simultaneous in-situ analysis of instabilities and first ......[k/(kbt)trc3]q is a scalar function...

Numerical investigation of a two-fluid model for shear-banding polymer solutions

Soroush Hooshyar and Natalie Germann

Fluid Dynamics of Complex Biosystems, School of Life Sciences Weihenstehpan,Technical University of Munich, Germany

ABSTRACTShear banding, the formation of localizedbands of differing shear rates, is a phenomenonobserved in many soft materials. Here we havedeveloped a novel thermodynamically consis-tent two-fluid model to study steady-state shearbanding in semi-dilute entangled polymer so-lutions. In this model, the formulation of theboundary conditions is straightforward, as thedifferential velocity is considered as a statevariable. The behavior of the model was ana-lyzed for a cylindrical Couette flow and a rec-tilinear channel flow. We found that stress-induced migration is the diffusive term respon-sible for the formation of shear bands. Thesteady-state solution is smooth and unique withrespect to different deformation histories anddifferent values of the diffusivity constant. Thesimplicity of this model makes it attractive foruse in more complex flows.

1 INTRODUCTIONShear banding occurs in soft materials such aswormlike micelles, semi-dilute entangled poly-mer solutions, and soft glassy materials. Understrong shearing deformations, these fluids candevelop inhomogeneous regions in the flow, in-cluding multiple localized bands with differentshear rates, known as shear bands. Typically,the shear stress of semi-dilute entangled poly-mers monotonically increases with the shearrate. Several one-fluid reptation models such asthe Rolie-Poly model1 can realistically capturethe flow curve and predict transient banding.However, there is experimental evidence for thepossibility that shear bands can also exist atsteady state.2, 3 Furthermore, experimental datasuggest that polymer solutions can form spa-

tially inhomogeneous concentration profiles.4, 5

Accordingly, it has been hypothesized that themechanism giving rise to shear banding is thecoupling between the polymer stress and con-centration through diffusion.6, 7

The two-fluid method is appropriate for in-vestigating diffusional processes in complexfluids. In this approach, it is assumed thatthe local gradients in concentration and vis-coelastic stress generate a difference betweenthe velocities of the polymer and the solventmolecules, which allows them to diffuse at dif-ferent speeds. Cromer et al6 developed a two-fluid model for semi-dilute entangled polymersolutions using kinetic theory. In their model,the diffusive terms were included in the timeevolution equation for the polymer concentra-tion. Therefore, to conserve the polymer con-centration, a no-flux condition at boundarieswas necessary. To construct the other boundaryconditions, the differential velocity vanishes atthe boundaries. To predict the polymer stressmore reliably at rapid deformations, the Rolie-Poly model was used to describe conforma-tional dynamics. Germann et al8, 9 recently de-veloped a two-fluid approach using the general-ized bracket approach of non-equilibrium ther-modynamics. The advantage of this new ap-proach is that the differential velocity is treatedas a state variable. Consequently, the additionalboundary conditions arising from the derivativediffusive terms can be imposed directly with re-spect to this variable. Furthermore, the descrip-tion can be easily extended to multiphase sys-tems as the total mass is conserved by the timeevolution equations themselves.

Cromer and coworkers studied shear band-ing in semi-dilute entangled polymer solutions

Simultaneous In-situ Analysis of Instabilities and First Normal StressDifference during Polymer Melt Extrusion Flows

Roland Kádár1,2, Ingo F. C. Naue2 and Manfred Wilhelm2

1 Chalmers University of Technology, 41258 Gothenburg, Sweden2 Karlsruhe Institute of Technology - KIT, 76128 Karlsruhe, Germany

ANNUAL TRANSACTIONS OF THE NORDIC RHEOLOGY SOCIETY, VOL. 24, 2016

ABSTRACTA high sensitivity system for capillaryrheometry capable of simultaneously de-tecting the onset and propagation of insta-bilities and the first normal stress differ-ence during polymer melt extrusion flowsis here presented. The main goals of thestudy are to analyse the nonlinear dynam-ics of extrusion instabilities and to deter-mine the first normal stress difference inthe presence of an induced streamline cur-vature via the so-called ’hole effect’. Anoverview of the system, general analysisprinciples, preliminary results and overallframework are herein discussed.

