Submitted, Journal of Rheology, August 20, 2001.

Non-homogeneous Transient

Uniaxial Extensional Rheometry

Jonathan P. Rothstein and Gareth H. McKinley

Dept. of Mechanical Engineering, Massachusetts Institute of Technology

Cambridge, MA 02139; USA

Abstract:

coming later

Keywords:	Filament stretching, extensional rheology, axisymmetric contraction, Boger fluids, stress-conformation hysteresis, constitutive models

1	Introduction

A great deal of recent research has focused on understanding the extensional behavior of polymeric fluids [McKinley and Sridhar 2002]. The dynamical response of polymeric fluids in extension, especially mobile dilute polymer solutions, is quite different than in simple shear. For this reason, extensional rheology is a topic of great interest and importance to the polymer community. It is widely recognized that to generate physically consistent constitutive models capable of predicting the behavior of these elastic liquids in complex flows such as the flow into an axisymmetric contraction-expansion, it is absolutely essential to first understand the behavior of these elastic fluids in homogenous uniaxial extension [Ghosh et al. 2001].

Numerous devices including the four roll mill, the spin-line rheometer, the opposed jet device and the capillary rheometer have been used in the past to investigate the extensional behavior of polymeric solutions. Unfortunately, each device is plagued by an unknown prestrain history and some degree of shearing in the flow field. The filament stretching rheometer has emerged as an accurate device for reproducibly measuring the response of a fluid to an imposed homogeneous uniaxial extensional flow field [Anna et al. 2001].

1.1	The Filament Stretching Rheometer (FiSER)

The filament stretching rheometer was originally introduced by Matta and Tytus [Matta and Tytus 1990] as a "falling plate device." By placing a small amount of fluid between two cylindrical plates and allowing the bottom cylinder to fall under gravity they were able to impose a nearly pure extensional flow field. Sridhar and coworkers [Sridhar and Gupta 1991; Sridhar et al. 1991; Tirtaatmadja and Sridhar 1993] extended the work of Matta and Tytus by creating a filament stretching device that, by controlling the velocity of the bottom plate, could impose a nearly homogeneous uniaxial extension on the fluid filament while simultaneously measuring the force on the top endplate. This allowed Sridhar and coworkers to measure the evolution in the transient extensional viscosity of the fluid filament from equilibrium to steady state extension. Many other research groups around the world have since developed filament stretching rheometers of various designs and operating spaces [Anna et al. 2001]. A recent paper by Anna et al. [2001] reported the results of an interlaboratory comparison of filament stretching rheometers in which several different instruments were used to study three dilute polystyrene-based elastic fluids. The results validated the technique by demonstrating that even though extensional rheology has proven to be difficult measurement to make, quantitative agreement can be achieved by instruments of different design.

The filament stretching device used to make all the measurements presented in this chapter, FiSER III, is shown in Figure 1. FiSER III was originally designed and built by Spiegelberg and Tripathi to measurement the extensional rheology of polymer melts and is the third generation of filament stretching
rheometers in Professor McKinley's lab, following very closely the successful design of Anna [2000]. In this device, an initially cylindrical filament of fluid is stretched between two circular endplates. The linear stage consists of two brushless linear DC motors which are used to impose a known velocity profile on the two platens to which the top endplate and the measurement equipment is mounted. The upper motor controls the velocity profile of the top endplate while the lower motor, traveling at half the velocity of the upper motor, ensures that an array of measurement devices is kept at the midpoint of the stretching fluid filament. The motion control system is manufactured entirely by Compumotor (model 6250). The system has a resolution of 200 m in its vertical positioning, a maximum travel of 100cm, a maximum velocity of 100cm/s and a maximum acceleration of 6000cm/s2. The imposed velocity profiles and the data acquisition are controlled entirely through Labview. A load cell (Futek Model L2338) is mechanically isolated from the linear stage and mounted to the lower endplate. The force transducer has a maximum measurable load of 10g with a resolution of ±5mg. The dynamic response time of this load cell was reported to be approximately 78ms by Anna et al. [2001]. The midpoint diameter is measured with a laser micrometer
(Omron Z4LA) mounted on the lower platen. The resolution of the laser micrometer is ±20 m. A flow induced birefringence (FIB) measurement optical train is also mounted to the lower platen. The FIB system is discussed in more detail in Section 1.2. Accurate knowledge of the fluid temperature is required to ensure accurate knowledge of the rheological properties of the fluids. Before each experiment, a thermocouple is placed within the fluid sample and the experimental temperature of the fluid is recorded. Due to the short duration of these experiments, the temperature is assumed to remain constant throughout the course of the experiment even though the ambient temperature can vary by several degrees over the course of an hour.

