The effect of temperature and concentration on N 1 ...

The effect of temperature and concentration on N 1 and tumbling in a liquid crystal polymer C.-M. Huang, J. J. Magda, and R. G. Larson Citation: Journal of Rheology (1978-present) 43, 31 (1999); doi: 10.1122/1.551037 View online: http://dx.doi.org/10.1122/1.551037 View Table of Contents: http://scitation.aip.org/content/sor/journal/jor2/43/1?ver=pdfcov Published by the The Society of Rheology Articles you may be interested in Rheometric investigation on the temporal shear thickening of dilute micellar solutions of C 14 , C 16 , and C 18 TAB NaSal J. Rheol. 52, 923 (2008); 10.1122/1.2933352 Photochemical switching behavior of azofunctionalized polymer liquid crystal/ Si O 2 composite photonic crystal Appl. Phys. Lett. 89, 153131 (2006); 10.1063/1.2363928 Development of wavy texture in startup flows of liquid crystalline polymer solution through a slit cell J. Rheol. 46, 881 (2002); 10.1122/1.1485276 Steady and transient rheological behavior of mesophase pitches J. Rheol. 42, 781 (1998); 10.1122/1.550912 An Attempt to Measure the First NormalStress Difference N 1 in Shear Flow for a Polyisobutylene/Decalin Solution “D2b” at Shear Rates up to 10 6 s 1 J. Rheol. 33, 821 (1989); 10.1122/1.550066

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The effect of temperature and concentration on N1 and tumbling in a liquid crystal polymer C.-M. Huang and J. J. Magdaa) Departments of Materials Science Engineering and Chemical Fuels Engineering, University of Utah, Salt Lake City, Utah 84112

R. G. Larson Department of Chemical Engineering, University of Michigan, Ann Arbor, Michigan 48109-2136 (Received 13 March 1998; final revision received 23 September 1998)

Synopsis The shear rate g˙ min at the relative minimum in the N1 flow curve is studied as a function of temperature and concentration for liquid crystalline ~hydroxypropyl!cellulose ~HPC!. For lyotropes, at least, g˙ min is the shear rate necessary to halt director ‘‘tumbling’’ and align the molecules. HPC is a convenient polymer for studying the relationship between lyotropic and thermotropic liquid crystalline polymers because it exhibits a pure thermotropic phase at elevated temperatures, and room temperature lyotropic phases at moderate concentrations in m-cresol. At the highest possible polymer concentration at which reliable rheology data can be obtained ~around 70 wt % polymer!, indirect evidence for director tumbling is observed, in that N1 retains a local minimum versus shear rate. For the highest concentrations this minimum N1 value is positive, rather than negative, as is the case at lower concentrations and as is predicted by the Doi theory. Empirical time–temperature and time–concentration shifting can be used to estimate g˙ min from measured values of the shear viscosity. © 1999 The Society of Rheology. @S0148-6055~99!00401-0#

I. INTRODUCTION The occurrence of director tumbling is well established for solutions of rod-like liquid crystal polymers ~LCP’s! at room temperature @Srinivasarao and Berry ~1991!, Burghardt and Fuller ~1991!, Magda et al. ~1991!, Larson and Mead ~1993!#. In such systems, at low shear rates, the average molecular orientation ~i.e., the ‘‘director’’! has no steady state, but rotates end-over-end or ‘‘tumbles’’ @Marrucci and Maffettone ~1989!, Larson ~1990!#. However, the experimental evidence for director tumbling in pure meltprocessable LCP’s ~thermotropes! is far less convincing @Cocchini et al. ~1991!, Guskey and Winter ~1991!, Cocchini et al. ~1992!, Winter and Wedler ~1993!, Han and Chang ~1994!, Langelaan and Gotsis ~1996!, Beekmans et al. ~1996!, Chang and Han ~1997!#. There are many possible explanations for this @Baek et al. ~1993a!#. For example, such polymers are never very rod-like; in fact, flexibility is often intentionally introduced into the polymer backbone in order to lower the melting temperature. One may hypothesize that at the elevated temperatures necessary for processing flows of a thermotrope, the a!

Author to whom all correspondence should be addressed; electronic mail: [email protected]