INTRODUCTIONCapillary rheometry is the preferredrheological characterisation method forpressure-driven processing applications,e.g. extrusion, injection moulding. Themain reason is that capillary rheometry isthe only method of probing material rheo-logical properties in processing-like condi-tions, i.e. high shear rate, nonlinear vis-coelastic regime, albeit in a controlledenvironment and using a comparativelysmall amount of material.1 Thus, it isof paramount importance to develop newtechniques to enhance capillary rheome-ters for a more comprehensive probing ofmaterial properties. Extrusion alone ac-counts for the processing of approximately35% of the worldwide production of plas-tics, currently 280⇥ 106 tons (Plastics Eu-rope, 2014). This makes it the most im-portant single polymer processing opera-

tion for the industry and can be found ina variety of forms in many manufacturingoperations. Extrusion throughput is lim-ited by the onset of instabilities, i.e. prod-uct defects. Comprehensive reviews on thesubject of polymer melt extrusion insta-bilities can be found elsewhere.4,6 A re-cent method proposed for the detectionand analysis of these instabilities is that ofa high sensitivity in-situ mechanical pres-sure instability detection system for cap-illary rheometry.8,10 The system consistsof high sensitivity piezoelectric transducersplaced along the extrusion slit die. In thisway all instability types detectable, thusopening new means of scientific inquiry. Asa result, new insights into the nonlinear dy-namics of the flow have been provided.9,14

Moreover, the possibility of investigatingthe reconstructed nonlinear dynamics wasconsidered, whereby a reconstructed phasespace is an embedding of the original phasespace.2,14 It was shown that a positive Lya-punov exponent was detected for the pri-mary and secondary instabilities in lin-ear and linear low density polyethylenes,LDPE and LLDPE,.14 Furthermore, it wasdetermined that Lyapunov exponents aresensitive to the changes in flow regime andbehave qualitatively different for the iden-tified transition sequences.14 It was alsoshown that it is possible to transfer thehigh sensitivity instability detection sys-tem to lab-sized extruders for inline ad-vanced processing control and quality con-trol systems.13

A very recent possibility considered

139

for Couette flows between parallel plates andconcentric cylinders.6, 7 Although the shearbanding behavior of these polymeric systemshas yet to be theoretically studied for Poiseuilleflows, numerical predictions by the Vasquez-Cook-McKinley (VCM) model are availablefor wormlike micellar solutions.10 It was nu-merically shown for wormlike micelles thatabove a critical pressure gradient, the velocityprofile exhibits a plug-like profile with a highshear band near the walls and a low shear bandnear the center of the channel. The kink sepa-rating these bands is related to the local maxi-mum in the profile of the first normal stress dif-ference. Furthermore, the volumetric flow rateundergoes a spurt as the pressure gradient is in-creased above a critical value.

Here we aimed to adopt the new two-fluidapproach to describe the diffusion banding insemi-dilute entangled polymer solutions withineither a cylindrical Couette flow or a Poiseuillechannel flow. The remainder of this article isorganized as follows. In Sec. 2, we describe thetwo-fluid method. The behavior of the model isanalyzed for the flows in Sec. 3 and the conclu-sion is drawn in Sec. 4.

2 POLYMER MODELIn this section, we present a new two-fluidmodel for semi-dilute entangled polymer solu-tions developed using the two-fluid frameworkof generalized bracket approach of nonequilib-rium thermodynamics.8, 9 The total system isconsidered to be closed, isothermal, and incom-pressible. This system consists of one poly-meric species and one viscous solvent. Forthe polymer, we define the following variables:the mass density rp, the momentum densitymp = rpvp, and the conformation tensor cp, de-fined as the second-moment of the end-to-endvector of the polymer chains. Furthermore, vp

denotes the velocity field and np = (rp/Mp)NAthe polymer number density, with Mp being themolecular weight of the polymer and NA theAvogadro constant. For the viscous solvent, thefollowing variables are defined: the mass den-sity rs and the momentum density ms = rsvs,where vs is the velocity field. The governing

time evolution equations are given as

r ∂v∂ t

= �rv ·———v�———p+——— ·sss , (1)

rprs

r

✓∂∂ t

+v ·———◆

(DDDv) =rsr

⇥�———

�npkBT

�+——— ·sss p⇤

�rp

r

h�———(nskBT )+hs———2vs

i� G0

DDDDv , (2)

∂np

∂ t= �——— ·

�vp np

�, (3)

