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An experimental and simulation study of dilute polymer solutions in exponential shear flow: Comparison to uniaxial and planar extensional flows Thomas C. B. Kwan Department of Chemical Engineering, Stanford University, Stanford, California 94305-5025

Nathanael J. Woo Scientific Computing/Computational Mathematics Program, Stanford University, Stanford, California 94305-9025

Eric S. G. Shaqfeha) Department of Chemical Engineering, Stanford University, Stanford, California 94305-5025 (Received 10 March 2000; final revision received 6 December 2000)

Synopsis The rheological properties of dilute polymer solutions in exponential shear, uniaxial, and planar extensional flows are compared using Brownian dynamics simulations of both freely draining, flexible bead-rod and bead-spring chains. We introduce a novel stress function for exponential shear which uses the extinction angle ␹ to take into account the orientation of the chain as it aligns in the flow. Comparing this new stress function during startup and relaxation in exponential shear with ␶ 11⫺ ␶ 22 in planar extensional flow and ␶ 11⫺ 21 ( ␶ 22⫹ ␶ 33) in uniaxial extensional flow, we find that for both models there is a quantitative agreement among the three different flows over a large range of Wi, strain, and chain length. Furthermore, the distributions of maximum extension show a microstructural equivalence between ensembles of chains in all three flows up to strains of 3 or 4 at all values of Wi simulated. Finally, we show three comparisons between experiment and simulation of the various flows: 共a兲 simulations and exponential shear experiments of a polyisobutylene/polybutene Boger fluid, 共b兲 simulations and uniaxial extension data from literature, and 共c兲 exponential shear and planar extensional data of a low-density polyethylene melt from literature. © 2001 The Society of Rheology. 关DOI: 10.1122/1.1346599兴

I. INTRODUCTION The study of the dynamics of an elongated polymer chain has been of great interest for many decades 关DeGennes 共1974兲, Keller and Odell 共1985兲, Larson 共1990兲, Sridhar 共1990兲, Hinch 共1994兲, Nguyen and Kausch 共1999兲兴. Although theories 关DeGennes 共1974兲, Hassager 共1974兲, King and James 共1983兲, Ryskin 共1987兲, Larson 共1990兲, Hinch 共1994兲兴 that make predictions on quantities such as chain microstructure and steady state viscosities are abundant, experimental techniques which can reliably measure extensional visa兲

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

© 2001 by The Society of Rheology, Inc. J. Rheol. 45共2兲, March/April 2001

0148-6055/2001/45共2兲/321/29/$20.00

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KWAN, WOO, AND SHAQFEH

cosities are comparatively scarce especially for dilute polymer solutions 关Keller and Odell 共1985兲, Tirtaatmadja and Sridhar 共1993兲兴. In the case of uniaxial extensional flow, the most popular and reliable technique to extract viscosity data is the filament stretching device. In such an apparatus, a polymer solution is trapped between two parallel plates which are then pulled apart at an exponentially increasing rate to generate uniaxial extension. Rapid stress increases of several orders of magnitude are observed, but a steadystate viscosity measurement can seldom be attained because of device length limitations. Nonetheless, experiments using the filament stretching device have provided many insights into the stress buildup and decay of a dilute polymer solution in a uniaxial extensional flow 关Tirtaatmadja and Sridhar 共1993, 1995兲, van Nieuwkoop and von Czernicki 共1996兲, Solomon and Muller 共1996兲, Doyle et al. 共1998兲, Orr and Sridhar 共1999兲兴. Obtaining extensional viscosity measurements for solutions in a planar extensional flow, on the other hand, is even more difficult than in uniaxial extensional flow. Most of the methods such as the sheet inflation, rotary clamp, and tube stretching techniques 关Denson and Crady 共1974兲, Denson and Hylton 共1980兲, Laun and Schuch 共1989兲, Dealy and Wissbrun 共1990兲, Meissner and Hostettler 共1994兲兴 are applicable to only melts because they all rely on the high elasticity of the macromolecules in the melt state. Polymer solutions, by virtue of being in a solution state, simply cannot sustain those kinds of deformations. The common method of creating a planar extension for polymer solutions is through the use of a stagnation point flow such as the four-roll mill or opposed jet device. The so-called four-roll mill was first invented by G.I. Taylor back in 1934 关Taylor 共1934兲兴 and later popularized by many groups 关Fuller and Leal 共1980兲, Dunlap and Leal 共1987兲, Lee and Muller 共1999兲兴 because of its facility for rheo-optical studies. Another such experimental apparatus is the opposed slot device, an adaptation of the opposed jet device 关Pope and Keller 共1978兲, Keller and Odell 共1985兲兴, which sucks fluid into a pair of opposite channels to create a planar extensional flow. However, both of these apparatuses suffer from several disadvantages 关Fuller and Leal 共1981兲, Schunk and Scriven 共1990兲, Higdon 共1993兲, Dontula et al. 共1997a兲, Lee and Muller 共1999兲兴. For example, the strain rate produced is not constant throughout the entire device, i.e., edge effects can be significant, thus creating a mixed shear and extension flow. Coupled with this problem is the fact that the residence time of a polymer in these apparatuses can vary dramatically depending on the streamline that it follows. As a result, full chain extension, indicated by a saturation in the birefringence signal, is seldom observed because of the limited strain that a polymer actually experiences 关Fuller and Leal 共1980兲, Dunlap and Leal 共1987兲, Lee and Muller 共1999兲兴. Furthermore, the stress in a polymer solution is rarely directly measured. Rather, it is inferred either from birefringence and light scattering data in the case of the four-roll mill or pressure drops in the opposed slots device. Hence, the field of experimental extensional rheometry remains a very active area of research. About a decade ago, Doshi and Dealy 共1987兲 and Zu¨lle et al. 共1987兲 proposed a novel type of time-dependent shear flow where the shear rate increases exponentially in time to mimic a steady extensional flow. With such a shear rate history, the principal elongation ratio of the flow grows exponentially in time, exactly analogous to a steady extensional flow. The advantages to such a setup are that extensional data could be generated using common shearing devices, stresses can be directly measured, and the strain rate is uniform throughout the device 共e.g., Couette cell or cone-and-plate rheometer兲. In a previous paper 关Kwan and Shaqfeh 共1998兲兴, we used Brownian dynamics simulations of bead-rod chains to investigate the efficacy of exponential shear in elongating single chain polymers. We found that not only do the normal and shear stresses grow by orders of magnitude 共and are therefore much greater than in simple shear flow兲, but also many chains become highly stretched during a certain period of strain before the onset of