© 1999 by The Society of Rheology, Inc. J. Rheol. 43~1!, January/February 1999

0148-6055/99/43~1!/31/20/$20.00

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HUANG, MAGDA, AND LARSON

polymer chain may be stiff enough for liquid crystallinity, but not stiff enough for director tumbling @Baek et al. ~1994!#. In order to explore this hypothesis, we study one of the few polymers known which exhibits liquid crystalline solutions at room temperature, and pure liquid crystalline melts at elevated temperature @Werbowyj and Gray ~1980!, Suto et al. ~1982!#. ~Hydroxypropyl!cellulose ~HPC! almost definitely exhibits director tumbling in room temperature liquid crystalline solutions, as inferred from the presence of a negative minimum in the value of the steady-state first normal stress difference N1 @Grizzuti et al. ~1990!, Baek et al. ~1993b!#. By systematically increasing the temperature and polymer concentration, one can experimentally approach the pure thermotropic state of HPC, and potentially observe the disappearance of director tumbling. In our previous work on HPC @Baek et al. ~1994!#, we studied liquid crystalline solutions in m-cresol between 20 and 55 wt % polymer, over a narrow temperature range between 25 and 50 °C. We also presented limited rheological data for the ‘‘pure’’ liquid crystalline melt containing antioxidant. The solution results were in generally good agreement with accurate numerical solutions of the Doi molecular theory @Doi ~1981!#, although surprising discrepancies were observed, particularly at lower temperatures. These discrepancies were tentatively attributed to viscous stresses, which may become increasingly important as the ‘‘free volume’’ available for packing of the semirigid molecules becomes small. In this work, we continue our investigation of the same binary system, but we first determine the phase diagram and then use it to study systematically thermal effects. A modified version of the principle of time–temperature equivalency is shown to yield qualitatively useful information. The validity of this principle could not be assumed at the outset, because a change in temperature ~or concentration! changes the order parameter of a liquid crystalline polymer, in addition to changing the frictional properties. Nevertheless, we find that time–temperature shift factors can be used to calculate the apparent flow activation energy, which is then used to look for evidence of drastic changes in polymer ‘‘free volume’’ at temperatures where large departures from the Doi theory were noted previously @Baek et al. ~1994!#. The phase diagram shows that the critical polymer concentration for formation of a liquid crystalline phase is about three times greater at 150 °C than at room temperature. According to the Flory theory @Flory ~1978!#, this implies that the persistence length ~a measure of chain rigidity! is three times smaller at 150 °C than at room temperature. We compare liquid crystalline phases at 150 °C with liquid crystalline phases at room temperature, to see if the threefold reduction in chain rigidity results in any qualitative changes in rheology. One possible qualitative change is a change in the shape of the N1 flow curve. For a tumbling liquid crystal polymer, N1 does not increase monotonically with shear rate, but rather shows a relative minimum at an intermediate shear rate g˙ min , at which N1 is often negative @Kiss and Porter ~1978!#. It is widely believed that the occurrence of a negative relative minimum is associated with a flow transition between director tumbling and flow-aligning behavior @Marrucci and Maffettone ~1989!, Larson ~1990!#. Here we will show that at high HPC concentrations, greater than around 60 wt %, the region of negative N1 disappears, but a local positive minimum in N1 is retained. We show that time– temperature and time–concentration shifting can be carried out smoothly without any apparent transitions that could be associated with the change in the sign of the minimum in N1. Hence we argue that the minimum in N1, whether positive or negative, suffices to reveal the existence of director tumbling. We then conclude that director tumbling exists


EFFECT OF TEMPERATURE AND CONCENTRATION

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in HPC solutions at concentrations up to at least 70 wt % polymer, and possibly into the thermotropic state itself.

II. MATERIALS AND EXPERIMENTAL METHODS Meta-cresol ~b.p. 203 °C! was purchased from the Sigma Chemical Company. HPC was taken from the same production lot used in a previous study @Baek et al. ~1994!# — Klucel E ~lot 8326! from the Hercules/Aqualon Corp. in Wilmington, Delaware. No ‘‘aging effects’’ @Leal et al. ~1995!# were observed in solutions which had been stored for up to 2 years. Unfortunately, near the end of this study, it became necessary to obtain a second production lot of HPC from the same supplier ~Klucel E, Lot FP10 10766!. Both HPC lots have the same nominal molecular weight ~80 000 g/mole!, but intrinsic viscosity measurements in ethanol give a higher molecular weight for the second lot ~106 000 vs 90 000 g/mole! using the available Mark–Houwink relation @Wirick and Waldman ~1970!#. The phase diagram for semiflexible HPC is known to be independent of molecular weight variations in this molecular weight range @Aden et al. ~1984!, Baek et al. ~1993b!#. Furthermore, rheological comparisons at higher concentrations suggest that the effective molecular weights of the two lots are closer than indicated by the intrinsic viscosity data. For example, a 45 wt % lyotropic solution using the second polymer lot has a viscosity ~at 1 s21! which is only 30% higher than a 45 wt % solution using the first lot, and the location of the maximum in the N1 flow curve is essentially the same @Huang ~1998!#. HPC was dried under a slight vacuum for 2 days at room temperature before preparing solutions. Measured amounts of HPC and m-cresol were placed in a closed container and allowed to interdiffuse for 2 months at 50 °C before rheological measurement. Pure HPC discolors within 20 min at melt temperatures ~150 °C!, which is insufficient time to complete the rheology experiments. However, when m-cresol concentrations above 10 wt % are added to the polymer, HPC degradation slows down considerably, with no change in rheological properties over a time period of at least 45 min at 150 °C @Huang ~1998!#. Polarizing optical microscopy ~POM! was performed using a Nikon hot stage with 300-fold magnification and a heating/cooling rate of 5–10 °C/min. Rheological measurements were made with a Weissenberg R-17 rheogoniometer with cone-and-plate geometry. The cone angle was 0.019 rad with a radius of 1.25 cm; although a more shallow cone was used in some special cases ~0.005 rad, 2.5 cm radius!. The rheogoniometer was modified since our last study @Baek et al. ~1994!# in order to access higher temperatures. Kapton flexible resistance heaters ~Omega Inc., model KH-206/5! were mounted on the inner surface of an environmental chamber which encloses the sample. A microprocessor-based, auto-tuning temperature controller ~Omega, Inc., model CN76033! was attached to the heaters, with the control signal supplied by a T-type thermocouple mounted near the rim of the rheometer plate. Temperature fluctuations rarely exceeded 0.1 °C during a rheology experiment. Samples of higher polymer concentration were either compression molded into disks, or preheated to the softening temperature before loading into the rheometer. No difference was observed in the decay of residual stresses after loading between the two methods. In most cases, the sample was heated up from room temperature to the temperature of interest, and then steady-state shear properties were measured after at least 100 strain units of preshearing. In order to test for thermal-history effects, the steady flow properties of a 55 wt % lyotropic solution were measured after preheating above the clearing temperature, and then cooling down into the liquid crystalline state @Huang ~1998!#. This preheating procedure was observed to have no effect on the steady flow properties; hence in lyotropic solutions there was no


