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The dynamics of two dimensional polymer nematics1
J. Non-Newtonian Fluid Mech., 76 (1998) 233 – 247
The dynamics of two dimensional polymer nematics1 T. Maruyama a, G.G. Fuller a,*, M. Grosso b, P.-L. Maffettone b a
Department of Chemical Engineering, Stanford Uni6ersity, Stanford, CA 94305 -5025, USA b Department of Chemical Engineering, Uni6ersity of Naples, Naples, Italy Received 5 August 1997; received in revised form 22 September 1997
Abstract The orientation dynamics of a monolayer of a nematic polymer solution were examined for both extensional and simple shear flows. In extensional flow, a ‘strong flow/weak flow’ criteria was investigated and found to occur at a critical rate of strain that was predicted by the simple molecular model of Doi and Hess. In simple shear flow, evidence of director wagging and flow alignment was obtained. © 1998 Elsevier Science B.V. All rights reserved. Keywords: Two dimensional polymer nematics; Simple shear flow; Orientation dynamics
1. Introduction Polymer liquid crystals are among the most difficult classes of complex liquids to understand rheologically. The complexity of these systems is revealed through a variety of nonlinear responses that include first and second normal stress differences that change sign, oscillatory shear stresses upon the start up of simple shear flow, and temporal responses that scale with the applied strain. These materials are further made complicated by the presence of a polydomain structure that results from the appearance of disclinations in the director field that defines their long range orientational order. In the case of polymer liquid crystals these defects are either very difficult to eliminate by hydrodynamic forces or are induced by flow. This is particularly thought to be the case when simple shear flow is applied, causing the director field to rotate in a manner that produces strong spatial distortions that generate large elastic stresses. For this reason, understanding the dynamics of polymer liquid crystals normally requires a recognition of the wide range of structural length scales that can contribute to their dynamics and rheology. Over the past 10–15 years, molecular and microstructural models have been advanced that offer insight into the basic mechanisms that are responsible for the non-Newtonian behavior * Corresponding author. Fax: +1 415 7257294. 1 Dedicated to the memory of Professor Gianni Astarita 0377-0257/98/$19.00 © 1998 Elsevier Science B.V. All rights reserved. PII S 0 3 7 7 - 0 2 5 7 ( 9 7 ) 0 0 1 2 0 - 1
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that was mentioned above. These models are due to Hess [1] and Doi [2] and capture the orientational flow behavior of lyotropic polymer liquid crystals through an extension of the classic rigid dumbbell model developed to describe rodlike polymers in solution. By adding a nematic potential function that accounts for interactions among the rods to the basic convection diffusion equation, simple but successful predictions of many of the important rheological responses of polymer liquid crystals have been possible. The resulting partial differential equation was first solved subject to simplifying assumptions by Doi in a manner that excluded the possibility of properly modeling effects such as director tumbling and changes in sign of the normal stress differences. These restrictions were removed by Marrucci [3] in a series of papers that examined the two dimensional form of this model which could be solved in a straightforward fashion. This approach produced a number of important predictions of real behavior and was able to anticipate numerous phenomena that have since been observed experimentally. When solved in this form the model correctly predicts director tumbling in transient simple shear flow and a transition from tumbling to ‘wagging’ and from wagging to flow alignment of the director as the shear rate is increased. These transitions were also demonstrated to be the source of the observed changes in sign of the first normal stress difference. The success of properly solving the model in two dimensions motivated Larson [4] to seek numerical solutions to the full three dimensional model and this led to predictions of changes of sign of the second normal stress difference, which were subsequently verified experimentally [5]. These models were restricted to descriptions of monodomain systems where defects are absent and it is evident that the presence of defects in the form of disclinations in the director are responsible for many rheological observations. For example, when in the tumbling regime, the Doi – Hess model does not predict steady solutions but leads to predictions of oscillatory phenomena in macroscopic properties. Several approaches to incorporate a polydomain structure into the basic model have been attempted [6,7] with the result that steady, time-independent behavior can be predicted. However, a theoretical understanding of disclinations and how to model their response to flow is presently an active research area. The vast majority of research has only considered simple shear flow, principally due to its predominance in rheometry and its importance to process flows. Consequently, the extensional flow behavior of polymer liquid crystals is largely unknown although it is recognized that such flows can dominate industrial processes such as spinning that are applied to lyotropic polymer liquid crystals. Furthermore, there are very few experimental results on systems that are free of defects that would enable a critical examination of the Doi–Hess model in the absence of this important complication. Recently, in the Stanford laboratory, progress has been made on the measurement of the orientation dynamics of monolayers of nematics formed from rodlike polymers [8,9]. The advantage of these systems are many fold. First, they can be used to directly test the two-dimensional form of the model, which is amenable to analytical solution. Second, it has been possible to largely eliminate defects in these layers and this makes it possible to test simpler, monodomain models. Finally, these systems are accessible to direct measurement of the order parameter through the use of simple methods in optical polarimetry. This latter advantage greatly simplifies both testing of models (since the calculation of the order parameter is much simpler than the calculation of the stress) and allows greater flexibility on the types of flows that can be studied. The first demonstration of the capability of making measurements directly on polymer monolayers examined the response to extensional flow and directly tested the Doi–Hess
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model [8,9]. The model was found to produce reasonable quantitative predictions of the experimental results involving two-dimensional extensional flow. In particular, good fits to the data were possible for experiments involving flow reversals and flow cessation. By fitting the model to the data, it was demonstrated that the two molecular parameters of the model, the rotational diffusion coefficient and the strength of the nematic potential, could be extracted from the data. In the present paper, the extensional flow properties of two-dimensional polymer nematics are further explored. Experimental data are offered where the concentration of the polymer is varied. In addition, a prediction of the model of a hysteresis effect in extensional flow is examined. This hysteresis is the result of a ‘weak flow/strong flow’ criteria that arises when the strength of hydrodynamic forces is either greater than or less than the strength of the nematic potential function. Finally, studies of the orientation dynamics for a layer subjected to transient simple shear flow are presented.