EXPONENTIAL SHEAR FLOW

323

tumbling. Thus, in the present paper, we compare in a quantitative manner exponential shear flow with uniaxial and planar extensional flows. As an aside, we will also be making a comparison between planar and uniaxial extensional flows, because, surprisingly, except for a few publications 关Petrie 共1979, 1990a, 1990b兲兴, as far as we know very little research has been focused on the relationship between the rheological responses of a single polymer in these two flows. However, before we can begin a meaningful comparison between these three flows, we need to find a proper comparative stress function. The subject of choosing the proper material function for exponential shear arises in many papers published about this flow 关Doshi and Dealy 共1987兲, Zu¨lle et al. 共1987兲, Samurkas et al. 共1989兲兴, and in the present paper, we will propose a new material function which not only captures the kinematics of the flow but also takes into account the polymer’s response to it. Finally, we will also compare our Brownian dynamics simulation data to available experimental data to assess the experimental feasibility of exponential shear as an alternative to uniaxial and planar extensional flows. II. BACKGROUND A. Bead-rod model The primary polymer model that we use in our Brownian dynamics simulations is the freely jointed bead-rod chain, or the Kramers chain 关Bird et al. 共1987兲兴, and the technique of Brownian dynamics employed in the present study follows closely the one used by Doyle et al. 共1997b兲 and that described in our previous publication 关Kwan and Shaqfeh 共1998兲兴. We shall only give a very brief description of the relevant forces in the present simulations, but we refer the interested reader to the above works for a complete discussion of the technique. The external forces acting on the N beads of a Kramers chain ␯ ␯ ), hydrodynamic (F hy, ), and constraint include the following: Brownian (F Br, i i con, ␯ (F i ). 共We have used indicial notation to represent a three-dimensional vector and Greek superscripts to denote bead numbers.兲 The Brownian force arises from the constant bombardment of the polymer by the surrounding solvent molecules and is treated stochastically in these simulations. The hydrodynamic force on a bead is simply the drag created by the relative motion of the bead and the solvent since we have neglected hydrodynamic interactions between beads. Finally, the constraint force physically represents the force necessary to hold two adjacent beads at a constant distance from each other, and it is manifested as a tension along the rigid rods. The N⫺1 tensions are calculated by the method of Lagrange multipliers following the algorithm by Liu 共1989兲. At the start of a simulation, an ensemble of chains is generated independently. The position of each successive bead along a chain is chosen from a random vector distributed over the surface of a sphere of radius a. Next, the ensemble of chains is allowed to equilibrate in the absence of flow for 106 – 107 time steps. The flow field is turned on at t ⫽ 0, and we begin to take the statistics of the chain trajectories at regular intervals. The typical size of the ensembles used in the present simulations ranges from 500 to 1000, and the time step size varies from 1⫻10⫺4 to 5⫻10⫺4 ␨ a 2 /kT depending on the flow strength. B. Bead-spring model In this paper, we will also be presenting results for a bead-spring chain model. In particular, we use the multibead finitely extensible nonlinear elastic 共FENE兲 chain model where two adjacent beads along the chain are connected by a FENE spring whose force law is given by the following 关Warner 共1972兲兴:

324


Fspr i ⫽

HQ i 1⫺ 共 Q/Q 0 兲 2

共1兲

,

where H is the Hookean spring constant, Q is the magnitude of the connector vector, and Q 0 is the maximum extension of the spring. Similar to the bead-rod model, we obtain the following Langevin equation of the chain: dr␯i dt

⫽

冋

␯ u⬁ i 共ri 共t兲兲⫹

␯ FBr, ⫹F spr i i

␨

册

,

共2兲

where the Greek superscript ␯ again refers to the bead number and varies from 1 to M ⫹1 where M is the number of springs in a chain. The integration scheme used to solve Eq. 共2兲 is a more efficient version of the one developed by Hur et al. 共2000兲 C. Characteristic scales and dimensionless groups All the results that we will present are in dimensionless form unless otherwise stated, and in this section we will introduce all the relevant characteristic scales and dimensionless groups needed for the discussion. First, in the bead-rod model, all lengths are scaled on the size of the Kuhn step, which is simply the distance a between two adjacent beads. Forces are scaled with kT/a, where kT is the thermal energy and stresses with n p kT, where n p is the number density of polymers in solution. The fundamental time unit for the smallest length scale a is the single bead diffusion time ␨ a 2 /kT, where ␨ is the drag coefficient. However, for a bead-rod chain there are many different time scales other than the single bead diffusion time because of the many length scales involved. The most important time unit is the longest relaxation time of the chain ␭. In a previous study, Doyle et al. 共1997b兲 found that for a N-bead chain, ␭ ⫽ 0.0142N 2 , where ␭ is in units of the bead diffusion time, and we have used this formula in this study to characterize the relaxation times for our 50-, 100-, and 200-bead chains. For the FENE chain model, a different set of characteristic scales is present. Length is scaled on the equilibrium spring extension, 冑kT/H, where H is the spring constant, and time is scaled on the Hookean relaxation time, ␨ /4H. After nondimensionalization, we are left with a single dimensionless extensibility parameter, b ⫽ HQ 20 /kT, which bears a one-to-one correspondence with the chain length in the bead-rod model, namely b ⫽ 3(N⫺1)/M , where M is the number of springs in the chain. Note that the overall chain extensibility is simply given by b chain ⫽ M b, but we will be referring to the extensibility b of the individual springs throughout our discussion. The most essential dimensionless group in the problem is the Weissenberg number Wi, which is a measure of the flow strength. Physically, it is the ratio of the longest relaxation of the macromolecule to some characteristic time scale of the flow. For the two extensional flows, we will define Wi in the usual manner, i.e., Wi ⫽ ␭ ⑀˙ , where ⑀˙ is the dimensionless rate of strain in the 1-direction, made dimensionless with the bead diffusion time. For exponential shear flow, we will define Wi ⫽ ␭ ␣ , where ␣ is the dimensionless exponential rate constant, again made dimensionless with the bead diffusion time. Note that the Pe´clet number used in other papers 关Doyle et al. 共1997b兲, Doyle and Shaqfeh 共1998兲兴 is the same as our dimensionless ⑀˙ and ␣.


325

D. Flow kinematics All three flows investigated in this paper are linear and can be expressed as follows: u⬁ i ⫽ ␬ijx j ,

共3兲

where ␬ i j is the velocity gradient tensor defining the flow type, viz.

␬ij ⫽

再

␥˙ 共t兲␦i1␦ j2 for shear flow ⑀˙ 关 ␦ i1 ␦ j1 ⫺ 21 共 ␦ i2 ␦ j2 ⫹ ␦ i3 ␦ j3 兲兴 ⑀˙ 关 ␦ i1 ␦ j1 ⫺ ␦ i2 ␦ j2 兴

for uniaxial extensional flow,

共4兲

for planar extensional flow

where ␥˙ and ⑀˙ are the scalar shear and strain rates, respectively. For the two extensional flows the strain rate ⑀˙ is simply a constant. On the other hand, in exponential shear, the shear rate increases exponentially in time as follows 关Doshi and Dealy 共1987兲兴:

␣ ␥˙ 共t兲 ⫽ A␣e␣t⫹ e⫺␣t, A

共5兲

where A is the strain scale factor and ␣ is the exponential rate constant. In our simulations and experiments, A has been set to unity, and ␣ is the sole parameter with which we change the strength of the flow. E. Polymer stress and material functions The total stress of a polymer solution consists of a solvent and a polymer contribution denoted by the superscripts s and p, respectively 关Bird et al. 共1987兲兴

␴ij ⫽ ␴sij⫹␴ijp ,

共6兲

␴ij ⫽ ␶ij⫺P␦ij ,

共7兲

␴ij ⫽ 共␶sij⫺Ps␦ij 兲⫹共␶ijp⫺Pp␦ij 兲,

共8兲

where P ⫽ P s ⫹ P p and ␶ i j ⫽ ␶ si j ⫹ ␶ ipj , and the stress tensor ␶ i j is zero for fluid at rest. For a Newtonian solvent, the solvent contribution to the stress tensor can be readily calculated as ␶ si j ⫽ ␩ s E i j , where ␩ s is the viscosity of the Newtonian solvent and E i j is the rate of strain tensor. In our simulations, the polymer contribution is calculated by using the Kramers–Kirkwood equation for the stress tensor 关Bird et al. 共1987兲兴 which in dimensional form, becomes: N

␶ijp

␯ 具R␯i Fhy, 兺 j 典, ␯⫽1

⫽ np

共9兲

where R ␯i is the coordinate of bead ␯ relative to the center of mass of the chain and 具.典 denotes an average over an ensemble of chain conformations. Because of the stochastic nature of the Brownian forces, straightforward calculation of the stress tensor leads to fluctuations of order 1/冑⌬t. Hence, we employ the novel technique of noise filtering developed by Grassia and Hinch 共1996兲 and Doyle et al. 共1997b兲. Now we turn to the problem of selecting a stress function to use for comparison among the three different flows. In the case of the extensional flows, the choice is unambiguous. For uniaxial extensional flow, a single tensile stress, ␶ 11⫺ 21 ( ␶ 22⫹ ␶ 33), uniquely defines the stress state of the fluid. For planar extensional flow, there are two

326


FIG. 1. A sketch of a single polymer in shear flow. A solution of these single chain polymers gives an effective orientation angle of ␹ and forms a vector p i from the flow direction.