34


FIG. 1. Phase diagram for ~hydroxypropyl!cellulose ~HPC! in m-cresol determined by polarizing optical microscopy. The dashed curve represents a boundary to the left of which the relative minimum of N1 is negative.

evidence of the residual crystallinity sometimes observed in thermotropic melts of HPC @Wilson ~1991!#. An increase in the noisiness of the rheological data was observed for polymer concentrations above 60 wt %, whether in the liquid crystalline or the isotropic state @Huang ~1998!#. The quality of the data was noticeably better when the same elevated temperatures were used with lower polymer concentrations. We speculate that local areas of solidification occur near the sample interface when high temperatures are coupled with high polymer concentrations, thereby increasing the noisiness of the normal force signal. Nonetheless, the consistency shown between shift factors from N1 data at high shear rates and viscosity data at low shear rates ~Sec. III! argues against the occurrence of any massive flow instabilities. One instability that could not be avoided is edge fracture @Macosko ~1994!#. However, in our previous work, we showed that we could always tell the correct sign of N1 even with edge fracture present @Baek et al. ~1994!#. In the Appendix of this article, we extend this result by showing that we can confidently determine the existence of a relative minimum in N1, whether positive or negative, even if edge fracture is occurring. However, shear-stress data are more sensitive to edge fracture than g˙ min data. Hence, in some data plots, the range of shear rates over which viscosity data are reported is reduced compared to that for N1 data. III. RESULTS AND DISCUSSION A. Phase diagram Figure 1 contains relevant portions of the phase diagram for HPC/m-cresol measured using hot stage POM. Horizontal paths through the crescent-shaped coexistence region in Fig. 1 correspond to lyotropic transitions; vertical paths through the coexistence region correspond to thermotropic transitions. Whenever two different values for a phase boundary location were observed due to hysteresis ~ ' 5–10 °C!, the value giving a larger coexistence region was chosen for Fig. 1. Also shown are reported values for the melt



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TABLE I. Solutions of HPC in m-cresol investigated using rheological measurements. Polymer Biphase wt % in mcresol M v ~g/mole! region ~°C! ¯ ¯ ¯ ¯ 33–67 ¯ ¯ ¯ 74–117

20 21 23 25 27 28 30 31 35 39 45

90 000 90 000 90 000 90 000 90 000 90 000 90 000 90 000 90 000 90 000 90 000

113–137

45 50 55

106 000 90 000 90 000

113–137 122–146 134–159

57 60 65

90 000 106 000 106 000

¯ 142–165 147–170

70

106 000

150–175

Temperature studied ~°C! viscosity 25 25 25 25 25 25,37 25,30,37 25 25,37 25,37 25,37,50,60,71, 80,100,135,140, 145 25 25,38,50,135 25,37,55,100, 120,147 25 70,80,100,120 70,75,85,100, 120,130 140,155

Temperature studied ~°C! N1 ¯ ¯ ¯ ¯ 25 25,37 25,30,37 ¯ 25,37 25,37 25,37,50,60,71, 80,100,135,140, 145 25 25,38,50,135 25,37,55,100, 120,147 25 70,80,100,120 70,75,85,100, 120,130 140 155

temperature (T m ) and the clearing temperature (T I ) of the pure polymer @Suto et al. ~1982!, Wilson ~1991!#. Table I lists the thermodynamic states which have been studied rheologically. The phase diagram provides an approximate way of comparing the order parameter and the persistence length (l p ) for various liquid crystalline states studied rheologically @Larson ~1988!#. The Doi theory @Doi ~1981!# which when solved accurately is quite successful for room-temperature lyotropes, assumes that the polymers are perfectly rodlike (l p 5 `). However, most thermotropes are semiflexible, and the persistence length decreases with increasing temperature @Conio et al. ~1983!, Aden et al. ~1984!#, and may possibly increase with increasing concentration @Lautenschlager et al. ~1991!, Tkachenko and Rabin ~1995!#. According to Flory’s thermodynamic model @Flory ~1978!#, for all states lying along the upper boundary of the coexistence region in Fig. 1,

fp ' ~8/x !~ 122/x ! ,

~1!

