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Introduction

A model is a quantitative abstraction of a physical process, in which the descrip-
tion of the process is represented by the solution to a set of mathematical equations
(1,2). The model equations represent the behavior of the real process to the extent
that the equations embody an accurate description of the underlying physical and
physicochemical phenomena. The mathematical formulation enables the model
to be used for a variety of purposes, including design, control, and exploration
of operating strategies; the effects of changes in process variables and material
properties can be inferred from the model without extensive experimentation.
Mathematical models have long been used for these purposes in the chemical and
petrochemical industries, and computer-aided design and computer-aided manu-
facturing (CAD/CAM) have been taking on growing importance in some polymer
processing operations.

The essential elements of any model of a physical process are threefold: the

geometry, the relevant laws of physical conservation (mass, momentum, and en-
ergy), and the specific constitutive relations. In polymer processing applications,
the last of these could include the stress-deformation equations for the material be-
ing processed (the material rheology) and the kinetic equations for phase change.
The mathematical complexity of a model depends on the process and the type of
information required. Qualitative information can often be obtained from a model
that greatly simplifies the physical phenomena to obtain mathematical simplic-
ity and consists only of one or more algebraic equations, for example, whereas
structural predictions typically require the consideration of details of stress and
flow fields and hence the solution of nonlinear partial differential or integral equa-
tions. Advances in computing technology have made the latter goal possible for
some processing applications (3,4).

The term computer model is often employed as a consequence of the exten-

sive use of computing technology in conjunction with mathematical models. This
terminology is unfortunate, because it confuses modeling and simulation. The
equations describing the physical phenomenon make up the model and incorpo-
rate the full understanding of the process. The model is independent of the means
that are used to solve the equations, which might be analytical or numerical, with
or without the use of computers. Application of the model to simulate some specific
situation requires the choice of a solution technique; numerical artifacts associ-
ated with the solution method could introduce errors that are not inherent in the
physical assumptions on which the model is based. Comparison of the predictions
of a model with experiments requires careful separation of the effects of modeling
assumptions and numerical techniques.

Polymer processing operations include polymerization reactors; devices for

mixing, conveying, and extruding molten polymer or polymer solutions; and de-
vices for forming shaped objects in the liquid state and post-forming solid-state
operations. The modeling of chemical reactors to predict conversions and prod-
uct distributions is a mature art (5). Efforts in this field have focused on pro-
cess control and the conditions leading to instabilities and “runaway” reactions.

Encyclopedia of Polymer Science and Technology. Copyright John Wiley & Sons, Inc. All rights reserved.

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Polymerization reactors form a subset of the subject of reaction engineering, dis-
tinguished only by the need to know the details of the specific reaction pathways
and kinetics, and perhaps the dependence of viscosity on conversion. Polymeriza-
tion reactions and reactors, including reaction injection molding, are treated else-
where, and this article is concerned only with the modeling of the mechanics and
structure development in mixing, conveying, extrusion, and forming processes.

Rheology

Models of the mechanics of polymer processing operations can be categorized by
the rheological description used. Low molecular weight liquids are Newtonian,
which means that the stress at any point in a deformation field is a linear function
of the instantaneous rate of strain and is independent of any prior history; the
stress in an incompressible Newtonian liquid is characterized by a single material
parameter, the viscosity (

η), which can be temperature- and pressure-dependent

but is independent of the deformation rate:

σ = − pI + τ

(1)

τ = η[∇v + (∇v)

T

]

(2)

Here,

σ is the total stress, p the isotropic pressure, I the identity (unit)

tensor, and

τ the extra stress (ie, the stress in excess of the isotropic pressure).

∇ is the gradient differential operator, and v is the velocity vector;

T

denotes the

transpose of a tensor. For a one-dimensional flow with a single velocity component
v, in which v varies in a single spatial direction y that is transverse to the flow
direction, equation 2 simplifies to the familiar form

τ = η dv/dy

(3)

When the flow is isothermal and the liquid has a viscosity that can be taken to be
independent of pressure (ie,

η is constant over the entire flow field), substitution

of equations 1 and 2 into the balance of linear momentum leads to the Navier–
Stokes equations, which have been the subject of intense study in classical fluid
mechanics for more than a century.

In contrast, the stress in a macromolecular liquid depends in a nonlinear

manner on the entire history of deformation. The nonlinearity is manifested by
phenomena such as the strain-rate dependence of the shear viscosity, strain hard-
ening in extension, and the transverse normal stresses in shear flow that cause
extrudate swell. The dependence on history is demonstrated by the transient
buildup and relaxation of stresses following changes in flow variables, and by the
existence of the dynamic storage modulus (G



) in oscillatory shearing. There have

been many routes to the development of appropriate stress constitutive equations
for use in processing flows of polymeric liquids (6–8), including general continuum
mechanics formulations and molecular and quasi-molecular theories. No consti-
tutive theory developed to date is adequate for quantitative predictions in all
processing flows, in some cases because the complexity of the equation prevents

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efficient calculation in complex flow geometries, but some stress equations have
been useful in particular classes of flows.

Purely viscous constitutive equations, which account for some of the non-

linearity in shear but not for any of the history dependence, are commonly used
in process models when the deformation is such that the history dependence is
expected to be unimportant. The stress in an incompressible, purely viscous liquid
is of the form given in equation 2, but the viscosity is a function of one or more
invariant measures of the strength of the deformation rate tensor, [

∇v + (∇v)

T

].

[An invariant of a tensor is a quantity that has the same value regardless of the
coordinate system that is used. The second invariant of the deformation rate ten-
sor, often denoted II

D

, is a three-dimensional generalization of 2(dv/dy)

2

, where

dv/dy is the strain rate in a one-dimensional shear flow, and so the viscosity is
often taken to be a specific function–a power law, for example–of (

1
2

II

D

)

1
2

.]

