Modeling of Polymer Processing and Properties

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MODELING OF POLYMER
PROCESSING AND PROPERTIES

Introduction

The properties of polymeric systems are strongly dependent on their structure
which is itself fully controlled by the processing conditions. Thus, it is now well
accepted that the mechanical strength of polymer ﬁbers steadily increases with
an improvement of the orientation of the macromolecular chains (1). This is the
reason why ﬁber spinning is typically followed by a drawing step with the purpose
of aligning and extending the chains along the ﬁber axis. In some applications,
the polymer must be processed from solution, and ﬁber formation itself requires a
coagulation step in order to consolidate the polymer into a solid ﬁlament suitable
for drawing. In that case, the solvent-coagulant exchange at the polymer interface
strongly controls the internal structure of the ﬁber (2), and hence its ultimate
tensile strength and elongation at break. In other techniques, ﬁber formation
from solution occurs upon rapid quenching to a lower external pressure and/or
temperature (3). In this case, the size and the porosity of the ﬁber is strongly
dependent on the polymer concentration and quenching conditions.

A theoretical description of all the factors controlling the effect of processing

conditions on polymer structure and properties is extremely complex because
of the need to consider a wide variety of different factors such as polymer con-
centration, molecular weight, as well as external variables (time, temperature,
pressure). In view of this complexity, this work has focused on the development
of kinetic Monte-Carlo lattice models. These models are mesoscopic in the sense
that the unit lattice length is of the order of the statistical segment length for the
polymer chain and atomic level details are omitted. The objective of this review
is to describe these models in some detail and show their value in getting a better

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understanding of the interplay between processing conditions, polymer structure,
and product properties.

Polymer Drawing

This section focusses on polymer deformation in the solid state. Models for ﬂow-
induced structure and orientation in the molten state have been presented in Ref-
erences 4 and 5. Most previous approaches (1,6,7) for describing the deformation
of solid polymer systems are essentially phenomenological and static representa-
tions which are incapable of providing a uniﬁed description of the various morpho-
logical changes that may occur depending on the dynamics and on the structure
at the molecular level. All these models are incapable of simultaneously providing
quantitative information on (1) stress–strain behavior, molecular extension, and
fracture; (2) morphological changes that occur during deformation; and (3) the
effect of testing variables, such as temperature and strain rate.

One model for the deformation of polymer solids has been described at length

in References 8 and 9. It is brieﬂy reviewed here. Figure 1 gives this model rep-
resentation of the entangled solid polymer network prior to deformation. The en-
tanglements, ie, knots formed between chains, are denoted by the heavy black cir-
cles. For polyethylene in the melt, the molecular weight between entanglements,

melt

= 1900, which corresponds to n = 14 statistical segments. For solution cast

polyethylene samples, M

is easily obtained from

= M

melt

/φ

(1)

in which

φ denotes the polymer concentration. The dotted lines in Figure 1 repre-

sent the attractive (van der Waals or hydrogen) intermolecular bonds connecting
sections either of the same chain or of different chains. Tensile drawing is assumed
to occur along the y-axis.

The network is deformed at a constant temperature T and a rate of elongation

ε. During deformation, the bonds between chains are allowed to break according
to the Eyring kinetic theory of fracture (7), ie, at a rate

= τexp[ − (U − βσ )/kT]

(2)

Here,

τ is the thermal vibration frequency, U and β are the activation energy

and volume, and

σ = M

(3)

is the local stress. In equation (3),

ε is the local strain and K is the elastic constant

for the bond. The breaking of these attractive bonds leads to a release of the chain
strands, which are now available to support the external load. Once broken, these
bonds are assumed not to reform.

As the stress of the “freed” chain strands increases, slippage through en-

tanglements sets in at a rate that has the same functional form as that for the
attractive bonds (eq. (2)) but with different values for the activation energy U and

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Entanglement

Molecular

Chain

Intermolecular

Attraction

Fig. 1.

Schematic representation of a model for an unoriented polymer network. The

heavy black circles represent the entanglement loci and the dotted lines denote the at-
tractive bonds between chains. Reprinted with permission from Ref. 8. Copyright (1987)
American Chemical Society.

volume

β. In eq. (2), σ now represents the difference in stress in two consecutive

chain strands that are separated by an entanglement. This stress difference is
calculated using the rubber elasticity theory (10). Thus, the stress on a stretched
chain strand having a vector length r is given by

σ = αkTL

− 1

/nl)

(4)

In equation (4), n denotes the number of statistical chain segments of length

l in the strand whereas L

− 1

is the inverse Langevin function and

α is a propor-

tionality constant.

Equations (2) and (3) for the attractive bond breaking process are executed

on the computer with the help of a Monte-Carlo lottery in which bond i breaks
according to a probability

= v

max

(5)

Here, v

is obtained from equation (2), whereas v

max

is the rate of breakage

of the most strained bond in the array. After each visit of a bond, the time t is

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incremented by 1/[v

max

n(t)] where n(t) is the total number of intact bonds at time

t. The simulation of chain slippage through entanglements is executed using a
similar technique in which n(t) now denotes the total number of entanglements
left at time t. The present model also allows for chain breaking when the local
draw ratio on a particular chain strand exceeds its maximum value given by the
square root of its number of statistical segments.

After a very small time interval

δt has elapsed, the attractive bond break-

ing, chain slippage, and fracture processes are halted and the network is elongated
along the tensile y-axis by a small constant amount determined by the rate of elon-
gation

ε. The network is then relaxed to its minimum energy conﬁguration, using

a series of fast computer algorithms, described in Reference 11, which steadily
reduce the net residual force acting on each entanglement point. Upon comple-
tion of network relaxation, the Monte-Carlo process of bond breakings and chain
slippage is restarted for another time interval

δt, and so on and so forth, until the

network fails.

