Selection of Thermotropic Liquid Crystalline Polymers for Rotational Molding Eric Scribben Dissertation submitted to the Faculty of Virginia Polytechnic Institute and State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY in Chemical Engineering Dr. Donald G. Baird, Chairman Dr. Richey M. Davis Dr. Garth L. Wilkes Dr. Peter Wapperom Dr. Scott Case Dr. Martin Rogers July 19, 2004 Blacksburg, Va Keywords: rotational molding, TLCP, coalescence, sintering
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Selection of Thermotropic Liquid Crystalline Polymers for Rotational
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Selection of Thermotropic Liquid Crystalline Polymers for
Rotational Molding
Eric Scribben
Dissertation submitted to the Faculty of
Virginia Polytechnic Institute and State University
in partial fulfillment of the requirements for the degree of
Table 2.2. Packing Fractions for Commercial Rotational Molding Powders 31
Table 2.3 Scaling Exponents According to Transport Mechanism 44
Table 2.4. McNeil Chart [2] 69
Table 2.5. Parameters Effecting Heating Time 71
Table 2.6. Comparison of Gas Transport Properties at 35°C of Vectra A900 and PAN
[162] 80
Table 3.1. Weight Average Molecular Weight, Polydispersity, and Melt Index 117
Table 3.2. Descriptions of the powder samples used in the densification study. 129
Table 4.1. Weight Average Molecular Weight, Polydispersity, and Melt Index 146
Table 4.2. Single Mode UCM Coalescence Model Parameters and Calculated Values
for the Deborah Number at 180°C. 151
Table 4.3. Multimode Upper Convected Maxwell Model Parameters at 180°C 153
Table 5.1. UCM coalescence model parameters and calculated values for the Deborah
Number. 184
Table 6.1. Descriptions of the powder samples used in the densification study. 210
Table 6.2. Apparent density of the sieved TLCP particles. 216
Table 6.3. The apparent density of the samples used in the densification study. 224
Table 6.4. Results of the density and tensile measurements for the densification study,
S3* represents the results from the extended cycle time. 225
Table 6.5. Comparison of the tensile strength and modulus of sample S3 when molded
in the presence of air. 227
Table 6.6. The average density, tensile strength, and tensile modulus from the
rotationally molded distribution D4 compared the results for the
distribution from the densification study. 230
1 Introduction 1
1 Introduction
1 Introduction 2
1 Introduction
1.1 Thermotropic Liquid Crystalline Polymers
Liquid crystalline behavior was first discovered in 1888 when Reinitzer observed
that upon heating, cholesteryl benzoate melted to form a turbid fluid then appeared to
melt again into a transparent phase at higher temperatures [25]. The first polymeric
liquid crystal† was reported in 1937 when it was observed that above a critical
concentration the tobacco mosaic virus formed two phases, of which one was birefringent
[3]. The first synthetic polymeric system was a poly (γ-benzyl-L-glutamate) solution,
which was reported in 1950 [8]. The first thermotropic LCPs were reported in the mid
70’s [9, 12, 28]. Since then a tremendous amount of research has been done from the
quantumchemical to the macroscopic level on a great number of systems [22]. The result
is that liquid crystalline polymer science has developed into its own discipline.
Liquid crystal (LC) describes a class of materials with long-range molecular order
somewhere between the crystalline state, which exhibits three-dimensional order, and a
disordered isotropic fluid [15]. Four classes have been identified to describe LC order:
nematic, cholesteric, smectic, and discotic [11]. Nematic describes a LC phase with only
one-dimensional long range ordering; it possesses long-range orientational order
(molecular alignment) but only short-range positional order (spatial ordering) [8]. A
cholesteric is very similar to a nematic, but it is periodically twisted along the axis
perpendicular to the long-range order axis. A smectic phase characterizes two-
† Liquid crystal was later renamed mesomorphic phase or mesophase. Mesomorphic is defined: of intermediate form [7]. Friedel found this more appropriate because these materials are not crystalline and may not even be liquids, it is a stable intermediate phase existing between the liquid and crystalline states.
1 Introduction 3
dimensional order. A discotic mesophase can occur when disc-like liquid crystalline
molecules align in columns. An illustration of these structures can be found in Figure
1.1.
Figure 1.1. Friedelian Classes: a. Nematic; b. Cholesteric; c. Smectic A; d. Discotic.
Liquid crystalline order originates, in polymeric systems, from nonflexible repeat
units, called mesogenic units, with an axial ratio greater than three [8, 21]. In dilute
solution the rigid molecules are capable of random arrangement. As concentration
increases, the molecules are forced to adopt an oriented conformation because of
a. b.
c. d.
1 Introduction 4
intermolecular repulsions or excluded volume interactions. Mesogenic units can be either
rod, disc, or lathe-like and may appear within the molecule backbone either randomly or
in a recurring rigid/ flexible structure. These are called main chain LCPs. The other
common type of LCP structure, side chain LCPs, occur as a rigid pendent group to a
flexible polymer backbone with orientation that can range from parallel to perpendicular
to the backbone [21]. Of course another less common possibility would be for the LCP
to contain both main chain and side chain units. This leads to a near infinite number of
possibilities ranging in structure of mesogenic group and arrangement. Most of the
property differences noted between side and main chain LCPs have been related to the
greater mobility of side chain mesogenic units as a result of increased backbone
flexibility [21].