The dependence of the stress on the strain-deformation history of macro-

molecular liquids can be incorporated in two ways. The stress constitutive equa-
tion can be formulated as a differential equation, in which the extra stress

τ is

the solution of an equation that is typically of the general form

A(

τ /G) ·

τ

G

+ λ

D(τ /G)

Dt

= λ[∇v + (∇v)

T

]

(4)

Here, A(

τ /G) is a nonlinear tensor function of the extra stress; the simplest

such form is a Maxwell liquid, in which A is equal to the unit tensor and the
first term in equation 4 simply becomes

τ /G. λ is the time constant for stress

relaxation, and G is the shear modulus, which is a material property; the shear
viscosity

η is equal to the product λG. D/Dt denotes a nonlinear time derivative that

accounts for the invariance of the physical quantities under changes of the frame
of reference of the observer;

D/Dt contains terms of the form τ · ∇v, but the precise

form depends on the particular constitutive theory. Equations of this type follow
from transient-network theories, which are based in part on concepts embedded in
the theory of rubber-like elasticity. Equations of this type also follow from “tube”
or “reptation” theories, in which the chain in an entangled melt or concentrated
solution is envisioned as being restricted to an imaginary tube made up of the
constraints imposed by the surrounding chains, which restrict chain motion that is
transverse to the molecular backbone. (Differential equation forms for tube models
usually require mathematical approximations that are not contained in the basic
model.) The relaxation time and shear modulus depend on the deformation rate
or stress in some theories, whereas in others they depend on a dynamical scalar
or tensor variable characterizing the structural state of the melt or solution. If the
spectrum of relaxation modes characterizing the linear viscoelastic response is to
be included, the stress is comprised of a sum of terms,

τ =

τ

i

(5)

where each term in the sum satisfies an equation of the form of equation 4, with a
spectrum of material parameters

{λ

i

, G

i

}. The values of the spectral parameters

in the limit of vanishingly small deformations are obtained from classical linear
viscoelastic experiments.

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The stress constitutive equation can also be formulated as an integral over

the history of the deformation. The most common form used for simulations is

τ (t) =

t



− ∞

m(t

− t



) h(I

C

,II

C

) C

− 1

(t

,t



) dt



(6)

Here, C

− 1

is the Finger measure of strain and m(t) is a memory function that

can be determined from the linear viscoelastic response; m(t) can be expressed in
terms of the spectral parameters

{λ

i

, G

i

} and is usually taken to be a sum of expo-

nentials. h(I

C

, II

C

) is a nonlinear “damping” function that depends on the first and

second invariants of the Finger strain. Other strain measures may also be used,
and the popular K-BKZ model includes a small second term proportional to the
Cauchy strain measure C. The time–strain separability indicated by the product
of the functions m and h fails experimentally at very short times (9). As with the
differential equation forms, constitutive models of the general form of equation 6
follow from general continuum mechanics formulations, from transient-network
formulations, and from tube theories. There is sometimes a one-to-one equiva-
lence between a differential and integral equation formulation, but in general
only approximate equivalences can be developed. Most tube models are naturally
formulated as integral equations.

There are advantages and disadvantages to both differential and integral

constitutive equations. Differential equations adapt naturally to the numerical
methods commonly used for the solution of the momentum and energy equations,
which are in differential form, and most process modeling has been carried out
using differential stress constitutive equations. It is often important to include
the relaxation spectrum in process calculations, however, and each term in the
spectrum (eq. 5) requires an additional nonlinear partial differential equation
in the solution set, which greatly increases the magnitude of the computational
problem. The entire spectrum enters naturally in the integral formulation through
the memory function m(t), but it now becomes necessary to track the history of the
strain with great accuracy along material element paths on a computational grid
that cannot be laid out to ensure that particle paths pass through the spatial nodes.

Finally, there are some constitutive models that cannot be expressed in closed

form as differential or integral equations, but require the solution of a Fokker–
Plank equation (or an equivalent set of stochastic differential equations) for the
orientation distribution of chain segments in order to compute the stress (Ref-
erence 4, pp. 338 ff; Reference 10 and references therein). This technique may
become more useful as computing power increases, but to date it has been used
only for viscometric flows and a few very simple non-viscometric geometries.

Classification

Classifications of viscoelastic flows are useful for analysis, mainly in determining
conditions under which a viscoelastic constitutive equation that accounts for fluid
memory can be replaced by a much simpler purely viscous equation.

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The broadest classification scheme that has been found to be useful in model-

ing is based on the Deborah number (11–13), which is loosely defined as the ratio
of the fluid time scale to the time scale of the process. Small Deborah numbers
correspond to situations in which stress transients occur rapidly compared to the
total processing time; hence the “memory” of the fluid is short and can be ne-
glected. Large Deborah number flows take place over a time scale in which stress
transients cannot be neglected, and a viscoelastic description like that in equation
4 or 6 is essential. The significance of this classification is apparent in a linear
viscoelastic frequency sweep, where the Deborah number is the product of the
relaxation time

λ and the frequency ω; for λω  1 the loss modulus G



dominates

and the response is that of a viscous liquid, whereas for

λω 1 the storage mod-

ulus G



dominates and the response is closer to that of an elastic solid. The fluid

time scale is unambiguous for polymeric liquids described by constitutive equa-
tions such as equation 4; if the spectrum of relaxation times is required, as with
equation 5 or 6, an average or maximum relaxation time would be used, and there
is some ambiguity in the choice. There is frequently ambiguity in the choice of
the process time scale; the residence time is the obvious choice in a unidirectional
flow, but the selection is not clear in more complex flow fields.

The concept of strong and weak flows (14) is one attempt to quantify the Debo-

rah number concept unambiguously. This classification is based on the notion that
shearing flows are an ineffective (weak) means of stretching polymer molecules,
whereas extensional flows can be effective (strong) in elongating molecules. The
physical basis for the concept is rooted in the coil-stretch transition; a macro-
molecule modeled by a Rouse chain with a maximum relaxation time

λ

m

will

be stretched out in an elongational flow with stretch rate

 when λ

m

 >

1
2

.

