,
using standard procedures described earlier and elsewhere(l). 1
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1
0
2
4
Experimental Procedures y Velocity gradient
L
^ Flow direction
Vorticity vector
ζ Figure 4. The shearflowgeometry between onefixedand one moving parallel plates. The design of shear flow cells for in-situ x-ray scattering must take account of both the three-dimensional nature of the shear flow field and the scattering geometry. Figure 4 shows a schematic of shear flow between parallel plates. The simplest scattering geometry is for the incident beam to lie parallel to the velocity gradient (VV) using relatively x-ray transparent material such as mica for the parallel plates. In this situation, the scattering vector Q is tilted out of the plane containing the flow vector ν and the vorticity vector. In fact, Q lies on the surface of a cone with a semiangle of 90-Θ about the incident beam as shown in Figure 5. Thus, for small values of Θ, the x-ray beam probes the structure in the plane defined by ν and the vorticity vector. However, when peaks in the wide-angle scattering regime are of interest, such as from semi-crystalline polymers, this out-of-plane tilt may make interpretation of
Cebe et al.; Scattering from Polymers ACS Symposium Series; American Chemical Society: Washington, DC, 1999.
394
the scattering with respect to the flow geometry a rather complex matter. For the materials studied here 2Θ ~ 5° and hence the x-ray scattering effectively probes the structure in a plane parallel to the parallel plates.
Flow direction
2
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]
1
Scattered beams
Figure 5. The relationship between the scattering vector, Q, and theflowgeometry
We utilize the azimuthual distribution of intensity I (a) to evaluate the level of preferred orientation. Each point on the circle, at constant 2Θ value but with varying a, corresponds to a specific orientation of the scattering vector Q. Inspection of Figure 5 shows that as the scattering vector is slightly tilted out of the plane, it is no longer possible to access the situation where the angle between the flow direction and Q is zero. This is a basic problem with fixed flat detectors such as image plates, film and CCD based detectors. For small scattering angles, as is the case here, the missing data can be safely ignored. However, in many cases, this issue must be addressed if reliable orientational data is to be obtained. Figure 6 shows a schematic of a shear flow x-ray cell developed at the University of Reading to facilitate the geometry described above (7). The system consists of two stainless steel parallel plates; one rotating and one fixed in which the separation is defined with a phosphor-bronze ring. The rotating plate has slotted windows covered in mica which allow transmission of the incident beam for ~ 90% of each revolution. The fixed plate has a single aperture also covered with a mica sheet which allows the scattered intensity to exit the shear cell for scattering angles < 40 degrees. Mica exhibits modest absorption at wavelengths ~ 1.5À, but has the particular advantage that the scattering from the mica is localized in single crystal type diffraction spots. These can be easily identified in the patterns. If they arise from the window on the moving plate, the intensity in the region of the diffraction spots must be obtained by extrapolation. It is not sufficient to subtract a "empty cell", since the diffraction spots will rotate with the window. The parallel plate arrangement results in a linear variation of the shear rate across the radius of the shear cell. The shear rate resolution of the x-ray scattering procedures, depends upon the effective beam size to the plate radius at the incident beam. In this cell, that particular radius is - 9mm and with a laboratory source the shear rate
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resolution is ~ 3% while with the focused beam available at a synchrotron source, the resolution is - 1%. The rotating plate is driven by a stepper motor and encoder system and provides for continuous, stepped and oscillatory flow fields. In previous work on lyotropic liquid crystal polymers (8) we utilized a similar arrangement in which the plate radius was ~ 32mm with an accompanying greater shear rate resolution. Similar rotating shear cells have been used by Romo-Uribe et al (9) and by Hongladarom et al (10). Picken developed an oscillating linear motion shear cell, which has advantages in terms of a uniform shear rate and single global flow direction, although there is an obvious limitation in the shear strain which can be applied before reversing the direction of flow (11).