where f p is the critical polymer volume fraction and x is equal to 2lp /d, d being the chain diameter. One may use Eq. ~1! in conjunction with the location of the coexistence region in Fig. 1 to obtain a relative comparison of l p values. For example, l p should be about four times larger for the coexistence state at room temperature ( f p ' 25%) than for the coexistence state at 200 °C ~i.e., the pure melt!. The dilute solution value of l p for HPC has been measured at 25 °C in various solvents ~but not m-cresol!, and is in the range of 10–20 nm @Werbowyj and Gray ~1980!, Conio et al. ~1983!, Aden et al. ~1984!, Shtennikova et al. ~1990!#. The HPC contour length is about 120 nm for the molecular weights studied here, and the diameter is about 1 nm. It should be emphasized that Eq. ~1! is not likely to be quantitatively accurate, and other explanations besides flexibility


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FIG. 2. N1 flow curves at various temperatures ~see legend! for a 45 wt % solution of HPC in m-cresol. The biphasic range is 113–137 °C ~from POM!.

change may be given for the increase in f p with T @Conio et al. ~1983!#. However, given the scarcity of persistence-length data in the literature for thermotropes, Eq. ~1! can be taken as a first estimate. B. N1 flow curves Consider a vertical path in Fig. 1 corresponding to a fixed polymer concentration of 45 wt %, and connecting a purely liquid crystalline state with a purely isotropic state. The variation of the N1 flow curve along this path is shown in Fig. 2. Figure 3 shows the variation of the N1 flow curve along a similar path at higher polymer concentration (c p 5 55 wt %). Despite the wide temperature range covered ~25–147 °C! corresponding to a threefold reduction in l p @see Eq. ~1!#, the various N1 flow curves are qualitatively similar in shape. In particular, each flow curve exhibits a relative minimum at a value of the shear rate denoted g˙ min . The single exception is the flow curve for the wholly isotropic state ~145 °C in Fig. 2!. For conventional polymeric liquids, or for nontumbling liquid crystals, N1 increases monotonically with increasing shear rate @Larson ~1988!#. Therefore occurrence of a relative minimum in the N1 flow curve may be indirect evidence for director tumbling, even if the value of N1 at the minimum is positive ~e.g., the 25 °C flow curve in Fig. 3!. Further evidence for the validity of this hypothesis will be presented later. Recall that the flexibility hypothesis discussed in Sec. I. Perhaps the most important new result in Fig. 3 is the indirect evidence that 55 wt % HPC solutions ‘‘tumble’’ at temperatures at least as high as 147 °C. This is hot enough for pure HPC to flow. The stiffness of the HPC chain should be no greater in a 55 wt % solution at 147 °C than in a pure melt at the same temperature. Therefore, if pure liquid crystalline HPC melts do not exhibit director tumbling ~a question which remains to be answered!, it is not because the chains are too flexible at melt processing temperatures.



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FIG. 3. N1 flow curves at various temperatures ~see legend! for a 55 wt % solution of HPC in m-cresol. The biphasic range is 134–160 °C ~from POM!. To avoid confusion, the data for 100 °C are omitted because they fall nearly on top of the data for 147 °C.

Unfortunately, it is impossible to use N1 to look for tumbling in the pure anisotropic melt or for any liquid crystalline solution of HPC above about 80 wt % polymer, because of the occurrence of an apparent yield normal stress. That is, after loading the sample, the normal forces do not relax back to zero in a reasonable length of time ~i.e., before the polymer begins to degrade!. Similar behavior has been reported by other laboratories for thermotropic polymers @Suto et al. ~1982!, Langelaan and Gotsis ~1996!#, and some authors have questioned the validity of rheology measurements on a sample with unrelaxed normal stresses @Cocchini et al. ~1992!, Han and Chang ~1994!#. For this reason, apparent N1 values measured on samples with unrelaxed normal stresses are not included in any of the figures in this article. However, it is interesting that the apparent N1 flow curve for the 90 wt % solution still has a positive relative minimum, although the depth of this minimum appears to be reduced relative to that observed at lower concentrations @Huang ~1988!#. Well before the onset of apparent yield stresses, the rheology of the liquid crystal begins to change qualitatively. For example, in Fig. 3, the room-temperature N1 flow curve is shifted upward with respect to the other flow curves, so that N1 is non-negative everywhere. In a previous article @Baek et al. ~1994!# this upward shift was tentatively attributed to a viscous contribution to the stress tensor ~neglected in the original Doi theory!. In Fig. 3, negative N1 values reappear at higher temperatures, perhaps because the chains become more flexible and less likely to jam, so that there is an increase in the available ‘‘free volume’’, which decreases the viscous stresses. However, at higher concentrations starting at 65 wt % HPC ~see Appendix!, one cannot raise the temperature high enough for reappearance of negative N1 because of the intervention of the liquid crystal/isotropic phase transition. The region of the phase diagram within which negative N1 can be observed is bounded on the left by the biphasic region, and on the right by the dashed curve in Fig. 1. This dashed curve is very steep, and the onset of apparent yield stresses occurs well to the right of this curve in Fig. 1.