(The “upper convected” Maxwell model is the continuum equivalent of a fluid
made up of Rouse chains, and the stresses in an elongational flow for this fluid
model become unbounded when

λ

m

 >

1
2

.) Let L be the velocity gradient, L

= ∇v,

and

α an eigenvalue of the matrix λ

m

L

−

1
2

I. A flow is strong if any eigenvalue

α

has a positive real part, in which case a Rouse chain will become fully extended
if the time in the flow is large relative to a relaxation time; this is equivalent to
the condition

λ

m

 >

1
2

for pure extension. A processing flow can be strong in some

regions and weak in others according to this classification.

Classification in terms of stress level has not been widely used, but stress

level is an important factor in process modeling. It is well known that process
behavior in confined flows changes dramatically when the wall stress in highly
entangled melts or concentrated solutions is comparable in magnitude to the shear
modulus, which is essentially equal to the plateau value of the storage modulus.
Flow instabilities like melt fracture (15) are observed, for example, and there is
an apparent failure of adhesion between the melt and the metal surface (15).
Low stresses are those for which

τ /G

p



≤ 1, where τ  denotes the magnitude

of the stress tensor and G

p



is the plateau modulus. This ratio is equivalent to

what is often called the recoverable shear and is proportional in channel flow to a
dimensionless group known as the Weissenberg number, We

= λv/d, where λ is a

characteristic relaxation time, v is the average velocity, and d is the diameter or
thickness of the channel. As discussed subsequently, the appropriate formulation
of wall boundary conditions for model equations is uncertain outside the low stress
regime.

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Flows must also be classified on the basis of whether the melt is isotropic or

anisotropic at rest, and whether the material is homogeneous or heterogeneous.
Liquid crystalline polymers are anisotropic at rest and heterogeneous in the melt,
containing highly oriented “domains” over micrometer-length scales. Even ele-
mentary flows of liquid crystalline polymers have not been successfully modeled,
because the continuum rheology is incompletely understood. Highly filled fiber
composites are heterogeneous and usually anisotropic in the melt even at rest
because of the absence of Brownian restoring forces. Modeling of shaping flows
for these systems is primitive, except in some idealized situations, because of the
need to account for nonlocal interactions in the rheology. The only well-developed
art is for homogeneous polymers that are isotropic at rest.

Numerical Methods

The equations describing polymer processing operations are usually coupled, non-
linear partial-differential or partial-differential-integral equations in which two or
three spatial directions and perhaps time appear as independent variables. Fully
three-dimensional problems can usually be solved only for purely viscous liquids,
and substantial simplification is usually required even for two-dimensional prob-
lems of viscoelastic liquids because of limitations of computer speed and mem-
ory. In some situations the geometry provides simplifications that lead to closed-
form analytical solutions, but these are rare, and numerical methods are usually
required in order to obtain process information. Numerical methods relevant to
polymer processing flows are discussed in (3,4,16,17). The following three broad
classes of numerical methods are in common use:

Finite difference methods are the traditional approach to the numerical so-

lution of partial differential equations. The spatial regime is divided into a grid,
as shown in Figure 1. Partial derivatives are approximated by differences; the
derivative

∂u/∂x at grid point (i,j) might be approximated as [u

i

,j

− u

i

−1,j

]/h, for

example. The partial differential equations are thus reduced to a set of algebraic
equations for the values of the dependent variables at the grid points. Iterative

Fig. 1.

Finite difference grid.

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269

Fig. 2.

Finite element grid.

methods are usually required, because the equations are generally nonlinear. Fi-
nite difference methods are used infrequently for numerical simulation of vis-
coelastic flows because of a perceived lack of versatility. Programming is difficult
unless the grid spacing is uniform, but accurate approximation of derivatives re-
quires closer spacing of grid points in regions of rapid change of the dependent
variables than in regions of little change; nonuniform spacing can be introduced,
usually through coordinate transformations, but the techniques for doing so have
not been widely adopted for viscoelastic simulation. Finite difference methods are
difficult to apply to complex shapes, especially with curved boundaries, although
this limitation too can be overcome. Finite difference is the approach of choice for
flow problems when inertial effects are significant, but this is rarely the case in
polymer processing.

Finite element methods are used in most commercial simulation programs

for polymer processing. In this approach the spatial regime is broken up into
small elements, usually of triangular or quadrilateral shape, as shown in
Figure 2. The solution is approximated over each element in terms of simple
functions, usually linear or quadratic, with unknown coefficients that must then
be evaluated. The Galerkin method of weighted residuals is the most common
method for evaluating the unknown coefficients; in this method, weighted aver-
ages of the equations, using the approximating functions themselves as weighting
functions, are required to be zero. The approximation leads to a set of algebraic
equations for the coefficients that are usually nonlinear and must be solved itera-
tively. The finite element method is adaptable to complex shapes, and the element
size can be easily changed throughout the regime to account for different rates of
change of dependent variables.

The Galerkin finite element method has a rigorous foundation for the type of

equation set that usually arises in creeping isothermal flow of Newtonian fluids
and in steady heat conduction, where the differential operator is elliptic and “self-
adjoint,” but the method is commonly used for other flows as well. Inertial terms
in the momentum equation, and the convective flow terms in the energy equation,
which are important even in the absence of inertia, cause the system to be non-
self-adjoint, however, whereas the equations describing viscoelastic flow are non–
self-adjoint even for creeping flow; the operator for viscoelastic flow can sometimes
change from elliptic to hyperbolic in a part of the flow domain. Hence, the stan-
dard tools may sometimes fail. “Streamline upwinding” and “discrete Galerkin”

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methods are sometimes used for the convective terms. Some codes, particularly
those that include heat transfer in the metallic surfaces, use hybrid methods in
which some parts of the problem are solved using finite elements and others using
finite differences.