LC order can exist in either solutions or melts. LC transition in solution is a
function of concentration and temperature and is referred to as lyotropic systems [5].
Melts, since concentration is fixed, are only temperature sensitive and are referred to as
thermotropics. The LC phase exists between the crystalline melting point, Tm (or in the
case where no crystalline state exists, the glass transition temperature, Tg), and the upper
transition temperature where the fluid reverts to an isotropic liquid, Tlc→i [20].
Unlike isotropic fluids, orientation is quantified to describe the state and dynamics
of liquid crystals. Molecules are preferentially oriented about an axis, an apolar unit
vector, n, called the director. The direction of the director is typically arbitrary but can
be uniformly aligned through imposing boundaries, applying external magnetic or
1 Introduction 5
electric fields, or inducing viscous flow [20]. Preferential orientation implies that
molecules actually possess a distribution of orientation about the director and therefore a
parameter is needed to describe that distribution. Assuming rigid rod molecules, the
order parameter tensor is defined as [7]:
( )∫
−= iijjiiij duuutufS δ
31, (1.1)
where ui is a unit vector which describes the orientation of a rigid rod molecule, δij is the
Kronecker delta, and f(ui,t) is the orientation distribution function.
The order parameter tensor has a few properties worth noting. It is deviatoric,
therefore its trace is equal to zero and it is symmetric. Its eigenvalues or principal values
(S1, S2, S3) which define the principal axes of orientation must also add to zero. If all
three are equal then S = 0 and the material is isotropic. If two of the eigenvalues are
equal then the system is axially symmetric (as a nematic) and Sij can be represented by:
−= ijjiij nnSS δ
31 (1.2)
where S is a certain scalar equivalent to the Hermans orientation function,
2
1cos3 2 −=
θS (1.3)
ni the projection of the director in the ith direction, while the brackets denote the system
average. Values of S range between, 1 > S > -1/2, with the values 1, 0, and –1/2
representing uniaxial orientation, random orientation, and biaxial orientation.
1 Introduction 6
Valid application of a single order parameter to describe orientation requires that
the material is uniform throughout or a “monodomain” with a single director. It is
possible for elastic distortions to induce slight continuous variation in the director of
monodomain systems [7]. This director variation is typically observed in low molecular
weight materials and becomes less likely as molecular weight increases because of steric
effects [8]. As elastic distortions become more difficult, free energy increases, and
director variation becomes discontinuous, resulting in the formation of defects and
polydomain textures.
Two types of defects have been identified for nematic liquid crystalline polymers
[7]. In thick samples it is possible to observe a system of dark flexible filaments (defined
as disclinations) that correspond to lines of singularity in molecular alignment and result
in the formation of multiple domain texture. The other defect occurs when the imposed
boundary conditions are continuously degenerate (no preferred axis in the plane of the
walls). A system of singular nodes (noyaux) form on the surface and the resulting
general texture is called a Schlieren texture.
Polydomain texture and defects are important to this particular project because of
their influence on rheology. As previously eluded to, the formation of defects is
associated with an increase in free energy. Under quiescent conditions the system
attempts to minimize excess free energy by combining neighboring disclinations with
opposite signs thus eliminating the pair and increasing domain size [13]. However,
mechanical energy can be stored during deformation by an increase in the number of
1 Introduction 7
defects. The development of texture and orientation during deformation can change the
rheological response to imposed stresses and strains.
Liquid crystals have a number of interesting and potentially useful properties.
Optically, the material can be birefringent, although this may be limited to a local scale
[8]. There are also a few polymeric liquid crystals that show nonlinear optical behavior
[29]. Many systems have anisotropic diamagnetic and dielectric properties [20]. The
modulus is often anisotropic, dependant upon the quality of alignment [19, 26]. An
interesting rheological feature is that the nematic phase has a lower viscosity (parallel to
the director) than the isotropic phase [8]. Negative normal stresses have also been
reported for some systems when subjected to steady shear [15]. LCPs also tend to have
low partial entropy of dissolution and therefore have a relatively high resistance to
solvents [8]. Gas transport studies have revealed that they have excellent barrier
properties because of their low gas solubility in their solid state [14]. Another notable
property is that structures can be molded with extremely accurate dimensions because of
their low or negligible coefficient of thermal expansion relative to flexible chain
polymers [5]. They also exhibit high modulus, strength, and impact properties [18].
These properties can be exploited to apply LCPs in applications where flexible chain
polymers perform inadequately.