∂c∂ t

= �vp ·———c+ c ·———vp +(———vp)T · c

� 1l1

(1�a)I+a K

kBTc�·✓

c� KkBT

I◆

+1l2

tr(

KkBT

c)�3�q✓

c� KkBT

I◆

+Dnonloc

⇣c ·———(——— ·sss p)+ [———(——— ·sss p)]T · c

⌘. (4)

Eq. (1) is the Cauchy momentum balance, witht being the time, r = rp +rs the total constantmass of the polymer solution, p the pressure,v the total average velocity of the polymer so-lution, and sss the total extra stress. Eq. (2) isthe time evolution equation for the differentialvelocity DDDv, where K is the modulus of elas-ticity, kB the Boltzmann constant, T the abso-lute temperature, hs the viscosity of the sol-vent, G0 the modulus of elasticity, D the dif-fusivity constant, and sss p the extra stress asso-ciated with the polymer. The divergence of sss p

accounts for the stress-induced migration andthe spatial gradients of the number densities de-scribe the Fickian diffusion. Eq. (3) is the timeevolution equation for the polymer concentra-tion. It is a material derivative that accountsfor the fact that the polymer concentration canvary locally. Eq. (4) is the time evolution equa-tion for the conformation of the polymer. Theleft-hand side and the first three terms on theright-hand side constitute the upper-convectedtime derivative. The fourth term on the right-hand side is the Giesekus relaxation account-ing for hydrodynamic interactions, with a be-ing the anisotropy factor. This term was addedto capture the overshoot of the shear stress dur-ing the start-up of a simple shearing flow trig-gering the shear band formation. The fifth termin Eq. (4) is a nonlinear relaxation term that weadded to capture the upturn of the flow curveat high shear rates. The power-law pre-factor

S. Hooshyar and N. Germann

140

[K/(kBT )trc�3]q is a scalar function of thetrace of the conformation tensor, which is a rel-ative measure for polymer extension. Note thatthis term closely resembles the term used in theRolie-Poly model to describe convective con-straint release including chain stretch.1 The lastterm of Eq. (4) controls the smoothness of theprofiles according to the value of the nonlocaldiffusivity Dnonloc. The time evolution equa-tions are closed by an explicit expression forthe extra stress

sss = sss p +hs

h———vs +(———vs)T

i

= np (Kc� kBT I)+hs

h———vs +(———vs)T

i, (5)

where the first term accounts for the viscoelas-tic stress associated with the polymer and thesecond term for the contribution of the viscoussolvent. The phase velocities associated withthe polymer and the solvent can be calculatedusing the total average velocity and the differ-ential velocity as follows:

vp =v+rs

rDDDv , (6)

vs =v�rp

rDDDv . (7)

In the new model, the differential velocity isconsidered as a state variable. The extra bound-ary conditions arising from the derivative termsappearing in Eq. (2) can be directly formulatedwith respect to this variable. For instance, no-slip and no-penetration conditions are recov-ered if the tangential and normal componentsof the differential velocity are set to zero atthe walls, respectively. Furthermore, the addi-tional stress-diffusive term appearing in Eq. (4)requires special treatment on the boundaries.We set the parameter Dnonloc equal to zero onthe solid walls since diffusion should vanish atthese locations within a distance less than theradius of the Gyration because of local surfaceeffects.

We worked with dimensionless quantities.Location is scaled by the characteristic lengthH, time is scaled by the characteristic relax-ation time t = t/l1, the extra stress and theconformation tensor are scaled as sss = sss/G0

and c = (K/kBT )c, respectively, where G0 =n0

pkBT . The dimensionless parameters with re-spect to these scalings are the elasticity num-ber E = G0l 2

1 /rH2, the ratio of the molecu-lar weight of the solvent to that of the poly-mer c = Ms/Mp, the polymer initial concentra-tion µ = npeq/(npeq + cnseq), where npeq andnseq denote the polymer and the solvent numberdensities at the equilibrium state of rest, respec-tively, the viscosity ratio b = hs/h0, the ratio ofthe characteristic relaxation times e = l1/l2,and the local and nonlocal diffusivity constantsD = Dl1/H2 and Dnonloc = Dnonlocl1/H2, re-spectively.

