Viscoelasticity

Materials such as plastics, rubber, glass, blood, and food stuff typically exhibit unusual behavior. Some of them show a large swelling (possibly a few times the outlet channel width) when they leave a die. Others present a strong increase in their resistance to deformations while stretched or blown. These phenomena and many others are related to the complex behavior of these VISCO-ELASTIC materials, which evolve partly as a viscous liquid and partly as an elastic solid within a particular time scale.  These fluids are also referred to as "memory fluids" because they tend to accumulate the history of prior deformations.  It is important to note, however, that this is a fading memory, since deformations in the recent past play a more important role than deformations in the distant past.

Until recently, these complexities were beyond the scope of CFD. Thanks to the improvements brought to the existing viscoelastic models, the numerical techniques, and the increasing power of today's computer hardware, viscoelastic effects can be simulated while designing an extruded product, a blown bottle, or a stretched film. In many cases, discarding viscoelasticity from the simulation would simply lead to unrealistic and useless results.

Modeling viscoelastic behaviors has been a strength of POLYFLOW software for more than two decades. POLYFLOW has, by far, the widest library of viscoelastic models available in any commercial code. Differential models, whether in 2D or 3D, for isothermal or non-isothermal, transient or steady state applications, are used commonly in POLYFLOW. Simple Upper-Convected Maxwell, White-Metzner or Oldroyd-B models and more sophisticated laws such as Phan Thien-Tanner, Giesekus, FENE-P or pom-pom[DCPP], involving a single or multiple relaxation mode, allow for a much more accurate description of the actual behavior of these complex materials. From these results, invaluable information is gained, leading to real breakthroughs in the understanding of the flow patterns and further improvements in the design of new products.

Extensional Behaviors

Molecular Considerations

In order to understand the macroscopic consequence of viscoelastic behavior, it is useful to go back to a simple and qualitative molecular description of the polymer. Contrary to other materials such as glass, water or gas, polymer molecules are very long chains of monomers; these macromolecules may also exhibit branches. At rest, these long chains are wound around themselves and entangled, as displayed in the figure below. When the polymer is stretched, either due to a pulling effort or because of an inflating pressure, the macromolecules are stretched and essentially oriented along a preferred direction.

In the case of fiber spinning, the resistance to deformation first increases due to the stretching and orientation of the macromolecules. Next, when the molecules become aligned with the pulling direction, the resistance to subsequent deformation will largely result from branching.  In the absence of branching, stretched and oriented macromolecules will mainly slip along each other and will not produce any further resistance to extension.  Branched macromolecules behave differently, a bit after the fashion of barbed wire: a macroscopic increase of the melt resistance is observed (before a possible fracture): this is the strain hardening.

The behavior in blow molding or thermoforming process can be more complex. Blowing either a parison or a sheet of plastic like a balloon creates a biaxial extensional deformation in the thin parison or sheet. In stretch blow molding, the pre-heated preform is first stretched by a plunger, and undergoes an initial deformation that is close to planar extension, before the biaxial blowing. These various kinematics are usually accompanied by differentiated melt behaviors.  The typical strain hardening of most plastic or elastomeric materials leads first to an increase of the parison or sheet resistance to deformation. A subsequent dramatic decrease of this resistance may be observed, which could induce instabilities in the process.

The POLYFLOW Solution

POLYFLOW has numerous viscoelastic models able to take these phenomena into account. Differential viscoelastic models have been proven successful in reproducing the experimental observations. The Olroyd-B, Giesekus, Phan Thien-Tanner and FENE-P models were used in order to simulate the 3D flow through a narrow flow channel, blow molding, filament stretching, extensional recovery, etc.

These non-linear viscoelastic models have an extensional viscosity varying non-linearly as a function of the extension rate. Contrary to other more simple models, such as the upper convected Maxwell or other White-Metzner models, the above-mentioned models still keep limited extensional viscosity for the whole range of extension rates.