1.2 Rotational Molding
Rotational molding, also known as rotomolding, is a process used to manufacture
hollow plastic products [6]. This process is comprised of several independent
1 Introduction 8
phenomena: particulate gravitational flow, conductive melting, sintering, and
densification [31]. Particulate gravitational flow, also referred to as granular flow, is
essential for material distribution during mold rotation. A firm understanding of
conductive melting is required because, as long as coalescence is possible, it is the most
time consuming step in the rotational molding cycle. Therefore, an accurate account of
heat transfer to the tumbling powder is required to optimize cycle times. Sintering and
densification are detrimental to the process because incomplete coalescence renders
rotational molding impossible.
The process begins when polymer in either powder, granular, or viscous liquid
form is loaded into a hollow mold which is then simultaneously rotated about two
principal axes. Heat applied to the external surface conducts to the tumbling powder,
which eventually melts and adheres to the mold surface. While heating continues, the
powder sinters into an evenly distributed layer and densifies eliminating trapped air
bubbles from the melt. The mold continues to rotate as it is cooled by water spray, forced
air, or a fog or mist spray before water spray to solidify the product. Once the plastic is
sufficiently rigid, rotation and cooling are halted and the product is removed [6].
Rotational molding has a number of desirable advantages over competing
processes such as blow molding, thermoforming, and injection molding. Parts retain
little residual stress because flow is driven by surface tension and gravity, which
produces much lower deformation rates than the competing processes [6]. Although
weld lines may appear from lack of inter particle diffusion, they are much smaller than
1 Introduction 9
those created by impinging flow fronts because material is continuously distributed
throughout the mold. Material distribution also contributes to uniform wall thickness and
strong corners [30]. Structures can be reinforced by inserts or by multiple wall
construction with multiple resins. Essentially no material is wasted because gates and
sprues are not used. Finishing work is minimized because inserts and high quality
graphics can be incorporated into the molding process [6]. Tooling cost is relatively low
because there is no need to withstand high pressures or manufacture cores to produce
hollow structures. The technique possesses a manufacturing advantage over competing
processes as product dimensions increase, not only because of the ease of physically
forming the product but also because of economics [31].
Rotational molding does have several notable limitations. Manufacturing times
are relatively long; cycles are on the order of minutes to hours while other processing
techniques finish within seconds or minutes. This requires the materials to remain stable
at processing conditions for extended periods of time. Currently, there are a limited
number of materials used in practice, primarily commodity polymers such as
polyethylene, polypropylene, polystyrene, polyvinyl chloride, and nylon 6, 11, and 12
[15]. Very few engineering and high performance polymers have been used [6]. There
also may be an increased material cost due to an added grinding step [30]. Finally,
product design is slightly limited because certain geometrical features, such as ribs, are
difficult to mold.
1 Introduction 10
Two problems have been identified with the rotational molding of TLCPs. As
previously explained, sintering and densification are a part of rotational molding. Any
problem prohibiting their success will, in turn, hinder rotational molding. The second
problem is that rotational molding thermoplastics must be either a powder or granular to
ensure ample material distribution occurs during molding. This implies that materials
must be ground. Grinding of TLCPs generally returns particulate with large aspect ratios
that clump together bearing low bulk density, poor granular flow, and insufficient
material distribution. Incomplete coalescence (ie. porous product) results and is the
primary obstacle preventing the application of TLCPs to rotational molding. However, if
this is overcome, TLCPs have the potential to deliver a level of chemical resistance and
structural integrity unavailable from current materials.
1.3 Polymer Sintering
The term ‘sintering’ refers to the process of forming a homogeneous mass from
particulate without melting [31]. The material processing community utilizes this term
interchangeably with coalescence despite the fact that coalescence is intended for use in
processes where material often exceeds the glass transition temperature, Tg, for
amorphous materials and the melt temperature, Tm, for semicrystalline materials. With
this in mind, when applied to the simplest system, sintering describes the process where
two particles or fluid drops are driven by surface tension (and resisted by viscous
dissipation) to coalesce into a single drop [4]. Frenkel [10] was the first to explain this
behavior for Newtonian fluids, he derived the following expression for growth of the
normalized neck radius:
1 Introduction 11
21
0
Γ=at
ax
η (1.4)
where x/a is the neck radius normalized by the instantaneous particle radius, Γ is the
surface tension, t is time, η is viscosity, and a0 is the initial particle radius.
Recently, a desire for a more thorough understanding of polymer coalescence has
developed because of the use of polymeric materials in processes such as fabrication of
Average Thickness The average diameter between the upper and lower surfaces of a particle at its most stable position of rest.
Average Length The average diameter of the longest chords measured along the upper surface of a particle in the position of rest.
Average Breadth The average diameter at right angles to the diameter of average length along the upper surface of a particle in its position of rest.
Chunkiness Reciprocal of elongational ratio.
Circularity Ratio of circumference of a circle with the same projected area to the actual circumference of the projected area.
Elongational Ratio The largest particle length to its largest breadth when the particle is in a position of rest.
External Compactness The square of the diameter of equal area to that of the profile, divided by the square of the diameter of an embracing circle.
Feret’s Diameter The diameter between the tangents at right angles to the direction of scan, which touch the two extremities of the particle in its position of rest.
Martin’s Diameter The diameter which divides the particle profile into two equal areas measured in the direction of scan when the particle is in a position of rest.