In Figure 2, a sketch of a typical fluid filament within the filament stretching rheometer is shown. In this figure, L(t) is the time evolving length of the filament, Rmid(t) is the midpoint radius of the filament, Fz(t) is the force exerted by the filament on the endplate,is the velocity of the endplate, and L0 and R0 are the initial length and radius of the fluid filament. The goal of extensional rheometry is to impose a motion with a constant stretch history. In other words, an initially cylindrical filament must remain cylindrical, thinning uniformly in the radial direction as it is elongated axially. As indicated schematically in Figure 2, the imposed flow field is not quite uniform. The initial fluid loading is not a perfect cylinder, Figure 2a, because the fluid sags under gravity. Gravity affects not only the initial filament shape, but may also affect the filament shape over the course of the entire stretch, causing an asymmetry in the fluid filament by draining some of the fluid from the top endplate. The result is a low extension rate limit below which the effect of gravity cannot be neglected [Anna et al. 2001]. The no-slip condition at the endplates results in a second nonuniformity. As shown in Figure 2b, as the filament is stretched, a thin film of fluid remains attached to the rigid endplates. The resulting filament shape deviates from the desired ideal cylinder, more closely resembling a classic wineglass shape. At short times and small initial aspect ratios, L0/R0, the no-slip condition at the endplate surface results in a shear dominated flow and eventually can lead to elastic endplate instabilities at large strain rates [Spiegelberg and McKinley 1996]. The operating space for a generic filament stretching rheometer is described in detail by Anna et al. [2001].

In an ideal homogeneous uniaxial extensional flow the ends of the cylinder separate exponentially in time [Bird et al. 1987],

whereis the imposed, constant stretch rate. However, as described above, the non-slip condition at the endplates does no allow the fluid filament to assume the shape of an ideal cylinder. The small volume of fluid that remains at the endplates does not deform appreciably and the central fluid filament sees a time-
dependent effective stretch rate based on the fluid radius,that is larger than the stretch rate imposed on the endplates,

In fact, a lubrication approximation can be used to demonstrate that the effective deformation rate is precisely equal to at small strains [Anna et al. 1999].

The total deformation of the system can be described in terms of a Hencky strain. For the case of the ideal filament, the Hencky strain can be expressed as

However, for the nonideal filament, the total Hencky strain becomes an integral of the effective strain rate over all time,

If a constant stretch rate is applied to the endplates, this test is classified as a Type II experiment [Kolte et al. 1997]. If the test is designed in such a way that the effective strain rate is constant than the test is classified as a Type III experiment. To accurately determine the desired material properties of the elastic fluid, Type III experiments must be performed. An effective technique for generating Type III tests from Type II master curves has recently been developed [Orr, 1999 #265; Anna, 1999 #163] and has been employed in all the experiments presented in this paper.

The strength of the extensional flow is characterized by the Deborah number which is the ratio of the longest characteristic relaxation time of the fluid, to the characteristic timescale of the kinematics,

Filament stretching rheometers have historically been used to measure the transient extensional viscosity of polymeric fluids in homogenous uniaxial extension

where is the principal elastic tensile stress difference generated within the filament. In a detailed force balance of the filament stretching rheometer, Szabo [Szabo 1997] showed that the principal elastic tensile stress difference generated within the filament could be calculated from the total force measured by the load cell, Fz , if the weight of the fluid, the surface tension and the inertia of the fluid are taken into account

whereis the surface tension of the fluid,is the density of the fluid andis the acceleration of the endplate. The last term of Equation 7 is due to the fluid inertia and is, in practice, negligibly small. In homogenous uniaxial extension a dimensionless extensional viscosity, the Trouton ratio, is often formed

However, if the extension rate is not constant, then the Trouton ratio may not be the best dimensionless form of the primary tensile stress. If we look at the form of the polymeric stress for the ith mode of a Hookean dumbbell model,

then a self consistent choice to nondimensionalize the stress is the elastic modulus, nkBT, where n is the number density of polymer chains in the solution and kB is the Boltzmann constant. This choice will become even more intuitive when flow induced birefringence measurements are discussed in Section 1-2.

The elastic modulus can be derived directly from physical arguments for a Hookean dumbbell model like the Rouse-Zimm model. For the 0.025wt% polystyrene solution, the elastic modulus becomes, nkBT = 0.28 Pa. However, to compare our experimental measurements to theory, we will use a single relaxation time constitutive model such as the Oldroyd-B or FENE-P model. To convert from a multimode model to a single mode model one must first determine the appropriate relaxation time to use. The simplest choice for the fluid relaxation time scale is the longest relaxation time, lz, however using the Zimm timescale does not accurately describe the entire relaxation spectrum for a single mode model. Alternatively, a viscosity weighted average or of the relaxation spectrum can be used