nonzero combinations, ␶ 11⫺ ␶ 22 and ␶ 22⫺ ␶ 33 , but the choice between the two is clear. The former represents the stress difference between the elongation and the compression axes, while the latter is the difference between the compression axis and the neutral direction. Thus, if we are comparing to the tensile stress in uniaxial extensional flow, the first principal stress difference, ␶ 11⫺ ␶ 22 , is clearly the proper one to choose because they both represent the stress difference between the elongation and compression axes. On the other hand, exponential shear, by virtue of being a shear flow, has three nonzero stress functions, namely the shear stress ( ␶ 12), the first ( ␶ 11⫺ ␶ 22) and second ( ␶ 22⫺ ␶ 33) normal stress differences, but none of these three combinations represent the stress difference between the elongation and compression axes. Furthermore, the location of the elongation and compression axes is also ill-defined. Although the flow kinematics itself prescribes the elongation axis at 45° from the flow direction 共and the compression axis is simply perpendicular to the elongation axis兲, the polymer chain does not actually align along this axis because the rotational nature of the shear flow constantly pulls the macromolecule toward the flow direction. The stress function that we are introducing takes into account not only the flow kinematics but also the polymer’s response to it. The essential idea is to follow the elongation axis provided by the chains themselves and calculate the stress difference relative to this axis instead of the one dictated by kinematics. As a chain is distorted by a shear flow, the initially isotropic coil structure becomes elongated, and the direction in which the chain points provides a natural axis of elongation. Realizing this, we can simply follow this axis as it moves toward the flow direction due to the vorticity component of the shear flow and calculate the stress difference along and perpendicular to it. Although the aforementioned scheme is readily amenable to a Brownian dynamics simulation, it is impossible to implement in an experiment because measuring the molecular orientation of individual polymers is not only extremely impractical in cases where one can actually ‘‘see’’ the orientation 共e.g., DNA兲 but also simply impossible for most common polymers 共e.g., polystyrene and polyisobutylene兲. However, the extinction angle from a birefringence experiment, which measures the orientation of the effective medium of anisotropic polymer chains, is a close approximation that we can both simulate and experimentally obtain. To mathematically express the stress function discussed above, we start by defining an orientation vector p i , which points along the extinction angle ␹ from the flow direction, i.e., 共see Fig. 1兲 pi ⫽ cos ␹␦i1⫹sin ␹␦i2 .

共10兲

The new stress function, which we shall denote as ␶ * , is simply the difference between the component of the stress along and perpendicular to this orientation vector, p i , viz.


327

␶* ⫽ pi␶ijp j⫺共␦˜ ij⫺pip j 兲␶ij ,

共11兲

where ␦˜ i j is the two-dimensional isotropic tensor ␦ i1 ␦ j1 ⫹ ␦ i2 ␦ j2 . Combining Eqs. 共10兲 and 共11兲 and simplifying, we obtain the following:

␶* ⫽ 共␶11⫺ ␶ 22兲 cos 2␹⫹2␶12 sin 2␹.

共12兲

From this equation, we see that to find ␶ * , experimentally we need a combination of mechanical and optical rheometry, the former for the two stresses, ␶ 11⫺ ␶ 22 and ␶ 12 , and the latter for the extinction angle ␹. We note that Eq. 共12兲 is very similar in form to the Mohr’s circle of stress in solid mechanics, since both formulas essentially give the stress along an arbitrary axis between the 45° and the ‘‘1’’ axis. Another quantity of interest, apart from the stress difference in Eq. 共12兲, is the strain rate along ␹. We can apply the same procedure above to determine this. Denoting the strain rate along ␹ as ⑀˙ * , we have,

⑀˙ * ⫽ piEijp j ,

共13兲

where E i j is the rate of strain tensor for exponential shear, 21 ␥˙ (t) ( ␦ i1 ␦ j2 ⫹ ␦ i2 ␦ j1 ). After simplifying, we obtain

⑀˙ * ⫽ 12␥˙ 共t兲sin 2␹

共14兲

for the strain rate along ␹. Note that ⑀˙ * is a time varying quantity unlike the value for planar and uniaxial extensional flows and it depends on time through both ␥˙ (t) and ␹ (t). The idea of taking into account the vorticity’s effect on the rate of strain experienced by the chain is not a new one. Other researchers have alluded to this issue 关TaksermanKrozer 共1963兲, Lumley 共1969兲兴, and in particular, Astarita 共1979兲 even proposed a method of calculating the relative rate of rotation of a particle with respect to the rate of strain’s principal axes at that particle as a way of classifying a mixed flow as strong or weak. Later, Schunk and Scriven 共1990兲 incorporated this method into a constitutive equation to study the mixed flow in opposed nozzles and slide coating. Having proposed a suitable stress function, we now introduce a new material function appropriate for exponential shear. By analogy with the extensional flows where the material function is the ratio of the stress difference along the elongation and compression axes to the constant strain rate, the material function for exponential shear is the ratio of the stress function defined above to the exponential rate constant, ␣, i.e.,

␩exp ⫽

共␶11⫺ ␶ 22兲 cos 2␹⫹2␶12 sin 2␹

␣

共15兲

.

The choice of ␣ in the denominator of the expression is not an arbitrary one, and the reasons for this choice will be discussed in detail in the following sections. The subject of defining a meaningful material function for exponential shear has been discussed many times in the past by others. For example, Doshi and Dealy 共1987兲 proposed the following two alternatives:

␩exp ⬅

冦

␶12共 t 兲 ␥˙ 共 t 兲 2

关共 ␶ 11⫺ ␶ 22兲 2 ⫹4 ␶ 12共 t 兲兴 1/2

2 ␥˙ 共 t 兲

.

共16兲

328


The top expression clearly does not account for the contribution from the normal stresses, which is quite significant for highly elastic fluids and the numerator of the second expression actually corresponds to our stress function if the principal axes of the stress and refractive index tensors are coincident, i.e., tan 2␹ ⫽

2␶12

␶ 11⫺ ␶ 22

.

共17兲

For the bead-rod simulation results that we shall present, we have verified that the stress calculated from Eq. 共12兲 and 关 ( ␶ 11⫺ ␶ 22) 2 ⫹4 ␶ 212兴 1/2 are identical, and hence, the principal axes of the stress and index of refraction tensors are, indeed, the same. The fact that these principal axes are the same for bead-rod single chain models in simple shear and extensional flows was originally demonstrated elsewhere 关Doyle et al. 共1997b兲, Doyle and Shaqfeh 共1998兲兴. However, in both expressions of Eq. 共16兲, the denominator does not reflect the strain rate that the polymer experiences. In fact, using the exponentially increasing shear rate grossly overestimates the actual strain rate experienced by the chain, and as a result, in all the experimental and analytic data provided by Doshi and Dealy and others using their material functions 关Samurkas et al. 共1989兲, Schieber and Wiest 共1989兲兴, no strain-hardening effects were observed unlike those often reported in extensional flows. Zu¨lle et al. 共1987兲 also proposed a simple material function, viz.

␩exp ⫽

␶12共 t 兲 ␣

.

共18兲

Although the numerator in Eq. 共18兲 suffers from the same drawback as the one mentioned above by Doshi and Dealy, Zu¨lle et al. used the exponential rate constant in the denominator, recognizing it as the ‘‘characteristic constant’’ of the flow. Using their material function, they observed strain hardening behavior in exponential shear. However, as Samurkas et al. 共1989兲 pointed out, with the material function above, even a Newtonian liquid would exhibit an unbounded stress growth because of the exponentially increasing shear rate. However, with our proposed stress function, a Newtonian fluid attains a constant value of 2 ␣ ␩ s , where ␩ s is the shear viscosity. The difference lies in the fact that the extinction angle ␹ above in our function approaches zero asymptotically. So, even though the shear stress will grow exponentially, sin 2␹ will tend to zero so that those two effects will be canceled out. Consider a circle of unit radius in the 1-2 plane 共flow-gradient plane兲. When subjected to a shear flow, the circle will deform into a ellipse that is rotated at an angle ␹ from the flow axis. In this scenario, ␹ is given by the following: 关Lodge 共1964兲兴

␹ ⫽ cot⫺1共 21共␥⫹冑␥ 2 ⫹4 兲兲 ,

共19兲

where ␥ ⫽ exp(␣t)⫺exp(⫺␣t) for exponential shear. For a Newtonian liquid, ␶ 11⫺ ␶ 22 ⫽ 0 and ␶ 12 ⫽ ␩ s ␥˙ . Combining all this into our stress function above, we find that the Newtonian value for the stress is actually 2 ␣ ␩ s . In a recent paper, Petrie 共1998兲 also discussed the fundamental nature of shear versus extensional flows 共in the context of constitutive equations for polymer melts兲. However, in his re-examination of the data collected by both Zu¨lle et al. and Samurkas et al., he was still not able to determine whether exponential shear is more akin to simple shear or planar extension.