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C. Time–temperature shifting Negative N1 values apparently do not occur at any shear rate for liquid crystalline states to the right of the dashed curve in Fig. 1. This dashed curve was determined empirically; its physical significance remains elusive. As discussed above, it may signify the onset of a large viscous contribution to the stress tensor. One might expect this to occur when the glass transition temperature (T g ) of the sample is approached @Baek et al. ~1994!#. Unfortunately, T g is difficult to detect for HPC using calorimetry @Suto et al. ~1982!, Huang ~1998!#. Instead we use values of the time–temperature shift factors (a T ), which are related by the well-known Arrhenius equation to the apparent activation energy for flow (E a ). The value of E a is expected to increase sharply as T g is approached @Ferry ~1980!#. According to the principle of time–temperature equivalency, an increase in temperature ~at fixed concentration! is equivalent to a decrease in the viscoelastic time scale, by a multiplicative factor which is denoted by a T . A well-known empirical estimate of a T for polymer melts is given by the low-shear-rate viscosity at temperature T, divided by the low-shear-rate viscosity at the reference temperature. This is reasonable for conventional polymers, because almost all molecular theories predict that the low-shear-rate viscosity is proportional to the mean relaxation time @Ferry ~1980!; Larson ~1988!#. Hence the ratio of the low-shear-rate viscosities at two different temperatures should be approximately equal to the ratio of the mean relaxation times of the polymer. However, since the liquid-crystal order parameter S is a function of temperature, there is in principle no reason to expect time–temperature shifting to be valid for LCPs. Indeed, in general from rigid-rod theories such as the Doi theory, we expect the zero-shear viscosity h 0 to be given by:

h0 5 nkTlf ~S!.

~2!

Here n is the number of rods per unit volume and l is the mean relaxation time ~related to the rotary diffusivity of the rigid rods!. Thus, like a conventional polymer, the viscosity of an LCP is predicted to be proportional to the mean relaxation time. However f (S), the function of S which follows the mean relaxation time in Eq. ~2!, has no counterpart for conventional polymers; one might expect its presence to invalidate the usual time– temperature shifting procedures. If S is a strong function of temperature, then Eq. ~2! implies that the ratio of h 0 at two different temperatures is not equal to a T , the ratio of the mean relaxation times. Nevertheless, far away from the isotropic-liquid crystalline boundary, changes to the order parameter with temperature are less pronounced, and we will see that time–temperature shifting appears to be a qualitatively valid procedure. Despite the complicating presence of the order parameter, many laboratories have used conventional time–temperature shifting procedures with viscosity data to estimate shift factors and E a for other LCP systems: poly~benzylglutamate! ~PBG! in m-cresol @Mewis and Moldenaers ~1987!, Walker et al. ~1995!#, HPC in dimethylacetamide @Suto et al. ~1986!#, HPC in water @Walker and Wagner ~1994!#, poly~1,4-phenylene-2,6benzobisthiazole! ~PBT! in methane sulfonic acid @Einaga et al. ~1985!#, and a thermotropic polyesteramide @Gonzalez et al. ~1990!#. For many LCPs, it is difficult to assign unambiguously a value to the low-shear-rate viscosity which appears in Eq. ~2!. Some authors have computed shift factors by comparing viscosity values at a fixed value of the shear rate. However, as pointed out by Walker and Wagner @Walker and Wagner ~1994!#, such a procedure underestimates E a due to shear thinning. Walker and Wagner take a T to be equal to the horizontal shift on a logarithmic scale needed to superpose shear stress



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FIG. 4. Viscosity flow curves at various temperatures ~see legend! for a 55 wt % HPC solution in m-cresol.

curves at fixed shear stress ~not fixed shear rate!. One can motivate this procedure by introducing a dimensionless factor H into Eq. ~2!:

s 5 nkTlg˙ f ~S!H~De! .

~3!

Here s is the shear stress, and H(De) is an unknown shear thinning function which is assumed to depend on the Deborah number (De 5 l g˙ ) but not on the order parameter. Consider the variation in the shear stress curves ~i.e., shear stress versus shear rate! with temperature at fixed polymer concentration. According to Eq. ~3!, if variations in S can be neglected, then one can equate the shift factor to the horizontal shift needed to superpose the shear stress curves. Time–temperature shift factors thus calculated using shear stress curves will be denoted a T, s . The validity of this procedure is investigated in Figs. 4–7. Figures 4 and 6 contain unshifted viscosity data at fixed concentrations c p 5 55 and 65 wt %, and Figs. 5 and 7 contain the composite viscosity curves created by shifting at fixed stress to the reference temperature ~25 °C for the 55% solution, and 70 °C for the 65% solution which is too viscous to study at room temperature!. The composite curve for 65 wt % HPC has two inflection points, and resembles in shape the well-known ‘‘three-region flow curve’’ @Onogi and Asada ~1980!#. The curve at 55 wt % lacks a clear region I over the shear-rate range covered, but it may be present at lower shear rates than those accessed @Burghardt et al. ~1995!#. In order to check the consistency between mean relaxation times derived from shear stress and N1 measurements, the N1 flow curves of Fig. 3 are plotted again in Fig. 8 after rescaling the abscissa by the shift factors derived from the shear stress data. The N1 flow curves are shifted by this procedure in such a way that the various minima coincide to within a factor of 3 in Fig. 8, a result which has also been observed for liquid crystalline solutions of PBG in m-cresol @Mewis and Moldenaers ~1987!#. Thus the shift in 1/g˙ min with temperature is roughly similar to the shift in the shear-stress curve. There is, however, a small, but systematic, shift of the minimum in N1 toward the left in Fig. 8 with increased temperatures, a shift also seen at other concentrations. This imperfection in time–temperature shifting may be due to a temperature dependence of the order parameter. Of course, if time–temperature shifting were rigorously valid for the samples, the


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FIG. 5. Composite viscosity flow curve for a 55 wt % HPC liquid crystalline solution, constructed by shifting at constant shear stress ~shift factor a T, s , reference temperature 25 °C!. The arrow locates the shear rate of the local minimum in the N1 flow curve at 25 °C.

minima in the shifted N1 curves would not only occur at the same shear rate, but all the curves would superimpose perfectly. If, as we suspect, there is a positive viscous contribution to N1 that becomes large at low temperatures, this might account for the behavior observed in Fig. 8. However, there might then also be a viscous contribution to the shear viscosity which could limit the validity of shifting of this quantity also. The good quality

FIG. 6. Viscosity flow curves at various temperatures ~see legend! for a 65 wt % HPC solution in m-cresol. The biphasic range is 147–170 °C ~from POM!.



