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Rheology of a liquid crystalline polymer dispersed in a flexible polymer matrix
J. Non-Newtonian Fluid Mech. 86 (1999) 3±14
Rheology of a liquid crystalline polymer dispersed in a flexible polymer matrix Brian L. Riisea,b, Nathalie Miklera,b, Morton M. Denna,b,* a
Material Sciences Division, Lawrence Berkeley National Laboratory, University of California, Berkeley, CA 94720-1462, USA b Department of Chemical Engineering, University of California, Berkeley, CA 94720-1462, USA Received 4 June 1998; received in revised form 25 August 1998
Abstract The addition of 10% of an aqueous lyotropic liquid crystalline dispersed phase to an isotropic polydimethylsiloxane melt increases the steady shear and normal stresses and the dynamic moduli, but the increase caused by the liquid crystalline dispersed phase is much less than expected from the Palierne and Doi±Ohta theories of blends. In contrast, the Palierne theory does an adequate job of predicting the linear viscoelastic behavior of a blend with a 20% LCP dispersed phase. # 1999 Elsevier Science B.V. All rights reserved. Keywords: Blend; Liquid crystalline polymer; Palierne model; Emulsion model; Doi±Ohta model
1. Introduction Blends containing small amounts of liquid crystalline polymers (LCP) in a flexible polymer matrix are of industrial interest because the small amount of LCP can lead to easier processing and improved mechanical properties (e.g., [1]). Studies carried out thus far, have concentrated on blends of thermotropic LCPs with molten flexible polymers. The rheology of such blends is complex, Kim and Denn [2], for example, found that adding a small amount of a thermotropic polyester to polyethylene terephthalate caused a minimum in the shear viscosity at very low shear rates, where the dispersedphase morphology was close to spherical, with a corresponding minimum in the linear viscoelastic complex viscosity. Thermotropic LCPs have to be studied at high temperatures, where chemical reactions, including ester exchange and degradation, can occur. In this work we examine the rheology of a model system of aqueous solutions of hydroxypropyl cellulose (HPC) dispersed in a matrix of polydimethylsiloxane ÐÐÐÐ * Corresponding author. E-mail: [email protected] 0377-0257/99/$ ± see front matter # 1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 7 - 0 2 5 7 ( 9 8 ) 0 0 1 9 9 - 2
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(PDMS). HPC is lyotropic, so it is possible to study blends in which the dispersed phase is either isotropic or liquid crystalline, and the rheological experiments can be carried out at or near room temperature. 2. Rheological models 2.1. The Palierne model The Palierne [3,4] model describes the linear viscoelastic behavior of dispersions of droplets of an incompressible viscoelastic liquid in an incompressible viscoelastic matrix. If the interfacial tension between the components is independent of shear and the polydispersity of the droplet distribution of the dispersed phase is less than 2.3, the model predicts the following expression for the complex shear modulus G*(!) of the blend in terms of the complex moduli Gm*(!) and Gd*(!) of the matrix and dispersed phases, respectively as 1 3 H ! 0 00 (1) G ! G ! iG ! Gm ! 1 ÿ 2 H ! where H !
4=Rv 2Gm ! 5Gd ! Gd ! ÿ Gm !16Gm ! 19Gd ! 40=Rv Gm ! Gd ! 2Gd ! 3Gm !16Gm ! 19Gd !
(2)
is the interfacial tension, the volume fraction of the dispersed phase, and Rv the volume-averaged radius of the dispersed phase. Intrinsic to the model is a characteristic time scale for droplet dynamics of order mRv/, where m is the viscosity of the matrix fluid. (There is a scale factor that depends on the viscosity ratio, but it is of the order of unity for the viscosities considered here.) The Palierne model has been shown to describe data for immiscible blends of isotropic polymers (e.g., [5±9]). The model does not describe the linear viscoelastic behavior of a blend of a thermotropic polyester LCP in an isotropic polyester melt [2] or that of highly concentrated blends with non-deformable dispersed phases [10]. 2.2. The Doi±Ohta model The Doi±Ohta [11] model describes the transient and steady shear rheology and interface evolution of a 1 : 1 blend of immiscible Newtonian liquids having the same viscosity and density. According to the theory, the steady-state shear stress and the first normal stress difference N1 are both proportional to the shear rate. This proportionality was found to hold for the excess stresses caused by the interface in blends of viscoelastic liquids of various compositions and viscosity ratios [6,12±14], where the excess stresses are defined as follows: _ _ _ ÿ d ÿ 1 ÿ m ;
(3a)
_ N1 _ ÿ N1d ÿ 1 ÿ N1m j j: _ N1
(3b)