Morphologies of Deformation.

Figure 2 shows a series of calculated

stress–strain curves for melt-crystallized polyethylene at three different molec-
ular weight values (M

= 1900, 9500, and 250,000). The lowest molecular weight

sample exhibits a low elongation at break and brittle failure. As the molecular
weight is increased (M

= 9500), the stress–strain curve exhibits a so-called yield

250,000

9,500

1,900

Draw ratio

Stress

, MP

Fig. 2.

Calculated stress–strain curves for three poylethylene samples of different molec-

ular weights. To convert MPa to psi, multiply by 145. Reprinted with permission from
Ref. 8. Copyright (1987) American Chemical Society.

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315

BRITTLE

FRACTURE

NECKING

MICRO-

NECKING

HOMOGENEOUS

DEFORMATION

Fig. 3.

Dependence of morphology on molecular weight. Top: Deformation morphologies

obtained from the model for melt-crystallized polyethylene at increasing values of the
molecular weight. From left to right: M

= 1900, M = 8500, M = 20,000, and M = 250,000.

Bottom: Experimental morphologies obtained at comparable molecular weights. Reprinted
with permission from Ref. 8. Copyright (1987) American Chemical Society.

point (the ﬁrst maximum in the stress–strain curve), followed by a ﬂat region
characteristic of neck formation. At draw ratio values

λ < 4, strain hardening is

observed. For the highest M

= 250,000, strain hardening occurs almost immedi-

ately past the yield point.

A set of computer-generated morphologies is presented in Figure 3 (top) for

different melt-crystallized polyethylene samples with increasing values (from left
to right) of the molecular weight. As was clearly exempliﬁed in the stress–strain
curve (Fig. 2), the low molecular weight sample exhibits brittle failure with a
sharp breaking surface. As the molecular weight is increased, a necking mode of
deformation is observed, ie, a region of highly deformed molecular chains bridging
two sections in which the macromolecules are still in a random coil conﬁgura-
tion. This necking region propagates along the sample by continuously drawing
more and more chains from the undeformed sections into the deformed region. At
M

= 20,000 a micro-necking morphology is observed in which numerous necks

are obtained. For still higher M

= 250,000, the number of micro-necks becomes

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Brittle Fracture (

= 0.004)

Necking (

= 0.02)

Micro-Necking (

= 0.1) Homogeneous Deformation

(

= 1)

Fig. 4.

Dependence of morphology on degree of dilution

φ (M = 500,000). Deformation

morphologies obtained from experiment (left) and from the model (right) for solution cast
polyethylene at increasing values of the entanglement spacing factor

φ. From left to right:

φ = 0.004, φ = 0.02, φ = 0.1, and φ = 1. In all cases, M = 475,000. Reprinted with permission
from Ref. 9. Copyright (1988) American Chemical Society.

so large that the deformation becomes essentially homogeneous. The bottom of
the ﬁgure shows experimental morphologies obtained at comparable molecular
weight values. An excellent agreement between theory and experiment is found.

A similar set of transitions can be obtained by keeping the molecular weight

constant and varying the entanglement spacing, as is obtained in solution cast
samples (see eq. (1)). Figure 4 shows a sequence of experimental (left) and calcu-
lated (right) morphologies for M

= 475,000 at increasing values of φ (9). As with

the results of Figure 3, model morphologies are in good qualitative agreement with
experimental observation. This illustrates that the present approach is capable
of handling the very complex issue of connecting events on the molecular level to
macroscopic properties and features.

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Importance of Drawing Conditions.

As noted in the introduction, the

mechanical properties of polymeric ﬁbers are strongly dependent on the length of
the macromolecular chains and also on their orientation with respect to the ﬁber
axis. Since orientation increases with draw ratio (1), it is of uttermost importance
to identify the factors controlling polymer drawability. A detailed study of the im-
portance of temperature and elongation rate on the maximum draw ratio of melt-
crystallized polyethylene, using the model described above, has been presented
in Reference 12. This work was motivated by early experimental investigations
which seemed to indicate that, for each molecular weight, there may exist an op-
timum draw temperature and/or rate of deformation (13,14). Figure 5a shows the

100

150

Rate = 6.25 min

−1

atio

Temperature,

°C

(a)

atio

0.1

100

Temperature = 130

°C

Rate, min

−1

(b)

Fig. 5.

Calculated dependence of the maximum draw ratio on testing conditions for melt-

crystallized polyethylene with M

= 143,000. (a) Dependence on temperature at constant

elongation rate (6.25 min

− 1

); (b) Dependence on elongation rate at constant temperature

(130

◦

C). Reprinted with permission from Ref. 12. Copyright (1988) American Chemical

Society.