Figure 6. Schematic of the shearflowcell used for x-ray rheology studies. Alternative geometries, which evaluate the structure in the planes defined by either y and VV, or W and the vorticity vector, are possible but are not so convenient to implement, particularly in the wide-angle scattering regime. One approach is to use a Couette cell as shown in Figure 7. In this geometry the sample is sandwiched between a fixed inner cylinder and a rotating outer cylinder. If the incident beam is directed along the centre line (Figure 7), the scattering probes the structure in the plane defined by y and the vorticity vector in a similar manner to the parallel plate geometry. However, if the incident beam is located at the tangent to the midpoint between the cylinders as shown in Figure 7, the scattering probes the plane defined by VV and the vorticity vector. Such an arrangement can be used to probe the three dimensional nature of the molecular organization under flow. For example it can confirm or exclude the possibility of a near uniaxial preferred orientation at high shear rates. We have developed such a cell for the study of structured melts in the wide-angle regime (12). The tangential configuration has the complexity of
Cebe et al.; Scattering from Polymers ACS Symposium Series; American Chemical Society: Washington, DC, 1999.
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anisotropic absorption corrections, if quantitative evaluation of the intensity data is to be made (12). We have also explored the use of a parallel plate system in which the incident beam lies in the plane of the plates (Figure 8) (13). By placing the incident beam along the center line, the scattering vector lies largely in the plane containing the VV and v, whilst directing the incident beam along the tangential line results in the scattering vector lying in the plane defined by V V and the vorticity vector as shown in Figure 8. This geometry has the particular advantage that by placing the incident beam at intervals between these two limiting configurations the complete three-dimensional nature of the molecular organization can be evaluated. However, a major disadvantage is one of absorption, either in terms of the path length or the anisotropy of the absorption correction as with the Couette cell.
rotating cylinder Figure 7. Theflowand scattering geometry for a Couette styleflowcell In this contribution, we focus on results obtained using the parallel plate geometry shown in Figure 6 in which the structure probed is that which lies in the plane defined by the flow vector and the vorticity vector. To record the x-ray scattering patterns in a time-resolving manner we have utilized a Photonic Science CCD based detector system coupled with a Data Translation frame grabber system. The detector was part of a larger integrated system which facilitated specific programmed sequence of flow and synchronized data collection (14). This enables data collection cycles of less than Is, although in this work, cycle times of ~ 10s were employed. The detector has an active face of ~ 50mm in diameter which allows the full azimuthual dependence of the scattering to be obtained with the incident beam centered on the detector face, although of course the IQI range was somewhat limited. In this work we only recorded the diffuse peak at IQI ~ 0.4 Â" . Some steady state measurements were made using a laboratory based x-ray source; namely a Cu targeted sealed tube running at 1.6kW with an incident beam monochromator and pinhole collimation. Time-resolving measurements exploited the intense x-ray beam available at the beamline 16.1 of the CCLRC Daresbury Synchrotron. This is a fixed wavelength beam-line mounted on a 6T wiggler with a bent Germanium focusing monochromator and a Quartz mirror. 1
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397
Figure 8. An alternativeflowcell geometry to explore the three-dimensional consequences of theflowfield
Results
Shear
Relaxation
0.30 0.25 0.20 0.15
1
0.10 0.05 0.00
4000
2000
0
t i m e /s
Figure 9 Values of
obtained for a sample of PPC at 65 °C and at particular times after the imposition of a flow field with a shear rate of 2.5s' and after the cessation of the flow field as indicate in the figure. 2
1
Figures 9 and 10 show the results from one experiment on PPC deep in the thermotropic phase at 65°C. A steady state shear flow of 2.5s" is applied and the level of global preferred orientation is evaluated from a series of x-ray scattering patterns recorded with a cycle time of ~15s over a period of ~ 1000s. Using the azimuthual variation of intensity of the diffuse peak at IQI ~ 0.4 A , values of the global orientation parameters
and
are obtained as a function of time. A value of
,
= 0 indicates a completely isotropic state, while
,
= 1 indicates a fully aligned system. A nematic liquid crystal systems exhibits microscopic order 1
1
2
2
4
4
2
4
Cebe et al.; Scattering from Polymers ACS Symposium Series; American Chemical Society: Washington, DC, 1999.