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FIG. 7. Composite viscosity flow curve for a 65 wt % HPC liquid crystalline solution, constructed by shifting at constant shear stress ~shift factor a T, s , reference temperature 70 °C!. The arrow locates the shear rate of the local minimum in the N1 flow curve at 70 °C.

of the superposition obtained in Fig. 7, however, as well as the correspondence of the shear rates at which N1 minima occur in Fig. 8 suggests, however, that such viscous stresses, if present in the shear viscosity, do not seriously affect time–temperature shifting.

FIG. 8. N1 as a function of shifted shear rate for a 55 wt % HPC liquid crystalline solution at various temperatures. Shift factors a T, s were calculated using the shear stress ~reference temperature 25 °C! and the symbols have the same meaning as in Fig. 3.


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FIG. 9. Natural logarithm of shift factor a T, s based on the shear stress vs inverse temperature, for HPC solutions at several concentrations ~see legend!. The reference temperature is 70 °C for the two upper curves, and 25 °C for the three lower curves. The vertical bars locate the biphasic region ~from POM!.

Figure 9 contains an Arrhenius plot of the shift factors a T, s obtained by shifting the shear stress curves at constant shear stress. Each curve corresponds to a different, fixed value of the polymer concentration. A reference temperature of 25 °C was used for the lower polymer concentrations, and 70 °C was used for the higher polymer concentrations, which are too viscous to study at room temperature. Figure 10 contains a similar plot of the shift factors a T,N derived from the N1 flow curve, where the shift factor in this case is defined as g˙ min at the reference temperature divided by g˙ min at temperature T. Thus the shift factors in both figures were calculated neglecting the temperature dependence of the order parameter. If the errors associated with this neglect were significant, the two figures should show systematic differences. However, in fact, the figures are quite similar. In addition, there is a relative minimum for both shift factors near the clearing temperature. This result is expected, since the rapid decrease in the order parameter that occurs near the isotropic–liquid crystalline transition as the temperature is increased or the concentration is lowered produces an increase in viscosity, which is in a direction opposite that normally observed @Suto et al. ~1986!; Suto et al. ~1989!#. The location of the coexistence region determined by POM is shown by the vertical lines in Figs. 9 and 10 and is in excellent agreement with the region of negative slope of shift factor versus 1/T. At temperatures far below the clearing temperature, the slopes in both figures converge on the same value, giving an asymptotic value of E a 5 8365 kJ/mole for the apparent flow activation energy. There is no indication that E a is significantly larger for liquid crystalline states to the right of the dashed curve in Fig. 1. Thus, if large viscous stresses are responsible for the upward shift in the N1 vs g˙ curve at low temperatures as seen in Fig. 3, these viscous stresses do not seem to be related to any nearby glass transition. The value reported here for E a is the same within experimental scatter for flow regions I and II; the value appears to be less in region III but this may be a consequence of edge



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FIG. 10. Natural logarithm of shift factor a T,N ~based on the shear rate at the relative minimum in N1) vs inverse temperature, for HPC solutions at several concentrations ~see legend!. The reference temperature is 70 °C for the two upper curves, and 25 °C for the three lower curves. The vertical bars locate the biphasic region ~from POM!.

fracture. For aqueous lyotropic HPC solutions, E a is also reported to have the same value in regions I and II, in this case 50.164.9 kJ/mole @Walker and Wagner ~1994!#. E a for water alone is about 17 kJ/mole; E a for m-cresol alone is about 50 kJ/mole. Thus for lyotropic systems of HPC in both water and m-cresol, E a is larger by 33 kJ/mole than that of the pure solvent, possibly due to the semiflexible nature of the polymer chain @Walker and Wagner ~1994!#. By contrast, nematic solutions of a more rodlike polymer, PBT, are reported to have an apparent activation energy close to that of the solvent alone @Einaga et al. ~1985!#. This is also true for lyotropic solutions of PBG in m-cresol at concentrations 37 wt % polymer and below, for which E a is 50.360.8 kJ/mole @Mewis and Moldenaers ~1987!, Walker et al. ~1995!#. At 40 wt % PBG in m-cresol, E a is significantly larger and depends strongly on the shear stress, but this is likely due to the occurrence of a recently discovered hexagonal phase transition @Ugaz et al. ~1997!#. D. Time–concentration shifting One may hypothesize that an increase in polymer concentration at fixed temperature is roughly equivalent to a shift in the polymer time scale @Han and McKenna ~1997!#. By analogy with a T , one can define the time–concentration shift factor a C to be equal to the ratio of the mean relaxation time at two different concentrations. For conventional polymers, a reduction in temperature and an increase in polymer concentration are often found to have similar effects on the mean relaxation time @Osswald and Menges ~1996!#. This is also found to be true for liquid crystalline HPC at concentrations above about 35 wt % polymer, as shown by the data in Fig. 11. In Fig. 11, the room-temperature value of g˙ min exhibits a relative maximum at C p ' 35 wt %. At this same temperature, the viscosity value at fixed shear rate exhibits a relative minimum at C p ' 28 wt % @Huang


44


FIG. 11. Inverse of the shear rate of the relative minimum in N1 vs polymer concentration at various temperatures ~see legend!. The vertical bars locate the biphasic region ~from POM!.