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calculated dependence of the draw ratio on temperature for monodisperse melt-
crystallized polyethylene with M

= 143,000. The ﬁgure reveals a sharp maximum

in the draw ratio at T

= 130

◦

C. A careful examination of the present model results

gives the following insight. At low temperature T

< 100

◦

C, all the samples break

at their “natural” draw ratio

λ = 5, as determined from the number of statistical

segments between entanglements in the melt. As the temperature is increased,
the model results reveal an onset of slippage of chains through entanglements,
which leads to a steady increase in drawability. There is, however, an upper limit to
that increase. Thus, at T

= 130

◦

C, the rate of chain slippage becomes comparable

to the rate of straining the overall sample and any further increase in tempera-
ture leads to a progressive decrease in drawability. Results (not reproduced) for
other monodisperse molecular weights show a similar behavior with different val-
ues for the optimum drawing temperature. These observations are of importance
since they predict that the sharp optimum of Figure 5a cannot be observed for a
polydisperse sample as every single molecular weight in the distribution has an
optimum temperature window which will be different from the others. Also, since
our model is based on a thermally activated rate theory, we expect that results sim-
ilar to those of Figure 5a will be obtained when keeping the temperature constant
and varying the elongation rate. A conﬁrmation of that model prediction is given in
Figure 5b.

A further test of our model results is presented in Figure 6, which displays

actual experimental samples drawn to their breaking point at different values of
the deformation temperature and at a constant elongation rate of 23 min

− 1

. The

samples are for NBS SRM 1484 linear polyethylene which has a close-to monodis-
perse molecular weight distribution with M

= 111,000 and M

= 125,000. It is

evident that, for this particular rate, the optimum drawing temperature is around
75

◦

C. Note that the location of the optimum temperature window is different from

that predicted from our model simulations (see Fig. 5a). A better ﬁt could have
been obtained by adjusting the activation energy for chain slippage, which has
not been attempted in the present calculations.

The results discussed above for monodisperse polyethylenes clearly demon-

strate that each molecular weight exhibits a different temperature or elongation
window within which optimum drawing occurs. The model reveals that, within
those windows, the rate of slippage of chains through entanglements reaches its
optimum value. These observations strongly point to the need for very accurate
control of the temperature and rate of deformation for optimum drawing, partic-
ularly for high molecular weight polymers.

Polymer Coagulation

Polymer coagulation deﬁnes the process by which a solution of a polymer in a
good solvent is quenched in a nonsolvent, leading to solvent–nonsolvent exchange
and polymer precipitation. This process is at the basis of a wide range of indus-
trial processes including wet spinning (3), ﬁbridation (15), and membrane for-
mation (16). A major drawback of the solution/coagulation process is that the
coagulant moves into the polymer gel by forming large tear-dropped macrovoids
(ﬁngers), which are usually very difﬁcult to eliminate. Several theories have been

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105

115

RATE

= 23 min

−1

Fig. 6.

Dependence of maximum draw ratio on temperature (

◦

C) (Polyethylene M

= 10

Actual melt-crystallized NBS SRM 1484 polyethylene samples drawn to break at different
temperatures. The elongation rate was 23 min

− 1

. Reprinted with permission from Ref. 12.

proposed for describing ﬁnger formation. These include linear stability analyses
of the solvent–nonsolvent exchange at the polymer interface (17,18) and studies
of the mass transfer paths before the onset of phase separation (19–21). All these
studies are based on the derivation of analytical equations for diffusion of the
polymer, solvent, and coagulant. In view of the great complexity of the problem,
all previous models had to resort to several simplifying assumptions: (1) the diffu-
sion process is purely one-dimensional, (2) thermodynamic equilibrium is always
preserved at the bath–polymer interface, (3) phase separation is not considered
and is assumed to occur through nucleation and growth. In fact, it is now well ac-
cepted that polymer coagulation is a nonequilibrium process, entirely controlled
by the rate of solvent–coagulant exchange through the interface.

In view of the great difﬁculties associated with analytical studies of the de-

velopment of polymer structure during coagulation, a computer simulation of the
ternary diffusion process is employed. One approach has been described at length
in Reference 22. In brief, the coagulation process is simulated on a lattice, whose
sites represent either a nonsolvent, a solvent, or a polymer aggregate. For sim-
plicity, all three components are assumed to have equal molar volume and the

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Polymer
Solution

Coagulation

Bath

(a)

(b)

ij,

∝

−3

−2

ij,

∝

−3

+ 6

−2

Fig. 7.

Schematic representation of a lattice model for diffusion of a coagulant into a

polymer solution. The polymer, the solvent, and the coagulant particles, are denoted by
symbols

•, ×,

and

◦,

respectively. The diffusion process is simulated through a series

of two-particle exchanges on nearest-neighbor lattice sites. The rates of exchange are ob-
tained from equation (6) in which E

,α

and E

,α

are the interaction energies of the pair

with nearest-neighbor particles, before and after the exchange (see Fig. 7b). As diffusion
proceeds, the coagulant–polymer interface (dashed line) moves upwards and, since the co-
agulation bath is assumed to be inﬁnite, the lattice sites below that interface are being
continuously replenished with coagulant particles (see text). Courtesy of Journal of Poly-
mer Science.

simulation is limited to a two-dimensional geometry similar to that encountered
in typical experimental studies. In these setups, a drop of polymer solution is
placed between two microscope slides whereas the coagulant is introduced near
the edge (2,23). A schematic representation of the two-dimensional model of poly-
mer coagulation is given in Figure 7a. The polymer solution is represented by a
lattice of sites which are ﬁlled-in, according to the polymer weight fraction, by
either a solvent or a polymer particle. A detailed study of the importance of the
polymer particle “size” can be found in Reference 24. The bottom ﬁve rows of the
lattice represent the inﬁnite bath ﬁlled-in with coagulant particles. Coagulant,
solvent, and polymer sites are referred to by indexes 1, 2, and 3 respectively. The
ternary polymer–solvent–coagulant diffusion process is simulated as follows. A
pair of nearest-neighbor—i and j—lattice sites are picked at random. Denoting by
α the local environment for that pair, the rate for an exchange ij→ji within α is
calculated from (25)

i j

→ ji,α

= τ

, j

− 1exp[β(E

i j

,α

− E

,α

)