398 parameters in the range of 0.3 to 0.7 and hence if there is a single director alignment, the global value should not exceed those values. As Figures 9 and 10 show, a steady level of preferred orientation is achieved within ~ 100s and this is maintained over some 1000s. In general it is found that a steady state is reached within 100-200 strain units (i.e. shear rate χ time) in line with mechanical rheological measurements. At a particular point, the shear flow is halted and the state of preferred orientation is monitored as a function of time following the cessation shear flow. The level of preferred orientation drops rapidly with time, exhibiting a single relaxation time behavior with τ ~ 100s. The values of
which are obtained during steady state shear are ~ 1/3 of the equivalent
values in line with simple models of nematic ordering.
and
represent two components of the orientation distribution and care must be taken not to draw specific conclusions from a single component. However, it is interesting to note that the value of
changes sign during the relaxation stage, possibly reflecting a change in the shape of the orientation distribution. The availability of higher order orientational parameters allows for the reconstruction of the orientational distribution (1) and for a more sensitive test of molecular theories of liquid crystal flow. Although we have made use of a uniaxial description of preferred orientation, we do not know how correct it is to make such an assumption about the molecular organization. To answer such issues we must make use of the alternative flow/scattering geometries shown in Figures 7 & 8. Preliminary measurements using the geometry shown in Figure 8 and lyotropic solutions of HPC suggest that the preferred orientation is uniaxial at high shear rates in that similar values of
are recorded when the orthogonal plane is evaluated (13). Figure 9 Values of
obtained for s sample of PPC at 65°C and at particular times after the imposition of a flow field with a shear rate of 2.5s" and after the cessation of the flow field as indicate in the figure 4
2
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2
4
4
2
2
1
0.15
ι
Shear
Relaxation
«.10 0.05 -J. 0.00 -0.05 -0.10 · 2000
time /s Figure 10 Values of
obtained for a sample of PPC at 65°C and at particular times after the imposition of a flowfieldwith a shear rate of 2.5s' and after the cessation of the flowfieldas indicate in the figure. 4
1
Cebe et al.; Scattering from Polymers ACS Symposium Series; American Chemical Society: Washington, DC, 1999.
399 By performing a series of measurements as in Figures 9 & 10 but for different shear rates and differing temperatures we can build up a map of the flow behavior. Figure 11 shows a plot of the values of
obtained in steady state flow for the shear rates shown at temperatures of 65, 85 and 110°C. Note that the shear rate is plotted on a logarithmic scale.
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2
Figure 11 Valuesof
obtained for PPC at 65 °C ( O), 85°C ( Φ) and 110°C ( V) during steady state shear 2
0.6 A
Q_ V
0.3 A
0.1
10
100
1000
1
Shear rate /s"
Figure 12 Values of
obtained for a 55% w/v aqueous solution of HPC at 25°C during steady state shear flow ( 16) 2
Cebe et al.; Scattering from Polymers ACS Symposium Series; American Chemical Society: Washington, DC, 1999.
400 These curves show three regimes. At low shear rates, there is almost no preferred orientation, whilst at high shear rates, there appears to be a plateau level of
~ 0.4. Between these two points there is a more or less linear increase of
with the logarithm of the shear rate. The data sets for three temperatures appear rather similar apart from an offset along the shear rate axis. 2
2
170°C 180°C 185°C 190°C 195°C
0.6
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0.4 V 0.2
é 9
4
0
V
•
i
s
•
V •
0.0 •π—
0.1
1
10 ι
100
shear rate /s
Figure 13 Values of
obtained for HPC in the thermotropic liquid crystal range during steady state shear flow (17). Temperatures corresponding to the different symbols are given in the inset. 2
Such data has similarities with earlier studies on lyotropic aqueous solutions of HPC (15,16). The equivalent orientational behaviour in steady state shear flow for a 55% w/v aqueous HPC solution is shown in Figure 12. These data also show three regimes although the level of preferred orientation at low shear rates is clearly greater for the aqueous solution, and the distinction between regimes 2 & 3 are blurred. Lyotropic HPC and thermotropic PPC have considerably different concentrations of liquid crystal forming molecule (55% and 100%), yet their responses to the flow field are similar. It is emphasized that we can make direct comparisons between these two materials since the derivative PPC and the lyotropic solutions were prepared from the polymer; Klucel Ε which was used to prepare the aqueous solutions. Figure 13 shows the equivalent data obtained for Klucel Ε in the thermotropic phase range (17). The level of preferred orientation is strongly dependent on the temperature. There appears to be no low orientation regime at low shear rates as observed for both the lyotropic HPC and the thermotropic PPC. Since the experiments on thermotropic HPC were performed close to the clearing temperature, the microscopic order parameters will change rapidly with temperature. At high shear
Cebe et al.; Scattering from Polymers ACS Symposium Series; American Chemical Society: Washington, DC, 1999.