329

FIG. 2. The product of the longest relaxation time of the chain and the strain rate along ␹ for N ⫽ 50 and Wi ⫽ 5, 10, and 20 from ␣ t ⫽ 0 – 5. Note that the strain rate increases very slowly from ␣ t ⫽ 0 to 3 共inset兲 and then increases rapidly thereafter.

III. RESULTS AND DISCUSSION A. Strain rate along the extinction angle In Sec. II E we presented both a new stress function and a new material function for exponential shear flow. Although we elaborated on the physical motivation behind the choice of stress function, we gave the material function, particularly ␣ as the overall strain rate, without proof. In this and the following section, we will justify this choice. In Fig. 2, we show the product of ␭, the longest relaxation time of the chain, and ⑀˙ * the strain rate along the extinction angle ␹ as a function of ␣ t for N ⫽ 50 chains at Wi ⫽ 5, 10, and 20. The strain rate is calculated according to Eq. 共14兲 using the extinction angle ␹ from simulation. In all three cases, for ␣ t Ⰶ 1, ␭ ⑀˙ * is approximately equal to Wi. This is because the chains initially align themselves along the 45° line, the axis of elongation in a shear flow. Furthermore, at short times, the time-dependent shear rate is approximately 2␣. Thus, using these two short time limits 共␹ ⬇ 45° and ␥˙ ⬇ 2 ␣ 兲 and Eq. 共14兲, we see that, indeed, the product of ␭ ⑀˙ * is simply ␭␣ or Wi. For ␣ t up to 3, we see that the strain rate is only slowly increasing, and this is the consequence of two competing effects, an exponentially increasing shear rate and a diminishing extinction angle. Figure 3 shows the extinction angle as a function of ␣ t at the same N and Wi as Fig. 2. We see that the extinction angle in all cases decreases from an initial value of 45° to approximately 4° at ␣ t ⫽ 3. Hence, from Eq. 共14兲, sin 2␹ decreases from 1 to about 0.14, while ␥˙ (t) increases from 2␣ to 20␣. These two opposite trends approximately offset each other such that the strain rate increases slowly from ␣ t ⫽ 0 to 3. Beyond ␣ t ⫽ 3, the extinction angle decreases much more slowly, and the growing exponential component of the shear rate dominates, resulting in a rapidly increasing strain rate along ␹. Therefore, to prescribe a constant elongation rate ⑀˙ for planar extensional flow that would correspond to an exponential shear flow of the same flow strength, we use the exponential rate constant ␣, which is simply the initial strain rate along ␹, as an approximation since the strain rate increases very slowly from ␣ t ⫽ 0 to 3. Further justification for this choice is given below.

330


FIG. 3. The extinction angle for the same chain length, Wi, and ␣t. These time dependent ␹ values are used to calculate the strain rate in Fig. 2.

B. Hookean dumbbell and Rouse theories Before we proceed to the results of the Brownian dynamics simulations of bead-rod and bead-spring chains, we present analytic predictions of the Hookean dumbbell and Rouse models to demonstrate that even these linear force models are capable of exhibiting the equivalency of the rheological response of a polymer in exponential shear and extensional flows. To calculate the polymer stress for the Hookean dumbbell in these two flows, we use the Kramers–Kirkwood form of the stress tensor in conjunction with the solution of the diffusion equation for 具 Q i Q j 典 , where Q i is the end-to-end vector of a dumbbell. The solution of the equations for the stress for planar and uniaxial extensional flows is given, respectively, by the following:

␶11⫺ ␶ 22 ⫽

4WiH

2 ⫺2WiH 1⫺4WiH

⫹

1 1⫹2WiH

再冋冉

1

1⫺2WiH

exp⫺

1 WiH

冊册冎

⫹2 ⑀˙ t

再

冋

exp⫺

1 WiH

册

⫺2 ⑀˙ t

共20兲

,

冋冉

冊册

1 3WiH 2 1 ␶11⫺ 共 ␶ 22⫹ ␶ 33兲 ⫽ ⫺WiH exp ⫺ ⫺2 ⑀˙ t 2 1⫺2WiH WiH 共 1⫺2WiH 兲共 1⫹WiH 兲 ⫹

1 1⫹WiH

冋冉

exp ⫺

1 WiH

冊册

⫹1 ⑀˙ t ,

共21兲

where stress has been nondimensionalized by n p kT and WiH ⫽ ⑀˙ ␨ /4H, or simply the Weissenberg number for a Hookean dumbbell based on the Hookean relaxation time ␭ H ⫽ ␨ /4H. For exponential shear flow, not only do we need to compute the three different components of the stress tensor in order to calculate ␶ * as given in Eq. 共12兲, but we also need to evaluate the extinction angle ␹, which is related to 具 Q i Q j 典 by the following:


331

FIG. 4. Results of Hookean dumbbell predictions for exponential shear and planar extensional flows at three different Wi: 共a兲 1.5, 共b兲 5, and 共c兲 10. The solid line denotes the stress in the asymptotic limit 1 Ⰶ ␣ t; 1 Ⰶ Wi. 共d兲 Comparison between the total stress 共filled symbols兲 and the contribution from linear viscoelasticity 共open symbols兲.

2␹ ⫽ arctan

冉

2具Q1Q2典

具Q1Q1典⫺具Q2Q2典

冊

共22兲

.

Combining Eqs. 共12兲 and 共22兲 and simplifying, we arrive at the following simple expression for the dimensional ␶ * for a Hookean dumbbell:

␶* ⫽ npH关4具Q1Q2典2⫹共具Q1Q2典⫺具Q2Q2典兲2兴1/2.

共23兲

The solutions for the different components of the stress tensor in exponential shear flow are given by the following:

␶12 ⫽ WiH

␶11⫺ ␶ 22 ⫽

2 8WiH 2 4WiH ⫺1

⫹

2WiH

e ⫺t/␭ H ⫹

再

再

e ␣ t ⫺e ⫺t/␭ H 1⫹WiH

4WiH 2 WiH ⫺1

WiH

WiH ⫹1 2WiH ⫹1

⫹

再冋冉冎

e ⫺ ␣ t ⫺e ⫺t/␭ H 1⫺WiH

exp ⫺ 1⫺

e 2 ␣ t ⫹WiH ⫹

1 WiH

2WiH

冎

共24兲

,

冊册冋冉再

␣ t ⫹exp ⫺ 1⫹ WiH

1⫺WiH 1⫺2WiH

1 WiH

冊册冎冎 ␣t

e ⫺2 ␣ t ⫹WiH .

共25兲 In the case of the Rouse model, conversion from bead coordinates to normal mode coordinates casts the problem into a series of equations with the same form as that for the Hookean dumbbells. Thus, we can simply superpose the properly weighted solutions of the Hookean dumbbell case.