This average relaxation time is known as the Oldroyd relaxation time of the fluid. For the Rouse-Zimm spectrum in which the time constant and viscosity of each relaxation mode is related to the longest or Zimm mode by a simple recursion relation [Bird et al. 1987] this becomes

where s -1.40 ( h* )0.78 and h* is the hydrodynamic interaction parameter [Bird et al. 1987]. However, for rheologically simple fluids, the predictions of the Rouse-Zimm model and the Oldroyd-B model must agree in the limit of zero shear rate

To satisfy these two equations, the value of the elastic modulus must be modified slightly from the value obtained through physical arguments. This is achieved through a slight modification to the number density of polymer chains, n. The new value of the elastic modulus, denoted (nkBT)Oldroyd, is give by

For the 0.025wt% polystyrene solution at T = 25°C,

This method for calculating the elastic modulus has been followed by several groups in the past [Doyle et al. 1998; Spiegelberg and McKinley 1996] and will be used throughout this manuscript.

1.2	Flow Induced Birefringence (FIB)

The molecular polarizability and hence the macroscopic refractive index of a typical linear homo-
polymer chain is different in the direction parallel to and normal to the chain backbone [Fuller 1995]. Flow- induced birefringence utilizes this refractive index mismatch to probe the microscopic deformation of polymer chains. Optical birefringence measurements are noninvasive and extremely sensitive to small changes in the local orientation and deformation of the polymer chain. For this reason, flow-induced birefringence measurements have been used quite extensively to examine steady and transient flows of polymeric fluids [Fuller 1995]. By passing light of a known polarization state and frequency through the polymeric fluid sample and measuring the resulting change in polarization state, flow-induced birefringence can be used to determine the local anisotropy in the conformation of the polymer chains [Fuller 1995]. The resulting expression for the birefringence is

where is the measured birefringence, C is the stress-optical coefficient, nkBT is the elastic modulus of the fluid, andis a measure of the anisotropy in the average conformation of the polymer chain. The polymer conformation is characterized by the dimensionless second moment tensor, A = <QQ>/Qeq2, where Q is the end-to-end vector of the polymer chain. The theoretical value of the stress optic coefficient is given by [Fuller 1995]

where isotropic index of refraction is n = 1.5468 and the difference in polarizability is ( 1 - 2) = −1.45×10−23 cm3 resulting in a value of C = −6.4×10−9 Pa−1 for the 0.025wt% PS/PS solution [Brandrup and Immergut 1989]. However, to be consistent with the values in the literature of polystyrene in an aromatic solvent, a stress-optic coefficient value of C = −5.0×10−9 Pa−1 will be used in all of our FIB calculations [Doyle et al. 1998].

Some of the earliest birefringence measurements in extensional flows of polymer solutions were performed by Keller and coworkers [Muller et al. 1988; Pope and Keller 1977; Pope and Keller 1978]. Flow induced birefringence measurements were performed in the four-roll mill, the opposed jet device and the cross-slot device and used to determine the degree of chain orientation in extensional flows. This work represents the first experimental documentation of the existence of a coil-stretch transition, observed at a Deborah number of De = 1. Talbott and Goddard [Talbott 1978; Talbott and Goddard 1979] developed a technique for measuring birefringence in slender fluid filaments drawn by a fiber spinner. By measuring the birefringence along the spinline and coupling it directly to simultaneous measurements of the tension within the fiber, Talbott and Goddard were able to demonstrate the breakdown of the stress-optic law for semi-dilute separan solutions at very low stresses, Dt 2kPa.

The first flow induced birefringence measurements of dilute and semi-dilute polymer solutions undergoing transient uniaxial elongation in a filament stretching device were reported by Doyle et al. [Doyle et al. 1998]. Experimental measurements and Brownian dynamic simulations revealed the existence of a 'stress-conformation' hysteresis [Doyle et al. 1998]. Doyle et al. found that in a strong stretching flow, the average configuration and resulting stress in a polymer chain evolve along different paths during stretching and relaxation. This hysteresis was shown to arise from non-equilibrium coupling between the macroscopic flow field and the internal degrees of freedom of the fluid microstructure. These experiments, for all their success, were not without their limitations. To avoid scattering and spreading of the polarized laser light, the initial diameter of the filament was increased and the initial aspect ratio greatly reduced. This resulted in a shear dominated flow at small strains. To minimize the effect of gravity, the filament was immersed within a Plateau tank containing a low viscosity, neutrally buoyant fluid. The experiments were further limited in the amount of total strain, that could be applied to the filament and were thus unable to reach the steady-state extension of the test fluid. Recently, Sridhar and coworkers [Sridhar et al. 2000] have developed an in-air flow induced birefringence system for their filament stretching rheometer which does not require immersion in a Plateau tank. Sridhar and coworkers were able to reach large strains, for both steady and unsteady uniaxial extension flows.