401 rates the material may be close to a monodomain texture and hence the recorded orientation parameters
are essentially the microscopic order parameters. Consideration of the
values shown in Figure 13 show that they do indeed reflect the changes expected from simple models of nematic ordering. Inspection of Figures 12 and 13 shows that the high shear rate values of
for lyotropic and thermotropic HPC are similar, it is the low shear rate behaviour which is different. 2
2
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2
We have considered the response to shear flow of three different polydomain textured liquid crystal polymers which have the same molecular length characteristics and might be expected to have similar molecular stiffness. Despite these molecular similarities, the response to shear flow is rather varied. From the viewpoint of mechanical rheology, the lyotropic HPC system differs from the thermotropic HPC and the thermotropic PPC, in that it exhibits two sign changes in the first normal stress difference (18). Such negative first normal stress differences are thought to be associated with the transition from a director tumbling regime to a director-wagging regime. The orientational behaviour of the lyotropic HPC is entirely consistent with such a model. At low shear rates there is extensive tumbling and an increasing level of defects which eventually lead the inhibition of tumbling (19) and the adoption of a flow-aligning regime. Baek et al (18) have speculated that the absence of negative first normal stress differences in thermotropic HPC may be due to increased polymerpolymer interactions which are inevitable in a concentrated system. Director tumbling is predicted from molecular theories of rigid rod systems subject to shear flow and depends critically on the characteristics of the microscopic order parameters S and S . The equation below shows the relationship for the tumbling parameter, β, proposed by Larson (20). 2
4
When β > 1 the system is flow aligning and if β < 1 the system is tumbling. Using the high shear rate experimental orientation parameters as indicative of the microscopic order parameters we derive values of β ~ 0.9 for the lyotropic HPC, β~1.2 for the thermotropic HPC and β~1.04 for a flow aligning low molar mass nematic system (21). The thermotropic PPC system does not seem to approach a monodomain texture at high shear rates, as may be deduced from the relatively low values of
and from the shear history independent relaxation behaviour (5), and hence we have not attempted to calculate a value for β from the data presented. We must treat these values of β with some caution, since we can not be certain that the director distribution is uniform. Moreover, the expression given for β is itself a simplification based on the first two components of a series representing the microscopic orientational order; values of S and S must be significant as indeed the x-ray scattering data shows. 2
6
8
There are many factors which can influence the response of a liquid crystal polydomain texture to shear flow. Rey (22) has recently calculated a comprehensive map showing the variation in orientation modes for an initially monodomain sample,
Cebe et al.; Scattering from Polymers ACS Symposium Series; American Chemical Society: Washington, DC, 1999.
402 subjected to shear flow, as a function of the Ericksen number, which is the balance between viscous and elastic stresses and a parameter describing the ratio of shortrange to long-range elasticity. The application of this model or other molecular theories, to the current experimental data, is hampered by the unavailability of a rigorous approach to realistic polydomain systems. We assume that, for the three materials considered here, the ratio of short-range to long-range elasticity is similar since the underlying molecular architectures are identical. As a consequence, variation in the response to shear flow for each material must arise from variations in the Ericksen number. We can obtain a measure of the relative viscosities by consideration of the relaxation behaviour following the cessation of shear flow. Figure 14 shows the variation in
for a sample of PPC at 65°C following the cessation of a shear rate of 2.5s" fitted with a single time constant exponential decay. The time constants obtained are independent of the previous shear rate but strongly depends on the temperature (5). We find that the ratios of these time constants for different temperatures are qualitatively in accord with the critical shear rates for the same temperatures (Figure 11) at which substantial global orientation development develops. This suggests that the similar but shear rate sifted orientation data for PPC (Figure 11) arise from variations in the relaxation times due to differences in temperature. It seems reasonable to view the major differences between lyotropic and thermotropic HPC in a similar manner. The low shear rate variation is not simply a matter of concentration, but reflects the differences in the balance between viscous and elastic stresses. 2
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1
0.35 -i
i
0.30
0.25 ί λ
αΓ'
° -
2
o.i5
0
!