Projected Area Diameter The diameter of a sphere having the same projected area as the particle profile in the position of rest.
Roundness Factor Ratio of the radius of the sharpest corner to the most round corner with the particle in a position of rest.
Specific Surface Diameter The diameter of a sphere having the same ratio of external surface area to volume as the particle.
Surface Diameter The diameter of a sphere having the same surface area as the particle.
Stokes Diameter The diameter of a sphere having the same terminal velocity as the particle.
Volume Diameter The diameter of a sphere having the same volume as the particle.
The quantities in
2 Literature Review 28
Table 2.1 as well several additional measures have been applied to image analysis
algorithms. Of these additional measures, the most notable has been the use of Fourier
analysis. The length of the radius vector from the particle’s center to its surface is
measured as a function of the angle between the radius and reference vectors [116]. This
function of radius length is then plotted against the angle and analyzed by the Fourier
cosine series to obtain a power spectrum for the particle. This technique is capable of
regenerating the original particle’s silhouette with a great degree of accuracy, providing
that the phase angle data are conserved. Unfortunately, extending the technique to the
prediction of behavioral aspects has proven to be extremely difficult [98].
An effective system for image analysis has not been identified because it is
unclear exactly what information is needed to describe powder behavior. Should a
general shape description be determined or only a measurement of shape features relevant
to a specific problem? Since particle-particle interaction is desired, the mixture of shapes
in the powder may be more relevant than individual particle shape. Ultimately, a
connection between shape and interaction must be developed. Presently, this feat cannot
be seriously considered because the state of particle characterization is still limited to
providing an effective means of shape identification [116].
Three philosophies are currently being explored in image analysis of particle
shapes [116]. The first pertains to quantifying only the relevant shape features (RSF) for
a given process or problem, but this assumes that the relevant features are known. The
second is to quantify particles according to a formation system because the formation
2 Literature Review 29
mechanism inherently places constraints upon particle shape. For example, roundness
can be used for particles formed from a shot tower, crystal shape can be used for particles
formed by crystal growth, and angularity for particles formed from brittle fracture. This
approach is the most fundamental and shows the most promise [115]. The third is to
generate a comparison particle from a given set of measured parameters (dimensions).
This approach is the most extensively used because data can be easily generated to
evaluate new particle characterizing algorithms.
2.1.1.3 Bulk Density
Bulk density is a measure of powder packing efficiency. Packing efficiency is
important because it governs the number of contact points between particles as well as
the number and size of voids between them. Typically, bulk density is inversely related
to powder flow. An increase in bulk density increases the flow rate, which indicates
better flow.
Bulk density is dependent upon particle shape, size, and size distribution.
Gaussian distributions tend to produce high packing density and intimate particle-to-
particle contact during the coalescence step of particle adhesion. The relationship
between distribution and density is more universally described by measuring the packing
fraction, defined as the ratio of the density of the powder bed to the material density.
Sometimes, void fraction is used, which is one minus the packing fraction. Packing
fraction can be understood by considering a powder composed of spheres having equal
diameter. If the spheres are packed in a body centered cubic mode the packing fraction
2 Literature Review 30
would be 0.534 [32]. Upon melting, assuming complete densification, the material will
occupy nearly half the volume of the original powder. There are numerous packing
arrangements (and accompanying packing fractions) that these spheres could take:
orthorhombic (0.605), tetragonal-spherical (0.698), and rhombohedral (0.740).
The coordination number is another way to describe packing. It represents the
number of contact points each particle has with neighboring particles. The previously
mentioned packing arrangements have the following coordination numbers: cubic 6,
† Data in figure were obtained by solving the open pore and M-S models; the M-S model results were shifted along the dimensionless time axis to coincide with the open pore model at a relative density of 0.942.
2 Literature Review 55
It was found that the closed pore model over predicts densification rate once pores
close. The result was attributed to the shrinkage of bubbles contained in a polymer melt
ceased to be controlled by viscous and surface free energy but by dissolution of air from
the bubble through the surrounding melt [79,88]. It was shown from a force balance on a
typical bubble in molten polymer that buoyancy forces can be neglected because the
apparent viscosity is so large [23]. A diffusion model was developed to predict
densification by simultaneously solving diffusion, conservation of momentum, and
continuity equations through a force balance around the encapsulated bubble [89]. For an
ideal gas bubble surrounded by a Newtonian fluid the shrinkage of the bubble radius is
given by:
( )[ ]
0
23 3
241
ar
g
fg
drdcDaa
RTP
dtd
aPPdtda
=
=
Γ−−=η
(2.23)
where a is the time dependant radius of the gas bubble, a0 is the initial bubble diameter, r
is radial position, Pg is the pressure inside the bubble, Pf is the system pressure, R is the
universal gas constant, T is the system temperature, D is the diffusion coefficient, and c is
gas concentration. The model was evaluated for several linear low density polyethylenes,
ethylene butyl acrylate, and ethylene vinyl acetate. Results were in good agreement with
experiment for relative densities above 0.942 and could be used in conjunction with open
pore models to describe the entire densification of a polymer melt.