/2]

(6)

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In equation (6),

− 1

, j

is related to the mutual diffusion coefﬁcient D

∞

, j

of i and

j particles at inﬁnite dilution through (26)

∞

, j

= τ

− 1

, j

(

δx)

(7)

where

δx denotes the unit lattice length. Note that, in equation (6), all the effects of

particle interactions are relegated to the Boltzmann exponent of (half) the energy
difference between initial and ﬁnal states for the exchanging pair. That functional
form is guided by a requirement of commutativity of the individual two-particle
exchanges (25). Using equation (6), a probability for exchange is then calculated
through

i j

→ ji,α

= v

i j

→ ji,α

max

(8)

in which v

max

denotes the highest rate of exchange among all the pairs on the

lattice (see also eq. (5)). Having determined the probability p for an exchange
ij

→ji (eqs. 6–8), a random number between 0 and 1 is generated and the move

is allowed if that number falls below p. After each visit of a pair and whether
the pair is allowed to exchange or not, the overall “time” t is incremented by
1/[v

max

n] in which n denotes the total number of pairs of sites on the lattice (note

the similarity with our Monte-Carlo process for polymer drawing, as described
earlier). In addition to two-particle exchanges, the model also allows for small
clusters to move through a series of one-lattice-step displacements of entire rows
or columns of polymer particles. Also, since the coagulant bath is assumed to
be inﬁnitely large, solvent particles diffusing into the bath to a depth of more
than 5 lattice units with respect to the polymer–bath interface are systematically
removed from the lattice and replaced by coagulant particles.

Skin and Finger Formation.

First attention is turned to the applica-

tion of the model to the coagulation of a solution of 20% Nomex

, an aro-

matic polyamide commercialized by DuPont, dissolved in N-methylpyrrolidinone
(NMP). An estimation of the range of values of the model parameters can be
found in Reference 22. Figure 8 shows the calculated morphology obtained af-
ter precipitation in water for a time t

= 0.02 s. Only the polymer particles have

been represented. The ﬁgure clearly reveals that the coagulant diffuses inside the
polymer gel through a series of ﬁnger-like structures. The latter have a strikingly
similar shape and originate through regularly spaced pores in a thin skin, in per-
fect agreement with experimental observation (2). Note the important shrinkage
of the polymer gel whose initial position was located at the bottom of the ﬁgure.
A careful analysis of our model results leads to the following insight into the ori-
gin of the skin/ﬁnger formation. Immediately upon immersion of the polymer gel
in the coagulation bath, the fast solvent-coagulant exchange across the interface
combined with the large repulsive forces between polymer and coagulant cause an
immediate precipitation at the interface. This process is too fast for any segrega-
tion of the Nomex particles into polymer-rich and polymer-poor domains and, as a
result, a thin skin starts to form. Since the polymer concentration is low, the skin,
in its early stages, is not homogeneous and it presents defects along its contour.
The latter form the initiation pores for ﬁngers, which then quickly grow inside
the polymer solution as it is faster for a solvent particle to exit through a pore

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= 0.02 s

Fig. 8.

Calculated coagulant–polymer interface for a 20% solution of Nomex in NMP

precipitated in water at t

= 0.02 s. The ﬁgure is for a square lattice of 450 × 600 sites; only

polymer particles (symbol

•) have been represented. The ﬁgure is for ε

–

= 0.77, ε

–

= 0,

and

–

= 2.8. Courtesy of Journal of Polymer Science.

than through a defect-free skin. In support of the proposed mechanism for ﬁnger
formation, experiments clearly reveal that ﬁngers do not develop when a skin is
absent or when the polymer volume fraction is too high, ie, when skin defects are
less probable.

Importance of the Solvent–Coagulant Miscibility.

To recall, the misci-

bility between solvent and coagulant is, in this model, controlled by the interaction
energy parameter

–

. A study of the effect of that parameter on the morphol-

ogy of the precipitated polymer is presented in Figures 9a–d, for approximately
the same coagulation time t

= 0.02–0.03 s. For perfect mixing (ε

–

= 0, Fig. 9a),

the polymer is seen to coagulate into large agglomerates of ca 0.3 mm in diam-
eter (dust-like structure). As the miscibility of solvent and coagulant decreases
(

–

= 0.5 and ε

–

= 0.77, Figs. 9b and 9c), the present model results clearly

reveal the development of ﬁnger-like structures which also become more regular
as

–

increases. Finally, for very immiscible solvent and coagulant (

–

= 1.5,

Fig. 9d), the formation of a sponge-like structure with spherical pores ﬁlled-in with

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323

(a)

(b)

(c)

(d)

Fig. 9.

Effect of the solvent–coagulant miscibility on polymer structure. (a)

–

= 0, (b)

–

= 0.5, (c) ε

–

= 0.77, and (d) ε

–

= 1.5. The structures are for a coagulation time t =

0.02 s, except (d) which is for t

= 0.03 s. All the other parameters are the same as in Figure

8. The extent of shrinking of the polymer–coagulant interfaces with respect to their initial
positions have not been represented. Courtesy of Journal of Polymer Science.

coagulant is observed. It is interesting to note that the sequence of precipitated
morphologies obtained in Figure. 9 through a decrease in solvent–coagulant mis-
cibility is also very similar to that observed experimentally (27) when increasing
the initial polymer concentration. Based on our understanding from the model,
this is not surprising since a decrease in solvent–coagulant miscibility effectively
makes solvent and polymer particles more alike as far as the coagulant is con-
cerned. A further investigation of the model results also reveals that the transition
from sponge to ﬁnger to dust-like structures is also accompanied by a dramatic
increase of the rate of coagulation, in perfect agreement with experimental obser-
vation. Thus, for the dust-like structure studied in Figure 9a, it is found that the
diffusion coefﬁcient for the penetration depth of the coagulation front is around
10

− 5

/s. For the sponge-like structure of Figure. 9d, a diffusion coefﬁcient of

ca 2

× 10

− 7

/s is found.