332


In Figs. 4共a兲–4共c兲 we show the analytic results for the polymer stresses of a Hookean dumbbell in the three flows at three different values of Wi, 1.5, 5, and 10. Note that since these values of Wi are all greater than 0.5, the critical value for coil-to-stretch transition in an extensional flow, we would not expect the calculated stress to attain a steady state. Rather, we would expect it to continue to grow for all time. Unless otherwise stated, for the remainder of the paper the term ‘‘polymer stresses’’ refers to the stress function introduced in Sec. II E for exponential shear, ␶ 11⫺ ␶ 22 for planar extensional flow, and ␶ 11⫺ 12 ( ␶ 22⫺ ␶ 33) for uniaxial extensional flow. Similarly, the flow strength parameter Wi is defined as ␭␣ for exponential shear and ␭ ⑀˙ for the two extensional flows, where ⑀˙ ⫽ ␣ as explained in the previous section. In addition, the stress results presented in this and the following sections have all been nondimensionalized with n p kT, and all the transient results have been plotted using strain units ( ␣ t ⫽ ⑀˙ t). First, comparing uniaxial and planar extensional flows, we find that the calculated stresses are essentially indistinguishable 关cf. Figs. 4共a兲–4共c兲兴. Upon closer examination of the analytic expressions for the stresses, we see that in fact they have the same functional forms and only differ slightly in the numerical constants. Hence, we should expect such good agreement, and we shall concentrate on the comparison between exponential shear and the extensional flows. At Wi ⫽ 1.5, we see that there is quantitative agreement between the stresses in exponential shear and the extensional flows up to ␣ t of 1.5. As the flow strength increases, the stresses coincide for larger strains until at Wi ⫽ 10 and greater 共not shown兲 where they agree quantitatively over the entire range of strain for which we solved the diffusion equation. Unlike the case of the two extensional flows, the analytic expressions for exponential shear and the two extensional flows are vastly different, but motivated by the quantitative agreement both in the small strain limit and the high Wi limit, we explore the possibility of an asymptotic equivalence in these two limits. Indeed, when we take the appropriate limits of the analytic expressions, we see that the stresses are asymptotically equivalent in two regimes, ␣ t Ⰶ 1 and ␣ t Ⰷ 1; Wi Ⰷ 1. For 1 Ⰶ ␣ t, the stress in exponential shear and planar extensional flows approaches 4 ␣ t and 4 ⑀˙ t, respectively, and for ␣ t Ⰷ 1; Wi Ⰷ 1, e 2 ␣ t and e 2 ⑀˙ t , respectively. In the previous section, we rationalized on a physical basis using ␣ in the material function for exponential shear by examining the strain rate along ␹. However, from the explanation above we could have arrived at the same conclusion on mathematical grounds by taking the appropriate limits of the solutions of a Hookean dumbbell in the two flows and realizing their asymptotic equivalence if we simply match ␣ with ⑀˙ . Note that it is very important to realize in reference to these results for the Hookean dumbbell and for all the results in this paper that the equivalence between our proposed stress function in exponential shear and that of planar or uniaxial extensional flow is not a consequence of linear viscoelasticity. Indeed, in all our calculations 共and in the experiments presented in Sec. III E兲 for ␣ t ⭓ 1 the normal stresses in the proposed stress function dominate. Of course in linear viscoelasticity for exponential shear flow, there would only be shear stresses. In Fig. 4共d兲 we show a comparison of our stress function for a Hookean dumbbell in exponential shear at Wi ⫽ 10 with that same function calculated under the conditions of linear viscoelasticity. Clearly, the value of the polymer stresses is up to 2 orders of magnitude greater than that predicted from linear viscoelasticity. Next, we use the simplest multibead spring model, or the Rouse chain, to determine whether the trends seen in Hookean dumbbells also hold for more complex models. Figures 5共a兲 and 5共b兲 show the polymer stress in exponential shear and planar extensional flows as predicted by the Rouse theory. From Fig. 5共a兲 we see that the stress in expo-


333

FIG. 5. 共a兲 Rouse chain with 1, 2, 5, 10, and 20 modes and we observe convergence in the stress after inclusion of 20 modes. 共b兲 20-mode Rouse chain at Wi ⫽ 20 and 501. We see quantitative agreement when ␣ t Ⰶ 1 and 1 ⬍ ␣ t Ⰶ Wi.

nential shear has converged after inclusion of 20 modes in a single chain at Wi ⫽ 20 共where Wi is based on the longest relaxation time of the chain兲 and that adding more modes increases the stress only negligibly. We have also verified this convergence at a much higher Weissenberg number of 501 and for planar and uniaxial extensional flows 共plot not shown兲. Thus for values of Wi on the order of 10–1000, a chain including 20 Rouse modes is enough to capture the stress growth during startup of these flows. Figure 5共b兲 shows the stresses for a 20-mode chain in planar extensional and exponential shear flows at Wi ⫽ 20 and 501. We again see a quantitative agreement for ␣ t Ⰶ 1, consistent with the asymptotic results found in the Hookean dumbbell case. The stresses in the three flows also agree well with each other in the other limit at ␣ t Ⰷ 1 and Wi Ⰷ 1, but the reason for the agreement is slightly different from that in the Hookean dumbbell case.

334


FIG. 6. Comparison between uniaxial and planar extensional flows for bead-rod chains: 共a兲 N ⫽ 50, 100, and 200 and Wi ⫽ 20; 共b兲 Wi ⫽ 0.5, 2, 10, and 20 and N ⫽ 100. Filled symbols represent uniaxial extensional flow; open symbols represent planar extensional flow.

Since the Rouse model introduces a spectrum of relaxation times, each mode of the chain, in turn, experiences a different Wi. In order for the stress in exponential shear to agree with that in the extensional flows, each individual mode has to approach the asymptotic limit ␣ t Ⰷ 1; Wip Ⰷ 1, where Wip is the Weissenberg number of the pth mode. For Wi ⫽ 20, the asymptotic limit is only attained for the first three modes or so because Wip ⫽ Wi/ p 2 . Beyond p ⫽ 3, Wip becomes too small for the limit to be valid. Although only the first three modes agree with each other, they contribute over 50% of the overall stress, and hence we still find a small region of agreement (0 ⬍ ␣ t ⬍ 2). On the other hand, for Wi ⫽ 501, significantly more modes attain the asymptotic limit, which then leads to a larger region (0 ⬍ ␣ t ⬍ 4) where the stresses from the different flows coincide. C. Bead-rod chain simulations In the previous section, we showed that even the linear models such as the Hookean dumbbell and the Rouse chain can predict an asymptotic equivalence in the stress response of a polymer in exponential shear and extensional flows. However, these models themselves are only valid in the very long chain limit. In this section, we present results from simulations of flexible bead-rod chains to explore the effects of finite extensibility. First, we focus our attention on the comparison between the startup of uniaxial and planar extensional flows. Figures 6共a兲 and 6共b兲 show a comparison of the polymer