2. Theoretical background The basic structural element of the Doi–Hess model of polymer liquid crystals is the rigid dumbbell, which is used to prescribe the orientation of an individual test rodlike polymer chain. Such a chain is assigned an orientation given by a unit vector, u, and the system of rods in solution is characterized by a distribution of orientations specified by the orientation distribution function, c(u). The time evolution of the orientation distribution function responds to three effects: rotational Brownian diffusion; a nematic potential; and the hydrodynamic forces arising from flow. These forces combine to give the following convection–diffusion equation:
n
(c ( (c c ( ( =D + V(u) − (u; c), (t (u (u kBT (u (u
(1)
where D is the rotational diffusion coefficient and V(u) is the nematic potential function. The parameter U = 2cl 2 scales this function where c is the surface concentration of the rods and l is their length. In Ref. [9], Eq. (1) was solved for a two dimensional flow, 6 =o; (x, − y, 0) where o; is the strain rate. Among the predictions of the model for this flow is the steady state value of the order parameter, S, as a function of dimensionless strain rate o; *. This is shown plotted in Fig. 1 for U =3.66. There are two curves shown in this plot, a solid, upper curve that represents stable solutions and a lower branch of unstable solutions drawn as a dashed curve. Whether a solution is found on a particular branch depends both on the initial condition of the director relative to the elongation direction of the flow and the magnitude of the dimensionless strain rate. If the director is initially oriented parallel to the elongation direction, the order parameter will follow the solid upper curve regardless of the value of o; *. As the strength of the flow is increased, the order parameter is predicted to monotonically increase, as expected. However, if the director is initially oriented in the direction orthogonal to the stretching direction, strain rates where o; *B o; *lim (o; *lim =0.977 in the case of Fig. 1) are not sufficiently strong relative to the nematic potential to re-orient the director towards the stretching direction in the flow. For such weak flows, the application of flow causes a diminution of the order parameter in the nematic. Above
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this critical velocity gradient, however, the hydrodynamic forces are sufficiently strong to re-orient the director in the flow direction regardless of the initial orientation of the director. In the case of simple shear flow, 6= g; (y, 0, 0), a variety of dynamical responses are possible, depending on the magnitude of the dimensionless shear rate, G=g; /D, and the value of the nematic potential parameter U. The transient response predicted by the model can be classified into one of two types, depending on U. When this parameter is larger than the critical value that sets the boundary between the isotropic and nematic states, but less than a critical value, U* = 2.78, the model predicts a flow-alignment behavior where the orientation of the director is monotonically driven towards a steady state direction. In Fig. 2, the orientation angle of the director relative to the flow direction is plotted as a function of G. In general, this angle is a decreasing function of shear rate and its ‘zero shear’ value is a decreasing function of U. Also plotted in this figure is the order parameter as a function of G. When U has values in excess of U*, the response to transient shear flow is qualitatively different and undergoes a behavior that shows dramatic transitions as a function of G. Three regimes of transient flow behavior are predicted to occur: director tumbling for 0 BGB G1; wagging of the director when G1 B G B G2; and flow alignment for G\G2. In the mode of director tumbling, the director undergoes an oscillatory response where its angle of orientation relative to the flow direction varies between 990 with a period that depends on the value of U. When wagging occurs, the response is still oscillatory, but the excursions in the orientation angle are markedly reduced in magnitude. In the flow-alignment regime, the orientation angle achieves steady, time-independent values. In Fig. 3, the ‘steady state’ values of the orientation angle and the order parameter are plotted for U= 3.66.
Fig. 1. The scalar order parameter, S, as a function of strain rate predicted by the model. The solid curve represents solutions that were generated by taking the director to be initially parallel to the stretching direction of the flow. The dash curve was determined from simulations where the director was initially perpendicular to the stretching direction. The value of U was 3.66.
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Fig. 2. The orientation angle of the director relative to the flow and the scalar order parameter plotted as functions of F. The value of U was taken to be 2.5, where the model predicts flow alignment of the director irregardless of the strength of the flow.
3. Experimental methods and materials
3.1. The two dimensional polymer solution The polymer used in this study was poly (phthalocyaninato siloxane) (PcPS) a so-called ‘hairy
Fig. 3. The ‘steady state’ values of the orientation angle and the order parameter versus G for U= 3.66. This value of U leads to prediction of director tumbling and for that reason steady state solutions are not predicted below G=6.8.
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rod’ polymer synthesized and generously provided by the research group of Professor Wegner of Mainz. This polymer had a weight averaged molecular weight of 59 500 but was quite polydisperse. The structure of this molecule is that of a highly rigid rod of length 17 nm and width 1.9 nm. This chain strongly absorbs light in the green region of the visible spectrum and does so in a highly dichroic manner such that light polarized perpendicular to the chain axis is much more strongly absorbed than light polarized parallel to this axis. The PcPS polymer forms stable monolayers on top of water but neat films of the polymer are too rigid to continuously deform as a liquid in the presence of flow. For that reason, the polymer is fluidized with the addition of a small molecule amphiphile, arachidyl alcohol. As a result, the monolayers studied here can be considered as two-dimensional polymer solutions. These were prepared by spreading the solution of the blend onto the surface of water using chloroform as a spreading solvent. The PcPS rods are also susceptible to aggregation and this effect has been observed in neat films using electron microscopy [10]. Although it would be expected that aggregation phenomena would be less severe for PcPS dispersed within arachidyl alcohol at lower concentrations, recent observations suggest that aggregation can also occur in these solutions (C.W. Frank, private communication, 1997). When present, aggregation of the rods will affect the ability of the model to represent the physical system in at least two ways. First, the number concentration of the rods will be affected. Second the length distribution will be altered in a manner that depends on the degree and nature of the aggregation. This latter effect refers to whether the rods primarily aggregate end-on-end or side-by-side. The experiments were performed on a KSV 5000 trough fashioned from Teflon and equipped with a Wilhelmy balance for the purpose of measurement of surface pressures.
3.2. The optical train and flow cells The optical train used for the measurement of dichroism has been described in detail in Ref. [8]. It consists of a polarization modulated arrangement of optical components consisting of a ‘green’ HeNe laser (wavelength of 543.6 nm), a polarizer, photoelastic modulator, quarter wave plate, and detector. The photodiode detector resides within the water subphase beneath the monolayer and the incident optical train. The signal from the detector was analyzed using lock-in amplifiers that provide sufficient information to obtain both the dichroism, Dn¦, and the orientation angle, u. The angle u defines the orientation of the principal axis of the imaginary part of the refractive index tensor of the sample (the difference in the principal eigenvalues of this tenser is the dichroism) and a laboratory axis. The laboratory axis is taken to be the flow direction. In the measurement of the dichroism, the observed quantity in the experiments is the extinction, d¦ =
2p Dn¦d , l
where d is the monolayer thickness and l is the wavelength of the light. The four roll mill used in this study was designed after the suggestions of Higdon [11] and has been described previously in Ref. [8]. This flow device is often used to generate an approxima-
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Fig. 4. Schematic diagrams of the flow cells used in this study. (a) The four roll mill; (b) the parallel band device.