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The subscripts d and m refer to the dispersed and matrix phases, respectively. The theory always predicts a positive deviation from a linear mixing rule for the steady-state stresses, in contrast to the results reported for blends of thermotropic LCPs and isotropic melts (e.g., [1,2]). The theory also predicts a characteristic length scale for the dispersed phase as : (4) L nm _ This scaling has been observed for isotropic viscoelastic blends with different viscosity and volume ratios by Vinckier and coworkers [6], but not by Lacroix and coworkers [8]. 3. Materials The matrix fluid for all experiments was a polydimethylsiloxane (Aldrich PDMS 200), while solutions of hydroxypropyl cellulose (Hercules Klucel EF, Lot 3577, nominal molecular weight 100 000) were used for the dispersed phase. Aqueous solutions of HPC form a liquid crystalline phase above a critical concentration; the liquid crystalline phase is likely to be cholesteric at very low shear rates, but to become nematic at higher rates [15]. We prepared HPC solutions of 35 and 50 wt.% by adding powdered HPC to distilled water and stirring intermittently for several days. All rheological measurements were carried out at 128C; according to the phase diagram prepared by Guido [16] for this HPC/water system, the 35% solution is isotropic and the 50% solution is fully liquid crystalline at 128C. (Our samples showed the same features with optical microscopy as those reported by Guido.) The densities of the PDMS and 35% and 50% HPC solutions are, respectively, 970, 1060 and 1090 kg/ m3. Blends were prepared in both a Brabender mixer and a kitchen blender, with little difference in droplet sizes. All blends used in the results reported here were prepared in the kitchen blender. A 10 wt.% blend of the 35% HPC solution (HPC-I) is denoted I-10 (for `isotropic'); 10 and 20 wt.% blends of 50% HPC (HPC-N) ae denoted N-10 and N-20 (for `nematic'), respectively. Detailed dropletsize distributions were determined for N-10 by using a CCD camera attached to relay lens and analyzing the images using NIH Image software. Visual observation through an optical microscope was employed to obtain characteristic droplet sizes for the I-10 and N-20 blends. The method of determining the interfacial tension is described in the Appendix. 4. Rheological measurement All rheological measurements were made on a Rheometrics Mechanical Spectrometer (RMS 800) with cone and plate geometry. A cone with 12.5 mm radius and 0.1 radian cone angle was used for most measurements; a 50 mm cone with a 0.04 rad cone angle was necessary to obtain reproducible torque measurements in steady shear for HPC-N and reproducible normal forces for HPC-N, HPC-I and PDMS. Torque measurements were made with either a 200 g cm or a 2000 g cm transducer, depending on the sensitivity required. The rheometer was equipped with a fluid bath for temperature control of 0.18C. After the material was loaded in the rheometer, a shear rate of 0.3 sÿ1 was applied for 300 s and excess material was removed from the 25 mm cone and plate by holding a curved spatula at the sample
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edge. Trimming was not possible with the 50 mm cone. Samples were left to relax for up to one hour after loading and between measurements. Reproducible normal stresses could only be obtained for N1 values greater than 20 Pa. Strain sweeps established that the response of the pure PDMS and the HPC solutions was linear at 5% strain, and this value was used for all dynamic measurements reported here. 5. Results 5.1. Rheology of pure components Steady shear data for the PDMS and the isotropic and liquid crystalline HPC solutions are shown in Fig. 1. The PDMS has a nearly constant viscosity and quadratic normal stresses over the attainable range. The isotropic HPC solution (HPC-I) exhibits some shear thinning, while the liquid crystalline HPC solution (HPC-N) shows a characteristic `three-region' viscosity curve. The viscosity ratios in all cases are within the range where droplet breakup of the dispersed phase is expected to occur for Newtonian fluids [17]. The normal stresses for the LCP are positive over the entire accessible range, in contrast to the results of Grazed and coworkers [18], although there is an inflection in the normal stress curve near the start of Region II. The normal stresses in the LCP are significantly larger than those in the isotropic HPC at low rates, while the situation is reversed at higher rates. Dynamic data for the PDMS and the isotropic and liquid crystalline HPC solutions are shown in Fig. 2. The PDMS and isotropic HPC show the typical approach to linear G00 and quadratic G0 at low frequencies. The liquid crystalline HPC appears to approach a non-zero value of G0 at low frequencies. A comparison between 2G0 and N1 for the three fluids is shown in Fig. 3; the agreement between the two rheological functions is very good for the isotropic fluids, whereas there is no agreement for the LCP. 5.2. Droplet size distribution A histogram of the droplet size distribution for blend N-10 (10 wt.% liquid crystalline HPC in PDMS) is shown in Fig. 4. The volume- (Rv) and number-averaged (Rn) radii are 7.1 and 3.4 mm,
Fig. 1. Steady shear viscosity (open symbols) and first normal stress difference (filled symbols) for PDMS, HPC-N and HPC-I at 128C.
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Fig. 2. Dynamic storage (filled symbols) and loss (open symbols) moduli for PDMS, HPC-N and HPC-I at 128C.
Fig. 3. Comparison of 2G0 and N1 for PDMS, HPC-N and HPC-I at 128C.
respectively, with a polydispersity Rv/Rn 2.1, which is within the range for which the Palierne model may be applied in the form given in Eqs. (1) and (2). Droplet distribution data following shearing at rates up to 1 sÿ1 and strains up to 900 are shown in Table 1. There is little effect of strain and strain rate on the droplet size or polydispersity, except that the distribution is somewhat narrower after shearing at
Fig. 4. Droplet size distribution for N-10 after mixing in the kitchen blender.