335

stresses of a bead-rod chain: 共a兲 across various chain lengths 共N ⫽ 50, 100, and 200 Kuhn steps兲 at Wi ⫽ 20 and 共b兲 across various flow strengths 共Wi ⫽ 0.5, 2, 10, 20兲 at N ⫽ 100 during the startup of the two flows. We see that as expected in strong flows, the stresses grow by at least an order of magnitude before saturation, but surprisingly, except at very low strain ( ⑀˙ t ⬍ 1) there is a quantitative agreement in the stress response for all Wi and N simulated. This is consistent with the results for the linear models presented in the previous section. Physically, this result indicates that the neutral direction in planar extensional flow in which the rate of strain of the flow is zero 共i.e., a fluid element is being neither compressed nor elongated兲 does not affect the polymer stretch to any significant degree. We note that in the past other groups have also compared uniaxial and planar extensional flows. For example, using polyethylene melts, Laun and Schuch 共1989兲 found that the planar extensional viscosity exhibited significant strain hardening behavior and its value fell only slightly below the uniaxial extensional viscosity at the same strain rate over the same strain. In addition, Petrie 共1990b兲 analytically solved for the uniaxial and planar extensional viscosities for many different fluid models 共the Phan– Thien–Tanner, Acierno–Marucci, Giesekus, and the FENE-P models兲 and found that all models predict the same steady-state uniaxial and planar extensional viscosities in the limit of very high strain rates. However, as far as we know, there has been no formal comparison between the transient rheological response of dilute polymer solutions in these two flows either by experiment or simulation. Now that we have established an equivalence between uniaxial and planar extensional flows, we turn to a comparison of these two shear-free flows with exponential shear. In Figs. 7共a兲–7共c兲, we show the stresses of a single bead-rod chain of various length 共N ⫽ 50, 100, and 200 Kuhn steps兲 during the startup of exponential shear and planar extensional flows with Wi ⫽ 5, 10, and 20. 关Since we have already shown that the stress in planar extensional flow is completely equivalent to that in uniaxial extensional flow, we did not plot the latter in Figs. 7共a兲–7共c兲 for clarity.兴 For all flow strengths and chain lengths, we see that the stress rapidly increases over several orders of magnitude in value and approaches saturation at around a strain of 5 which is the end of the startup run. Furthermore, there is quantitative agreement between the stresses from the two flows at all flow strengths for N ⫽ 50 and 100. For N ⫽ 200 at Wi ⫽ 5, the stress from exponential shear lies slightly above that for planar extension at intermediate strains (2 ⬍ ␣ t ⬍ 4), but the agreement becomes quantitative as Wi increases. This suggests that the asymptotic region 共␣ t Ⰷ 1; Wi Ⰷ 1兲 found in the previous section also holds true for bead-rod chains but that chain size also plays a role as well. As the chain size increases, we need to increase to higher Wi to achieve the asymptotic equivalence. Hence, we see that there is a striking correspondence in the mechanical response of not only the simple linear models but also the more fine-grained bead-rod model. Physically, the correspondence between the three flows when the proper stress function is used in each case shows a fundamental similarity in the way they elongate a single polymer chain. From examining the different kinematics of the three flows, we see that their main differences are that: 共1兲 exponential shear contains a vorticity component, and 共2兲 planar extension and exponential shear contains a neutral direction with no compression or extension. However, it seems that these differences have little effect at least in terms of the stress response during startup, i.e., the elongational component of exponential shear is able to fully extend a polymer before the chain tumbles due to vorticity, and the neutral direction does not offer any relief for an elongated chain because the pulling motion along the axis of elongation ‘‘squeezes out’’ fluctuations in the neutral direction. What seems to be the important criterion is the presence of a principal elongation ratio that increases exponentially in time.

336


FIG. 7. Comparison between exponential shear and planar extensional flow for bead-rod chains at Wi ⫽ 5, 10, and 20; 共a兲 N ⫽ 50; 共b兲 N ⫽ 100; 共c兲 N ⫽ 200.

However, parity in polymer stress is not a sufficient condition for parity in polymer microstructures. To this end, Figs. 8共a兲 and 8共b兲 show the distributions of maximum extension for an ensemble of 500 chains at Wi ⫽ 20 at four different strains and N ⫽ 100. Note that only the distributions for exponential shear and planar extensional flows at a single chain length are shown because the distributions look similar across different chain sizes. For uniaxial and planar extensional flows, the distributions look identical and only the magnitudes differ slightly. So, we have simply shown a representative distribution at N ⫽ 100 from the latter. For exponential shear, we see that the time evolution of the distribution follows the same few stages. At ␣ t ⫽ 1 共not shown兲, the distribution is essentially that at equilibrilium, but by the next strain unit, the Gaussian peak from the previous strain has been ‘‘swept’’ towards various higher extensions with a small number of chains already attaining their maximum extension, the chain length N⫺1. Next, the distribution exhibits a distinct sharp peak at N⫺1 at a strain between 3 and 4. We recall that in our previous communication 关Kwan and Shaqfeh 共1998兲兴 we had inferred that the 50-bead chains are in their most extended state at ␣ t ⫽ 3 because the shear stress is at its maximum. We now see that our original inference was, in fact, quite accurate. Contrasting these distributions which have well-defined peaks at the maximum


337

FIG. 8. Distribution of chain extensions for Wi ⫽ 20 and N ⫽ 100 at strain ⫽ 2, 3, 4, and 5. The chain extension has been normalized by the maximum extension: 共a兲 exponential shear; 共b兲 planar extension.

extension with those from steady shear flow 关Hur et al. 共2000兲兴 which are more or less uniform in their in their distribution of extensions, we can unambiguously identify the elongational capability of exponential shear. Finally, at a strain of 5, the peak is once again swept back towards smaller extensions. The evolution of stretch for the two extensional flows is also straightforward. At small strains, the Gaussian distribution is found. As strain increases, the chains begin to unravel under the flow field, and the original Gaussian peak becomes broadened and then a much narrower peak emerges at N⫺1, i.e., the maximum extension. By strains of 3 or 4, a sharp peak is formed at N⫺1 indicating a large number of chains are already stretched to their maximum. At still higher strains, the peak continues to grow until as ⑀ → ⬁ 共not shown兲 the entire distribution approaches a single delta function at N⫺1. Comparing the distributions from the three different flows, we again find a quantitative agreement for strains up to 3 or 4. Hence, up to a strain of 3 or so, the observed parity in the macroscopic stress is, in fact, a consequence of the chains’ microstructural equivalence in the three flows. At strains higher than 3, a more complex situation arises. From the previous plot of the polymer stress, we see that there is a quantitative agreement throughout the entire startup run 共from 0 to 5 strain units兲 between exponential shear and planar extension. However, from the distributions of the chains’ maximum extension, we find that the clear peak at

338


the strain of 3 continues to grow in planar extensional flow, while it slowly disperses into a broad distribution in exponential shear. The reason for this difference lies in the rotational character of shear flows as opposed to the irrotational nature of extensional flows. Physically, a chain in uniaxial and planar extensional flows simply stretches along the axis of elongation, which is coincident with the flow direction. In the absence of external fields and depending on initial configuration and strain rate, the maximum extension of the chain monotonically approaches some steady-state value. On the other hand, in shear flows, the axis of elongation is 45° from the direction of flow. Hence, an extending chain must rotate away from its axis of elongation to align with the flow. However, once it rotates from the axis of elongation, the strain rate that it experiences changes. In the case of a simple shear flow, the strain rate along the axis of elongation is 21 ␥˙ where ␥˙ is the steady shear rate, and it decreases monotonically to zero at the flow axis. Thus, a chain rotating away from the 45° line in simple shear will experience a drop in strain rate which allows the chain to release its tension and retract somewhat 关Babcock et al. 共2000兲兴. However, in exponential shear, the strain rate along the axis of elongation increases exponentially in time like 21 ␣ (e ␣ t ⫹e ⫺ ␣ t ). So even though the chain is rotating away from the axis of elongation, the strain rate that it samples does not decrease. In fact, it increases dramatically as seen in Fig. 2. This surge in strain rate keeps the tension high along the chain leading to the observed large stresses. At the same time, the chain is also affected by the rotational component of the flow and begins to ‘‘tumble’’ as described by Doyle et al. 共1997b兲 and Hur et al. 共2000兲 for simple shear flow. During this tumbling period, the chain assumes a variety of folded configurations, which gives rise to a more or less uniform distribution of maximum extension. However, unlike in simple shear where much of the stress along the chain is released, the folded configurations in exponential shear still maintain very high tension along the straight segments. In fact, within the freely jointed bead-rod chain model, turns and folds along the chain backbone do not relieve stress because only the links hold stress. So long as tension is maintained in the links themselves, the macroscopic stress will be high regardless of the number of folds in the chain. To illustrate the qualitative difference between chain microstructures at strains of 3 and 5, Fig. 9 shows sample configurations from 100-bead chains. Although we have not shown the entire ensemble of configurations, the figures above represent the most common configurations at that strain. As anticipated, we observe that at ␣ t ⫽ 3 straight chains dominate the population while at ␣ t ⫽ 5 a wide array of folded configurations prevails.