tion to two-dimensional extensional flow over a region of fairly large spatial extent that exists in the interior region existing within four identical rollers that are set on the corners of a square. A schematic of the device is offered in Fig. 4a. Also shown in Fig. 4a is a side view of the rollers showing a notched profile at the upper surface. This notch was used to ‘pin’ the interface to the upper surface of the rollers by immersing the rollers to a depth such that the interface containing the PcPS monolayer coincided with the notch. In this manner, a flatter and more stable interface could be achieved. The simulations by Higdon assume the liquid being subjected to the flow field of the device is Newtonian in its rheology and can be used to establish the linear relationship between the angular velocity of the rollers, v, and the strain rate, o; . In the case of the flow of the PcPS interfaces of interest in this study, such a Newtonian response cannot be assumed and it is necessary to experimentally establish the relationship between o; and v. As found previously [8], the PcPS monolayers under consideration here are highly viscous interfaces that couple directly to the motion of the rollers of the four roll mill. For this reason, the motion of the underlying water subphase can be neglected when one considers the fluid dynamics of the monolayers in response to transient flow conditions. This assumption can be verified through direct observation of the kinematics of the interface by either optical microscopy or by Brewster angle microscopy (BAM). This latter microscopy utilizes polarized light reflected from the surface at the Brewster angle of the water subphase and offers a convenient method of visualizing flow-induced distortions of an interface. It can be found described in
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detail in Ref. [14].Using BAM, we observed that the motion of the interface during transient flow conditions revealed that the PcPS monolayers responded over time scales that were substantially faster than the time-dependent dynamics of the water subphase. For example, upon abrupt cessation of the roller velocities, it was found that the PcPS monolayers appeared to arrest their motion almost instantaneously in spite of the fact that the underlying water took some much longer period of time to decelerate and become motionless. BAM images were also used to establish the relationship between o; and v. This was done by tracking thc motion of small particles (presumably dust) that are invariably left at the interface. An example of such a map of particle trajectories for a PcPS solution of 7.38 mol% subjected to v = 0.314 rad s − 1 is shown in Fig. 5. From this figure it is evident that the streamlines of the flow are very close to the expected hyperbolic shape. Also note the existence of a center stagnation point at the geometric center of the flow where the velocities tend to zero but where the velocity gradients are finite. The optical measurements of dichroism were taken with the light sent through the stagnation point. The velocity data extracted from plots such as those shown in Fig. 5 were differentiated as a function of position and used to determine the strain rate for several different values of the roller velocities. The relationship between o; and v is shown in Fig. 6. From this plot it was found that o; = 0.212v. When compared to similar data from unnotched rollers, it was found that using notched rollers led to relatively larger velocity gradients for the same values of the roller velocities. Simple shear flow was generated using a parallel band cell pictured in the schematic of Fig. 4b. A detailed description of this cell can be found in Ref. [12]. This device consists of two belts of PET that were held between pulleys that were coupled in a manner that led to the motion shown in the figure when one pulley was driven by a motor. In the interior region between the belts, a linear simple shear flow is established except for regions near the two ends of the belts. To reduce the importance of the end effects, the gap between the belts was made smaller than
Fig. 5. The trajectories of particles at the interface generated by the roll mill at roller velocities of v = 0.314 rad s − 1 for the 7.38 mol% PcPS solution. Note the hyperbolic shape of the streamlines.
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Fig. 6. The rate of strain calculated from differentiation of the data in Fig. 5 plotted as a function of the roller velocities.
the distance between the pulley ends by a factor of 5.4. The influence of the end effects was further minimized by making the optical measurements by placing the laser beam used in the dichroism experiments at the geometric center of the flow cell. At this location, a line of flow stagnation exists where the velocity of the fluid tends to zero. For this flow cell, it was assumed that the velocity field in the interior region was a linear function of position and the shear rate of the simple shear flow was calculated as g; =2VB/w where VB was the linear velocity of the belts and w is the gap between them. This flow cell was immersed into the fluid to a depth that the upper surface of the belts met the interface. In this manner, the interface could be ‘pinned’ to the top surface of the belts. By properly adjusting the depth of the belts, the curvature produced by the meniscus at the belts could be manipulated to produce as flat an interface as possible.
4. Results and discussion
4.1. Extensional flow results The rotary diffusion constants and scalar order parameters were measured on polymer solutions of four different concentrations and are shown plotted in Figs. 7 and 8. This quantity was determined using the four roll mill and performing flow reversal experiments that were followed by flow cessation. The rotary diffusion constants were determined by fitting exponential curves to the dichroism data acquired during the relaxation of flow-induced orientation once the flow was arrested. Such data can be found plotted in detail in Ref. [9]. The concentration of 4 mol% was found to produce an isotropic layer with zero order parameter and a diffusion constant was not measured. The three higher concentrations were found to be anisotropic at
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Fig. 7. The rotary diffusion constant versus roller speed for two polymer concentrations ( for 5 mol% and for 7.38 mol%).
rest, but the scalar order parameter did not monotonically increase, as one would expect. The value of S was measured to be lower for the highest concentration than was found for the two smaller concentrations. Although defects in the monolayers could not be directly seen using BAM, it is thought that this decrease in orientation could result from the presence of domains of different orientation that cannot be annealed by hydrodynamic forces. Certainly it is known that as the concentration of PcPS increases, the monolayers ultimately become quite brittle and resist flow-induced deformation. Another possible explanation for the decreased orientation at
Fig. 8. The scalar order parameter S versus PcPS concentration.