Effect of Additives in the Coagulant.

In industrial applications, the

coagulant power and coagulant–solvent miscibility are often varied through the
incorporation of additives into the coagulant. These additives may consist of pure
polymer solvent or of large amounts of inorganic salts. The presence of an addi-
tive in the coagulant can be easily accounted for in the lattice model (28). Figures
10a–c show the effect of adding CaCl

to water during coagulation of a 20% solu-

tion of Nomex in NMP. The ﬁgures clearly reveal a transition from a ﬁnger-like to
a dense polymer structure at

∼50% CaCl

concentration. Further investigation of

the model results reveals that the transition is not associated with any decrease
in the rate of coagulation, which stays at high values around 10

− 5

/s. The

approach, therefore, clearly conﬁrms the advantages of using high salt concen-
trations in the coagulant. Further study of the model results also reveals that

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(a)

(b)

(c)

Fig. 10.

Effect of adding increasing amounts of CaCl

to the aqueous coagulant. The

amounts of CaCl

are as follows: (a) 35%, (b) 40%, and (c) 50%. All three ﬁgures are for a

coagulation time t

= 0.015 s. Reprinted from Ref. 28. Copyright (1995), with permission

from Excerpta Medica Inc.

the success of that technique rests upon the use of a salt that is poorly miscible
with the polymer solvent. This observation is in line with the previous ﬁnding
(see Fig. 9) that a decrease in solvent–coagulant miscibility leads to more dense
polymer structures. The effect of adding high concentrations of the solvent to the
aqueous bath has also been studied. At 35–49% solvent, the model results clearly
reveal a transition from a ﬁnger-like morphology into a more uniform structure,
as observed experimentally (29,30). However, in contrast to the case of a salt ad-
dition, that transition is also associated with a dramatic decrease in the rate of
coagulation. Thus, with 49% solvent in the coagulant, no polymer has yet precipi-
tated after t

= 0.02 s and coagulation becomes an extremely slow process entirely

controlled by particle nucleation and growth.

Admittedly, these results are for short coagulation times (t

< 0.03 s) which

has allowed the study of structures no larger than a few microns in depth. There-
fore, it could be argued that, being for small coagulation times t

< 0.03 s, the results

of the present paper are not representative of the fully coagulated structures that
sometimes require up to a few minutes to obtain. In support of these results, it

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MODELING OF POLYMER PROCESSING AND PROPERTIES

325

should be noted, however, that actual micrographs of coagulated morphologies
obtained at different times, ranging from a few seconds to several minutes, do not
reveal any evidence for a change in polymer structure with time. Thus, a change
in these calculated morphologies at later times remains improbable and the gross
morphology of the ﬁnal polymer structure should be adequately described by the
present model results.

Polymer Quenching

The previous section presented a model study of the coagulation of a polymer solu-
tion brought upon by diffusion of a nonsolvent. The present section focuses on the
faster process of phase separation inside a binary polymer–solvent system induced
by an abrupt change in external temperature and/or pressure. This problem is of
obvious importance in polymer processing techniques based on thermally induced
phase separation (31) or on rapid expansion of supercritical solutions (32). A typi-
cal example of the latter is the ﬂash spinning process. Here, a polymer solution is
ﬁrst heated under high pressure and then rapidly released through a spinnerette,
leading to the formation of a ﬁbrillar polymeric network.

Equilibrium properties of polymers in dilute solutions are reasonably well

understood (33) and polymer–solvent phase equilibria can be easily obtained
through the use of the classical Flory-Huggins mean-ﬁeld lattice model (34) or,
alternatively, from more recent computer simulations of Gibbs ensembles (35,36).
On the other hand, the development of polymer morphology inside the coexistence
curve is only poorly understood and previous work has been restricted essentially
to kinetic studies of the collapse of a single polymer chain (37–39). For that reason,
attention is turned to multichain systems and focus is on the very early stages
(t

1 s) of spinodal decomposition, during which equilibrium concentrations and

morphologies are being established. The purpose of developing such a multichain
model is to get a better understanding of the dependence of polymer structure on
molecular weight, polymer concentration, and quenching conditions.

In this model (40), ﬁrst an array of polymer chains at inﬁnite temperature

is constructed for a given value of the chain volume fraction V

and chain length

N (henceforth in statistical segment units). This is obtained by randomizing an
ordered array of chains by a series of Monte-Carlo moves of two kinds. The ﬁrst is
a nonlocal reptation which randomly moves a bond from one end of a chain to the
other (36). The second incorporates local random moves such as end-ﬂip, 2-bond
kink jump, and 3-bond crankshaft motion (41). Upon completion of that process, an
interaction energy

ε < 0 is imposed between pairs of nonbonded nearest-neighbor

chain segments and the system is quenched to a ﬁnite temperature T. The new
state of the polymer chains is then obtained, as a function of time, by using the
local Monte-Carlo moves described above (ie, nonlocal reptation is excluded) which
are made to occur at a rate (see also eq. (6))

= τ

− 1

exp[

− β(U

− U

)

/2]

(9)

in which U

and U

denote the local energy in the ﬁnal and initial states, respec-

tively, whereas

τ corresponds to the reorientation time of a statistical segment,

326

MODELING OF POLYMER PROCESSING AND PROPERTIES

Vol. 3

0.4

0.2

0.0

0.6

0.8

1.0

itical temper

ature

Polymer volume fraction

Fig. 11.