The optical path for the polarization modulated flow birefringence system employed in this research is shown in Figure 3a and is similar in design to the system pioneered by Frattini and Fuller [Frattini and Fuller 1984]. The kernel of the current birefringence system was developed at NASA's Glenn Research center for use in filament stretching experiments in microgravity. The rigors of unmanned sounding rocket experimentation require that the system be robust, compact and sensitive. These requirements prove advantageous in earth's gravity as well. The system is light, insensitive to mechanical vibration and can be successfully mounted to the lower platen of the filament stretching rheometer. The operation and calibration of this system with its unconventional dual-crystal electro-optical modulator have been presented previously by Mackey and coworkers [Mackey et al. 1999]. The polarized laser light is generated by a 4.5 mW, 635nm laser diode (Thorlabs CPS43AP05ME). The light is first passed through a linear polarizer (New Focus 5524) oriented at 0° from the vertical axis of the filament stretching rheometer. The polarized light is then sent through a dual-crystal electro-optical modulator oscillating at 44kHz and oriented at 45°. After passing through the filament, the light reaches a second linear polarized (Newfocus 5524), known as the analyzer, which is oriented at −45°. Finally, the light passes through a biconvex collecting lens (Newport KBX39) and a mask before impinging on a photodetector (Thorlabs PDA55).

There are several major difficulties associated with successfully taking optical measurements of a thinning fluid in a filament stretching rheometer. The radial curvature of the filament acts like a cylindrical lens, fanning the laser light and dramatically reducing the intensity of the light reaching the photodetector. As the filament shrinks, the magnitude of the light spreading will increase. Depending on the axial strain imposed, the filament may become smaller than the cross section of the incoming laser beam and allow laser light which has passed around the filament to reach the photodetector. To correct both of these problems, a lens and a mask were arranged as shown in Figure 3b. A similar idea was presented and analyzed by Talbott [Talbott 1978]. The lens is placed such that the diverging light exiting the filament (which acts as a cylindrical lens) is refocused onto the photodetector. The light that passes around the filament is not spread and is blocked by the mask which is composed of a black cylindrical rod of diameter, dmask = 2.0mm. By mounting the mask vertically, directly behind the collecting lens, the minimum amount of light which has passed through the filament is blocked. Alternatively, the mask can be reduced in size and placed at the focal point of the lens [Talbott 1978]. A simple ray tracing analysis by Talbott [Talbott 1978] showed that the error associated with assuming that the path length of the laser light through the fluid is equal to the filament diameter is negligible. At the onset of stretching the filament, the no slip condition at the endplates and the sagging due to gravity result in a strong axial curvature. The degree of off-axis vertical deflection of the laser beam is a complicated function of the stretch rate, fluid properties and the initial loading of the sample. For a strain hardening fluid, the filament will approach an ideal cylinder at moderate strains. The effects of axial curvature are thus only present at early times. However, as the filament thins and elongates, the lateral position of the filament can shift due to air currents, slight misalignment of the rheometer or mechanical vibration. To lessen the impact of both these problems a large area photodetector (3mm × 3mm) was used. The advantage of this large area photodetector is a reduced sensitivity to filament location and axial curvature. The disadvantage is an enhanced sensitivity to ambient light which must then be minimized. Even with this large area photodetector, short-time birefringence measurements were rendered impossible by the axial curvature of the filament. Fortunately, at small strains, the polymer coils have not been extended far enough to generate an appreciable birefringent signal and the interesting physics is easily observable at larger strains.

The output from the photodetector is split and fed through a low pass filter (EG&G 5113) to determine the DC intensity and a pair of lock-in amplifiers (EG&G 7260) to measure the amplitude of the signal in phase with the fundamental frequency and out of phase with first harmonic of the dual-crystal modulator. These measurements are recorded on a PC through a data acquisition card interfaced with Labview. The intensity the light entering the photodetector is again given by:

where M34 and M32 are the components of the Mueller matrix corresponding to the optical train shown in Figure 3a and Ac is the amplitude of the electro-optical modulation calibrated such that the Bessel function of the first kind of order zero is equal to J0(Ac) = 0 [Frattini and Fuller 1984]. The advantage of using a modulation technique is that it is possible to calculate both the retardation,and the extinction angle, simultaneously,

where d is the optical pathlength equal to d = 2Rmid(t) and light is the wavelength of the laser light. The direction of the stretch is aligned with the optics at 0° from vertical. The extinction angle, and therefore the I2 signal, should be vanishingly small. Assuming the small retardation approximation, << 1, is valid and = 0

and Equation 19 simplifies to

where the squareroot appears in the denominator of the expression for the retardation because in practice, IDC is an RMS voltage. The validity of these assumptions will be confirmed in Section 2.1.