3 RESULTS AND DISCUSSIONThe model was solved for a cylindrical Couetteflow and a Poiseuille channel flow. The modelparameters were determined by fitting to thesteady shear rheology of a 10 wt/wt% (1.6M)polybutadiene solution.11 The flow problemwas solved using a Chebyshev pseudospectralprocedure with 200 discretization points as de-scribed in Refs.8, 9, 12

For the cylindrical Couette flow, we usedthe cylindrical coordinate system as the refer-ence frame. The characteristic length is givenas H = Ro �Ri, where Ro and Ri are the radii ofthe outer and inner cylinders, respectively. Sub-sequently, we take the normalized coordinater⇤ = (r � Ri)/H to indicate the location in thecylindrical gap. The outer cylinder is kept sta-tionary, whereas the inner cylinder rotates withthe azimuthal velocity specified as in Refs.:8, 9

vq (r⇤ = 0, t) = Wi tanh(at) . (8)

Here, a denotes the dimensionless ramp rateof the rheometer and Wi = l1V/H the Weis-senberg number with V being the dimensionalangular velocity.

Fig. 1 shows the steady-state profile of thepolymer number density with and without thestress-induced migration. It is obvious thatstress-induced migration is responsible for theoccurrence of the shear bands. The horizontalline indicates that the polymer number densityis nearly uniform across the cylindrical gap ifthe stress-induced term is ignored.










141

0.0 0.2 0.4 0.6 0.8 1.00.997

0.998

0.999

1.000

1.001

1.002

n p

r

With stress-induced migration Without stress-induced migration

*

Figure 1. The number density of the polymeris calculated for the model with and without

the term corresponding to stress-inducedmigration. The values of the model parameters

used in the calculation were a = 0.73,e = 0.0025, q = 1.46, Wi = 6,

b = E�1 = 10�5, µ = c = 10�1, andD = Dnonloc = 10�3.

Figure 2(a) and 2(b) display the effect of theWeissenberg number on the steady-state pro-files of the velocity and polymer number den-sity, respectively. Two distinct shear bands canbe seen for 3 < Wi < 63. If we increase the Wiparameter, the kink separating the bands movesfrom the rotating inner wall to the stationaryouter wall. The profile of the polymer concen-tration is banded, which is in contrast to thepredictions of standard polymer models. Thecurvature of the geometry justifies the curvedshape of this profile.

Figure 3 shows how the deformation historyaffects the steady-state profile of the polymernumber density. This was determined by ramp-up and ramp-down tests. We started the ramp-up test from rest and the ramp-down test fromthe steady-state solution at Wi = 100. A uniquesteady-state solution was obtained independentof the deformation history.

Figure 4 shows the effect of the nonlocaldiffusivity constant Dnonloc on the steady-statevelocity profile. The profile is smoother forlarger values of Dnonloc.

For the Poiseuille flow in a rectilinear chan-

0.0 0.2 0.4 0.6 0.8 1.0

0

2

4

6

8

10

0.0 0.2 0.4 0.6 0.8 1.00.996

0.997

0.998

0.999

1.000

1.001

1.002

v θ

Wi=1 Wi=2 Wi=6 Wi=8 Wi=10

(a)

*r

n p

Wi=1 Wi=2 Wi=6 Wi=8 Wi=10

(b)

*r

Figure 2. The profiles of (a) the velocity and(b) the polymer number density calculated for

different Weissenberg numbers. The othermodel parameters are the same as those given

in the legend of Figure 1.

nel, we used the cartesian coordinate systemwith the origin at the centerline. The walls arekept stationary whereas a nonzero dimension-less pressure gradient Px = DpH/LG0 is ap-plied in the x-direction. To avoid unnecessarycomputations, we solved the model for half ofthe channel.

Figure 5 shows the effect of the value ofthe pressure gradient on the steady-state pro-files of the velocity and the polymer concentra-tion across the gap. Here, y denotes the locationin the channel width, with y = 0 and 0.5 corre-sponding to the centerline and the wall, respec-tively. The velocity profile forms a low shearrate band near the center and a high shear rate


142

0.0 0.2 0.4 0.6 0.8 1.00.996

0.997

0.998

0.999

1.000

1.001

1.002

n p

vθ(t)=Wi tanh(αt)

vθ(t)=100(1-tanh(αt))+Wi tanh(αt)

*rFigure 3. Steady-state profiles of the polymer

number density calculated using differentdeformation histories. The model parametersare the same as those given in the legend of

Figure 1.