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Fig. 9. The steady state values of the scalar order parameter versus the dimensionless strain rate for the 7.38 mol% solution ( for measured values larger than the steady value, which was 0.52, and for measured values smaller than that).
higher concentrations is that aggregation of the chains may be occurring to a greater extent compared with more dilute systems. It should be pointed out that the value of the parameter Utr = 2.11 at which the isotropic to nematic transition takes place compares favorably to the value of 2.35 predicted by the model. The general trend of the rotational diffusion coefficient was to increase with concentration, and this behavior is in accordance with the polymer liquid crystal theory [13]. It predicts that at higher concentrations in the nematic regime polymer molecules can rotate more easily due to higher orders of molecular alignment. For the 5 mol% solution the diffusion coefficient was independent of the applied strain rate. The 7.38 mol% solution was, however, found to have a diffusion coefficient that was an increasing function of the strain rate. This increase in diffusion coefficient could also result from the enhanced order of molecular alignment due to applied flow. For this solution the lower limiting value, D=0.027 s − 1 was adopted to nondimensionalize the strain rate and the shear rate. The steady state values of the scalar order parameter are plotted in Fig. 9 as a function of the dimensionless strain rate for the 7.38% solution. As motivated by the ‘weak-strong flow’ effect pictured in Fig. 1, two sets of experiments were conducted; one with the initial orientation of the director parallel to the stretching direction of the flow, and one with the director initially oriented perpendicular to this axis. As expected, when the director was initial set parallel to the stretching direction, the application of extensional flow induced increases in the order parameter monotonically with the strain rate. On the other hand, when the alignment of the director was set perpendicular to the outgoing axis of the flow, it was found that a critical velocity gradient existed below which the order parameter was diminished with flow. Above this strength of flow, the order parameter was again an increasing function of strain rate. Furthermore, the orientation angle of the director relative to the flow direction did not change for ‘weak’ flows but
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changed by 90 degrees upon the application of ‘strong’ flows. The dimensionless value of the critical velocity gradient that defines the boundary between strong and weak flows was determined to be 0.407, which is close to the value predicted by solving Eq. (1), which is 0.977. One would expect that the lower order parameter states measured for weak flows would be unstable to fluctuations that would cause the system to slowly evolve towards the upper stable branch of solutions. Instead, it was found that the low orientation states were remarkably stable for long periods of time. One explanation for this finding is that the application of weak flows onto systems where the director is orthogonal to the stretching direction cause the appearance of defects in the monolayer that ‘trap’ low orientation states in the system. Indeed, very often two different values of the order parameter were determined for the same value of the velocity gradient in the weak flow regime for separate experiments. It is believed that the existence of multiple values is also a manifestation of a polydomain structure that may be set up in weak flows that do not have sufficient strength to reorient the director in the straining direction.
4.2. Simple shear flow results Simple shear flow measurements were taken on the 7.35% sample. It was found that this flow type was much more difficult to study compared with the four roll mill flows. This is most likely due to the inherent weakness of simple shear flow relative to extensional flow and the greater difficulty in maintaining a flat, time-independent interface with the parallel band device. An additional difficulty with simple shear flow is that it is difficult to identify the precise level of offset in the measured signals compared with the possibility of removing this source of error in the four roll mill flow. As discussed in Ref. [8], the signal at the detector is used to generate the following two pieces of information: Iv = tanh d¦ sin 2u + e1
(2)
I2v = tanh d¦ cos 2u+ e2
(3)
where e1 and e2 represent unavoidable error offsets. These two values, Iv, and I2v, are produced by forming the ratios of the outputs from the lock-in amplifiers with the ‘DC’, time-independent part of the signal. In the case of the four roll mill, the orientation angle, u, is constrained by symmetry to have a value of either 0 or 90°. For that reason, only the ratio I2v is required. Furthermore, during a flow reversal experiment in the four roll mill, u is switched between 0 and 90° for strong flows as the director re-orients itself. For this reason, a flow reversal can be used to uniquely identify the offset, e2, in this flow. In the case of simple shear flow, the orientation angle can assume any value between −45 and 45° relative to the flow direction and a flow reversal induces the transformation, u −u. From equation Eq. (2), it is evident that a reversal in flow direction causes a sign change in Iv and such a procedure can be used to determine the offset e1. However, a flow reversal will not reverse the sign of I2v, and for an anisotropic sample, e2 is left unknown. For this reason, the data taken in shear flow were analyzed by assuming that the equilibrium scalar order parameter in the absence of flow for a particular solution was the same as that found using the four roll mill cell. This additional information is sufficient to determine e2.
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Fig. 10. The scalar order parameter and the orientation angle at steady state plotted as functions of the dimensionless shear rate. The vertical, dashed lines define the separation of regions of tumbling, wagging, and flow alignment for U =3.66.
The 7.35% solution was subjected to transient shear flow where the flow was reversed in direction and then stopped. This protocol was chosen in order to reveal the existence of phenomena such as director tumbling and wagging. In Fig. 10, the scalar order parameter and the orientation angle at steady state are plotted as functions of the dimensionless shear rate, G. The vertical, dashed lines in Fig. 10 define the theoretically predicted separation of regions of htmbling, wagging, and flow alignment for U =3.65, which is appropriate for the 7.35% solution.
Fig. 11. Iv (the lower curve) and I2v, (the upper curve) as a function of time for the 7.35 mol% solution at a dimensionless shear rate of 5.54. The curves are properly shifted so that they do not overlap.
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It was not possible to acquire reliable data for shear rates lower than G =2.77 where tumbling is expected to occur. The single data point at G=5.54 is shown to have an orientation angle of zero degrees. This particular shear rate produced a time-dependent response shown in Fig. 11 where S and u are seen to oscillate in time in a manner that resembles wagging of the director. For this reason, the average orientation angle was taken to be zero when plotted in Fig. 10. In the region of the strongest flows, the data tended towards steady, time-independent values. In particular, the orientation angle tended towards values in the range of 14°. These values of the orientation angle are of a reasonable magnitude, but larger than the model prediction of 3 –4°.
5. Conclusions The phenomena of ‘weak-strong’ extensional flows in polymer liquid crystals predicted by the simple Doi-Hess model has been verified experimentally. It can be remarked that recent experiments in our laboratory have identified a similar effect for three dimensional systems composed of thin but macroscopic layers of solutions of colloidal, rodlike sepiolite particles residing on top of a dense, immiscible oil. The experiments in simple shear flow were unable to access flows that were sufficiently weak to place them in the region of director tumbling. However, limited evidence of director wagging was observed in addition to the phenomena of flow alignment of the director.
Acknowledgements This work was supported by the Center for Polymer Interfaces and Macromolecular Assemblies (NSF-MSERC). Partial support was obtained from the Petroleum Research Fund of the ACS. TM thanks Bridgestone Corporation for a graduate fellowship. PLM was partially supported by the EEC, TMR Rheology of Liquid Crystals, C.N. FMRX-CT96-0003.