Dependence of the critical temperature

−k

ε on polymer volume fraction.

Results obtained by the present approach are denoted by open symbols

◦, N = 20; ,:

= 60; and : N = 1000. Both local and nonlocal (ie, reptation) moves have been used in

our Monte-Carlo simulations. Filled-in symbols are reproduced from Reference 35 (

•, N =

100) and Reference 36 (

, N = 64). The curves are drawn to guide the eye. Reprinted with

permission from Ref. 40. Copyright (1997) American Chemical Society.

which is of the order of 10

− 8

s. As in all previous models, after each attempted

move, the overall “time” t is incremented by 1/[v

max

n] in which v

max

represents the

highest rate among all n possible moves on the lattice.

The polymer–solvent phase equilibria are obtained from a study of the be-

havior of the speciﬁc heat

= [ ν

− ν

]

/[(N+1)k

]

(10)

in which

ν denotes the number of nearest neighbor nonbonded polymer segments.

The critical temperature

−k

ε for phase separation is obtained from the loca-

tion of the maximum in a plot of C vs T. Model results for the dependence of that
critical value on the polymer volume fraction V

are presented in Figure 11 for

chains of length N

= 20 (symbol ), 60 (), and 1000 ( ). The ﬁgure reveals a

pronounced asymmetry of the curves and a shift of their maxima toward higher
T

and lower V

with an increase in N, in perfect agreement with experimen-

tal observation. Also represented are theoretical predictions obtained from two
previous approaches: (1) the slab geometry model (Ref. 35; symbol

•) and (2) the

Gibbs ensemble simulations (Ref. 36; symbol

). Both sets of results are in good

qualitative agreement with model predictions.

Vol. 3

MODELING OF POLYMER PROCESSING AND PROPERTIES

327

(b)

(a)

Fig. 12.

Polymer morphologies obtained at

−k

ε = 2.5 and 1.0 for two different simula-

tion times: t

= 5 × 10

τ (top) and t = 15 × 10

t (bottom). The representations are for cubic

lattices of 70 unit lengths (

∼70 nm) in each direction. Reprinted with permission from Ref.

40. Copyright (1997) American Chemical Society.

Effect of Quench Depth.

Now attention is turned to an analysis of the

structure development upon quenching inside the binodal curves of Figure 11.
The results are presented in Figure 12 for two (reduced) temperatures

−k

ε =

2.5 (Fig. 12a) and

−k

ε = 1.0 (Fig. 12b). The top set of morphologies is for a

short time t

= 5 × 10

τ. A bicontinuous sponge-like morphology is observed, in

agreement with experimental observation (31,42) and with a recent viscoelastic
model study (43). High temperatures (left) are seen to lead to a coarser morphol-
ogy, whereas lower ones (right) form a lacy structure with ﬁne and dense ﬁbrils.
Further study of the model results indicates that, within that early stage, spin-
odal decomposition occurs to quickly establish equilibrium concentrations and
the polymer forms a continuous percolated structure whose ﬁne features are ex-
acerbated at very deep quenches. For large simulation times t

= 10

τ (bottom

set of morphologies in Fig. 12), the ﬁgures reveal a substantial coarsening of the
structures and a progressive breaking up of the continuous polymer gel into iso-
lated spherical aggregates. This is the so-called percolation-to-cluster transition.
Within that regime, it was also found that the rate of coarsening increases with a
decrease in quenching temperature.

328

MODELING OF POLYMER PROCESSING AND PROPERTIES

Vol. 3

The Case of Mixed Solvent Systems.

There has been a recent interest

in the use of mixed solvent systems for polymer processing and isolation. It is
now common in industrial processes for polymer coagulation to employ solvent
mixtures which are designed to bring the polymer solution close to its solubility
limit. The formulation of these systems is often empirical and the behavior of
polymer chains in mixed solvents is still poorly understood. The present study
focuses on a 70/30 blend of chains of two different lengths N

= 4000 and N

50 (in statistical segment units). The blend is dissolved at 3.3% concentration in
two different solvents: (1) a mixture of 80% good solvent (repulsive interaction
energy with the polymer

ε = 1) and 20% bad solvent (ε = 41) and (2) a single

solvent system with average solvent power. The study proceeds as follows (44).
First the two polymer solutions are equilibrated at high temperatures. These
solutions are then quenched inside the two-phase region at the same temperature
T until thermodynamic equilibrium is achieved. The polymer structures obtained
in the two solvents for a quenching temperature T

= 40 are depicted in Figures

13a and 13b. In the average solvent case (Fig. 13a), the polymer blend is seen
to precipitate in a dense spherical globule. For the solvent mix (Fig. 13b), on
the other hand, the precipitated structure is more diffuse and a fraction of the
chains remains soluble in the solvent phase. Further investigation of the model
results reveals that those chains, which represent about 5% of the total polymer
mass, are made exclusively of the short length component. The presence of that
soluble fraction is due to preferential adsorption of the good solvent by the low
molecular weight chains (44–46). These results are of great importance as they
indicate that quenching a polydisperse system below the cloud point may induce
molecular weight segregation between the two phases: the longer chains, which
precipitate out ﬁrst, tend to populate the polymer-rich phase whereas the shorter

(a)

(b)

Fig. 13.