~1998!#. In the following, we focus on polymer concentrations above 35 wt %, for which g˙ min at fixed temperature decreases sharply with increasing concentration. It must be emphasized that for liquid crystalline PBLG the opposite trend is observed: g˙ min increases sharply with increasing concentration, as predicted by the Doi theory @Baek et al. ~1993a!#. In Sec. III C, reasonable success was achieved in calculating shift factors while neglecting the possible temperature dependence of the order parameter. Let us therefore ignore the possible concentration dependence of the order parameter as well. In this case, time–concentration shift factors a C,N can be calculated from N1 data as before: namely as the inverse of the ratio of the g˙ min values. However, the analysis of shear stress data to yield a C, s values is more complex than the analysis to yield a T, s values. According to Eq. ~3!, in order to superpose shear-stress curves measured at two different concentrations, one needs a horizontal shift equal to the ratio of the mean relaxation times, and a vertical shift equal to the ratio of the polymer concentrations. The vertical shift was not required when calculating a T, s values. Thus calculation of a C, s values is a sterner test of the validity of Eq. ~3!. We now choose the reference state to be 25 °C, 55 wt % HPC, and calculate shifts in l with respect to this state independently using N1 and s data, assuming that the value of l is the same at the reference state whether calculated from N1 or s. Figure 12 compares the shift factors calculated from N1 and s data for the isotherm at 25 °C. The ordinate is the ratio of the shift factor a C,N obtained from g˙ min to a C, s obtained from the shear stress; the abscissa is the reduced concentration C p /C * . Here C * is the maximum polymer concentration at which the isotropic phase is stable, estimated to be about 25 wt % at 25 °C from Fig. 1. If the analysis were exact and the data perfect, the ordinate would equal unity at every concentration. In practice, the ordinate varies from 1 to 2



45

FIG. 12. Shift factor calculated from the relative minimum in N1 (a C,N ) divided by shift factor calculated from the shear stress (a C, s ), plotted against reduced polymer concentration along the isotherm at 25 °C.

along the isotherm in Fig. 12. The shift factor based on g˙ min , a C,N , varies by a factor of 50 along the same isotherm, and that based on shear stress a C, s varies by a factor of 90. Therefore estimates of the shift in l based on N1 and s data are fairly consistent. One might wonder if better consistency could be achieved by including order-parameter effects, but unfortunately exact solutions to the Doi theory ~i.e., nonflow aligning! giving the shear viscosity against order parameter have not been published. E. Overall comparison of shift factors from s and N1 In the preceding sections, typical results for the quality of shifting have been presented for a few illustrative concentrations. In this section, we compare shift factors from the shear stress and the first normal stress difference for all of the liquid crystalline states listed in Table I, using both time–temperature and time–concentration shifting. There is some interest in being able to use the shear viscosity curve to predict the value of g˙ min @Mewis and Moldenaers ~1987!; Fried et al. ~1994!; Grizzuti et al. ~1990!; Dadmun and Han ~1994!#. For example, the viscosity flow curve may exhibit a ‘‘hesitation’’ or two nearby changes in concavity at a shear rate near g˙ min @Grizzuti et al. ~1990!; Marrucci and Maffettone ~1990!, Cidade et al. ~1995!#. Arrows have been placed in Figs. 5 and 7 in order to locate g˙ min with respect to the viscosity flow curve, so that the reader may judge the accuracy of this prediction. Here we consider an alternative procedure which does not require the full three-region viscosity flow curve. We make a quantitative test of the conjecture that the shift factors a T, s and a C, s can be used to estimate g˙ min . The chosen reference state is once again 25 °C, 55 wt % polymer. Shifts in the mean relaxation time with respect to the reference state are calculated from shear-stress curves using the same procedures discussed in the preceding sections, and are denoted as a T,C, s in Fig. 13. Alternatively, shifts in the mean relaxation time ~denoted a T,C,N ) can also be calculated as g˙ min,r /g˙ min , where g˙ min is the shear rate at which N1 is minimum for the liquid crystalline state of interest, and g˙ min,r is the shear rate at which N1 is minimum in


46


FIG. 13. Temperature–concentration shift factor a T,C,N calculated from the relative minimum in N1 vs shift factor a T,C, s calculated from the shear stress, for 24 different liquid crystalline states of HPC in m-cresol. Included are states with positive and states with negative values for the relative minimum in N1. The line shown is drawn to guide the eye only; it has slope equal one and passes through ~1,1!.