D. Bead-spring chain simulations Although in the previous section we varied the chain length from 50 to 200 Kuhn steps in order to study the effects of increasing chain size, a span of 150 Kuhn steps in total chain length is still rather limited in scope, but computational costs preclude us from feasibly simulating longer bead-rod chains. Furthermore, most industrially relevant polymers usually have molecular weights on the order of millions of repeat units. For example, a polystyrene chain of molecular weight 2⫻105 g/mol has almost 2900 Kuhn steps 关Doyle et al. 共1998兲兴, a number well beyond our simulation capability for bead-rod chains. Thus, we turn to bead-spring models to further explore the effects of chain size and to quantitatively simulate a large polymer chain using only molecular parameters. To demonstrate that the FENE chain model also produces equivalent stresses in exponential shear and extensional flows, we show in Fig. 10 the stress response during startup at M ⫽ 20, b ⫽ 30, and Wi ⫽ 5, 10, and 20. We see that again the stresses are in quantitative agreement with each other just as in the bead-rod simulations. Further-


339

FIG. 9. Sample configurations of 100-bead chains at strain ⫽ 3 and 5 at Wi ⫽ 20. Note that mostly stretched out chains exist at strain ⫽ 3 while more folded ones dominate at strain ⫽ 5.

more, from the distribution of chain extensions 共Fig. 11兲 at four different strains, we find that the evolution of the distribution through time is analogous to that found from the bead-rod model for both exponential shear and the extensional flows, i.e., the initial Gaussian peaks first broaden, then a sharp peak appears at the maximum extension at a strain of 3 or 4, and finally the distribution broadens again for exponential shear. Besides chain extensions, we also investigate the distribution of individual spring extensions Q. Unlike bead-rod chains where the polymer stress is a complicated function of the tensions

340


FIG. 10. Comparison of the stress for FENE chains during startup for the three flows at M ⫽ 20, b ⫽ 30, and Wi ⫽ 5, 10, and 20. We again see a quantitative agreement in all cases consistent with the bead-rod chain results.

along the rods, the stress in bead-spring chains is simply related to the individual spring extensions, and thus studying the distribution of Q can offer insights into the trends seen in the stress curves. In Fig. 12 we plot the distribution of spring extensions Q at the same parameter values as in Fig. 11. Recall that in both bead-rod and bead-spring models, we observe that the stress for exponential shear slightly exceeds that for extensional flow at intermediate strains at small Wi and large N while they are the same for small and large strains. Although the distributions in the two flows look almost identical, we find that the springs are on average at a slightly higher extension in exponential shear than in extensional flow at the same intermediate strain. At higher strains, both distributions become identical again with a distinct peak at the maximum b. From the above, the distributions for chain extensions and individual spring extensions seem to suggest different regions where the various flows elicit a similar rheological response. On the one hand, the distribution of chain extensions shows that the flows are identical until at high strains when chain tumbling begins in exponential shear. On the other hand, the distribution of spring extensions indicates that they are the same at high and low strains but different at intermediate strains. They are, in fact, offering complementary information on the dynamics of bead-spring chains. The scenario that fits these two pieces of evidence is the one that we have already proposed earlier when we discussed the distributions for the bead-rod chains. Although the onset of chain tumbling introduces numerous folded configurations into the ensemble, thereby shifting the distribution to lower chain extensions at high strains, the individual spring extensions of these


341

FIG. 11. Distribution of chain extensions for M ⫽ 20, b ⫽ 30, and Wi ⫽ 10 for exponential shear and planar extensional flows at strain ⫽ 2, 3, 4, and 5.

folded configurations do not diminish because the shear rate is increasing rapidly. In fact, the spring extensions achieved in exponential shear are identical to those in the extensional flows leading to the same stress at high strains. Thus, by studying the individual spring extensions, we have shown the fundamental reason why the stress in exponential shear continues to be identical to that in the extensional flows at high strains even though the chain configurations are no longer the same. E. Comparison between experiments and simulations In this section, we compare our experimental results and those available in the literature with our simulation findings. In the first comparison, we perform exponential shear on a polyisobutylene 共PIB兲/kerosene/polybutene Boger fluid of concentration 0.25 wt %

342


FIG. 12. Distribution of individual spring extensions for M ⫽ 20, b ⫽ 30, and Wi ⫽ 10 for exponential shear and planar extensional flows at strain ⫽ 2, 3, 4, and 5.

PIB (MW ⫽ 4.6⫻106 g/mol). Standard rheological characterizations are first carried out on the fluid to ensure that it exhibits the usual Boger fluid characteristics and to find the longest relaxation time for the fluid. Indeed, a plot 共not shown兲 of the shear viscosity shows that its values are constant for almost 2 decades of shear rates 共from 0.1 to 10 s⫺1兲. From a plot of the complex viscosity across a range of frequencies, we deduced the viscosity of the solvent component. However, as an alternative, we also prepared a solution of the pure kersone/polybutene solvent of the same concentration as in the Boger fluid and find that they give the same viscosity value of about 160 P. This value will be important later when we calculate the polymer contribution to the shear stress. Finally, the longest relaxation time of the solution is measured by running a steady shear experiment and then stopping the flow while observing the decay of the normal stress. By fitting a single exponential decay to the relaxation curve, we find a longest relaxation time of 11.1 s for the fluid. For comparison, if we take ⌿ 1 /2␩ p 共from a steady shear rate sweep


343

FIG. 13. 共a兲 First normal stress difference and 共b兲 polymer shear stress 共in units of Pa兲 vs ␣ t at Wi ⫽ 5 and 10 for PIB/kerosene/PB Boger fluid, 20-mode FENE chain, and 20-mode Rouse chain. The experimental results represent an average of 4–5 different runs to ensure their reproducibility.

experiment兲, we obtain a relaxation time of 3.3 s. In this study, we will be using the former as the longest relaxation of the polymer. All the parameters needed for an exponential shear experiment, namely Wi and ␣ t end , can be calculated from this experimental relaxation time. For our experiments, we use Wi ⫽ 5, and 10 to compare to the simulations, which corresponds to ␣ ⫽ 0.450 45 and 0.9009 s⫺1, respectively. As for ␣ t end , ideally we would like to run for as long as possible to observe stress saturation, but unfortunately the rheometer that we employed 共Rheometric Scientific RMS800兲 has a limit on the total strain that we can impose. As a result, the maximum ␣ t that we can attain in 2.5. At the start of an experiment, the pure solvent is first subjected to the exponential shearing at the same Wi and ␣ t end as desired for the Boger fluid. This way, the solvent contribution to the total shear viscosity of the polymeric sample can be readily subtracted. Alternatively, we could have calculated the solvent contribution by knowing the viscosity of the pure solvent and the prescribed time-dependent shear rate, but by measuring the pure solvent’s response to the exponential shear flow, we can also verify that the motor is faithfully creating the exponential shear program desired. We have found the solvent contribution using both methods, and they show a quantitative agreement with each other except at very small time. Figures 13共a兲 and 13共b兲 show the first normal stress difference and shear stress in dimensional units 共Pa兲 versus ␣ t at Wi ⫽ 5 and 10 for the PIB/kersosene/PB Boger fluid

344


described above. We can see that through 2.5 units of ␣ t, both the shear and normal stresses increase by almost 2 orders of magnitude. Again, the shear stresses shown are the polymeric contribution only, and so this dramatic increase in the shear stress is not because of the Newtonian solvent’s response to the exponentially increasing shear rate. Also shown on Figs. 13 are the simulated stresses of a 200-bead FENE chain and the calculated stresses of a 20-mode Rouse chain at the same Wi. The spring extensibility b for the FENE chain is chosen to match the molecular weight of the PIB via the following 关Spiegelberg and McKinley 共1996兲兴: b⫽