2	Results and Discussion

2.1	Steady Stretch Rates

In order to validate our flow induced birefringence experimental setup and measurement technique, a series of experiments were performed in the filament stretching rheometer at a constant imposed stretch rate. The spectrum of stretch rates and final Hencky strains were chosen to facilitate comparison with the literature [Doyle et al. 1998; Sridhar et al. 2000].

The viscoelastic test fluid used in all the experiments presented in this paper consists of an 0.025 wt% solution of monodisperse polystyrene (PS) (Scientific Polymer Products, Inc.) with a molecular weight of 2.25×106 g/mol and polydispersity of 1.03 dissolved in oligomeric styrene (Hercules). The solution is a dilute Boger fluid with c/c* = 0.24 [22]. The shear rheology of this fluid has been published previously and will not be reproduced here [Rothstein, 1999 #53; Rothstein, 2001 #195].

In Figure 4, the DC intensity, IDC, of the birefringence system and the two ratios, R = M34 and R2 = M32, are plotted as a function of time for a Deborah number of De = 4.1 a final Hencky strain of = 5. An inherent advantage of this ratiometric technique is the insensitivity of the two ratios to variations in the pathlength and the amount of scattering of the laser light, both of which lead to dramatic changes in the DC intensity over the course of the stretch. At early times, the signal is weak because of the axial curvature of the filament. In the present configuration, reliable birefringence measurements are not possible until a strain of = 1. At these small strains, the filament is subject to a reverse-squeeze shear flow which results in a very weak, positive birefringence signal. Recall that the retardation and thus the birefringence can be calculated directly from measurements of R using Equation 21. As time progresses and the accumulated Hencky strain increases to values greater than > 2, the polymer coils begin to orient with the flow and unravel. The birefringence signal grows increasingly negative until the flow is stopped. Observation of the polymer coil relaxation process is then possible until the filament fails or is deflect out of the laser's path as it drains under gravity. The results in Figure 3 also demonstrate the validity of the two assumptions made in the previous section. The first assumption, that the stretch direction is aligned with the 0° orientation of the optical train, is confirmed by the vanishingly small values of R2 throughout the entire stretch. The small angle approximation used to derive Equation 21 is also proven to be justifiable. The magnitude of R never grows larger than 0.05.

In Figure 5, the polymeric contribution to the tensile stress and the polymer coil conformation are plotted against the Hencky strain for the experiment shown in Figure 4, a Deborah number of De = 4.0 a final Hencky strain of = 5. Also included in Figure 5 is the theoretical prediction of the FENE-PM model. The polymer contribution to the tensile stress is calculated by subtracting the viscous stress from Equation 8

The measurements are plagued by noise at small strains (e < 2) where the extensional stress is dominated by the Newtonian response of the oligomeric polystyrene solvent and no appreciable deformation in the polymer coils has been accumulated. At larger strains, the agreement between the predictions of the FENE-

PM model and the experimental measurements is quite good. With the onset of strain hardening in the experimental measurements, the FENE-PM model begins to significantly underpredicts the amount of elastic stress accumulated by the fluid, but continues to accurately predict the polymer coil conformation. These observations are in agreement with the literature [Doyle, 1998 #74].

If the elastic stress is plotted against the polymer coil conformation as in Figure 6, a stress- conformation hysteresis is observed similar to that first noted by Doyle et al. [Doyle et al. 1998]. During the start-up of the stretch, the data follows the left-hand side of the loop while during relaxation the data follow the right-hand side of the hysteresis loop. The FENE-PM model cannot even qualitatively capture the hysteresis loop because it does not contain sufficient physical information about the internal configurations of the chains to determine the polymer stress. In the FENE-P model the stress is a single valued function of the polymer conformation and thus the birefringence [Doyle et al. 1998] is also single valued. As expected, the FENE-P model does agree quite well with the experimental data during the latter stages of relaxation. If a conformation dependent FENE dumbbell model [Doyle et al. 1998] or the FENE-G model [Ghosh et al. 2001] is used, then the hysteresis loop can be qualitatively predicted. A quantitative prediction which properly predicts the transient behavior of this experimentally observed hysteresis loop in extensional flow has yet to be achieved with any known constitutive model.

Similar hysteresis loops can be generated for filament stretching experiments performed at the same strain rate, but to different final Hencky strains. One way to appreciate how quickly the stress, birefringence and the magnitude of the hysteresis loop grow with increasing total Hencky strain is to plot several different experiments in a single figure. This is done in Figure 7, where the results from experiments performed at a Deborah number of De = 4.1 and a final Hencky strains of e = 3, 4 and 5 are presented. As before, the predictions of the FENE-P model are included in these plots. At these scales, the = 3 experiment is almost unperceivable. The magnitude of the hysteresis loop decreases with final strain. At these lower strains, the FENE-P model predictions represent the data well during the initial onset of the stretch. The experimental data does not begins to deviate from the FENE-P model predictions until a value of approximately n /C 250 Pa and a stress of p 250 Pa have been reached. These values represent the upper limit for the validity of the stress-optic rule for this dilute polymer solution. The stress-optic rule is commonly used to convert measurements of birefringence to stress in polymer melt experiments where it is valid up to much larger stresses, typically Dt 105 Pa [Janeschitz-Kriegl 1983].