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Table 1 Droplet distribution data for 10 wt.% blend of liquid crystalline HPC in PDMS Shear rate (sÿ1) No preshear 0.1 0.3 0.5 1.0 0.1 0.3 0.5 1.0 1.0
Total strain
Rv (mm)
Rn (mm)
Rv/Rn
0 60 60 60 60 540 540 540 540 900
7.1 8.7 7.6 8.2 6.6 8.8 7.6 7.1 7.6 6.6
3.4 3.8 3.7 3.8 3.2 3.6 3.8 3.6 3.8 4.5
2.1 2.3 2.1 2.2 2.1 2.5 2.0 2.0 2.0 1.5
1 sÿ1 for 900 strain units. This insensitivity to shear is inconsistent with the scaling predicted by the Doi±Ohta model, Eq. (4). Detailed droplet size distributions were not obtained for the other blends. From observations through the optical microscope the characteristic size of the 10% blend of isotropic HPC was similar to that of the N-10, while the characteristic droplet size for the 20% blend of liquid crystalline HPC was about 15 mm. These values are likely to be closer to number-averaged than volume-averaged radii. 5.3. Blend rheology The viscosities and primary normal stresses for the matrix fluid and the three blends are shown in Figs. 5 and 6, respectively. No overshoot or undershoot was observable during step-rate experiments for any of the blends at the lower rates, with only a slight overshoot at the higher rates; undershoot was never observed. The viscosities are plotted on a linear scale, so the apparent large shear thinning for the N-20 blend is deceptive; the change is only about 30% over two decades of shear rate. The excess stresses calculated using Eqs. (3a) and (3b) are plotted in Fig. 7. The excess shear stresses for the two 10% blends are approximately linear, consistent with the extension of the Doi±Ohta theory; there appears to be a deviation from linearity at the higher rates for the 20% blend with an LCP dispersed
Fig. 5. Steady shear viscosity of PDMS, N-10, I-10 and N-20 at 128C.
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Fig. 6. Primary normal stress difference for PDMS, N-10, I-10 and N-20 at 128C.
phase. Due to the substantial scatter it is difficult to draw definitive conclusions about the excess normal stresses, but it appears that the 10% blends again have a linear dependence on the shear rate, while the 20% blend seems closer to a quadratic dependence. It is notable that both the excess shear and normal stresses in the 10% blend with an isotropic dispersed phase (I-10) are threefold those for the 10% LCP dispersed phase (N-10). The storage moduli for the pure PDMS and the two 10% blends are shown in Fig. 8. The interfacial tension is of the order of 10 mN/m (see Appendix); the characteristic time scale in the Palierne model is of order mRv/, which for m 120 Pas, Rv 7 mm and 10 mN/m corresponds to 0.08 s, so the influence of droplet deformation is expected at frequencies less than about 10 rad/s. There is a small deviation in the suspension data from the pure PDMS in this range. Dynamic data for the 10% blend with an isotropic dispersed phase (I-10) are shown in Fig. 9, together with calculated curves from the Palierne theory (Eqs. (1) and (2)) for various values of /Rv. /Rv 5000 and 1400 Pa correspond to volume-average drops of 2 and 7 mm, respectively, which is within the likely range based on the microscopic observation. None of the curves provides a really good overall fit, but the trend is correct; the phase angle is close to 908 at the lower frequencies, leading to considerable uncertainty in the G0 measurements. Dynamic data for the 10% blend with a liquid crystalline dispersed phase (N-10) are shown in Fig. 10. Here, the Palierne model with the measured values of and Rv deviates substantially from the
Fig. 7. (a) Excess interfacial shear stress and (b) excess interfacial primary normal stress for N-10, I-10 and N-20.
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Fig. 8. Storage modulus for PDMS and 10% HPC dispersions.
Fig. 9. Storage and loss moduli for 10 wt.% dispersion of isotropic HPC (I-10) compared with the Palierne model.
data, well outside any experimental uncertainty; indeed, the value of G0 at low frequencies is substantially below the computed curve for /Rv 0. Dynamic data for the 20% blend with a liquid crystalline dispersed phase (N-20) are shown in Fig. 11. Here, the Palierne model does a reasonably good job of describing the data, including
Fig. 10. Storage and loss moduli for 10 wt.% dispersion of liquid crystalline HPC (N-10) compared with the Palierne model.
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Fig. 11. Storage and loss moduli for 20 wt.% dispersion of liquid crystalline HPC (N-20) compared with the Palierne Model.
Fig. 12. Effect of preshearing on the storage modulus for 20 wt.% dispersion of liquid crystalline HPC (N-20).