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S.Z Hess, Naturforsch. Teil A31 (1976) 1034. M Doi, J. Polym. Sci. Polym. Phys. Ed. 19 (1981) 229. G. Marrucci, P.L Maffettone, Macromolecules 22 (1989) 4076. R.G Larson, Macromolecules 23 (1990) 3983. J.J. Magda, S.G. Baek, L. de Vries, R.G Larson, Macromolecules 24 (1991) 4460. R.G. Larson, M Doi, J. Rheol. 35 (1991) 539. G. Marrucci, P.L Maffettone, J. Rheol. 34 (1990) 1231. M.C. Friedenberg, G.G. Fuller, C.W. Frank, C.R Robertson, Macromolecules 29 (1996) 705. P.L. Maffettone, M. Grosso, M.C. Friedenberg, G.G Fuller, Macromolecules 29 (1996) 8473. J. Wu, G. Lieser, G Wegner, Adv. Mater. 8 (1996) 151.
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The dynamics of two dimensional polymer nematics1 T. Maruyama a, G.G. Fuller a,*, M. Grosso b, P.-L. Maffettone b a
Department of Chemical Engineering, Stanford Uni6ersity, Stanford, CA 94305 -5025, USA b Department of Chemical Engineering, Uni6ersity of Naples, Naples, Italy Received 5 August 1997; received in revised form 22 September 1997
Abstract The orientation dynamics of a monolayer of a nematic polymer solution were examined for both extensional and simple shear flows. In extensional flow, a ‘strong flow/weak flow’ criteria was investigated and found to occur at a critical rate of strain that was predicted by the simple molecular model of Doi and Hess. In simple shear flow, evidence of director wagging and flow alignment was obtained. © 1998 Elsevier Science B.V. All rights reserved. Keywords: Two dimensional polymer nematics; Simple shear flow; Orientation dynamics
1. Introduction Polymer liquid crystals are among the most difficult classes of complex liquids to understand rheologically. The complexity of these systems is revealed through a variety of nonlinear responses that include first and second normal stress differences that change sign, oscillatory shear stresses upon the start up of simple shear flow, and temporal responses that scale with the applied strain. These materials are further made complicated by the presence of a polydomain structure that results from the appearance of disclinations in the director field that defines their long range orientational order. In the case of polymer liquid crystals these defects are either very difficult to eliminate by hydrodynamic forces or are induced by flow. This is particularly thought to be the case when simple shear flow is applied, causing the director field to rotate in a manner that produces strong spatial distortions that generate large elastic stresses. For this reason, understanding the dynamics of polymer liquid crystals normally requires a recognition of the wide range of structural length scales that can contribute to their dynamics and rheology. Over the past 10–15 years, molecular and microstructural models have been advanced that offer insight into the basic mechanisms that are responsible for the non-Newtonian behavior * Corresponding author. Fax: +1 415 7257294. 1 Dedicated to the memory of Professor Gianni Astarita 0377-0257/98/$19.00 © 1998 Elsevier Science B.V. All rights reserved. PII S 0 3 7 7 - 0 2 5 7 ( 9 7 ) 0 0 1 2 0 - 1
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that was mentioned above. These models are due to Hess [1] and Doi [2] and capture the orientational flow behavior of lyotropic polymer liquid crystals through an extension of the classic rigid dumbbell model developed to describe rodlike polymers in solution. By adding a nematic potential function that accounts for interactions among the rods to the basic convection diffusion equation, simple but successful predictions of many of the important rheological responses of polymer liquid crystals have been possible. The resulting partial differential equation was first solved subject to simplifying assumptions by Doi in a manner that excluded the possibility of properly modeling effects such as director tumbling and changes in sign of the normal stress differences. These restrictions were removed by Marrucci [3] in a series of papers that examined the two dimensional form of this model which could be solved in a straightforward fashion. This approach produced a number of important predictions of real behavior and was able to anticipate numerous phenomena that have since been observed experimentally. When solved in this form the model correctly predicts director tumbling in transient simple shear flow and a transition from tumbling to ‘wagging’ and from wagging to flow alignment of the director as the shear rate is increased. These transitions were also demonstrated to be the source of the observed changes in sign of the first normal stress difference. The success of properly solving the model in two dimensions motivated Larson [4] to seek numerical solutions to the full three dimensional model and this led to predictions of changes of sign of the second normal stress difference, which were subsequently verified experimentally [5]. These models were restricted to descriptions of monodomain systems where defects are absent and it is evident that the presence of defects in the form of disclinations in the director are responsible for many rheological observations. For example, when in the tumbling regime, the Doi – Hess model does not predict steady solutions but leads to predictions of oscillatory phenomena in macroscopic properties. Several approaches to incorporate a polydomain structure into the basic model have been attempted [6,7] with the result that steady, time-independent behavior can be predicted. However, a theoretical understanding of disclinations and how to model their response to flow is presently an active research area. The vast majority of research has only considered simple shear flow, principally due to its predominance in rheometry and its importance to process flows. Consequently, the extensional flow behavior of polymer liquid crystals is largely unknown although it is recognized that such flows can dominate industrial processes such as spinning that are applied to lyotropic polymer liquid crystals. Furthermore, there are very few experimental results on systems that are free of defects that would enable a critical examination of the Doi–Hess model in the absence of this important complication. Recently, in the Stanford laboratory, progress has been made on the measurement of the orientation dynamics of monolayers of nematics formed from rodlike polymers [8,9]. The advantage of these systems are many fold. First, they can be used to directly test the two-dimensional form of the model, which is amenable to analytical solution. Second, it has been possible to largely eliminate defects in these layers and this makes it possible to test simpler, monodomain models. Finally, these systems are accessible to direct measurement of the order parameter through the use of simple methods in optical polarimetry. This latter advantage greatly simplifies both testing of models (since the calculation of the order parameter is much simpler than the calculation of the stress) and allows greater flexibility on the types of flows that can be studied. The first demonstration of the capability of making measurements directly on polymer monolayers examined the response to extensional flow and directly tested the Doi–Hess
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model [8,9]. The model was found to produce reasonable quantitative predictions of the experimental results involving two-dimensional extensional flow. In particular, good fits to the data were possible for experiments involving flow reversals and flow cessation. By fitting the model to the data, it was demonstrated that the two molecular parameters of the model, the rotational diffusion coefficient and the strength of the nematic potential, could be extracted from the data. In the present paper, the extensional flow properties of two-dimensional polymer nematics are further explored. Experimental data are offered where the concentration of the polymer is varied. In addition, a prediction of the model of a hysteresis effect in extensional flow is examined. This hysteresis is the result of a ‘weak flow/strong flow’ criteria that arises when the strength of hydrodynamic forces is either greater than or less than the strength of the nematic potential function. Finally, studies of the orientation dynamics for a layer subjected to transient simple shear flow are presented.