Densification theory was extended to include composite systems of non-sintering
particles in a packing of sintering particles in two and three dimensions [68, 69]. For
2 Literature Review 56
example, a few percent of elastic material can greatly alter the viscosity and sintering rate
of the composite packing. Also, it is generally known that the geometrical arrangement
of the inclusions plays a crucial role in determining the effective properties of the
composite [67, 142]. One of the most interesting experimental results is the existence of
an inclusion volume fraction threshold where sintering is incapable of proceeding. This
cessation is thought to coincide with the formation of a percolating cluster of hard
particles. The model uses a network of points (representing particle centers) as well as
links connecting these points (representing contacts between particles) to replace the
packing. Sintering mechanics and packing densification are then inferred by observing
the deformation of the network.
Two models were devised for network deformation: the truss and the beam
models. The truss model imposed only force equilibrium on each particle while the beam
model required equilibrium of both forces and moments. The beam model appeared to
do a better job because the truss model does not contain particle rotation in predicting
deformation and rearrangement. The method was compared to several of the previously
mentioned constitutive models, which apply a mean field assumption implying that the
macroscopic deformation field gives the relative motion of two particles [65, 70]. The
assumption results in predicted properties that actually represent the upper bound of the
true properties [65]. The constitutive models suffice at low concentrations but fail to
capture clustered inclusion behavior.
2 Literature Review 57
BEM was applied to the two-dimensional multiply connected domains including
shrinking pores [158]. Various aggregate arrangements were considered: circular hole
centered and not centered in a circular disk, elliptic hole, three to six cylinders stacked
systematically, such that center to center distances were minimized, and several 4 x 4
arrays of cylinders with different contact radii. From the series of regular particle
packings it was concluded that cylinder sintering could be reasonably described by the
coalescence of two equal cylinders thus supporting the use of unit problems in the theory
of sintering. The cylindrical arrays were used to study interior pore formation and
evolution in truly multiply connected systems. For the various initial neck radii the pores
proceed towards the interior of the fluid as time progresses. Interestingly, the final pore
location for initial neck radii of 0.095 and 0.3 are approximately the same, meaning the
pores move faster in the latter case. This also implies that pore location in non-composite
powders is dependent upon geometrical features, not viscosity and surface free energy.
The main limitation was that it was computationally demanding, although it was also
incapable of handling cases where boundaries of neighboring particles, not initially in
contact, touch during simulation. In a later publication the BEM was reformulated to
handle axisymmetric three-dimensional objects such as opposing cones, stacked rings,
sphere and ring, and coupled spheres [159]. Results look promising from a geometrical
progression point of view but no experimental evidence is available for confirmation.
Coupling compaction and sintering by combining unit problems has been used to
describe densification [63, 64, 65, 66]. The two unit problems were external traction
induced compaction of an infinite line of spheres and two spheres with surface tension
2 Literature Review 58
driven sintering, which was computed by using the previously discussed FEM solution
for viscous flow. The approach assumed that the contact area alone was sufficient to
describe the kinetic part of the constitutive response. Predictions for a two dimensional
packing of mono sized particles compared well with experiment as long as the wetting
factor (introduced in the compaction unit problem) was arbitrarily chosen properly.
Interestingly, some of the voids grew while others shrank. It was found that the wetting
factor accounting for interaction with the surface could completely change the nature of
the velocity field. Friction with the surface introduces a dissipation term that depends on
the absolute velocity, which leads to length scale dependence in the energy balance. So
defects larger than a certain size will dilate even though the entire packing will shrink.
2.1.2.4 Equilibration
Equilibration refers to the relaxation of molecular structure at the contact
interface. A body formed from the coalescence of smaller particles must experience
various molecular relaxation processes to become a fully homogeneous, history
independent material. Often, neck growth and densification may occur faster than these
relaxation processes. This can become exaggerated by compositional inhomogeneity at
particle interfaces as with the breakdown of hydrophilic bilayers at interfaces between
densely packed latex particles [111]. Still, the contribution of molecular interdiffusion
and stress relaxation at compositionally homogeneous interfaces may be equally
dramatic.
2 Literature Review 59
Perera and Vanden Eynde [125, 126] displayed the extent that stress relaxation in
bulk polymer may lag behind the deformation by examining the development of
mechanical stresses in latex paints during and after film formation. It was shown that
without coalescing aids stresses reached a maximum of approximately 1 MPa but took
several weeks to relax to approximately 0.2 MPa, even with the addition of coalescing
aids. Although the magnitude of the peak was suppressed, the residual stresses continued
to remain after many days.