\
0.10 0.05
4
0.00
i 0
2000
1000
3000
time /s
Figure 14 Values of
obtained for a 55% w/v aqueous solution of HPC at 25°C following the cessation of shearflow(5). The solid line represents a single time constant exponential decay. 2
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403
Summary
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We have shown that X-ray rheology procedures can provide useful structural information on the influence of shear flow on the textures of liquid crystal polymers. It is clear that such reorganization is complex and there will be considerable value in developing techniques, which can more directly yield information on the threedimensional nature of the flow and the changes in molecular organization. The timeresponses available using a synchrotron source coupled with a CCD based detector are ideally suited to this type of measurement. Although the three material systems studied here were based on the same underlying hydroxypropylcellulose architecture, the response to shear flow is varied. These variations are not simply related to the whether the material is lyotropic or thermotropic and we attribute the major part of these variations to the changing balance between viscous and elastic stresses.
Acknowledgements This work was supported by the Engineering and Physical Research Council. The synchrotron studies were performed at the CCLRC Daresbury SRS. We thank Jim Woodcock and Clark Balague for the essential design and construction of the flow stages and Aqualon Ltd for the provision of the samples of Klucel E.
References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.
11. 12. 13. 14.
Mitchell G.R. and Windle A.H. in Developments in Crystalline Polymers-2 1988, Editor D.C.Bassett Elsevier:London Chapter 3 Werbowwji R.S. and Gray D.G. Macromolecules 1980 13 69 Onogi A. and Asada T. in Rheology Vol 1 edited Astarita G., Marrucci G. and Nicolais L. Plenum Press New York 1980 p127 Shimamura K., White J.L and Fellers J.F J.Appl. Polym. Sci. 1981 26 2165 E.M.Andresen and G.R.Mitchell submitted to Macromolecules Keates P.A., Peuvrel E. and Mitchell G.R. Polymer Communications 1992 33 3298 G.R.Mitchell, J.A.Pople, E.M.Andresen and P.G.Brownsey Advances in X-Ray Analysis 1997 41 in press Keates P., Mitchell G.R., Peuvrel-Disdier E., Riti J.B., and Navard P. J. NonNewtonian Fluid Mechanics 1994 52 197 Romo-Uribe A. and Windle A.H. Macromolecules 1996 29 6246 Hongladsrom K., Ugaz V.M., Cinader D.K., Burghardt W.R., Quintana J.P., Hsiao B.S., Dadmun M.D., Hamilton W.A. and Butler P.D. Macromolecules 1996 29 5346 Picken S.J., Aerts J., Visser R. and Northolt Macromolecules 1990 23 3849 Mitchell G.R. Rheologica Acta 1999 Andresen E.M., Pople, J.A. and Mitchell G.R. unpublished work Pople J.A., Keates P.A. and Mitchell G.R. J. Synchrotron Rad. 1997 4 267
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404
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15. Keates P.A., Mitchell G.R., Peuvrel-Disdier E., and Navard P., Polymer 1993 34, 1316 16. Keates P.A., Mitchell G.R., Peuvrel-Disdier E., and Navard P., Polymer 1996 37, 893 17. Andresen E.M. and Mitchell G.R. Europhysics Letters 1998 43 296 18. Baek S.-G., Magda J.J., Larson R.G. and Hudson S.D. J Rheology 1994 38 1473 19. Marrucci G. and Greco F Adv. Chem,. Phys. 1993 86 331 20. Larson R.G. and Mather P.T. in Theoretical Challenges in the Dynamics of Complex Fluids edited by McLeish T.M. Kluwer Dordrecht 1997 21. Pople J.A. and Mitchell G.R. Liquid Crystals 1997 23 467 22. Rey A.D. Phys Rev Ε 1998 57 5609
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