Boundary element methods can be applied to a limited class of problems in

which Green’s theorem equating volume and surface integrals can be exploited;
this usually requires that the differential operator in the equations be the Lapla-
cian. The differential equations can be partially integrated and converted to a
set of equations in which the dependent variables enter only through integrals
along the boundary. In such a case the computational problem is substantially
reduced; instead of approximations over a two-dimensional area, for example, ap-
proximation is required only along a one-dimensional boundary curve. The method
of approximation along the boundary is usually similar to that used in finite ele-
ment methods, hence the name boundary element. The boundary element method
has been adapted to the solution of several transient and free-surface processing
problems, as well as to the calculation of the orientation of rodlike suspensions in
Newtonian fluids, but general-purpose codes are not widely available.

Continuous Low Deborah Number Processes

The most successful process modeling has been carried out on continuous low Deb-
orah number processes (18–20). This category includes calendaring, the metering
region in melt extrusion, and many coating operations. The fluid deformation in
these flow fields is closely approximated by combined drag- and pressure-driven
flow between nearly parallel surfaces, and the lubrication approximation (21,22),
developed by Reynolds for the analysis of lubrication flows, is applicable as a
first approximation. The lubrication approximation is broadly based on the as-
sumption that the flow between nearly parallel surfaces is locally the same as
the fully developed flow between parallel walls, for which exact solutions to flow
problems exist. (The isothermal lubrication flow of a Newtonian fluid would have
a locally parabolic velocity profile, for example, but with the length scale defining
the parabola changing slowly with position in the flow.) The lubrication equa-
tions are sometimes integrated across the gap in flows in which there are two
primary flow directions and a narrow gap in the transverse direction, leading
to a set of equations for local gap-averaged velocities and the pressure distri-
bution; this averaged formulation is sometimes referred to as the Hele–Shaw
approximation.

The flow of a viscoelastic liquid between infinite parallel walls is a visco-

metric flow, or a flow with constant stretch history. The velocity profile for a fluid
that is isotropic at rest is determined in such a flow only by the shear viscos-
ity, although the stress distribution depends on the viscoelastic parameters. A
nearly parallel flow for which the Deborah number is low, and stress growth and
relaxation is not important, can be treated as though the local flow were that be-
tween infinite parallel walls; in that case the viscoelasticity is not important for
determining the flow field and the process can be analyzed with the lubrication
or Hele-Shaw equations as though the polymer were purely viscous. Effects at-
tributable to the viscoelastic parameters (eg, interface movement in co-extrusion)

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can then be determined using the computed flow field. The main complicating
factors in such a calculation are the correct incorporation of the energy balance
equation, since the temperature field is generally developing throughout the flow,
and the pressure dependence of the viscosity. Calculation of velocity and temper-
ature distributions, residence times, and pressure drops is straightforward; some
closed-form analytical solutions are available for simple processing geometries,
and some commercially available computer codes can routinely handle two- and
three-dimensional nonisothermal creeping flows of inelastic liquids. Good heat
transfer coefficient data are limited, however, and little attention has been paid
to the effect of high pressure on physical properties (22).

The procedure outlined above has long been the standard approach. The

simulation of coating thickness in blade coating, for which experimental data are
available for a range of liquids of increasing viscoelasticity, is a good example
of the application and limitation of these approximations. The process geometry
is shown schematically in Figure 3, together with a finite element mesh used
for computation. Data for the coating thickness as a function of gap spacing are

Fig. 3.

(a) Blade coating; (b) surfaces requiring boundary conditions; and (c) finite-

element mesh used in calculations. From Ref. 23. Courtesy of the American Institute of
Chemical Engineers.

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Fig. 4.

Coating thickness as a function of gap thickness for a blade angle of 10

◦

, showing

agreement between lubrication theory (—–), finite element calculations (– – –), and experi-
mental data for aqueous solutions of 1.25% carboxymethylcellulose and (

, 0%;

ⵧ

, 0.005%;

䊊

, 0.020%) polyacrylamide. From Ref. 24. Courtesy of the American Institute of Chemical

Engineers.

shown in Figure 4 for three shear-thinning polymer solutions: an aqueous solution
of 1.25% carboxymethylcellulose, and the same solution with an additional 0.005%
and 0.02% of a high molecular weight polyacrylamide; all three liquids have nearly
the same shear viscosity functions, with a power-law index of 0.6 in the power-
law region, but the polyacrylamide solutions are more viscoelastic. The solid line
is the lubrication approximation to the flow. The dashed line is a finite-element
numerical solution of the full two-dimensional purely viscous equations without
the nearly parallel approximation. The lubrication approximation is acceptable for
small gap spacings but deviates systematically from the numerical solution of the
full equation set for larger spacings. The solution to the purely viscous equations
agrees well with the data for the least elastic solution, but deviates substantially
for the most elastic liquid; the coating flow with the latter is clearly not a low
Deborah number process.

Periodic Low Deborah Number Processes

Mold filling, which is a repetitive transient process, can often be treated as a low
Deborah number process when the fill time is long relative to a characteristic
time for stress relaxation, so that only purely viscous rheology need be consid-
ered. The fluids are usually of very high viscosity, and the characteristic gap size
is usually small; inertial effects are thus usually negligible compared with vis-
cous stresses. The transient terms in the momentum balance are inertial, so the

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273

Fig. 5.

Experimental and computed front movement during filling of a compression mold

with a silicone liquid. (—–) experimental; (– – –) predicted (FEM); (-----) predicted (BEM).
Reproduced from Ref. 27. Courtesy of the Society of Plastics Engineers.

momentum equation becomes quasi-static if inertial effects can be neglected. Mold
filling under these conditions is simulated by solving the steady-state momentum
equation for an inelastic liquid, with the front movement calculated at each time
step from the normal velocity at the front. The numerical methods used for con-
tinuous processes can thus be applied, with (conceptually) minor modification to
follow the moving front (24–26). Special techniques must be implemented to follow
temperature transients and any solidified regions.