capturing the distinct shoulder in G0 at frequencies between 1 and 10 rad/s, although there is a systematic deviation at the higher frequencies. The importance of interfacial effects is clearly seen, with a large positive deviation in G0 from the computed curve for /Rv 0. The curve for /Rv 667 Pa corresponds to the measured interfacial tension and a volume-average radius of 15 mm; the fit is better, especially for G0 , if /Rv 1000 (Rv 10 mm). There was no significant difference in the values of G0 for experiments without preshear and with 900 strain units of preshear at frequencies above 3 rad/s; at lower frequencies there was a small increase in G0 with preshearing, roughly of the order of the difference between the curves for /Rv 667 and / Rv 1000 (Fig. 12). Similar behavior was observed for the 10% solutions. 6. Discussion The addition of 10% of the liquid crystalline dispersed phase to the isotropic matrix fluid increases the steady shear and normal stresses and the dynamic moduli, in contrast to the low-rate behavior observed with polyesters, but the increase caused by the liquid crystalline dispersed phase is much less
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than what might be expected. The excess stresses associated with the interfacial tension from the 10% isotropic dispersed phase are several times larger than those from the 10% LCP dispersed phase; similarly, the effect of the LCP dispersed phase on the dynamic storage modulus is considerably less than expected from the Palierne theory. In contrast, the Palierne theory does an adequate job of predicting the linear viscoelastic behavior of the blend with a 20% LCP dispersed phase, and the contribution to the interfacial steady stresses is an order of magnitude greater than that of the 10% LCP dispersed phase. The experimental observations for the 10% LCP dispersed phase seem to require an interfacial tension that is substantially less than 10 mN/m for consistency with the theories for blends, and in fact the storage modulus data at low frequencies, where the presence of a dispersed phase with a finite interfacial tension should have the greatest effect, lie below the theoretical curve for /Rv 0. Guido [16] reported a characteristic length scale for texture for the liquid crystalline HPC system of about 5 mm, which is comparable to the characteristic droplet size in the 10% system (and indeed larger than most droplets); i.e., on the average, the droplets are likely to contain a very small number of `domains'. By contrast, the 15 mm droplets in the 20% system are more than 25 times larger in volume than a `domain' and are likely to be characteristic of the bulk material. There is no obvious reason why the domain texture would affect the interfacial properties, and if so, why texture would result in a reduced interfacial tension for a monodomain, but the issue is worth further investigation. It is generally believed that liquid crystalline HPC solutions are cholesteric at rest but nematic under shear [15]; this structural change and the differences in texture may also play a role in the observed behavior (e.g., small droplets experience less deformation than large droplets), but it is not obvious how or why. HPC solutions are shear sensitive; Gentzler and Miller [19], for example, found a 20% decrease in the GPC-measured molecular weight of Klucel EF solutions following repeated passes through a 4 : 1 contraction, with a slight increase in the polydispersity. Some degradation of the HPC could have occurred during blending, although we did not expect degradation to be the problem in the relatively dilute blends studied here. We have repeated all calculations for the Palierne model assuming a tenfold drop in the modulus of the dispersed phase, with little change in behavior; this is because G00 is not sensitive to the dispersed phase and the major effect of the dispersed phase on G0 is from the interfacial term.
Fig. 13. Estimation of interfacial tension by placing a drop of PDMS on HPC.
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Fig. 14. HPC-N droplet in contact with PDMS.
The relative insensitivity of the droplet-size distribution of N-10 to the degree of shear seems to indicate a substantial deviation from the predictions of the Doi±Ohta theory. It may be that shearinduced coalescence of the small droplets created by the high-intensity blending was too slow a process to be observed in our blends, which were all far from the 1 : 1 composition assumed in the theory. The small change in the size distribution for shearing of 900 strain units and the small increase in G0 with preshearing at frequencies where droplet deformation dynamics are significant both suggest some effect of shear on the droplet size distribution, but of such small magnitude that it can be discounted as a major factor in the overall observations. 7. Conclusions The behavior of a polymer blend with a 10% liquid crystalline dispersed phase in an isotropic matrix does not follow the predictions of the Palierne and Doi±Ohta theories for small droplets, but a 20% blend with larger droplets is consistent with the former theory. In the absence of a theory for blends with an LCP dispersed phase, the experimental observations stand without explanation, but several interesting issues emerge. The first is the difference between the behavior of the 10% blends with isotropic and LCP dispersed phases, where the latter seems to respond in part like a system with a sharply reduced interfacial tension. Similarly, the contrast between the 10% and 20% LCP blends is striking, where the only obvious physical difference between the two concentrations is that the size scale of droplets in the 10% blend is of the order of one domain, while droplets in the 20% blend are of the order of 25 domains. Finally, the insensitivity to shear and the apparent absence of coalescence is unexpected. The underlying physical issues seem to be the nature of the interface between the liquid crystalline and isotropic phases and the deformation of liquid crystalline droplets, and this is where research should be focused. Acknowledgements This work was supported in part by the Director, Office of Energy Research, Office of Basic Energy Sciences, Materials Science Division of the US Department of Energy under Contract No. DE-AC07376SF00098.
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Appendix A Interfacial tension measurement If we put a droplet of PDMS on a layer of HPC, a surface force balance (Fig. 13) yields the following HPC=air PDMS=HPC cos 1 PDMS=air cos 2 :
(5)
The surface tension of PDMS at room temperature was reported to be 19.8 mN/m [20]. We used the pendant drop method to estimate the room temperature surface tension of the 50 wt.% HPC, which we found to be approximately 30 mN/m. From photographs of PDMS droplets on HPC-N (e.g., Fig. 14) we found 1 88 and 2 98, leading to a value of PDMS/HPC of around 10 mN/m at room temperature. Tsakalos and coworkers [21,22] have reported an interfacial tension of about 13 mN/m for the HPC-PDMS system for both isotropic and liquid crystalline HPC using several different methods. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22]
A.A. Handlos, D.G. Baird, J. Macromol. Sci. Rev. Macromol. Chem. Phys. C35 (1995) 183. W.-N. Kim, M.M. Denn, J. Rheol. 36 (1992) 1477. J.F. Palierne, Rheol. Acta 29 (1990) 204. J.F. Palierne, Rheol. Acta 30 (1991) 497. D. Graebling, R. Muller, J.F. Palierne, Macromolecules 26 (1993) 320. I. Vinckier, P. Moldenaers, J. Mewis, J. Rheol. 40 (1996) 613. S. Kitake, A. Ichikawa, N. Imura, Y. Takahashi, I. Noda, J. Rheol. 41 (1997) 1039. C. Lacroix, M. Grmela, P.J. Carreau, J. Rheol. 42 (1998) 41. H. Yamane, M. Takahashi, R. Hayashi, K. Okamoto, H. Kishihara, T. Masuda, J. Rheol. 42 (1998) 567. M. Bousmina, R. Muller, J. Rheol. 37 (1993) 663. M. Doi, T. Ohta, J. Chem. Phys. 95 (1991) 1242. Y. Takahashi, S. Kitade, N. Kurashima, I. Noda, Polym. J. 26 (1994) 1206. Y. Takahashi, N. Kurashima, I. Noda, M. Doi, J. Rheol. 38 (1994) 699. I. Vinckier, P. Moldenaers, J. Mewis, J. Rheol. 41 (1997) 705. K. Hongladarom, W.R. Burghardt, Rheol. Acta 37 (1998) 46. S. Guido, Macromolecules 28 (1995) 4530. H.P. Grace, Chem. Eng. Commun. 14 (1982) 225. N. Grazed, S. Cavella, P. Cicarelli, J. Rheol. 34 (1990) 1293. M. Gentzler, C. Miller, U.C. Berkeley, 1998, personal communication. S. Wu, J. Adhesion 5 (1973) 39. V. Tsakalos, D. Rusu, P. Navard, E. Peuvrel-Disdier, Synthetic Polymer J. 1 (1994) 133. V.T. Tsakalos, These de Doctorat, Ecole des Mines de Paris, Sophia-Antipolis, 1995.