2. Theoretical background The basic structural element of the Doi–Hess model of polymer liquid crystals is the rigid dumbbell, which is used to prescribe the orientation of an individual test rodlike polymer chain. Such a chain is assigned an orientation given by a unit vector, u, and the system of rods in solution is characterized by a distribution of orientations specified by the orientation distribution function, c(u). The time evolution of the orientation distribution function responds to three effects: rotational Brownian diffusion; a nematic potential; and the hydrodynamic forces arising from flow. These forces combine to give the following convection–diffusion equation:
n
(c ( (c c ( ( =D + V(u) − (u; c), (t (u (u kBT (u (u
(1)
where D is the rotational diffusion coefficient and V(u) is the nematic potential function. The parameter U = 2cl 2 scales this function where c is the surface concentration of the rods and l is their length. In Ref. [9], Eq. (1) was solved for a two dimensional flow, 6 =o; (x, − y, 0) where o; is the strain rate. Among the predictions of the model for this flow is the steady state value of the order parameter, S, as a function of dimensionless strain rate o; *. This is shown plotted in Fig. 1 for U =3.66. There are two curves shown in this plot, a solid, upper curve that represents stable solutions and a lower branch of unstable solutions drawn as a dashed curve. Whether a solution is found on a particular branch depends both on the initial condition of the director relative to the elongation direction of the flow and the magnitude of the dimensionless strain rate. If the director is initially oriented parallel to the elongation direction, the order parameter will follow the solid upper curve regardless of the value of o; *. As the strength of the flow is increased, the order parameter is predicted to monotonically increase, as expected. However, if the director is initially oriented in the direction orthogonal to the stretching direction, strain rates where o; *B o; *lim (o; *lim =0.977 in the case of Fig. 1) are not sufficiently strong relative to the nematic potential to re-orient the director towards the stretching direction in the flow. For such weak flows, the application of flow causes a diminution of the order parameter in the nematic. Above
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this critical velocity gradient, however, the hydrodynamic forces are sufficiently strong to re-orient the director in the flow direction regardless of the initial orientation of the director. In the case of simple shear flow, 6= g; (y, 0, 0), a variety of dynamical responses are possible, depending on the magnitude of the dimensionless shear rate, G=g; /D, and the value of the nematic potential parameter U. The transient response predicted by the model can be classified into one of two types, depending on U. When this parameter is larger than the critical value that sets the boundary between the isotropic and nematic states, but less than a critical value, U* = 2.78, the model predicts a flow-alignment behavior where the orientation of the director is monotonically driven towards a steady state direction. In Fig. 2, the orientation angle of the director relative to the flow direction is plotted as a function of G. In general, this angle is a decreasing function of shear rate and its ‘zero shear’ value is a decreasing function of U. Also plotted in this figure is the order parameter as a function of G. When U has values in excess of U*, the response to transient shear flow is qualitatively different and undergoes a behavior that shows dramatic transitions as a function of G. Three regimes of transient flow behavior are predicted to occur: director tumbling for 0 BGB G1; wagging of the director when G1 B G B G2; and flow alignment for G\G2. In the mode of director tumbling, the director undergoes an oscillatory response where its angle of orientation relative to the flow direction varies between 990 with a period that depends on the value of U. When wagging occurs, the response is still oscillatory, but the excursions in the orientation angle are markedly reduced in magnitude. In the flow-alignment regime, the orientation angle achieves steady, time-independent values. In Fig. 3, the ‘steady state’ values of the orientation angle and the order parameter are plotted for U= 3.66.
Fig. 1. The scalar order parameter, S, as a function of strain rate predicted by the model. The solid curve represents solutions that were generated by taking the director to be initially parallel to the stretching direction of the flow. The dash curve was determined from simulations where the director was initially perpendicular to the stretching direction. The value of U was 3.66.
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Fig. 2. The orientation angle of the director relative to the flow and the scalar order parameter plotted as functions of F. The value of U was taken to be 2.5, where the model predicts flow alignment of the director irregardless of the strength of the flow.
3. Experimental methods and materials
3.1. The two dimensional polymer solution The polymer used in this study was poly (phthalocyaninato siloxane) (PcPS) a so-called ‘hairy
Fig. 3. The ‘steady state’ values of the orientation angle and the order parameter versus G for U= 3.66. This value of U leads to prediction of director tumbling and for that reason steady state solutions are not predicted below G=6.8.
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rod’ polymer synthesized and generously provided by the research group of Professor Wegner of Mainz. This polymer had a weight averaged molecular weight of 59 500 but was quite polydisperse. The structure of this molecule is that of a highly rigid rod of length 17 nm and width 1.9 nm. This chain strongly absorbs light in the green region of the visible spectrum and does so in a highly dichroic manner such that light polarized perpendicular to the chain axis is much more strongly absorbed than light polarized parallel to this axis. The PcPS polymer forms stable monolayers on top of water but neat films of the polymer are too rigid to continuously deform as a liquid in the presence of flow. For that reason, the polymer is fluidized with the addition of a small molecule amphiphile, arachidyl alcohol. As a result, the monolayers studied here can be considered as two-dimensional polymer solutions. These were prepared by spreading the solution of the blend onto the surface of water using chloroform as a spreading solvent. The PcPS rods are also susceptible to aggregation and this effect has been observed in neat films using electron microscopy [10]. Although it would be expected that aggregation phenomena would be less severe for PcPS dispersed within arachidyl alcohol at lower concentrations, recent observations suggest that aggregation can also occur in these solutions (C.W. Frank, private communication, 1997). When present, aggregation of the rods will affect the ability of the model to represent the physical system in at least two ways. First, the number concentration of the rods will be affected. Second the length distribution will be altered in a manner that depends on the degree and nature of the aggregation. This latter effect refers to whether the rods primarily aggregate end-on-end or side-by-side. The experiments were performed on a KSV 5000 trough fashioned from Teflon and equipped with a Wilhelmy balance for the purpose of measurement of surface pressures.