Effect of solvent characteristics on precipitated polymer morphologies. The ﬁgure

is for a blend precipitated in (a) the average solvent and (b) a mixture of 80% good solvent
(

ε = 1) with 20% bad solvent (ε = 41). Total polymer concentration is 3.3% and quenching

temperature is set at T

= 40 (in k

units). The representation is for a simple cubic lattice

of 70

× 70 × 70 sites. Courtesy of Journal of Polymer Science.

Vol. 3

MODELING OF POLYMER PROCESSING AND PROPERTIES

329

(a)

(b)

(c)

Fig. 14.

Effect of quench depth, T, on precipitated polymer morphologies. (a) T

= 20, (b)

= 10, and (c) T = 2 (in k

units). The ﬁgure is for a 32/68 blend of chains of lengths N

4000 and N

= 10, dissolved in a mixture of 80% good solvent (ε

–

= −0.25) with 20% bad

solvent (

− 3

= 45). Total polymer concentration is 3.3%. Courtesy of Journal of Polymer

Science.

chains, having greater solubility, remain in the solvent phase. The effect of quench
depth on the soluble polymer fraction has also been studied. The model results
are shown in Figure 14 for a 32/68 blend of chains of lengths N

= 4000 and N

= 10, precipitated at increasing quench depths: (1) T = 20; (2) T = 10, and (3) T

= 2 (in k

units). The model reveals that the soluble fraction ﬁrst decreases and

then increases with quench depth. This increase is due to a higher preferential
adsorption at low temperatures. It should be noted also that this effect seems to
be indicative of a solvent mixture in which the good solvent has a tendency to form
a complex with the polymer (

–

= −0.25).

Thermally Bonded Nonwovens

Mechanical Properties.

The processing of ﬁbers into planar ﬁbrous ma-

terials such as papers, nonwovens, and wood ﬁber boards has received a lot of
attention over the years because of their great versatility and their ease of manu-
facturing into complex shapes for a wide range of applications. In spite of its im-
portance, little attention has been paid to the fracture behavior of ﬁbrous sheets.
Previous theories are based on random lattice models with various types of dis-
order (47,48). The ﬁbrous nature of the networks has been included in more re-
cent approaches (49,50). These models, however, are conﬁned to two-dimensional
representations and, as such, they are incapable of describing actual sheet struc-
tures in which bonding between layers plays a major role. They are also based
on simpliﬁed fracture criteria and provide only static calculations of stress–strain
curves. Again, analytical approaches that attempt to tackle all the aspects of pa-
per deformation are clearly impossible and, for that reason, one needs to turn to
computer modeling. Here the focus is on nonwoven sheets, such as Tyvek

, which

are formed by pressing spunbonded ﬁbrous webs (layers) between two smooth
heated rolls. These webs are created through ﬂash spinning of a solution of high
density polyethylene through a spinneret containing a single hole.

In the model (51), a spunbonded web (henceforth referred to as “layer”) is rep-

resented by an array of ﬁber strands connected on a two–dimensional x-y lattice.

330

MODELING OF POLYMER PROCESSING AND PROPERTIES

Vol. 3

Fig. 15.

Model construction of a ﬁbrous layer. E

and G

denote the elastic tensile and

shear modulus for the ﬁber. Reprinted from Ref. 51. Copyright (1997), with permission
from Excerpta Medica Inc.

Fig. 16.

Consolidation into a sheet structure. G

denotes the shear modulus of the bonds

introduced during consolidation. Reprinted from Ref. 51. Copyright (1997), with permission
from Excerpta Medica Inc.

This array is built in a sequence of steps. At each step, a lattice site is chosen at
random and three ﬁber strands are constructed from that site until they reach
either the edge of the lattice or a previously constructed strand (see Fig. 15). Each
of the strands within a layer is modeled as a ribbon, made itself of a series of
lattice bonds. The sheet structure is then obtained by piling up a series of layers
on top of each other along the z-axis and consolidation is realized through bonding
nearest-neighbor ﬁber sites between layers (see Fig. 16). Upon tensile deforma-
tion, the individual ﬁber bonds are stretched and their local stresses combined
with thermal activation rate theory (see eq. (2)) deﬁne probabilities for bond rup-
ture. As in previous models, at regular time intervals, the individual lattice sites
are relaxed toward mechanical equilibrium with their neighbors by a systematic
sequence of operations which steadily reduce the net residual force acting on each
site.

Vol. 3

MODELING OF POLYMER PROCESSING AND PROPERTIES

331

(a)

(b)

Fig. 17.

Actual computer representations of two 3-layer sheets having different ﬁber cross

sections. The top structure is for a square cross section (aspect ratio

α = 1), whereas the

bottom is for ribbon-like ﬁbers (cross-section aspect ratio

α = 7). Both structures are for a

37% ﬁber volume fraction. Reprinted from Ref. 57. Copyright (1998), with permission from
Excerpta Medica Inc.

The results, to be presented below, will be for application to Tyvek sheets.