Rheology of a liquid crystalline polymer dispersed in a flexible polymer matrix Brian L. Riisea,b, Nathalie Miklera,b, Morton M. Denna,b,* a
Material Sciences Division, Lawrence Berkeley National Laboratory, University of California, Berkeley, CA 94720-1462, USA b Department of Chemical Engineering, University of California, Berkeley, CA 94720-1462, USA Received 4 June 1998; received in revised form 25 August 1998
Abstract The addition of 10% of an aqueous lyotropic liquid crystalline dispersed phase to an isotropic polydimethylsiloxane melt increases the steady shear and normal stresses and the dynamic moduli, but the increase caused by the liquid crystalline dispersed phase is much less than expected from the Palierne and Doi±Ohta theories of blends. In contrast, the Palierne theory does an adequate job of predicting the linear viscoelastic behavior of a blend with a 20% LCP dispersed phase. # 1999 Elsevier Science B.V. All rights reserved. Keywords: Blend; Liquid crystalline polymer; Palierne model; Emulsion model; Doi±Ohta model
1. Introduction Blends containing small amounts of liquid crystalline polymers (LCP) in a flexible polymer matrix are of industrial interest because the small amount of LCP can lead to easier processing and improved mechanical properties (e.g., [1]). Studies carried out thus far, have concentrated on blends of thermotropic LCPs with molten flexible polymers. The rheology of such blends is complex, Kim and Denn [2], for example, found that adding a small amount of a thermotropic polyester to polyethylene terephthalate caused a minimum in the shear viscosity at very low shear rates, where the dispersedphase morphology was close to spherical, with a corresponding minimum in the linear viscoelastic complex viscosity. Thermotropic LCPs have to be studied at high temperatures, where chemical reactions, including ester exchange and degradation, can occur. In this work we examine the rheology of a model system of aqueous solutions of hydroxypropyl cellulose (HPC) dispersed in a matrix of polydimethylsiloxane ÐÐÐÐ * Corresponding author. E-mail: [email protected] 0377-0257/99/$ ± see front matter # 1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 7 - 0 2 5 7 ( 9 8 ) 0 0 1 9 9 - 2
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(PDMS). HPC is lyotropic, so it is possible to study blends in which the dispersed phase is either isotropic or liquid crystalline, and the rheological experiments can be carried out at or near room temperature. 2. Rheological models 2.1. The Palierne model The Palierne [3,4] model describes the linear viscoelastic behavior of dispersions of droplets of an incompressible viscoelastic liquid in an incompressible viscoelastic matrix. If the interfacial tension between the components is independent of shear and the polydispersity of the droplet distribution of the dispersed phase is less than 2.3, the model predicts the following expression for the complex shear modulus G*(!) of the blend in terms of the complex moduli Gm*(!) and Gd*(!) of the matrix and dispersed phases, respectively as 1 3 H ! 0 00 (1) G ! G ! iG ! Gm ! 1 ÿ 2 H ! where H !
4=Rv 2Gm ! 5Gd ! Gd ! ÿ Gm !16Gm ! 19Gd ! 40=Rv Gm ! Gd ! 2Gd ! 3Gm !16Gm ! 19Gd !