0.0 0.2 0.4 0.6 0.8 1.00

2

4

6

v θ

Dnonloc=0.1 Dnonloc=0.01 Dnonloc=0.001 Dnonloc=0.0001

*rFigure 4. Effect of the nonlocal diffusivityconstant on the steady-state profile of the

velocity. The other model parameters are thesame as those given in the legend of Figure 1.

band near the wall. A striking difference to thepredictions of the VCM model is that the transi-tion between the shear bands is smooth even inthe case of zero stress diffusion. Furthermore,the plug-like profile is observed over a widerrange of the dimensionless pressure gradients(i.e., for 1 < Px < 200). The concentrationbands are also predicted for the same range.For larger pressure gradients, the polymer con-

centration is more homogeneous with the kinkcloser to the centerline. It is worth noting thatin contrast to the VCM model, this solution isunique for all pressure gradients and no hys-teresis is observed in ramp up/down tests.

0.0 0.1 0.2 0.3 0.4 0.5

0

20

40

0.0 0.1 0.2 0.3 0.4 0.50.988

0.990

0.992

0.994

0.996

0.998

1.000

1.002

Px=-1 Px=-1.6 Px=-1.8 Px=-2

v x

(a)

(b)

y

y

Px=-1 Px=-2 Px=-5 Px=-20

n p

Figure 5. Influence of the pressure gradient onthe steady-state profiles of the (a) velocity and(b) polymer concentration across the channel.

The values of the model parameters used in thecalculation were a = 0.73, e = 0.0025,

q = 1.46, b = E�1 = 10�5, µ = c = 10�1,and D = Dnonloc = 10�3.

4 SUMMARY AND CONCLUSIONSUsing the generalized bracket approach ofnonequilibrium thermodynamics, we have de-veloped a two-fluid model to study shear band-ing in semi-dilute entangled polymer solutions.Unlike in previous two-fluid models, the dif-ferential velocity is treated as a state vari-










143

able, which makes the specification of the ex-tra boundary conditions straightforward. Weused the Giesekus relaxation in the polymerconformation equation to account for hydrody-namic interactions and to capture the overshootof the shear stress. We added a nonlinear re-laxation term to predict the upturn of the flowcurve at high shear rates. This term is simi-lar to the term used in the Rolie-Poly model,which describes convective constraint releaseand chain stretch. Moreover, a stress-diffusiveterm was added to the conformation equationto control the smoothness of the profiles. Wefound that the stress-induced migration is thediffusive term responsible for the formation ofthe shear bands. The steady-state profiles aresmooth and unique with respect to applied de-formation history and the value of the diffusiv-ity constant. Further boundary conditions suchas slip conditions can be easily investigated ifthe differential velocity is treated as a state vari-able. The simplicity of our model makes it at-tractive for the study of complex flows.

ACKNOWLEDGMENTThe authors gratefully acknowledge the finan-cial support of the Max Buchner ResearchFoundation.

REFERENCES1. A. E. Likhtman and R. S. Graham. Simple

constitutive equation for linear polymer meltsderived from molecular theory: Rolie–polyequation. J. Non-Newt. Fluid Mech., 114(1):1–12, 2003.

2. M. Harvey and T. A. Waigh. Optical co-herence tomography velocimetry in controlledshear flow. Physical Review E, 83(3):031502,2011.

3. S. Ravindranath, S. Q. Wang, M. Olech-nowicz, and R. P. Quirk. Banding in simplesteady shear of entangled polymer solutions.Macromolecules, 41(7):2663–2670, 2008.

4. M. J. MacDonald and S. J. Muller. Ex-perimental study of shear-induced migrationof polymers in dilute solutions. J. Rheol.,40(2):259–283, 1996.

5. A. B. Metzner, Y. Cohen, and C. Rangel-Nafaile. Inhomogeneous flows of non-Newtonian fluids: generation of spatial concen-tration gradients. J. Non-Newt. Fluid Mech.,5:449–462, 1979.6. M. Cromer, M. C. Villet, G. H. Fredrick-

son, and L. Gary Leal. Shear banding in poly-mer solutions. Phys. Fluids, 25(5):051703,2013.7. M. Cromer, G. H. Fredrickson, and L. Gary

Leal. A study of shear banding in polymer so-lutions. Phys. Fluids, 26(6):063101, 2014.8. N. Germann, L. P. Cook, and A. N. Beris.

Investigation of the inhomogeneous shear flowof a wormlike micellar solution using a ther-modynamically consistent model. J. Non-Newt.Fluid Mech., 207:21–31, 2014.9. N. Germann, L. P. Cook, and A. N. Beris.

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simultaneous in-situ analysis of instabilities and first ......[k/(kbt)trc3]q is a scalar function...

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