Figure 7 also demonstrates the repeatability of these filament stretching experiments and the flow induced birefringence system in general. All three experiments superimpose quite nicely, following the same path in stretch and then again in the last stages of the relaxation process.	If the stretch rate is now changed, but the final Hencky strain held constant a new set of observations can be made. In Figure 8, the results of two filament stretching experiments terminated at a Hencky strain of = 4 are shown. The first experiment, at a Deborah number of De = 4.1 has been shown previously in Figure 7. The second experiment is conducted at a Deborah number 100% larger than the first, De = 8.2. The polymer coil conformation in each experiment grows to a value ofbefore the stretch is stopped. The amount of chain orientation and unraveling, is found to be independent of rate of deformation and only dependent on the total amount of accumulated strain. The stress, however, is approximately linearly dependent on the strain rate. This same observation has been made in the literature where the Trouton ratio was found to be independent of Deborah number above a Deborah number of De = 1 [Anna et al. 2001]. Both experiments, do, however, relax along the same path suggesting that there is a universal relaxation process independent of strain and strain rate.

It has been hypothesized in the literature that this stress-conformation hysteresis may significantly contribute to the enhanced pressure drop through axisymmetric-contractions and the enhanced drag observed in other complex flows [Rothstein, 1999 #53; Rothstein, 2001 #195] [Chmielewski et al. 1990; Solomon and Muller 1996]. However, taking a 4:1:4 contraction-expansion for example, the total accumulated Hencky strain along the centerline is only = 2.77. At these low strains, a constant stretch rate experiment shows very little hysteresis as indicated in Figure 7. However, it is possible that a nonhomogeneous uniaxial extension created to reproduce the strain rate profile a fluid element experiences as it travels along the centerline upstream of an axisymmetric contraction-expansion would demonstrate more substantial hysteresis and reveal some interesting new piece of the physical puzzle.

2.2	4:1:4 Contraction-Expansion Centerline Profiles

In the previous chapter, measurements of the axisymmetric flow induced birefringence along the centerline of the 4:1:4 contraction-expansion were reported for the flow of the 0.025wt% polystyrene Boger fluid. As expected, these birefringence measurements demonstrated significant orientation and elongation of the polymer chain upstream of the contraction-expansion. However, to gain more physical insight into this complex flow, it would be informative to simultaneously measure the stress and birefringence in the fluid passing through the contraction-expansion. Unfortunately, the only available measure of stress in the
contraction-expansion is the global pressure drop which is, in fact, the quantity into which we hope to gain some physical insight. Instead of measuring the stress and birefringence of a fluid element within the contract-expansion flow, we can isolate that fluid element and expose it to the identical time sequence of strain rates in the filament stretching rheometer.

The fluid element traveling along the centerline of the contraction-expansion is exposed to a transient nonhomogeneous extensional flow. To determine the exact strain rate profile required, the velocity profiles presented in Section 3.3.5 are analyzed. The centerline LDV measurements upstream of the 4:1:4 axisymmetric contraction-expansion at a Deborah number of De = 1.5 are shown in Figure 9. As in Section 3.3.5, the velocities are nondimensionalized by the average velocity within the contraction,, the distance upstream is nondimensionalized by the radius of the throat, and the time is
nondimensionalized by the shear rate, An Eulerian velocity profile of the form

is found to fit the data quite well for all Deborah numbers up to De = 4.5. Recall that a fluid element in the Sampson flow theory [Happel and Brenner 1965], experiences a velocity profile given by In Equation 23, A, B and C are constants. The value of A corresponds to twice the average upstream dimensionless velocity, A = 2/b3. However for the purpose of converting Equation 23 to a Lagrangian velocity profile analytically, we have chosen set to A to zero, A = 0, and refit the data to determine new values of B and C. The effect of this change is to limit the distance upstream for which the exponential fit is valid. Table 6-1 lists the values of the constants used to fit the velocity profiles at various Deborah numbers.