Fill rates and front movements are usually well described by Hele-Shaw

calculations for purely viscous fluids. Simulations of isothermal mold filling of a
silicone oil in a compression mold using both finite element (FEM) and boundary
element (BEM) methods are shown in Figure 5, together with experimental front
locations. Simulation of front movement in the injection molding of polypropylene
with a series of short shots, including the development of the weld line, is shown in
Figure 6. The polypropylene was treated as a purely viscous power-law fluid with
temperature-dependent physical properties, using a hybrid finite element-finite
difference technique to solve the Hele-Shaw equations.

Details of the kinematics within the mold, which would be required for the

computation of stress distributions, require solution of the full (quasi-static) mo-
mentum equation with an appropriate stress constitutive equation. This can be
done routinely for purely viscous fluids for flows with two velocity components

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Fig. 6.

Filling of a double-gated injection mold with polypropylene. (a) Experiment using

a series of short shots; and (b) simulation.

+++++++ = Computed weld line. Reproduced

from Ref. 28. Courtesy of Elsevier Science Publishers.

(two-dimensional or axisymmetrical) and for three-dimensional flows in relatively
simple mold geometries. Complex geometries can be addressed by linking together
Hele-Shaw solutions in regimes where the transverse velocity component can be
neglected with full three-dimensional calculations in other portions of the mold.
The three-dimensional flow near a moving front is rarely treated, although the
“fountain flow” near the front can be approximated by neglecting the larger of
the two radii of curvature of the front. A typical calculation of the flow near the
front is shown in Figure 7; numerous studies modeling these kinematics indi-
cate that the agreement with experiments on front movement is very good, and
that the computed flow field is insensitive to the particular choice of constitutive
equation, although the stress field is not. The implication of this observation for
mold filling is quite important, since low Deborah number computations, in which
viscoelasticity can be neglected, are much simpler than computations with vis-
coelastic constitutive equations. In many mold filling applications the flow field
can be computed first, treating the melt as purely viscous, and the viscoelastic
stress computed afterward. Since stress and orientation are linked in noncrys-
talline polymers through the stress-optical law, such a decoupled approximation
is efficient in computing molecular orientation distributions.

The computational problem is considerably more difficult if solidification is

to be taken into account. The assumption is often made that solidification is in-
stantaneous at a specified temperature. Some modeling studies have been carried

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275

Fig. 7.

Calculated tracer line formation as a function of time during mold filling. Repro-

duced from Ref. 29. Courtesy of the American Institute of Chemical Engineers.

out in which crystallization kinetics are included in the mass and energy balances
in order to estimate the degree of crystallization and the morphology when the
motion is frozen. Most such studies have used quiescent crystallization data. A
reliable theory of crystallization rates during deformation at very high stress lev-
els is still a subject of research (30,31), as is a complete theory of the rheology
of the crystal/melt mixture, and commercial codes do not address this issue. The
ultimate goal is the prediction of stress and morphology following molding (32).

Free-Surface Processes

The modeling of polyester fiber spinning from the melt is well developed for take-
up speeds below about 4,000 m/min, where crystallinity is low (33); commercial
and in-house computer codes have long been available for the prediction of tem-
perature and diameter profiles along the spinline and the radial birefringence
distribution (through a stress-optical coefficient) at take-up, and they have been
used widely to analyze process performance and to evaluate modifications. The
major challenge has been the incorporation of crystallization, which is an impor-
tant factor in the spinning of nylon at all take-up speeds and of polyesters at high
speeds. Spinline models that account for stress-induced crystallization and the
effect of the crystalline phase on the rheology (34) are now available .

Most spinline models are based on thin-filament equations, in which the

essentially elongational character of the deformation is exploited and the conser-
vation and constitutive equations are averaged over the diameter or, in a math-
ematically equivalent process, expanded in a small scaling parameter. The thin-
filament approximation is similar in approach to the lubrication approximation
for confined flows, but the low Deborah number assumption need not be made. It
is straightforward to include a solution of the energy equation that incorporates
the radial dependence of temperature in this formalism. (There is a region near
the point of extrusion where the melt undergoes a transition from the shear flow
in the spinnerette to the extensional flow on the spinline, and the thin-filament
approximation cannot apply in this region. Finite-element solutions of the flow

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Fig. 8.

Comparison of model calculations of diameter attentuation, birefringence, and

temperature profiles with pilot plant data for melt spinning of 0.65 IV poly(ethylene ter-
phthalate) at a take-up speed of 5947 m/min. Reproduced from Ref. 33. Courtesy of the
Society of Rheology.

in the transition region indicate that the thin-filament equations become valid
within a few spinneret diameters.) Full two-dimensional axisymmetric models are
available but add little additional physical understanding. Most spinline models
assume that the polymer is a Newtonian fluid, but differential viscoelastic models
are easily incorporated and there is no reason to use the Newtonian approximation
when rheological data are available. A comparison of the prediction of diameter at-
tenuation, birefringence, and mean temperature profiles for high-speed spinning
of poly(ethylene terephthalate) with pilot plant spinning data is shown in Figure 8.
The viscoelastic model incorporates stress-induced crystallization and success-
fully captures the rapid diameter attenuation and increase in birefringence at
the onset of crystallization. Predictions of the temperature profile and the final
stress level tend to be insensitive to stress constitutive assumptions, despite the
sensitivity of the diameter attenuation profile to the constitutive assumptions.
Limited dynamic simulations to study disturbance propagation through spinlines
have been less successful, probably because of inadequate treatment of the solid-
ification process (35).

The proper representation of macroscopic transport properties, particularly

the heat transfer coefficient, is a major problem in the predictive modeling of
spinning and other free-surface processing flows. Heat transfer coefficients are
typically obtained from experiments on nondeforming wires, and the extension
to a deforming surface with a variable cross section is not obvious. Data ob-
tained on real spinlines require either infrared or intrusive contact temperature

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277

Fig. 9.