Interfacial Strength
From a number of observations (i.e. the dependence of peel strength on time,
temperature, molecular weight, and plasticizers) it was hypothesized that the diffusion of
polymer chains across an interface should be the critical step for the development of
homogeneous bulk mechanical strength when two surfaces of the same polymer are
brought into contact. Advances in chain dynamics and recent developments in
experimental methods to measure polymer diffusion and mechanical properties have
stimulated studies examining the relationship between interfacial diffusion and
mechanical properties [78]. One such study was of the fracture toughness of PMMA and
poly(styrene-co-acrylonitrile) in tension. Samples were fractured below Tg then the crack
interface was allowed to ‘heal’ for various periods at temperatures from 5 to 15° above
Tg. The critical stress intensity factor for crack propagation, KIC, was measured below
Tg. It was observed that KIC increased with 41
t until the time to produce a monolithic
molded specimen was surpassed. The same rate law was observed in lamination of
2 Literature Review 60
polished surfaces. It was also found that molecular weight has little effect on the initial
equilibration rate [74, 75].
Another study demonstrated that correlations exist between molecular structure,
diffusion, and the development of strength by measuring changes in the tensile properties
of annealed poly(n-butyl methacrylate) latex films that were crosslinked with varying
amounts of methallyl methacrylate [172]. All of the films fabricated at 23°C, noting that
the material’s Tg is 29°C, showed brittle fracture in tension at strains less than 0.5. A
series of films with up to 2% crosslinking agent were formed at 90°C for a various times.
Samples with less than 2% of the crosslinking agent displayed plastic yielding behavior
and an increase in ultimate strain to approximately 3. Those same samples showed that
the annealing time required to produce ductile failure increased as the crosslink density
increased. The 2% samples all showed brittle behavior regardless of processing
conditions. The fracture energies (integrated area of stress vs. strain curve) increased by
more than 10 times over the transition from brittle to plastic failure. It was also found
that additional annealing time, after ductile fracture developed, could only increase
fracture energy up to 1.5 times. The diffusion rate was measured between the
uncrosslinked deuterated and protonated latex particles by SANS. The penetration depth
at the interface grew linearly with 21
t . It was shown that the time required to equilibrate
the fracture energy corresponded to penetration depths of 40 nm, which was comparable
to the radius of gyration for the selected chains (Mw ~ 5x105) [52]. It was concluded that
brittle fracture reflected the interfacial energy of densely packed domains without
polymer chains diffused across the interface and ductile facture was indicative of chains
2 Literature Review 61
crossing the interface, which could be restricted to a negligible sol fraction and free ends
of the gel. When the mean length of the sol fraction and end segments were greater than
the length necessary for entanglements ductile fracture occurred, otherwise brittle
fracture was produced. This was supported by observing that the crosslink density for the
network chain lengths to become equivalent was about 2%, and therefore sol fraction and
end segments at that density were too short.
Diffusion
There have been considerable advances in understanding the conformation and
motion of chains confined at an interface. Conventional Fickian diffusion in liquids
corresponds to a three-dimensional random walk with a mean square displacement,
2r∆ , that is proportional to time. The entanglements found in polymer melts act as
intermolecular conformational constraints that reduce the rate of diffusion. It has been
suggested that at short times chain segment motion is localized and corresponds to Rouse
modes which predicts 212 tr ∝∆ . Rouse behavior eventually yields to a one-
dimensional random walk along a contour defined by its entanglements as predicted by
reptation theory 412 tr ∝∆ . At some longer time, dτ , after gRr ∝∆ 2 (where Rg is
the radius of gyration) the chain will have disengaged from its initial entanglements.
After disengagement normal Fickian diffusion is restored. There is also a possibility that
the chains will begin to relax at some time before dτ , which restricts the length of the
reptation regime and introduces 212 tr ∝∆ at intermediate times.
2 Literature Review 62
A number of experiments have been performed to test the validity of these
predictions at interfaces. One such experiment was performed on deuterated polystyrene,
where their concentration near the interface was measured as a function of annealing time
[135]. Results confirmed that Fickian diffusion occurred at long times but, even though
the time exponent was less than one, experimental uncertainty made it impossible to
validate theory at shorter times. Another study used selective deuteration of polystyrene
chains to enable the middle of the chains to be distinguished from the ends [136]. The
redistribution of labels during annealing time was measured by secondary ion mass
spectrometry (SIMS). The experiment was able to confirm that end segments cross the
interface before the middle segments as would be expected from reptation theory. SANS
was used to measure the amount of interdiffusion that occurs in two systems, one having
a weight average molecular weight, Mw, of 250,000 and the other being 2,000,000 [170].
It was found that the radial penetration distance was greater than the radius of gyration
for the lower molecular weight sample with a time dependence of 21
t . The radial
penetration distance of the high molecular weight sample was less than the Rg and the
exponent for time dependence was slightly less than ½. A correlation between tensile
strength and annealing time was also attempted. The lower molecular weight sample
appeared to increase with 41
t while the higher molecular weight increased with 32
t . The
time dependence for the higher molecular weight sample is questionable since the particle
radius was actually less than Rg so the entropic driving force to expand the coil
dimensions should accelerate chain transport.
2 Literature Review 63
Actually, various methods have been proposed to relate mechanical strength of a
partially equilibrated interface with interfacial diffusion [111]. Griffith’s model for
fracture mechanics states that a crack will propagate when the imposed mechanical
energy per unit crack area equals the nominal energy per unit area, Gc, of crack surface.