(2)
is the interfacial tension, the volume fraction of the dispersed phase, and Rv the volume-averaged radius of the dispersed phase. Intrinsic to the model is a characteristic time scale for droplet dynamics of order mRv/, where m is the viscosity of the matrix fluid. (There is a scale factor that depends on the viscosity ratio, but it is of the order of unity for the viscosities considered here.) The Palierne model has been shown to describe data for immiscible blends of isotropic polymers (e.g., [5±9]). The model does not describe the linear viscoelastic behavior of a blend of a thermotropic polyester LCP in an isotropic polyester melt [2] or that of highly concentrated blends with non-deformable dispersed phases [10]. 2.2. The Doi±Ohta model The Doi±Ohta [11] model describes the transient and steady shear rheology and interface evolution of a 1 : 1 blend of immiscible Newtonian liquids having the same viscosity and density. According to the theory, the steady-state shear stress and the first normal stress difference N1 are both proportional to the shear rate. This proportionality was found to hold for the excess stresses caused by the interface in blends of viscoelastic liquids of various compositions and viscosity ratios [6,12±14], where the excess stresses are defined as follows: _ _ _ ÿ d ÿ 1 ÿ m ;
(3a)
_ N1 _ ÿ N1d ÿ 1 ÿ N1m j j: _ N1
(3b)
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The subscripts d and m refer to the dispersed and matrix phases, respectively. The theory always predicts a positive deviation from a linear mixing rule for the steady-state stresses, in contrast to the results reported for blends of thermotropic LCPs and isotropic melts (e.g., [1,2]). The theory also predicts a characteristic length scale for the dispersed phase as : (4) L nm _ This scaling has been observed for isotropic viscoelastic blends with different viscosity and volume ratios by Vinckier and coworkers [6], but not by Lacroix and coworkers [8]. 3. Materials The matrix fluid for all experiments was a polydimethylsiloxane (Aldrich PDMS 200), while solutions of hydroxypropyl cellulose (Hercules Klucel EF, Lot 3577, nominal molecular weight 100 000) were used for the dispersed phase. Aqueous solutions of HPC form a liquid crystalline phase above a critical concentration; the liquid crystalline phase is likely to be cholesteric at very low shear rates, but to become nematic at higher rates [15]. We prepared HPC solutions of 35 and 50 wt.% by adding powdered HPC to distilled water and stirring intermittently for several days. All rheological measurements were carried out at 128C; according to the phase diagram prepared by Guido [16] for this HPC/water system, the 35% solution is isotropic and the 50% solution is fully liquid crystalline at 128C. (Our samples showed the same features with optical microscopy as those reported by Guido.) The densities of the PDMS and 35% and 50% HPC solutions are, respectively, 970, 1060 and 1090 kg/ m3. Blends were prepared in both a Brabender mixer and a kitchen blender, with little difference in droplet sizes. All blends used in the results reported here were prepared in the kitchen blender. A 10 wt.% blend of the 35% HPC solution (HPC-I) is denoted I-10 (for `isotropic'); 10 and 20 wt.% blends of 50% HPC (HPC-N) ae denoted N-10 and N-20 (for `nematic'), respectively. Detailed dropletsize distributions were determined for N-10 by using a CCD camera attached to relay lens and analyzing the images using NIH Image software. Visual observation through an optical microscope was employed to obtain characteristic droplet sizes for the I-10 and N-20 blends. The method of determining the interfacial tension is described in the Appendix. 4. Rheological measurement All rheological measurements were made on a Rheometrics Mechanical Spectrometer (RMS 800) with cone and plate geometry. A cone with 12.5 mm radius and 0.1 radian cone angle was used for most measurements; a 50 mm cone with a 0.04 rad cone angle was necessary to obtain reproducible torque measurements in steady shear for HPC-N and reproducible normal forces for HPC-N, HPC-I and PDMS. Torque measurements were made with either a 200 g cm or a 2000 g cm transducer, depending on the sensitivity required. The rheometer was equipped with a fluid bath for temperature control of 0.18C. After the material was loaded in the rheometer, a shear rate of 0.3 sÿ1 was applied for 300 s and excess material was removed from the 25 mm cone and plate by holding a curved spatula at the sample
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edge. Trimming was not possible with the 50 mm cone. Samples were left to relax for up to one hour after loading and between measurements. Reproducible normal stresses could only be obtained for N1 values greater than 20 Pa. Strain sweeps established that the response of the pure PDMS and the HPC solutions was linear at 5% strain, and this value was used for all dynamic measurements reported here. 5. Results 5.1. Rheology of pure components Steady shear data for the PDMS and the isotropic and liquid crystalline HPC solutions are shown in Fig. 1. The PDMS has a nearly constant viscosity and quadratic normal stresses over the attainable range. The isotropic HPC solution (HPC-I) exhibits some shear thinning, while the liquid crystalline HPC solution (HPC-N) shows a characteristic `three-region' viscosity curve. The viscosity ratios in all cases are within the range where droplet breakup of the dispersed phase is expected to occur for Newtonian fluids [17]. The normal stresses for the LCP are positive over the entire accessible range, in contrast to the results of Grazed and coworkers [18], although there is an inflection in the normal stress curve near the start of Region II. The normal stresses in the LCP are significantly larger than those in the isotropic HPC at low rates, while the situation is reversed at higher rates. Dynamic data for the PDMS and the isotropic and liquid crystalline HPC solutions are shown in Fig. 2. The PDMS and isotropic HPC show the typical approach to linear G00 and quadratic G0 at low frequencies. The liquid crystalline HPC appears to approach a non-zero value of G0 at low frequencies. A comparison between 2G0 and N1 for the three fluids is shown in Fig. 3; the agreement between the two rheological functions is very good for the isotropic fluids, whereas there is no agreement for the LCP. 5.2. Droplet size distribution A histogram of the droplet size distribution for blend N-10 (10 wt.% liquid crystalline HPC in PDMS) is shown in Fig. 4. The volume- (Rv) and number-averaged (Rn) radii are 7.1 and 3.4 mm,
Fig. 1. Steady shear viscosity (open symbols) and first normal stress difference (filled symbols) for PDMS, HPC-N and HPC-I at 128C.
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Fig. 2. Dynamic storage (filled symbols) and loss (open symbols) moduli for PDMS, HPC-N and HPC-I at 128C.