Figure 17 shows actual computer representations of two 3-layer sheets having dif-
ferent ﬁber cross sections. The lhs structure is for a square cross section (aspect
ratio

α = 1), whereas the rhs is for ribbon-like ﬁbers with cross-section aspect ratio

α = 7. Both structures are for a 37% ﬁber volume fraction. Note that the sheet of
Figure 17b has a fewer number of pores but, more larger voids. Figure 18 shows
calculated stress–strain curves for polyethylene sheets for two different values of
the shear modulus G

of the ﬁber. The two curves show the presence of two distinct

regions characterized by different moduli. Within the ﬁrst region, which extends
to

∼9% elongation, computer results reveal an extensive deformation of the con-

solidating bonds. These bonds start to break at 2–4% elongation, which results in
a continuous bending down of the stress–strain curve. At higher elongations, one
enters the second region which is characterized by a gradual increase in the load
supported by the ﬁbers. As a result, the slope of the curve within that region is
seen to be strongly dependent on the ﬁber shear modulus G

. The latter observa-

tion is in line with the previous model ﬁnding (52) in that the tensile deformation
of structures reinforced with randomly oriented ﬁbers is fully controlled by the
shear modulus of the ﬁbers.

Permeability.

Nonwoven ﬁbrous media are also being extensively used

in ﬁltration and barrier applications. Several analytical approaches have been
proposed for describing the dependence of their permeability on the shape and
volume fraction of ﬁbers (53–56). For simplicity, all these studies assumed that the
ﬁbrous media is made of a regular array of equidimensional cells, each consisting
of a ﬁber segment surrounded by air. This is certainly not the case experimentally
as ﬁbrous media are made from a deposition process in which ﬁbers are laid into
a dense and compact mat. In one approach (57), the permeability of the sheet
to diffusional ﬂow is studied by a Monte-Carlo process. Here, the sheet is put in
contact with a large external bath of small particles which diffuse through the

332

MODELING OF POLYMER PROCESSING AND PROPERTIES

Vol. 3

0.3

0.2

0.1

0.0

Elongation, %

aper stress

, g/d

= 10 g/d

= 5 g/d

Fig. 18.

Calculated stress–strain curves for two polyethylene sheets with G

= 10 g/d and

= 5g/d (10 g/d∼

=1 GPa). In reference to the units commonly used in the paper industry,

a stress of 0.1 g/d is equivalent to a load of 10 lbs/in. on a 7 mil sheet. The curve is drawn
to guide the eye. Reprinted from Ref. 51. Copyright (1997), with permission from Excerpta
Medica Inc.

structure by hopping between nearest-neighbor lattice sites. This model allows
one to study in detail the importance of the sheet density, ﬁber shape, and the role
of interstitial voids between layers.

Model results for the dependence of diffusivity D (see Ref. 57) on ﬁber vol-

ume fraction V

are presented in Figure 19. For simplicity, the data have been

normalized by the diffusivity D

at V

= 0. First, square ﬁbers are examined (α =

1, Fig. 9a). Calculated data have been represented by symbols

◦ and • for sheets

of 5 and 10 layers, respectively. For both sheets, the diffusivity shows a monotonic
decrease with ﬁber volume fraction up to V

= 0.4. Also represented in the ﬁgure

(symbol

+) are calculated data from Reference 53 The predictions are higher than

the present study because, as stated earlier, previous approaches assume each
ﬁber to be surrounded by air so that the pores always belong to the continuous
phase, except at V

= 1. Ribbon-like ﬁbers with α = 7, are shown in Figure 19b.

Comparison with the previous ﬁgure shows a strong decrease in diffusivity with
an increase in cross-section aspect ratio

α. These observations may seem surpris-

ing in view of the large pore areas previously observed in Figure 17b at large
α = 7. At constant V

, diffusivity is, however, mainly controlled by the tortuosity

of the diffusion path across the sheet. The length of that path obviously increases
with

α, as wide ﬁbers act as large obstacles which force a diffusing particle to

wander within a plane perpendicular to ﬂow. Also represented in Figure 19b are
predictions from previous theories (54,56) for

α = 6. As in Figure 19a, these ap-

proaches lead to an overestimation of the diffusivity at high V

because they ne-

glect contacts between ﬁber elements.

Vol. 3

MODELING OF POLYMER PROCESSING AND PROPERTIES

333

.01

.001

.0001

0.0

0.2

0.4

0.6

0.8

1.0

(b)

.01

.001

.0001

0.0

0.2

0.4

0.6

0.8

1.0

Fiber volume fraction

(a)

Fig. 19.

(a) Normalized diffusivity D/D

for sheets of ﬁbers with

α = 1. Symbols are as

follows:

and •, data calculated with our model for 5- and 10-layer sheets; +, model data

taken from Reference 53. (b) Same as Figure 19a but, for sheets of ﬁbers with

α = 7. Model

data taken from Reference 56 for

α = 6 are denoted by the symbol +. Other model data

obtained in Reference 54 fall along the same line. Reprinted from Ref. 57. Copyright (1998),
with permission from Excerpta Medica Inc.

334

MODELING OF POLYMER PROCESSING AND PROPERTIES

Vol. 3

Conclusions

In summary, a series of kinetic Monte-Carlo models for the study of the effect of
processing conditions on polymer structure and properties have been presented.
These models are mesoscopic in the sense that the unit lattice length is of the
order of the statistical segment length for the polymer chains and atomic level de-
tails are omitted. There studies have clearly revealed the importance of molecular
weight, molecular weight distribution, and density of entanglements in control-
ling the drawability of ﬂexible polymers in the solid state. The approach has also
allowed description in detail of the processes of coagulation and quenching of a
polymer solution and reproduction the wide variety of different polymer struc-
tures that can be obtained. Finally, turning to the processing of ﬁbers into planar
ﬁbrous materials such as papers and nonwovens, it has been shown that kinetic
Monte-Carlo models can be equally successful at predicting their mechanical and
permeability properties.

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