For polymeric materials Gc includes a significant contribution from energy that is
dissipated by plastic deformation away from the surface. One method proposed that Gc is
proportional to the average distance, x∆ , the chains have diffused across the interface
and to establish equilibrium, the entanglement density at the interface must become equal
to that of the homogeneous melt. If diffusion is Fickian, it can be shown that the kinetics
for critical stress intensity scale as [74, 75]:
43
41
21
21 −
∝∆∝∝ MtxGK cIc (2.24)
which is consistent with experiments. For a particular polystyrene system it was
estimated that at complete equilibration nmx 2≅∆ , much less than Rg for the weight
average component but comparable to the Re (Rg for the chain segments between
entanglements.) While eRx ≤∆ applying reptation theory produces the following
kinetics.
81
21
txK Ic ∝∆∝ (2.25)
A second method suggested that Gc is proportional to the number of chains that
cross the interface per unit area instead of the number of entanglements per unit volume.
It was found that the predicted growth of Gc according to reptation theory depends
strongly on the initial arrangement of chain ends near the interface [37, 38]. For
2 Literature Review 64
randomly distributed chain ends it was found that 23
21 −
∝ MtGc and for chain ends
initially concentrated at the interface 41
41 −
∝ MtGc . The randomly distributed case
appeared to agree with experiments because 21
cIc GK ∝ .
In a third model it was assumed that xK Ic ∆∝ and the kinetics were determined
by applying reptation theory along with two initial conditions [84, 167, 168]. Chains
adjacent to the interface were restricted to nongaussian conformations and there were no
conformational restraints imposed on end segments at the interface. The result was that
the critical stress intensity is proportional to 41
41 −
Mt . All three models predict the same
time dependence, 41
t , but each has a unique description of the dependence on molecular
weight. Even with their agreement upon time dependence there is still doubt about its
generality because some experiments have shown 21
t [111].
Stress Relaxation
At least a fraction of the driving force for elastic or quasi-elastic neck growth is
retained as residual stress after coalescence has occurred. The relaxation of these stresses
may have either a direct (weaken interface) or indirect (influence diffusion properties
across interface) effect on material properties. Despite its importance, the quest for
complete understanding has been neglected by both theorists and experimentalists. This
is primarily because the effects of residual stress can be difficult to isolate and quantify.
In the case of amorphous polymers, residual stresses are the result of non-equilibrium
2 Literature Review 65
chain conformations, and therefore stress relaxation and diffusion may not be separable
events. This is one of the benefits of accurately simulating viscoelastic neck growth; it
could provide much needed insight into the evolution of stress fields.
Observations have been made on the development of strength and failure
morphology for neck growth of a sphere coalescing with a slab of the same material
(Lucite-40 acrylic). The spheres were partially coalesced with the previously annealed
slab for various times and at several temperatures, after which they were cooled below
their Tg. The sphere was then broken off and the structure and properties of the
remaining ‘stump’ were examined.
Two extremes are anticipated, elastic and viscous recovery. When only elastic
neck growth occurs, failure is expected as brittle fracture along the interface where axial
stresses have concentrated (JKR, compressive in the center and tensile around the
periphery) and leave a shallow crater. Residual axial stresses are distributed along the
failure surface and can remain indefinitely while gTT < . If the temperature surpasses the
glass transition temperature elastic compliance will increase enough for elastic recovery
to occur, which should manifest as a purely axial stump recovery. Residual stresses do
not develop during viscous neck growth and failure can occur either along the contact
interface or at some other point. If failure occurs along the contact interface (brittle),
then the remaining stump should be relatively stress free. Otherwise ductile fracture will
transpire, generating stresses along the fracture surface. Heating to above Tg should
allow surface forces concentrated at the periphery to induce viscous flow, spreading the
2 Literature Review 66
stump out on the slab. Both of these responses were observed with the Lucite system,
with evidence of contributions from both mechanisms at intermediate times. This
suggests that the time required for ductile failure to develop is comparable to the time for
viscous flow to occur, times greater than the terminal relaxation time.
2.1.3 Processing Considerations
In this section, the various processing variables associated with rotational molding
equipment are identified and their potential impact upon the product is discussed.
2.1.3.1 Mold Rotation
A common feature of rotational molding is that the mold is rotated about two
perpendicular axes [24]. An illustration of the two axes is shown in Figure 2.6. A
number of labels have been used to identify these axes. The major axis is also referred to
as the primary, arm, or polar axis. The minor or secondary axis may also be called the
plate or equatorial axis.
2 Literature Review 67
Major Axis
Minor Axis
Major Axis
Minor Axis
Figure 2.6. Rotation Axes
Mold rotation rates are typically slow. It has been shown that there is little benefit
in increasing above 10 rpm for a number of systems [149]. Slow rotation rates reduce
fluidization, ensuring that the plastic powder spends the majority of its time in the bottom
of the mold as a pool. Mold rotation is also important because it dictates the frequency
and duration that each point of the mold dips into the powder pool, which ultimately
controls the uniformity of part thickness. Therefore, it is essential to have uniform speed
throughout the entire rotation.