Fig. 3. Comparison of 2G0 and N1 for PDMS, HPC-N and HPC-I at 128C.
respectively, with a polydispersity Rv/Rn 2.1, which is within the range for which the Palierne model may be applied in the form given in Eqs. (1) and (2). Droplet distribution data following shearing at rates up to 1 sÿ1 and strains up to 900 are shown in Table 1. There is little effect of strain and strain rate on the droplet size or polydispersity, except that the distribution is somewhat narrower after shearing at
Fig. 4. Droplet size distribution for N-10 after mixing in the kitchen blender.
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Table 1 Droplet distribution data for 10 wt.% blend of liquid crystalline HPC in PDMS Shear rate (sÿ1) No preshear 0.1 0.3 0.5 1.0 0.1 0.3 0.5 1.0 1.0
Total strain
Rv (mm)
Rn (mm)
Rv/Rn
0 60 60 60 60 540 540 540 540 900
7.1 8.7 7.6 8.2 6.6 8.8 7.6 7.1 7.6 6.6
3.4 3.8 3.7 3.8 3.2 3.6 3.8 3.6 3.8 4.5
2.1 2.3 2.1 2.2 2.1 2.5 2.0 2.0 2.0 1.5
1 sÿ1 for 900 strain units. This insensitivity to shear is inconsistent with the scaling predicted by the Doi±Ohta model, Eq. (4). Detailed droplet size distributions were not obtained for the other blends. From observations through the optical microscope the characteristic size of the 10% blend of isotropic HPC was similar to that of the N-10, while the characteristic droplet size for the 20% blend of liquid crystalline HPC was about 15 mm. These values are likely to be closer to number-averaged than volume-averaged radii. 5.3. Blend rheology The viscosities and primary normal stresses for the matrix fluid and the three blends are shown in Figs. 5 and 6, respectively. No overshoot or undershoot was observable during step-rate experiments for any of the blends at the lower rates, with only a slight overshoot at the higher rates; undershoot was never observed. The viscosities are plotted on a linear scale, so the apparent large shear thinning for the N-20 blend is deceptive; the change is only about 30% over two decades of shear rate. The excess stresses calculated using Eqs. (3a) and (3b) are plotted in Fig. 7. The excess shear stresses for the two 10% blends are approximately linear, consistent with the extension of the Doi±Ohta theory; there appears to be a deviation from linearity at the higher rates for the 20% blend with an LCP dispersed
Fig. 5. Steady shear viscosity of PDMS, N-10, I-10 and N-20 at 128C.
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Fig. 6. Primary normal stress difference for PDMS, N-10, I-10 and N-20 at 128C.
phase. Due to the substantial scatter it is difficult to draw definitive conclusions about the excess normal stresses, but it appears that the 10% blends again have a linear dependence on the shear rate, while the 20% blend seems closer to a quadratic dependence. It is notable that both the excess shear and normal stresses in the 10% blend with an isotropic dispersed phase (I-10) are threefold those for the 10% LCP dispersed phase (N-10). The storage moduli for the pure PDMS and the two 10% blends are shown in Fig. 8. The interfacial tension is of the order of 10 mN/m (see Appendix); the characteristic time scale in the Palierne model is of order mRv/, which for m 120 Pas, Rv 7 mm and 10 mN/m corresponds to 0.08 s, so the influence of droplet deformation is expected at frequencies less than about 10 rad/s. There is a small deviation in the suspension data from the pure PDMS in this range. Dynamic data for the 10% blend with an isotropic dispersed phase (I-10) are shown in Fig. 9, together with calculated curves from the Palierne theory (Eqs. (1) and (2)) for various values of /Rv. /Rv 5000 and 1400 Pa correspond to volume-average drops of 2 and 7 mm, respectively, which is within the likely range based on the microscopic observation. None of the curves provides a really good overall fit, but the trend is correct; the phase angle is close to 908 at the lower frequencies, leading to considerable uncertainty in the G0 measurements. Dynamic data for the 10% blend with a liquid crystalline dispersed phase (N-10) are shown in Fig. 10. Here, the Palierne model with the measured values of and Rv deviates substantially from the
Fig. 7. (a) Excess interfacial shear stress and (b) excess interfacial primary normal stress for N-10, I-10 and N-20.
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Fig. 8. Storage modulus for PDMS and 10% HPC dispersions.
Fig. 9. Storage and loss moduli for 10 wt.% dispersion of isotropic HPC (I-10) compared with the Palierne model.
data, well outside any experimental uncertainty; indeed, the value of G0 at low frequencies is substantially below the computed curve for /Rv 0. Dynamic data for the 20% blend with a liquid crystalline dispersed phase (N-20) are shown in Fig. 11. Here, the Palierne model does a reasonably good job of describing the data, including
Fig. 10. Storage and loss moduli for 10 wt.% dispersion of liquid crystalline HPC (N-10) compared with the Palierne model.
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Fig. 11. Storage and loss moduli for 20 wt.% dispersion of liquid crystalline HPC (N-20) compared with the Palierne Model.