To calculate the proper velocity profile of the filament stretching rheometer endplates, we must first determine the strain rate profile of a fluid element traveling along the centerline of the contraction-expansion. By integrating Equation 23, the Lagrangian position of a fluid element can be determined as a function of time

where t0 is a dimensionless time constant related to the initial dimensionless upstream position of the fluid element, z0,

Throughout the course of these experiments, z0 will be set such that Equation 24 is then differentiated to determine the fluid velocity as a function of time

Deborah Number	B

C

t0	z0
0.500	1.694	0.849	4.697	-2.250
1.000	1.587	0.922	4.470	-2.037
1.500	1.665	1.048	4.000	-1.854
2.500	1.585	0.752	5.204	-2.427
3.500	1.750	0.557	7.468	-3.562
Table 1	Values for the exponential fit to the centerline velocity profiles upstream of a 4:1:4 axisymmetric contraction-expansion.

The strain rate of the fluid as a function of time thus becomes

while the total accumulated strain as a function of time is given by

From Equation 27 it is evident that the shear rate diverges, as This occurs well downstream of the throat, z = 0, and therefore does not pose a problem for our analysis.

Recall that in the filament stretching rheometer, the total Hencky strain for an ideal stretch is given by

Substituting into Equation 28 and exponentiating both sides, we can calculate the appropriate endplate separation

and the velocity profile of the endplate

required to simulate the transient nonhomogeneous uniaxial extension experienced by a fluid element traveling along the centerline upstream of a 4:1:4 axisymmetric contraction-expansion. The calculated profiles corresponding to a Deborah number of De = 1.5 are shown in Figure 10 in dimensional form. Note that the strain rate monotonically increases with time. The experiments are terminated when the Hencky strain equals the total strain accumulated within the contraction. For the 4:1:4 contraction- expansion, Hence, the total time for the stretch, tf, can be expressed as a function of the contraction ratio b and the time constant t0

Upon completion of the stretch, the platens are stopped and the relaxation of the fluid filament is observed. Although the strain rate profile during stretch will mimic the conditions upstream of the contraction-expansion, the relaxation process does not correspond to the region of biaxial compression downstream of the expansion. Theoretically, the downstream compression could be imposed by the filament stretching device, however, in practice the slender fluid filament will buckle under such a compressive loading.

2.3	Transient Measurement of Extensional Flow into a 4:1:4 Contraction-Expansion

Using the velocity profiles generated in Section 2.2 and the 'master curve' inversion procedure [Orr, 1999 #265; Anna, 1999 #163] it is now possible to directly observe the stress growth and polymer coil conformation of a fluid element traveling along the centerline into an axisymmetric contraction-expansion. The aim of this research is to explain the additional pressure drop observed in the flow of elastic liquids through energy loss argument associated stress-conformation hysteresis. Therefore, all the filament stretching results presented in this section will compare the polymeric tensile stress with the average conformation of the polymer chain.

In Figure 11, the results of the experiments described in the previous section are shown. The strain profile is equivalent to that observed upstream of a 4:1:4 contraction-expansion at a Deborah number of De = 1.5. The chains are aligned with the flow direction, but not significantly extended.. Note that the maximum conformation measured for the experiment is only A = 250, just 3% of the maximum extension of the polymer coil. Fortuitously, a small hysteresis loop exists. These stress vs. conformation plots are analogous to the stress vs. strain plots often used to investigate material properties in solid mechanics. In both cases, the area enclosed by the hysteresis loop has units of energy per unit volume. In solid mechanics, this energy represents the amount of work required to plastically deform an elastic solid. For a viscoelastic fluid, it represents the work done by a purely dissipative stress coupled to the non-equilibrium kinematics of the extensional flow whose physical origins are not well understood. To quantify the energy dissipation per unit volume, the polymeric tensile stress is numerically integrated over the entire hysteresis loop

The energy dissipation per unit volume is made dimensionless by dividing the elastic tensile stress by the elastic modulus of the test fluid. If the experimental data does not continue to zero conformation during the relaxation process, the hysteresis loop is closed using the FENE-P model predictions from the last data point to the origin. For the Type II experiment, the dimensionless dissipation energy is equal to ED = 1.1×107 while for the Type III experiment ED = 1.6×105, a difference of two orders of magnitude.

The series of experiments outlined in Table 1 were performed for the strain rate profiles equivalent to those observed along the centerline upstream of a 4:1:4 contraction-expansion at Deborah numbers between 0.5 ≤ De ≤ 3.5. Representative plots of the stress-conformation hysteresis loops for each of these Deborah numbers is presented in Figure 12. At a Deborah number of De = 0.5, Figure 12a, very little polymeric tensile stress or polymer stretch is generated. The dissipative energy loss associated with this stretch profile is minimal. As the Deborah number is increased, the stress, conformation and dissipative energy loss all increase. The initial stretch rates of the low Deborah number experiments are not large enough to result in significant chain orientation or unraveling. For these filament stretching experiments, the applied extension rates grow with increasing Deborah number, but the total accumulated Hencky strain is fixed at D = 2.77. Beyond a Deborah number of De = 1.5, a plateau of A 400 in the polymer chain deformation is observed even as the stress continues to grow. This is analogous to the observations made in Section 6.2.1 for constant stretch rate profiles.