Comparison of model calculations of bubble radius and film velocity with blown

film data for a low density polyethylene, 3.84-cm radius

× 0.8-mm thickness die, 4.1-kg/hr

throughput. Data of Tas. Experiment 12: inflation pressure

= 118 Pa, take-up force = 7.6

N,

× radius, (

) predicted radius,

䊉

velocity, (

) predicted velocity; experiment

15: 108 Pa, 7.7 N,

+ radius, (– – –) predicted radius,

䉬

velocity, (——) predicted velocity; ex-

periment 18: 95 Pa, 6.8 N,

∗ radius, (

) predicted radius,

velocity, (

) predicted

velocity. Reproduced from Ref. 38. Courtesy of the Society of Rheology.

measurements, neither of which is reliable for the accurate determination of a heat
transfer coefficient. Model calculations indicate that the diameter attenuation
is highly sensitive to the heat transfer coefficient; small uncertainties can mask
other factors and make extrapolation to new processing conditions questionable.

The blown film process is kinematically similar to fiber spinning, and thin-

sheet equations analogous to the thin-filament equations have been employed in
process modeling (36–38). As with the thin-filament equations, temperature vari-
ation in the film thickness direction can be incorporated in a straightforward way.
Figure 9 compares bubble radius and velocity profile data from three experiments
on a blown film line with a thin-sheet calculation that incorporates a viscoelastic
constitutive equation, a transverse temperature variation, and a crystallization
model (38). The calculation was based on an empirical heat transfer coefficient
that is not formulated in terms of scalable dimensionless groups. The operating
characteristics of film lines depend on details of the placement of cooling air, but
this feature cannot be incorporated into existing blown film models without a
better understanding of the heat transfer.

Modeling of the blown film process illustrates characteristics that are very

different from those of fiber spinning, even with primitive representations of the
heat transfer (37). These differences are a consequence of the hoop stress in the
bubble. Multiple steady states are computed for fixed operating conditions, for
example, and steady axisymmetric bubbles can vanish suddenly following small
changes in operating parameters; both phenomena are seen in practice. The

278

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Vol. 11

models also show important differences in the stability characteristics of the two
processes.

Numerics

The numerical realization of a model requires the availability of algorithms that
can solve the model equations accurately. Modeling of the detailed flow and stress
fields in confined geometries has been limited historically by the fact that iterative
numerical algorithms for solving the momentum and continuity equations to-
gether with the nonlinear constitutive equations typically fail to converge to a
solution above a critical value of the recoverable shear or Weissenberg number.
This convergence failure, which is often called the “high Weissenberg number
problem,” appears to be a numerical artifact of the techniques in use, which prop-
agate the large stresses generated near corners, lips, and other areas of stress
concentration into the bulk and degenerate the solution quality; precise identi-
fication of the cause is difficult, because the strength of the stress singularity
near a corner is an outstanding mathematical problem for the flow of most vis-
coelastic fluids, for which only partial solutions have been obtained (Reference 4,
pp. 68 ff).

Stress constitutive equations with the structure of equation 4 are hyperbolic

when the flow field is specified. Experience in areas where hyperbolic equations
commonly arise, such as compressible gas dynamics, has guided the development
of special techniques, including operator decompositions and “upwinding,” that
are implemented in some available computer codes. Use of these methods has
extended the range over which convergence can be obtained to the lower end of
practical processing conditions. Reviews can be found in References 4,16, and 17. A
simulation that is representative of the state of the art of computations for highly
viscoelastic melts is shown in Figure 10. Calculated isochromatic fringes, which
are contours of the principal stress [(

τ

xx

− τ

yy

)

2

+ τ

xy

2

]

1
2

, are superposed on the

Fig. 10.

Isochromatic fringe patterns for flow of a low density polyethylene (M

w

= 1.55 ×

10

5

, M

w

/M

n

= 11.9) past an offset cylinder of diameter 2.5 mm in a plane channel at 190

◦

C

with a mean upstream velocity of 8.53 mm/s. The mean relaxation time of the melt is
1.5 s. From Ref. 39. Courtesy of Elsevier Science Publishers.

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PROCESSING, MODELING

279

experimental fringes at 190

◦

C for the flow of a low density polyethylene through

a planar region containing an offset cylindrical insert. Characteristic Deborah
numbers based on mean residence times for flow in the small and large gaps
were, respectively, 0.7 and 11.5, indicating the likelihood of strong elastic effects.
The constitutive equation, which was fit to linear viscoelastic and steady shear
data, is a special form of equations 4 and 5 often used for simulations known
as the Phan–Thien/Tanner equation, with four terms in the linear viscoelastic
spectrum. The computations are in qualitative agreement with the experiments
and show the development of an asymmetric wake in which the stresses are very
high downstream of the aft stagnation point. The experimental wake extends
farther in the downstream direction than the calculated wake. The fringes appear
to be more densely packed in the lower upstream region close to the cylinder in
the experiments than in the calculations, indicating that the experimental stress
gradients were stronger than the computed gradients.

Boundaries

The tradition in fluid mechanics, dating to the mid-nineteenth century, has been
that a fluid adheres to a solid boundary during flow, and that the fluid at the
boundary therefore takes on the velocity of the solid. This “no-slip” condition has
been used routinely in the modeling of polymer flows, despite occasional hints in
the processing literature as early as the 1960s that there might be “slip” at the wall.
It is now generally accepted that the composition and state of the boundary affect
the flow (15); the properties of a polymer extrudate can be controlled by changing
the materials of construction of the metal die-face, for example, thus establishing
that the boundary is not simply a passive element in the flow field. Figure 11
shows flow curves for a linear low density polyethylene extruded through two

Fig. 11.

Flow curves for a linear low density polyethylene (M

w

= 114,000, M

n

/M

w

= 3.9)

at 200

◦

C in identical 1

× 20 mm 304 stainless steel () and CDA-464 naval brass (

䊊

)

capillaries. Reproduced from Ref. 40. Courtesy of the Society of Rheology.