3.2. The optical train and flow cells The optical train used for the measurement of dichroism has been described in detail in Ref. [8]. It consists of a polarization modulated arrangement of optical components consisting of a ‘green’ HeNe laser (wavelength of 543.6 nm), a polarizer, photoelastic modulator, quarter wave plate, and detector. The photodiode detector resides within the water subphase beneath the monolayer and the incident optical train. The signal from the detector was analyzed using lock-in amplifiers that provide sufficient information to obtain both the dichroism, Dn¦, and the orientation angle, u. The angle u defines the orientation of the principal axis of the imaginary part of the refractive index tensor of the sample (the difference in the principal eigenvalues of this tenser is the dichroism) and a laboratory axis. The laboratory axis is taken to be the flow direction. In the measurement of the dichroism, the observed quantity in the experiments is the extinction, d¦ =
2p Dn¦d , l
where d is the monolayer thickness and l is the wavelength of the light. The four roll mill used in this study was designed after the suggestions of Higdon [11] and has been described previously in Ref. [8]. This flow device is often used to generate an approxima-
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Fig. 4. Schematic diagrams of the flow cells used in this study. (a) The four roll mill; (b) the parallel band device.
tion to two-dimensional extensional flow over a region of fairly large spatial extent that exists in the interior region existing within four identical rollers that are set on the corners of a square. A schematic of the device is offered in Fig. 4a. Also shown in Fig. 4a is a side view of the rollers showing a notched profile at the upper surface. This notch was used to ‘pin’ the interface to the upper surface of the rollers by immersing the rollers to a depth such that the interface containing the PcPS monolayer coincided with the notch. In this manner, a flatter and more stable interface could be achieved. The simulations by Higdon assume the liquid being subjected to the flow field of the device is Newtonian in its rheology and can be used to establish the linear relationship between the angular velocity of the rollers, v, and the strain rate, o; . In the case of the flow of the PcPS interfaces of interest in this study, such a Newtonian response cannot be assumed and it is necessary to experimentally establish the relationship between o; and v. As found previously [8], the PcPS monolayers under consideration here are highly viscous interfaces that couple directly to the motion of the rollers of the four roll mill. For this reason, the motion of the underlying water subphase can be neglected when one considers the fluid dynamics of the monolayers in response to transient flow conditions. This assumption can be verified through direct observation of the kinematics of the interface by either optical microscopy or by Brewster angle microscopy (BAM). This latter microscopy utilizes polarized light reflected from the surface at the Brewster angle of the water subphase and offers a convenient method of visualizing flow-induced distortions of an interface. It can be found described in
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detail in Ref. [14].Using BAM, we observed that the motion of the interface during transient flow conditions revealed that the PcPS monolayers responded over time scales that were substantially faster than the time-dependent dynamics of the water subphase. For example, upon abrupt cessation of the roller velocities, it was found that the PcPS monolayers appeared to arrest their motion almost instantaneously in spite of the fact that the underlying water took some much longer period of time to decelerate and become motionless. BAM images were also used to establish the relationship between o; and v. This was done by tracking thc motion of small particles (presumably dust) that are invariably left at the interface. An example of such a map of particle trajectories for a PcPS solution of 7.38 mol% subjected to v = 0.314 rad s − 1 is shown in Fig. 5. From this figure it is evident that the streamlines of the flow are very close to the expected hyperbolic shape. Also note the existence of a center stagnation point at the geometric center of the flow where the velocities tend to zero but where the velocity gradients are finite. The optical measurements of dichroism were taken with the light sent through the stagnation point. The velocity data extracted from plots such as those shown in Fig. 5 were differentiated as a function of position and used to determine the strain rate for several different values of the roller velocities. The relationship between o; and v is shown in Fig. 6. From this plot it was found that o; = 0.212v. When compared to similar data from unnotched rollers, it was found that using notched rollers led to relatively larger velocity gradients for the same values of the roller velocities. Simple shear flow was generated using a parallel band cell pictured in the schematic of Fig. 4b. A detailed description of this cell can be found in Ref. [12]. This device consists of two belts of PET that were held between pulleys that were coupled in a manner that led to the motion shown in the figure when one pulley was driven by a motor. In the interior region between the belts, a linear simple shear flow is established except for regions near the two ends of the belts. To reduce the importance of the end effects, the gap between the belts was made smaller than
Fig. 5. The trajectories of particles at the interface generated by the roll mill at roller velocities of v = 0.314 rad s − 1 for the 7.38 mol% PcPS solution. Note the hyperbolic shape of the streamlines.
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Fig. 6. The rate of strain calculated from differentiation of the data in Fig. 5 plotted as a function of the roller velocities.
the distance between the pulley ends by a factor of 5.4. The influence of the end effects was further minimized by making the optical measurements by placing the laser beam used in the dichroism experiments at the geometric center of the flow cell. At this location, a line of flow stagnation exists where the velocity of the fluid tends to zero. For this flow cell, it was assumed that the velocity field in the interior region was a linear function of position and the shear rate of the simple shear flow was calculated as g; =2VB/w where VB was the linear velocity of the belts and w is the gap between them. This flow cell was immersed into the fluid to a depth that the upper surface of the belts met the interface. In this manner, the interface could be ‘pinned’ to the top surface of the belts. By properly adjusting the depth of the belts, the curvature produced by the meniscus at the belts could be manipulated to produce as flat an interface as possible.
4. Results and discussion
4.1. Extensional flow results The rotary diffusion constants and scalar order parameters were measured on polymer solutions of four different concentrations and are shown plotted in Figs. 7 and 8. This quantity was determined using the four roll mill and performing flow reversal experiments that were followed by flow cessation. The rotary diffusion constants were determined by fitting exponential curves to the dichroism data acquired during the relaxation of flow-induced orientation once the flow was arrested. Such data can be found plotted in detail in Ref. [9]. The concentration of 4 mol% was found to produce an isotropic layer with zero order parameter and a diffusion constant was not measured. The three higher concentrations were found to be anisotropic at
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Fig. 7. The rotary diffusion constant versus roller speed for two polymer concentrations ( for 5 mol% and for 7.38 mol%).
rest, but the scalar order parameter did not monotonically increase, as one would expect. The value of S was measured to be lower for the highest concentration than was found for the two smaller concentrations. Although defects in the monolayers could not be directly seen using BAM, it is thought that this decrease in orientation could result from the presence of domains of different orientation that cannot be annealed by hydrodynamic forces. Certainly it is known that as the concentration of PcPS increases, the monolayers ultimately become quite brittle and resist flow-induced deformation. Another possible explanation for the decreased orientation at
Fig. 8. The scalar order parameter S versus PcPS concentration.