Swelling

Molecular Considerations

In order to understand the macroscopic consequence of viscoelastic behavior, it is useful to go back to a simple and qualitative molecular description of the plastic. Contrary to other materials such as glass, water or gas, polymer molecules are very long chains, which may sometimes exhibit branches. At rest, these long chains are wound round themselves and entangled, as displayed in the figure below. When a polymer melt pushed through a narrow flow section, let's say a narrow outlet channel, the macromolecules undergo a shear stress.  Under a shear field, macromolecules receive a preferred orientation and are also stretched.

When the melt leaves the flow domain after the die lip, the macromolecule is free to rewind around itself, free to evacuate all stress stored during the journey through the narrow outlet channel. This is the stress relaxation phenomenon. In a free jet (extrudate), the microscopic recoil mechanism is accompanied by a transverse expansion, leading to swelling of the free jet and a slow-down of the flow.

The POLYFLOW Solution

POLYFLOW has numerous viscoelastic models able to take these phenomena into account. Differential viscoelastic models have been proven successful in reproducing the experimental observations. The Giesekus, Phan Thien-Tanner and pom-pom models were used in order to simulate the 3D flow through a narrow flow channel (shown below for a low flow rate (first picture) and high flow rate (second picture)) where a swelling of more than 100% is observed. Sometimes, in exceptional cases, larger swelling (800% and more) has also been observed.

These non-linear viscoelastic models have a first normal stress difference increasing with shear rate. In addition, their second normal stress difference is non vanishing, contrary to other simple models such as the upper convected Maxwell model or the White-Metzner model.

Recirculation

Recirculation or vortex creation is a typical phenomenon observed when dealing with viscoelasticity. The best known example published in the literature is the 4:1 contraction. In the entrant corner of the flow domain, just before the narrow channel, the particles are trapped in a large recirculating motion. Such a phenomenon can have dramatic consequences in industrial processes. The particles trapped in the vortex may remain in the extrusion die, or any other equipment, for a quite long time. The viscosity can evolve as a function of the residence time, leading to a hard kernel that would create a defect in the final product. Furthermore, if the local temperature is high in this region, the combination of both long residence time and intense heating often leads to the degradation of the resin grade, hence the quality of the final product. Other reactions, either chemical or physical may also be an undesired effect of these recirculation zones. This is why the designer will usually try to avoid such regions as much as possible.

In a contraction flow, vortices mainly result from elongation.  Secondary motions are also observed in flows through non-circular stratght channels: they originate from the second normal stress difference.  A nice example, illustrated by The Dow Chemical Company, shows slow transverse motions produced in the coextrusion flow of two LDPE melts through a channel with a square cross-section.  Such a simple experiment reveals the possible difficulties likely expected in coextrusion processes, where the control on fluid interface is easily lost.

Another illustration of vortices can be found in the well known WEISENBERG effect, where due to the stress concentrated around a rod rotating in a viscoelastic fluid, it appears that the material is climbing up along the rotating rod. This phenomenon is contrary to what is usually observed for Newtonian fluids, where the material tends to move away from the rod.

Rheological Models in POLYFLOW

Constant Viscosity

Generalized Newtonian

• Power law
• Bird-Carreau
• Cross
• Carreau-Yasuda
• Bingham
• Hershel-Bulkley
• KBKZ
• Doi-Edwards

Differential Viscoelastic

• Maxwell
• Olroyd-B
• White-Metzner
• PhanThien-Tanner
• Giesekus
• Fene-P
• pom-pom[DCPP]

Integral Viscoelastic (2-D Steady state only)

• Doi-Edwards
• Lodge Maxwell
• KBKZ (3-D transient for shell as well)

Those models can be combined with any of the following temperature dependence laws:

• Arrhenius
• Fulcher
• WLF

Through user-defined functions, more advanced behaviors, such as pressure dependence, ageing, etc ; can also be simulated.

Except for the integral viscoelastic models, the Generalized Newtonian models and differential viscoelastic models can be used in 2D and 3D, steady state or transient.