6M w sin2关tan⫺1共&兲兴 MC⬁M 0

,

共26兲

where M w is the molecular weight, M is the number of springs in the chain, C ⬁ is the characteristic ratio which is about 6.5 for PIB, and M 0 is the molecular weight of a monomer which is 56 g/mol. With these parameters, b is 258.3. We can see that both the Rouse and the FENE chains capture the qualitative and for ␣ t ⬎ 1.2 quantitative behavior of the Boger fluid well. The fact that even a Rouse chain can model the startup behavior indicates that the chains in solution are not near their maximum extensibility. However, this comparison does indicate the predictive power of the FENE chain 共and the Rouse chain at short time兲 in capturing the stress growth behavior of PIB in this flow. Throughout this paper, we have shown simulation evidence of the correspondence between exponential shear and planar and uniaxial extensional flows. Although reliable planar extensional viscosity measurements for dilute polymer solutions are scarce, measurements in uniaxial extensional flow do exist from experiments conducted using filament stretching devices. From our simulations, we have already seen that the stress growth in planar and uniaxial extensional flows are equivalent at all reasonable values of Wi. So, comparison of stress growth in either flow to exponential shear should be useful. Figures 14共a兲 and 14共b兲 show two sets of data 共Trouton ratio versus strain兲 from Sridhar’s filament stretching device and from Li and Larson’s Brownian dynamics simulations of bead-spring chains 关Li et al. 共2000兲兴. Superposed on those are our simulation results for bead-spring chains at the same De and M in both exponential shear and uniaxial extensional flows. 关Note that Li, Larson, and Sridhar characterized their extensional flow strength by the Deborah number 共De兲, whereas we have called this parameter the Weissenberg number because we used the exponential rate constant ␣ to define it for exponential shear flow.兴 First we see that our bead-spring simulations in uniaxial extensional flow are very close to those by Li and Larson. The slight difference in intermediate strains is probably due to the fact that we have used a FENE spring as opposed to Li and Larson’s Pade´ approximation to the inverse Langevin function. Comparing between the Trouton ratio from exponential shear and uniaxial extensional flow, we see that there is a close correspondence between the two at both values of De simulated. Furthermore, the simulated exponential shear curve also exhibits the same qualitative behavior as the extensional measurements in Sridhar’s polystyrene solution. This shows once again that exponential shear has great potential in generating extensional viscosity data. As a final comparison, we will compare our results for exponential shear flow with experiments which examine polymers in the melt state. Although our development of the new stress and material functions is predicated on single chain considerations only, we simply would like to test the range of applicability of these ideas. Note in this context, that the comparison presented below is only an exploratory one and much more experimental work must be accomplished before firm conclusions can be reached. We use the exponential shear and planar extensional data from a recently published paper 关Venerus


345

FIG. 14. Trouton ratio vs strain at: 共a兲 De ⫽ 16.3 and 共b兲 De ⫽ 72.7 for Sridhar’s polystyrene Boger fluid, Li and Larson’s bead-spring chain simulations in unaxial extensional flow with 20 beads, and our bead-spring chain simulations in exponential shear and uniaxial extensional flows with 20 and 100 beads.

共2000兲兴 for a common branched low-density polyethylene melt. In his paper, Venerus presents both normal and shear stress measurements of the melt in exponential shear and compares them to the planar extensional stress measurements of another group using one of the material functions proposed by Doshi and Dealy 关see the second line of Eq. 共16兲兴. However, as we have remarked previously, the use of the exponentially increasing shear rate as the strain rate experienced by the polymer may grossly overestimate the actual strain rate at least from single molecule considerations. In this context, Venerus finds that the extensional viscosity as defined by Doshi and Dealy’s material function does not exhibit the strain-hardening behavior observed in planar extensional flow and concludes that exponential shear is a weak flow. Using the normal and shear stress data from Venerus, we calculate our proposed stress and material functions 关via Eqs. 共12兲 and 共15兲, respectively兴. Note that the formulas require the extinction angle ␹, which is not available in the paper. However, if we assume that the principal axes of the stress and index of refraction tensors are coincident 共see Sec. II E兲, then Eq. 共12兲 simplifies to the following:

␶* ⫽ 冑共 ␶ 11⫺ ␶ 22兲 2 ⫹4 ␶ 212.

共27兲

Figures 15共a兲 and 15共b兲 show the dimensional stress and material functions calculated using our formulas for exponential shear and compared to the planar extensional data

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FIG. 15. 共a兲 Stress 共in Pa兲 and 共b兲 extensional viscosity 共in Pa s兲 vs strain for a low-density polyethylene melt 共data taken from Venerus兲. The filled symbols represent exponential shear, and open symbols represent planar extensional data. The strain rates are 0.003, 0.01, 0.03, and 0.1 s⫺1.

from the paper. We see that for all four strain rates, there is a quantitative agreement between the stresses 共and material functions兲 from the two different flows at small strain. Furthermore, the agreement between the two flows at higher strains becomes better as the strain rate increases, and at the largest strain rate 共0.1 s⫺1兲, there is a quantitative agreement for all strains up to 2.5. These observed trends seem to be in agreement with those predicted previously by the Hookean dumbbell and Rouse models and confirmed by the bead-rod and bead-spring simulations. In particular, through the solutions of the Hookean dumbbell and Rouse models, we had found two asymptotic regimes 共␣ t Ⰶ 1 and ␣ t Ⰷ 1;Wi Ⰷ 1兲 in which exponential shear and planar extensional flows are expected to produce the same rheological response. The first asymptotic regime is certainly consistent with the experimental results, and although the exact Wi for the melt is not known, the qualitative aspect of the second regime, i.e., better agreement with increasing flow strength, is still reflected in the plot of the stress. This apparent applicability of our new stress and material functions developed for dilute polymer solutions to melt flows clearly warrants further study. However, without further investigation, we cannot exclude the possibility of other melt effects coming into play and giving rise to the observed trends.


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IV. CONCLUSION Extensional rheometry remains an important field of research with continued experimental challenges because of the difficulty in obtaining a reliable steady state measurement of the extensional viscosity of dilute polymer solutions in both uniaxial and planar extensional flows. In this study, we have presented a novel way to combine mechanical and birefringence measurements of a dilute polymer solution in a simple, time-dependent shear flow to create a rheological response that corresponds to extensional data both macroscopically and microscopically. From solutions of the Hookean dumbbell and Rouse models, we find that the polymer stress as prescribed by our method for exponential shear and the extensional stress for the two steady elongational flows are equivalent asymptotically in two regimes, ␣ t Ⰶ 1 and ␣ t Ⰷ 1;Wi Ⰷ 1. Furthermore, there is essentially no difference between the rheological response of these models in planar and uniaxial extensional flow at all strains and strain rates. Brownian dynamics simulations of flexible bead-rod and bead-spring chains of various sizes at different values of Wi also confirm the findings from the linear spring models. There is a quantitative agreement with the stress of all three flows during startup and relaxation through a strain of 5. By examining the distribution of chain extensions between the flows, we find that although the macroscopic stresses show a quantitative agreement up to a strain of 5, the chain microstructures begin to show differences when comparison is made between exponential shear and the extensional flows at a strain of 3 or 4. We explain this as a consequence of the rotational component of the shear flow which causes the chains to ‘‘tumble,’’ thereby reducing the chain extension. By directly observing the configurations of the chains, we find that indeed numerous folded configurations are present at a strain of 5 compared to the predominantly straight configurations seen at a strain of 3. However, although the chains are folded, the stress held in the links and springs continues to approach saturation unlike in simple shear Hur et al. 共2000兲 because of the rapidly increasing shear rate in exponential shear flow. Experiments using a PIB/kerosene/PB Boger fluid in exponential shear show that the polymer shear stress and the first normal stress difference grow by almost 2 orders of magnitude. Furthermore, the entire startup behavior is well represented by both the 20mode FENE and Rouse models since the ␣ t applied is not large enough to see the effects of finite extensibility. However, comparison with the experimental data of Sridhar in uniaxial extensional flow shows a very good qualitative and nearly quantitative agreement between our simulated stress in exponential shear and the observed stress in uniaxial extensional flow over 6 strain units. Finally, an exploratory application of our results to polyethylene melt data reported in the literature appears to demonstrate that in certain flow parameter regimes the suggested material functions derived from single molecule considerations are applicable to melts as well. Further experimentation and analysis are required before firm conclusions can be reached in the case of melt flows. However, all these experimental results further support our initial assertion that exponential shear can elicit the same rheological response from dilute polymer solutions as steady planar and uniaxial extensional flows. ACKNOWLEDGMENTS E.S.G.S. would like to thank the National Science Foundation for supporting this work under the CPIMA Cooperative Agreement No. DMR-9400354-2. T.C.B.K. would like to thank the National Science Foundation for funding through a National Science Foundation Graduate Research Fellowship. We would also like to thank Professor S. J. Muller for allowing us to use her rheometer to carry out the experimental portion of the paper.

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