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PROCESSING, MODELING

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geometrically identical capillaries, one constructed of 304 stainless steel and the
other of CDA-464 naval brass (40). The data for the steel capillary show the range
of pressure fluctuations in the “spurt” regime. The polymer extruded through the
brass capillary experienced significantly lower stresses at the same throughput
(“apparent slip”) and was free of surface defects that appeared on the extrudate
from the steel die at a stress just above 0.17 MPa.

Most experimental evidence indicates that deviations from the no-slip condi-

tion are unimportant up to a wall stress of the order of 0.2 MPa (29 psi). Since the
plateau modulus of most polymers at processing conditions is of the same order,
this observation suggests that no-slip can be used with confidence under condi-
tions where

τ /G

p



< 1. The appropriate boundary condition at higher stresses

is not known, although some data indicate that the tangential velocity at the
wall is proportional to a power of the wall shear stress; the slip law proposed
by Navier in the early nineteenth century assumes a power of unity. It is not
straightforward to impose a slip boundary condition in a numerical code and
simultaneously satisfy mass conservation at a sharp corner, however. Slip con-
ditions have therefore been implemented only sparingly. The range of conver-
gence of numerical schemes is extended when a slip condition is used, since the
high stresses that degenerate the solution are moderated. The effect of wall slip
in complex geometries is relatively unexplored. Analytical solutions for flow of
shear-thinning inelastic liquids between converging planes using a Navier slip
law indicate substantial departure from the expected radial streamline pattern
(41), which, as discussed subsequently, could have a large impact on the stability
of die entry flows.

Mixing

Mixing in polymer melt processing takes place in the laminar regime. Laminar
mixing, which is related to the theory of chaotic dynamical systems, has been
intensely studied since the late 1980s (42), and techniques have been developed
for visualizing the evolution of apparent disorder in strictly deterministic systems.
Application to the simulation of continuous mixers for immiscible materials with
viscosities of comparable magnitude requires the use of an efficient mapping pro-
cedure for tracking fluid elements as they progress through the three-dimensional
flow (43), where the local kinematics and pressure profile are calculated with a
finite-element solution. A simulation of a cross section of identical black-and-white
inelastic shear-thinning Carreau-Yasuda fluids with a power-law exponent of 0.1
following laminar flow through a static mixer is shown in Figure 12. Simula-
tions of this type can be used in principle to optimize mixer design, but they have
been limited to inelastic fluids with identical viscosity functions and no interfacial
tension.

Droplet breakup and coalescence are the primary physical processes in the

mixing of liquids with very different viscosities. Computational tools for the
breakup of individual Newtonian droplets are well developed; Figure 13 shows
a boundary element calculation of the sequence of shapes of a polydimethylsilox-
ane droplet in a polyisobutylene of nearly the same viscosity, together with ex-
perimental data. Computational tools for breakup with viscoelastic constitutive

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PROCESSING, MODELING

281

Fig. 12.

Simulation of the mixing pattern for two identical shear-thinning inelastic liquids

in a static mixer after passing six blades oriented at 140

◦

. Reproduced from Ref. 44. Similar

images for Newtonian liquids may be found in Ref. 45.

equations are not well developed. Emulsion models for the rheology of immiscible
blends with a dispersed droplet phase do an excellent job of describing the lin-
ear viscoelasticity, and models for nonlinear deformations of blends based on an
understanding of droplet dynamics are under development (47).

Fig. 13.

Deformation and breakup of a polydimethylsiloxane droplet in a polyisobutylene

of nearly the same viscosity; viscosity

≈ 100 Pa · s, interfacial tension ≈ 2.4 mN/m. Images

are experimental data and profiles are simulations using slightly adjusted parameters to
ensure a good fit at pinch-off. Numbers are experimental total strain (time multiplied by
shear rate), whereas numbers in parentheses are computed total strain for the same droplet
shape. Reproduced from Ref. 46. Courtesy of the Society of Rheology.

282

PROCESSING, MODELING

Vol. 11

Stability and Sensitivity

Polymer processes are subject to a variety of instabilities that limit throughput
or degrade product quality. Dynamic models are routinely used in engineering
applications to study conditions under which stable operation cannot be main-
tained, and idealized models of some polymer process have been employed to gain
insight into operability. The most common technique for studying process stability
is linear stability theory (48), in which the dynamical response to infinitesimal dis-
turbances is obtained by solution of a set of approximate linear equations through
an eigenvalue analysis. Complex flows can be analyzed through the eigenvalues
of a matrix that arises in the course of a finite element solution. If infinitesi-
mal disturbances die out, the steady mode of operation is stable; if infinitesimal
disturbances grow, the steady mode of operation cannot be maintained (since no
process can be kept free of infinitesimal disturbances), and the system is unstable.
Linear stability theory establishes absolute instability but only relative stability;
a process that is unstable to infinitesimal disturbances is unstable to all distur-
bances, but a process that is stable to infinitesimal disturbances might still be
unstable to some finite disturbances.

Dynamic modeling has been quite successful in describing the onset of a flow

instability known as draw resonance, which occurs in fiber spinning and extrusion
coating (1,19,33). Draw resonance is characterized by regular oscillations in the
fiber diameter or coating thickness, with a period that is slightly greater than
the residence time in the melt zone. Some subtleties associated with the role of
heat-transfer mechanisms remain, and the effect of the spinneret flow (as trans-
mitted through the initial conditions used in the thin-filament equations) has not
been completely quantified, but the mechanism is well understood and the models
have been very useful in discriminating between the effects of rheology, cooling,
gravitational forces, and inertia. Dynamic modeling to describe the propagation
of disturbances through a melt spinline under stable spinning conditions is much
less well developed (35).

Spinline failure appears to occur through a cohesive mechanism, which has

received little theoretical attention (49,50). A simple scaling model that predicts
a critical value of the recoverable strain as a function of stretch rate captures the
qualitative features of the limited data on melt failure in extension (50,51).