Fig. 12. Effect of preshearing on the storage modulus for 20 wt.% dispersion of liquid crystalline HPC (N-20).
capturing the distinct shoulder in G0 at frequencies between 1 and 10 rad/s, although there is a systematic deviation at the higher frequencies. The importance of interfacial effects is clearly seen, with a large positive deviation in G0 from the computed curve for /Rv 0. The curve for /Rv 667 Pa corresponds to the measured interfacial tension and a volume-average radius of 15 mm; the fit is better, especially for G0 , if /Rv 1000 (Rv 10 mm). There was no significant difference in the values of G0 for experiments without preshear and with 900 strain units of preshear at frequencies above 3 rad/s; at lower frequencies there was a small increase in G0 with preshearing, roughly of the order of the difference between the curves for /Rv 667 and / Rv 1000 (Fig. 12). Similar behavior was observed for the 10% solutions. 6. Discussion The addition of 10% of the liquid crystalline dispersed phase to the isotropic matrix fluid increases the steady shear and normal stresses and the dynamic moduli, in contrast to the low-rate behavior observed with polyesters, but the increase caused by the liquid crystalline dispersed phase is much less
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than what might be expected. The excess stresses associated with the interfacial tension from the 10% isotropic dispersed phase are several times larger than those from the 10% LCP dispersed phase; similarly, the effect of the LCP dispersed phase on the dynamic storage modulus is considerably less than expected from the Palierne theory. In contrast, the Palierne theory does an adequate job of predicting the linear viscoelastic behavior of the blend with a 20% LCP dispersed phase, and the contribution to the interfacial steady stresses is an order of magnitude greater than that of the 10% LCP dispersed phase. The experimental observations for the 10% LCP dispersed phase seem to require an interfacial tension that is substantially less than 10 mN/m for consistency with the theories for blends, and in fact the storage modulus data at low frequencies, where the presence of a dispersed phase with a finite interfacial tension should have the greatest effect, lie below the theoretical curve for /Rv 0. Guido [16] reported a characteristic length scale for texture for the liquid crystalline HPC system of about 5 mm, which is comparable to the characteristic droplet size in the 10% system (and indeed larger than most droplets); i.e., on the average, the droplets are likely to contain a very small number of `domains'. By contrast, the 15 mm droplets in the 20% system are more than 25 times larger in volume than a `domain' and are likely to be characteristic of the bulk material. There is no obvious reason why the domain texture would affect the interfacial properties, and if so, why texture would result in a reduced interfacial tension for a monodomain, but the issue is worth further investigation. It is generally believed that liquid crystalline HPC solutions are cholesteric at rest but nematic under shear [15]; this structural change and the differences in texture may also play a role in the observed behavior (e.g., small droplets experience less deformation than large droplets), but it is not obvious how or why. HPC solutions are shear sensitive; Gentzler and Miller [19], for example, found a 20% decrease in the GPC-measured molecular weight of Klucel EF solutions following repeated passes through a 4 : 1 contraction, with a slight increase in the polydispersity. Some degradation of the HPC could have occurred during blending, although we did not expect degradation to be the problem in the relatively dilute blends studied here. We have repeated all calculations for the Palierne model assuming a tenfold drop in the modulus of the dispersed phase, with little change in behavior; this is because G00 is not sensitive to the dispersed phase and the major effect of the dispersed phase on G0 is from the interfacial term.
Fig. 13. Estimation of interfacial tension by placing a drop of PDMS on HPC.
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Fig. 14. HPC-N droplet in contact with PDMS.
The relative insensitivity of the droplet-size distribution of N-10 to the degree of shear seems to indicate a substantial deviation from the predictions of the Doi±Ohta theory. It may be that shearinduced coalescence of the small droplets created by the high-intensity blending was too slow a process to be observed in our blends, which were all far from the 1 : 1 composition assumed in the theory. The small change in the size distribution for shearing of 900 strain units and the small increase in G0 with preshearing at frequencies where droplet deformation dynamics are significant both suggest some effect of shear on the droplet size distribution, but of such small magnitude that it can be discounted as a major factor in the overall observations. 7. Conclusions The behavior of a polymer blend with a 10% liquid crystalline dispersed phase in an isotropic matrix does not follow the predictions of the Palierne and Doi±Ohta theories for small droplets, but a 20% blend with larger droplets is consistent with the former theory. In the absence of a theory for blends with an LCP dispersed phase, the experimental observations stand without explanation, but several interesting issues emerge. The first is the difference between the behavior of the 10% blends with isotropic and LCP dispersed phases, where the latter seems to respond in part like a system with a sharply reduced interfacial tension. Similarly, the contrast between the 10% and 20% LCP blends is striking, where the only obvious physical difference between the two concentrations is that the size scale of droplets in the 10% blend is of the order of one domain, while droplets in the 20% blend are of the order of 25 domains. Finally, the insensitivity to shear and the apparent absence of coalescence is unexpected. The underlying physical issues seem to be the nature of the interface between the liquid crystalline and isotropic phases and the deformation of liquid crystalline droplets, and this is where research should be focused. Acknowledgements This work was supported in part by the Director, Office of Energy Research, Office of Basic Energy Sciences, Materials Science Division of the US Department of Energy under Contract No. DE-AC07376SF00098.
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Appendix A Interfacial tension measurement If we put a droplet of PDMS on a layer of HPC, a surface force balance (Fig. 13) yields the following HPC=air PDMS=HPC cos 1 PDMS=air cos 2 :
(5)
The surface tension of PDMS at room temperature was reported to be 19.8 mN/m [20]. We used the pendant drop method to estimate the room temperature surface tension of the 50 wt.% HPC, which we found to be approximately 30 mN/m. From photographs of PDMS droplets on HPC-N (e.g., Fig. 14) we found 1 88 and 2 98, leading to a value of PDMS/HPC of around 10 mN/m at room temperature. Tsakalos and coworkers [21,22] have reported an interfacial tension of about 13 mN/m for the HPC-PDMS system for both isotropic and liquid crystalline HPC using several different methods. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22]
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