In Figure 14, the dissipative energy loss is plotted against Deborah number for the filament stretching experiments of the centerline strain rate profile upstream of a 4:1:4 contraction- expansion. The results for each Deborah number were repeated several times and the average value of dissipative energy loss is plotted along with error in these measurement which is typically on the order of ±15%. This error is the caused by noise in the birefringence and force measurements as well as the profile error associated with the Type II to Type III 'master curve' inversion procedure. The results are quite conclusive. Recalling the pressure drop measurements across a 4:1:4 contraction-expansion presented in Section 3.3.2, no additional pressure drop was observed until a Deborah number of De 0.5 was reached. This critical Deborah number agrees quite well with the onset conditions for significant energy dissipation. Figure 14 also includes a fit to the experimental data of the form ED = 1.58×106 (De - 0.48)0.5. The fit is in excellent agreement with the experimental data, however, it does not match the predictions of recent constitutive models. Take, for example, the Rallison dissipative stress model [Rallison 1997]. The form of the model is similar to a FENE- P model except the polymeric stress term now has an additional purely dissipative component which relaxes immediately after secession the flow

For strong deformations, the dissipative component of the polymeric stress in Equation 34 and therefore the total energy dissipation increases linearly with Deborah number. This scaling was verified for the stretch profiles used in these filament stretching experiments through numeric integration of Equation 34.

The discrepancies between the experimental observations and the constitutive model predictions at low Deborah numbers could be the result of several things. First, the Rallison dissipative model is known to uniformly overpredict the transient extensional viscosity, but especially at small strains. The model predicts very little difference in the maximum conformation for any of the stretch rate profiles, Amax(De=0.5)/ Amax(De=3.5) 0.8 whereas the experiments clearly demonstrate a strong dependence on the stretch profile for the low Deborah number experiments, Amax(De=0.5)/ Amax(De=3.5) 0.25. Another possible cause of the discrepancy may be gravitational sagging in the experimental measurements. Anna et al. [Anna et al. 2001] demonstrated that a critical Deborah number for gravitational sagging could be determined from the ratio of the Bond number to the Capillary number

which characterizes the relative importance of gravitational forces to viscous forces throughout the stretch. When this ratio is order Bo/Ca 1, gravitational sagging will influence the filament stretching results. From this dimensionless group a critical strain rate for gravitational sagging to become important

when Bo/Ca = 1. Finally, a critical Deborah number can be defined

For the current filament stretching experiments and 0.025wt% PS/PS solution, Desag = 3.43 at T = 25°C. In the worst case, the strain rate profile for a Deborah number De = 0.5, only the first = 0.5 strain units are affected by gravitational sagging. Beyond that point, and gravitational effects can be ignored. Thus, it is unlikely that gravitational sagging is the cause of the discrepancy between experiments and theory or the reason why the energy dissipation decreases so quickly as the Deborah number decreases to De = 0.5.

By tailoring the strain rate profile within the filament stretching rheometer to reproduce the extensional flow along the centerline upstream of a 4:1:4 contraction-expansion, it is possible to simultaneously measure the stress and conformation of a polymer coil in a prototypical complex flow. The resulting measurements of stress-conformation hysteresis lead us to conclude that the energy loss of a polymeric solution in extension is directly related to the additional pressure drop and drag coefficients observed for these same fluids in complex flows. If one hope to accurately model a polymer solution in complex flows, it is therefore essential that the chosen constitutive model can quantitatively reproduce the stress and polymer conformation in transient extensional flows. With the current state of the art bead-spring constitutive models this is not yet possible. Additional microstructural information is required to capture the physics of a polymer coil unraveling in a strong flow. The challenge is creating such a model that is also computationally tractable. A significant innovation has recently been put forth by Ghosh et al. [2001] who incorporated a rate dependent finite extensibility parameter into a standard FENE model. Their results are encouraging in transient extensional flows, but more work and further refinement of their idea remains.

In this chapter the concepts of extensional rheology and the filament stretching rheometer (FiSER) are introduced. The technique of flow induced birefringence measurements in extensional flows is presented and discussed. The results of homogenous uniaxial elongation are first presented to validate the experimental setup and measurement techniques. The results of nonhomogeneous uniaxial elongation in which the elongational strain rates experienced by a fluid element entering a 4:1:4 axisymmetric contraction- expansion is reproduced by the filament stretching rheometer are then presented. Simultaneous measurements of stress and conformation demonstrate the presence of a stress-conformation hysteresis as seen in homogeneous uniaxial elongational experiments. The magnitude of the additional energy dissipation is quantified for the strain rate profiles observed in the 4:1:4 axisymmetric contraction at various Deborah numbers. The dissipation is found to monotonically increase with Deborah number.