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Fig. 9. The steady state values of the scalar order parameter versus the dimensionless strain rate for the 7.38 mol% solution ( for measured values larger than the steady value, which was 0.52, and for measured values smaller than that).
higher concentrations is that aggregation of the chains may be occurring to a greater extent compared with more dilute systems. It should be pointed out that the value of the parameter Utr = 2.11 at which the isotropic to nematic transition takes place compares favorably to the value of 2.35 predicted by the model. The general trend of the rotational diffusion coefficient was to increase with concentration, and this behavior is in accordance with the polymer liquid crystal theory [13]. It predicts that at higher concentrations in the nematic regime polymer molecules can rotate more easily due to higher orders of molecular alignment. For the 5 mol% solution the diffusion coefficient was independent of the applied strain rate. The 7.38 mol% solution was, however, found to have a diffusion coefficient that was an increasing function of the strain rate. This increase in diffusion coefficient could also result from the enhanced order of molecular alignment due to applied flow. For this solution the lower limiting value, D=0.027 s − 1 was adopted to nondimensionalize the strain rate and the shear rate. The steady state values of the scalar order parameter are plotted in Fig. 9 as a function of the dimensionless strain rate for the 7.38% solution. As motivated by the ‘weak-strong flow’ effect pictured in Fig. 1, two sets of experiments were conducted; one with the initial orientation of the director parallel to the stretching direction of the flow, and one with the director initially oriented perpendicular to this axis. As expected, when the director was initial set parallel to the stretching direction, the application of extensional flow induced increases in the order parameter monotonically with the strain rate. On the other hand, when the alignment of the director was set perpendicular to the outgoing axis of the flow, it was found that a critical velocity gradient existed below which the order parameter was diminished with flow. Above this strength of flow, the order parameter was again an increasing function of strain rate. Furthermore, the orientation angle of the director relative to the flow direction did not change for ‘weak’ flows but
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changed by 90 degrees upon the application of ‘strong’ flows. The dimensionless value of the critical velocity gradient that defines the boundary between strong and weak flows was determined to be 0.407, which is close to the value predicted by solving Eq. (1), which is 0.977. One would expect that the lower order parameter states measured for weak flows would be unstable to fluctuations that would cause the system to slowly evolve towards the upper stable branch of solutions. Instead, it was found that the low orientation states were remarkably stable for long periods of time. One explanation for this finding is that the application of weak flows onto systems where the director is orthogonal to the stretching direction cause the appearance of defects in the monolayer that ‘trap’ low orientation states in the system. Indeed, very often two different values of the order parameter were determined for the same value of the velocity gradient in the weak flow regime for separate experiments. It is believed that the existence of multiple values is also a manifestation of a polydomain structure that may be set up in weak flows that do not have sufficient strength to reorient the director in the straining direction.
4.2. Simple shear flow results Simple shear flow measurements were taken on the 7.35% sample. It was found that this flow type was much more difficult to study compared with the four roll mill flows. This is most likely due to the inherent weakness of simple shear flow relative to extensional flow and the greater difficulty in maintaining a flat, time-independent interface with the parallel band device. An additional difficulty with simple shear flow is that it is difficult to identify the precise level of offset in the measured signals compared with the possibility of removing this source of error in the four roll mill flow. As discussed in Ref. [8], the signal at the detector is used to generate the following two pieces of information: Iv = tanh d¦ sin 2u + e1
(2)
I2v = tanh d¦ cos 2u+ e2
(3)
where e1 and e2 represent unavoidable error offsets. These two values, Iv, and I2v, are produced by forming the ratios of the outputs from the lock-in amplifiers with the ‘DC’, time-independent part of the signal. In the case of the four roll mill, the orientation angle, u, is constrained by symmetry to have a value of either 0 or 90°. For that reason, only the ratio I2v is required. Furthermore, during a flow reversal experiment in the four roll mill, u is switched between 0 and 90° for strong flows as the director re-orients itself. For this reason, a flow reversal can be used to uniquely identify the offset, e2, in this flow. In the case of simple shear flow, the orientation angle can assume any value between −45 and 45° relative to the flow direction and a flow reversal induces the transformation, u −u. From equation Eq. (2), it is evident that a reversal in flow direction causes a sign change in Iv and such a procedure can be used to determine the offset e1. However, a flow reversal will not reverse the sign of I2v, and for an anisotropic sample, e2 is left unknown. For this reason, the data taken in shear flow were analyzed by assuming that the equilibrium scalar order parameter in the absence of flow for a particular solution was the same as that found using the four roll mill cell. This additional information is sufficient to determine e2.
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Fig. 10. The scalar order parameter and the orientation angle at steady state plotted as functions of the dimensionless shear rate. The vertical, dashed lines define the separation of regions of tumbling, wagging, and flow alignment for U =3.66.
The 7.35% solution was subjected to transient shear flow where the flow was reversed in direction and then stopped. This protocol was chosen in order to reveal the existence of phenomena such as director tumbling and wagging. In Fig. 10, the scalar order parameter and the orientation angle at steady state are plotted as functions of the dimensionless shear rate, G. The vertical, dashed lines in Fig. 10 define the theoretically predicted separation of regions of htmbling, wagging, and flow alignment for U =3.65, which is appropriate for the 7.35% solution.
Fig. 11. Iv (the lower curve) and I2v, (the upper curve) as a function of time for the 7.35 mol% solution at a dimensionless shear rate of 5.54. The curves are properly shifted so that they do not overlap.
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It was not possible to acquire reliable data for shear rates lower than G =2.77 where tumbling is expected to occur. The single data point at G=5.54 is shown to have an orientation angle of zero degrees. This particular shear rate produced a time-dependent response shown in Fig. 11 where S and u are seen to oscillate in time in a manner that resembles wagging of the director. For this reason, the average orientation angle was taken to be zero when plotted in Fig. 10. In the region of the strongest flows, the data tended towards steady, time-independent values. In particular, the orientation angle tended towards values in the range of 14°. These values of the orientation angle are of a reasonable magnitude, but larger than the model prediction of 3 –4°.
5. Conclusions The phenomena of ‘weak-strong’ extensional flows in polymer liquid crystals predicted by the simple Doi-Hess model has been verified experimentally. It can be remarked that recent experiments in our laboratory have identified a similar effect for three dimensional systems composed of thin but macroscopic layers of solutions of colloidal, rodlike sepiolite particles residing on top of a dense, immiscible oil. The experiments in simple shear flow were unable to access flows that were sufficiently weak to place them in the region of director tumbling. However, limited evidence of director wagging was observed in addition to the phenomena of flow alignment of the director.
Acknowledgements This work was supported by the Center for Polymer Interfaces and Macromolecular Assemblies (NSF-MSERC). Partial support was obtained from the Petroleum Research Fund of the ACS. TM thanks Bridgestone Corporation for a graduate fellowship. PLM was partially supported by the EEC, TMR Rheology of Liquid Crystals, C.N. FMRX-CT96-0003.
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