The blown film process undergoes a variety of processing instabilities. Mod-

eling of these is less advanced than for fiber spinning, in large measure because of
the sensitivity to details of the heat-transfer mechanisms and the treatment of the
solidification region. A draw resonance-like instability is predicted to occur under
some conditions (37). One striking result predicted by the dynamic model is that
the process stability is highly sensitive to the manner in which process control
is effected; under some conditions, a bubble controlled by air-flow adjustment to
maintain a constant diameter is predicted to be stable, whereas the same bubble
controlled by a feedback system utilizing the internal pressure is predicted to be
unstable.

Melt extrusion is characterized by at least two distinct instabilities that

cause surface defects, generally known as melt fracture, or gross melt fracture,
and sharkskin, or sharkskin melt fracture (15). Sharkskin, a high frequency,

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PROCESSING, MODELING

283

Fig. 14.

Short shots with two-color injection molding of a filled polypropylene compound.

Reproduced from Ref. 55. Courtesy of the Society of Rheology.

small-amplitude surface distortion, appears to occur only in highly entangled lin-
ear polymers and is believed to be associated with a material failure resulting
from the stress concentration at the die exit. Melt fracture, in which the distor-
tions may be regular or irregular, typically occurs when the wall shear stress is
comparable to the plateau modulus (recoverable shear of order unity) and appears
to originate in the die land or at the die entry. There have been numerous linear
stability analyses of fully developed pressure-driven viscoelastic flow in long slits
and capillaries; the general conclusion has been that the flow is stable to infinites-
imal disturbances in the absence of inertia, but a linear instability may exist for
some constitutive equations (52). A weakly nonlinear (“subcritical”) instability
has been reported (53,54). Whether model calculations can predict the instability
that is observed in the actual process remains an open question.

Flow mark surface defects on injection-molded parts were first reported in

the early 1960s and have been considered by some authors to be an indication of
wall slip in the neighborhood of the advancing front (15). Figure 14 shows a series
of short shots of injection molding of a filled polypropylene compound in which a
thin strip of black polymer was placed along the centerline. With stable fountain
flow, as shown in Figure 7, the black polymer will coat both mold surfaces. Instead,
the black strip is first carried to the bottom of the mold, then to the top. A swirling
flow near the melt front is predicted by linear stability theory combined with a
finite element analysis for one of two constitutive equations studied (55). It thus
remains an open question as to whether an instability in the neighborhood of the
front is the cause of the flow marks, although this is the likely mechanism.

Interface stability in co-extrusion has been the subject of extensive analysis.

There is an elastic driving force for encapsulation caused by the second normal
stress difference (56), but this is probably not an important mechanism in most
coprocessing instabilities. Linear growth of interfacial disturbances followed by
dramatic “breaking wave” patterns is observed experimentally. Interfacial insta-
bilities in creeping multilayer flows have been studied for several simple constitu-
tive equations (57–59). Instability modes can be traced to differences in viscosity
and normal stresses across the interface, and relative layer thickness is important.

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Aside from the examples cited here, the experimental literature on viscoelas-

tic flow instabilities in the absence of inertia has emphasized dilute polymer solu-
tions. These stability studies include the flow between rotating concentric cylin-
ders, torsional cone-and-plate flow, flow in a “lid-driven” cavity, stagnation flow,
and flow past a cylinder in a channel, all of which have been analyzed for simple
constitutive equations (60). It is likely that these phenomena have relevance to
polymer melt processing, but direct connections have not been made. The com-
mon feature of these inertialess “elastic” instabilities seems to be the coexistence
of normal stresses and curved streamlines, and the experimental and theoreti-
cal onsets of instabilities for this class of flows correlate with a criterion of the
form

λU/( H)

1
2

> M

crit

(61,62).

λ is a mean relaxation time, U the characteris-

tic velocity, H the length scale defining the shear rate, and

the length scale

defining the curvature of the streamlines. M

crit

is a constant characteristic of the

specific flow. The critical value can be interpreted as the geometric mean of the
Deborah and Weissenberg numbers. Substantial streamline curvature predicted
in converging flow of shear-thinning fluids with wall slip (41) suggests a possible
die entry mechanism for melt fracture.

Anisotropic Materials

Anisotropic materials, such as liquid crystalline polymers and fiber-filled melts,
present special difficulties because of the incomplete state of development of
constitutive models that account for the spatial distribution of the orientation
and stress fields. In nearly all cases the constitutive equations are derived from
microscopic theories of the orientation of rodlike elements, and the statistical
mechanics requires a closure approximation that expresses the fourth moment
of the orientation distribution in terms of the second moment (63,64). The clo-
sure approximation can affect the details of the evolution of the orientation
distribution. The models for both types of anisotropic liquids are very similar in
structure, differing in the way in which structural interactions are incorporated.
Liquid crystalline polymer models incorporate interaction potentials between
neighboring rigid molecules and include Brownian forces. Simulations using con-
tinuum models of liquid crystalline polymers in simple geometries show orienta-
tional structures and layering in rectilinear shear flows, but they are unable to
replicate the complex defect-driven textures characteristic of real nematic poly-
mers, where the correlation lengths for nematic order are independent of the
geometric scale of the system (63,65,66). Models for fiber-filled systems are based
on theories for noninteracting and slightly interacting non-Brownian rods, and
they do not rigorously account for coordinated fiber motions; the latter are dealt
with empirically by introducing an apparent diffusion term with a coupling co-
efficient that correlates with the product of fiber aspect ratio and volume frac-
tion (67). Simulations of both liquid crystalline polymer and fiber-filled systems
often assume that the flow field can be calculated independently of the orien-
tation distribution, but full coupling is usually required to obtain even quali-
tatively correct flow and orientation field solutions to the constitutive and field
equations (63,64,68,69).

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285

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M

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The Levich Institute,
City College of the City University of New York