the reference state. These alternative ways of computing the shift factor are compared in Fig. 13 for 24 different liquid crystalline states over a range of temperatures from 25 to 130 °C and polymer concentrations from 35 to 65 wt %. The various liquid crystalline states encompass a wide portion of the HPC phase diagram, with the shift factor varying by over 3 orders of magnitude. Obviously there is a very strong correlation between the mean relaxation time calculated using shear stress data and from the location of the relative minimum in the N1 flow curve. Roughly the same correlation appears to be valid whether the relative minimum corresponds to either a positive or negative N1 value. The data points are scattered about a line of slope unity. The shift in relaxation time from the shear stress is within a factor of about 2–5 of the shift factor calculated from the relative minimum of N1 over a range of over 3 decades in shift factors. Berry and co-workers have already shown how to do time–temperature and time– concentration shifting on the viscosity flow curves of a lyotropic LCP using the lowshear-rate recoverable compliance @Einaga et al. ~1985!; Berry ~1991!#. However, Fig. 13 appears to be the first demonstration that a correlation exists between viscosity and g˙ min values when both temperature and concentration changes are included. Mewis and Moldenaers have shown that variations in h 0 with respect to temperature can be used to predict variations in g˙ min with respect to temperature at a single concentration of liquid crystalline PBG @Mewis and Moldenaers ~1987!#. However, until now, no one has demonstrated a similar ability to predict variations in g˙ min with respect to concentration over a wide concentration range. It would be interesting to see if the rough correlation of Fig. 13 between shear stress and g˙ min is similar for liquid crystalline PBG when the concentration varies, especially since g˙ min for PBG shows the opposite trend with concentration as HPC @Baek et al. ~1993a!#.



47

FIG. 14. The shear stress ~top! and the first normal stress ~bottom! at two shear rates near the expected location of the minimum N1 value. Each different symbol pair represents a fresh sample; lines are drawn to guide the eye only (T 5 130 °C, c p 5 65 wt %).

IV. CONCLUSIONS Based on the presence of a relative minimum in a plot of first normal stress difference N1 against shear rate ( g˙ min), semiflexible HPC is a tumbling liquid crystal in shear flow, at least for polymer concentrations up to about 70 wt %, which is the highest concentration for which reliable N1 data can be obtained. A dividing line has been located in the phase diagram of HPC, to the left of which the minimum in N1 is negative, and to the right of which the minimum is positive. The apparent activation energy for flow is the same on both sides of this dividing line, indicating that no fundamental change in flow mechanism occurs as the concentration increases, up to the threshold of the thermotropic state. This suggests that thermotropic HPC also tumbles, or if it does not, it is not because the polymer chains are too flexible at temperatures required to melt the thermotropic material.


48


A procedure has been developed for extending experimental shear rheology to longer time scales and higher polymer concentrations using approximate time–temperature and time–concentration shift factors. Shift factors calculated using shear-stress curves and calculated using the shear-rate location of the minimum ~positive or negative! in the N1 flow curve are the same over 3 decades of shift factor to within a factor of 2–5 scatter in the data. For densely packed HPC, unlike rod-like PBG, variations in the liquid-crystal order parameter can evidently be neglected when calculating shift factors. However, since shear-thinning effects need to be accounted for, temperature shift factors should be calculated at fixed shear stress, and concentration shift factors should be calculated at a constant ratio of shear stress to concentration. Appropriate calculation of the shift factors makes it possible to use viscosity data measured at low shear rates before onset of edge fracture to estimate the shear rate necessary for flow alignment. ACKNOWLEDGMENTS The authors are grateful to Professor Mohan Srinivasarao for his help with the optical microscopy measurements. APPENDIX—EDGE INSTABILITIES Difficulty in controlling the sample edge condition is a common experimental problem in high-shear-rate rheometry @Macosko ~1994!#. In our previous work, we showed that we could always determine the correct sign of N1, if not the correct magnitude, even in the presence of edge fracture @Baek et al. ~1994!#. Here we extend this result by showing that we can confidently determine the existence of a relative minimum in N1, whether positive or negative, even if edge fracture is occurring. Figure 14 contains results which are typical of those observed at higher polymer concentrations, in this case c p 5 65 wt %, T 5 130 °C. At this state point, based on extrapolation from lower concentrations, we expect g˙ min to be about 300 s21, and N1 to be positive at the minimum. Unfortunately, the sample fractures at a shear rate near 100 s21. Therefore, we repeat the stress measurements four times, each time using a fresh, unfractured sample. Each time, the apparent N1 value is lower at g˙ 5 300 s21 than at g˙ 5 30 s21, whereas the reverse is true for the apparent shear stress. Furthermore, if we reduce the shear rate on the fractured sample back to 30 s21, the value of N1 increases ~values not shown!. Therefore, the relative minimum which is observed for N1 is in all likelihood due to director tumbling, not edge fracture. Also note that there is no evidence of negative N1 values in Fig. 14, nor at measurements made at nearby shear rates. At this polymer concentration ~65 wt %!, the clearing temperature from POM is about 145 °C. At this polymer concentration, N1 measurements were made at temperatures between 70 and 140 °C, and no clear evidence for negative N1 values was observed at any shear rate @Huang ~1998!#. Therefore we conclude that the relative minimum in N1 is most likely always non-negative at this polymer concentration, or at the least is non-negative in all but a very narrow shear rate/temperature range.

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