200 ppm PAAm solution, the motion of a single tracer (Fig. 5) is qualitatively different from that observed at n L 3 = 5. One obvious difference is the frequency of the fluctuations. The half period of the oscillations for all the single tracer experiments was obtained by counting the number of excursions from a stable orientation and dividing the time by the number of excursions. Although this method will not be able to

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Overall distribution Ructuation t..

I

Ructuatlon

-

0

40

.-t Q. 0 Q.

I

-90

0

90

Q

Fig. 8. Schematic representation of the cau~ of a dip of the orientation distribution in the vorticity direction for ~= 5 in 200 and 500 ppm solutions.

nL

pick up fluctuations with periods shorter than the sampling time interval of 30 s, the results are interesting and are listed in Table 3. We note that the periods of the fluctuations in the n L 3 = 10 and 20 suspensions in the 200 ppm PAAm solution are roughly half as long as those in the other suspensions listed for the 200 and 500 ppm fluids. These short periods are comparable with the periods observed using the same fiber concentration in a 100 ppm fluid. They are also close to the Jeffery period (176 s) for the rotation of a single fiber in a Newtonian fluid. It is not surprising that the period for the suspensions made using the 100 ppm fluid would be comparable with the Jeffery period, because a single fiber is observed to spiral through a sequence of approximate Jeffery orbits in this fluid [4]. A single fiber in the 200 ppm fluid is confined to the flow-vorticity plane. However, it may be seen in Fig. 6 that fiber interactions 2.5

® C

¢x

¢

1.5

tO

ea

1

0

O.

0.5

0

25

50

75

Fig. 9. Comparison of orientation distributions: n L 3 = 5 in 500 ppm, :?: ppm, L; n L 3 = 20 in 200 ppm, --" n L 3 = 5 in 100 ppm, 7.

100

nL ~ =

5 in 200 ppm, ©;

nL ~ =

10 in 200

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lead to much greater tendency for a fiber to rotate out of this plane as the fiber concentration is increased from n L 3 = 5 to 20 in the 200 ppm fluid. The Newtonian rotation increases relative to the rotation due to non-Newtonian stresses as the projection of the fiber orientation in the gradient direction is increased; this may be seen from Eqs. (4)-(6) of Part 1. Thus, the suspensions with n L 3= 10 and 20 in the 200 ppm fluid have distributions very similar to that observed in the weakly elastic, 100 ppm fluid with n L 3 = 5. On the other hand, the n L 3= 5 suspension in the 200 ppm fluid where the fiber remains close to the flow vorticity plane has a distribution similar to that observed at the same fiber concentration in the more elastic 500 ppm fluid. The effect of increasing fiber concentration on the orientation distribution is quite different in the 500 ppm fluid from that in the 200 ppm fluid described above. Even at the highest concentration n L 3 = 20, the fiber remains within the flow-vorticity plane most of the time (see Fig. 6(c)). Thus, the behavior does not resemble that of lower elasticity fluids. Instead, the fiber interactions induce a rocking motion within the flow-vorticity plane between the two preferred orientations at ~ = _+40° (see Fig. 7). Since the fiber spends a relatively long time near the preferred orientation before rocking back to the other side, the orientation distributions shown in Fig. 3(c) exhibit peaks near 40 °. We have seen that the fiber orientation distributions in highly elastic polymer solutions exhibit a more complex set of behaviors (summarized in Table 4) than can be predicted from the current theoretical work [5,6]. The orientation distributions for fiber suspensions in Newtonian fluids are peaked near the flow direction [8], because this is the region where fibers in most of the Jeffery orbits spend a majority of their time. As seen in Part 1, the addition of a small concentration of polymer leads to a competition between the spiraling toward the vorticity axis predicted from weak elasticity theories and the randomization of fiber orientation due to hydrodynamic interactions. Thus, in this weakly elastic regime, increasing fiber concentration moves the orientation distribution towards the flow direction and increasing polymer concentration moves the peak towards the vorticity direction. However, in the highly elastic 2000 ppm fluid studied here, the polymer stress acts to confine the fiber to the flow direction. Although the peak of the orientation distribution is in the flow direction for both the 2000 ppm solution and the Newtonian fluid, the width of the peak is much smaller in the 2000 ppm fluid. At intermediate polymer concentrations, one sees a more subtle array of phenomena. As one goes from 2000 to 1000 ppm, the fixed orientation for a single fiber at the flow direction remains but a second fixed orientation at :t = 70 ° is also observed and the orientation distribution is bimodal. At 500 and 200 ppm, the only fixed orientation observed in the single fiber experiments is near ~ = 40 °. Finally, in the 200 ppm fluid, we see evidence for a competition between elastic stresses, which tend to keep the fiber orientation within the flow-vorticity plane, and hydrodynamic interaction, which tend to disturb the orientation out of this plane.

Acknowledgments The authors gratefully acknowledge the Cornell Injection Molding Program and the National Science Foundation (grant DDM-9212582) for supporting this study. Y.I. also acknowledges financial support from the Furukawa Electric Co., Ltd. Joseph Kukura, William Weber, Kris

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Korovae, Richard Muisener, Li Kun Tu, William Rieke, James Sorhagen, Bunchan Sombunsakdikun, Miguel Maldonado and Nelson Quim have performed numerous experiments and analyses and their help was invaluable. Dr. John Lawler at Hoechst Celanese Co., Ltd. has helped with the rheological characterization of some of the polymer solutions.

References [1] [2] [3] [3] [4] [5] [6] [7] [8] [9] [10] [I 1] [12] [13] [14] [15] [16]

E. Bartram, H.L. Goldsmith and S.G. Mason, Rheol. Acta, 14 (1975) 776. F. Gauthier, H.L. Goldsmith and S.G. Mason, Rheol. Acta, 10 (1971) 344. C.A. Stover and C. Cohen, Rheol. Acta, 29 (1990) 192. C.A. Stover and C. Cohen, Rheol. Acta, 29 (1990) 192. Y. lso, D.L. Koch and C. Cohen, J. Non-Newtonian Fluid Mech., 62 (1996) 115. L.G. Leal, J. Fluid Mech., 69 (1975) 305. O.G. Harlen and D.L. Koch, J. Fluid Mech., 252 (1993) 187. S.J. Johnson, A.J. Salem and G.C. Fuller, J. Non-Newtonian Fluid Mech., 34 (1990) 89. C.A. Stover, C. Cohen and D.L. Koch, J. Fluid Mech., 238 (1992) 277. Y. Iso, Orientation in simple shear flow of semi-dilute fiber suspensions in viscoelastic liquids, Ph.D. Thesis, Cornell University, 1994. G. Williams and D.C. Watts, Trans. Faraday Sot., 66 (1970) 80. G.D. Patterson, Adv. Polym. Sci., 48 (1983) 126. S.J. Muller, R.G. Larson and E.S.G. Shaqfeh, Rheol. Acta, 28 (1989) 499. R.G. Larson, E.S.G. Shaqfeh and S.J. Muller, J. Fluid Mech., 218 (1990) 573. E.S.G. Shaqfeh, S.J. Muller and R.G. Larson, J. Fluid Mech., 235 (1992) 285. S.J. Muller, E.S.G. Shaqfeh and R.G. Larson, J. Non-Newtonian Fluid Mech., 46 (1993) 315. A.J. Salem, Small angle light scattering as a probe of flow-induced particle alignment in non-Newtonian suspending fluids, Ph.D. Thesis, Stanford University. 1987.