2 Literature Review 68
Mold position relative to the axes is important because it is possible that rotation
can induce differential acceleration of powder as it passes over the mold surface. So, it is
suggested not to orient the mold such that the mold surface is on the centerline of the
minor axis of rotation [133]. Along the centerline, material is only affected by major axis
rotation with maximum acceleration occurring at the ends of the mold. Because minor
axis speeds are relatively slow, variation in radial acceleration across the mold can be
great enough to have a significant effect on the distribution of thickness. As this
suggests, the distribution of thickness throughout the part is dependent upon not only the
speed but also the ratio of the major/ minor axes speeds. Originally, the machines had a
fixed major-minor rotation rate ratio of 4:1, which is still used as a starting point for
preliminary trials with new molds [132]. Modern machines allow independent control of
the major and minor axes to accommodate molding odd shaped parts.
It is important to recognize that the ratios of rotation speeds are often given in
terms of a speed ratio. Rotation speeds are obtained from tachometers located on the
major and minor drive shafts. Since the minor drive shaft is located inside the major
shaft, the measured minor rotation speeds are actually the total of the major and minor
rotation rates. The speed ratio is defined as:
rpmMajorrpmMinorrpmMajor
RatioSpeed−
= (2.26)
Selecting an appropriate speed is not a precise science. Actual values are
typically determined, to a large extent, by trial-and-error. The main reference for setting
2 Literature Review 69
speed ratios is the McNeil chart, shown in Table 2.4 [2]. The table provides guidelines
for various mold shapes and settings. It should also be mentioned that appropriate speed
ratios may depend on factors like mold position with respect to the major and minor axes
and the ability to uniformly deliver heat to the mold surface.
Table 2.4. McNeil Chart [2]
Arm Plate8 to1 8 9
5 to 1 5 64.5 to 1 8 9.753.3 to 1 10 12.25
12 14.54 to 1 8 10
10 12.5
2 to 1 6 98 12
10 15
12 18
1 to 2 5 15
1 to 3 4 156 22.5
1 to 4 4 205 256 30
1 to 5 4 24
ShapesSpeed Ratio
Typical Axis Speeds
Any shape showing overlapping lines of rotation at 4 to 1
Rectangular boxes, horses with bent legs
Oblongs (horiz. mounted)Straight tubesSome defroster ductsBalls or globes
Cubes, balls, odd shapes
Rings, tires, balls
Any rectangle which shows two or more thin sides when run at 4 to 1Picture frames, manikins, round flat shapesHorses with straight legs
Balls whose sides are thin at 4 to 1, vertically mounted cylindersVertically mounted cylinders
Auto crash pads (vert. mount)Parts which should run at 2 to 1 but show thin side wallsFlat rectangles (gas tanks, suit cases, tote bin covers)Tires, curved air ducts,pipe angles, flat rectangles
2 Literature Review 70
2.1.3.2 Molding Cycle Time
For rotational molding to remain competitive against industrial blow molding and
emerging technologies such as twin sheet thermoforming and gas assisted injection
molding, cycle times must be reduced to a fraction of what they are today [24]. The
length of the molding cycle depends on each of the four cycle stages: loading, heating,
cooling, and unloading. Cycle times vary according to how the rotational molding
machine performs each of these steps [30]. Some machines have multiple arms, each
possessing a mold and occupying a step. If this is the case, each stage will require the
same amount of time, which is dictated by the longest step (typically the heating stage).
Although this is the most continuous approach to molding, it does not provide the most
control over each stage and large amounts of dead time are inherent in the shorter steps.
A single arm machine allows for the optimization of each stage. However, each stage is
not being used simultaneously, so production rates are lower. Industry’s solution was to
develop an independent arm machine that was capable of combining stage controllability
of a single arm unit with the more continuous operation of a multiple arm machine by
incorporating independent arm control and several holding areas.
Heating
The heating time can be divided into three parts: induction, fusion, and
densification [1]. Induction time refers to the time it takes the mold, from the onset of
heating, to reach the tack temperature. The tack temperature is identified as the
temperature where particles begin to adhere to the mold surface and therefore, nearly
coincides with the melt temperature. There are a number of parameters, listed in order of
2 Literature Review 71
influence in Table 2.5, that affect induction time [110]. Although induction does include
the elastic deformation step of coalescence, changing the length of induction time does
not influence final part properties. Fusion and densification times refer to the time
required to complete neck growth and densification. Parameters influencing their
duration are also presented in Table 2.5. Once again, altering the particle sintering time
will not have a significant impact final part properties. Densification time is strongly
dependent on gas diffusivity and, in addition to the parameters listed in the table, can also
be manipulated by adjusting air pressure inside the mold. The effects of pressure will be
discussed in section 2.1.3.3, though it is worth noting here that increasing pressure
increases the internal temperature and decreasing pressure decreases the internal
temperature [149]. When changing densification time it is important remember that
sufficient time should be allotted to complete densification as incomplete densification
has a dramatic effect on final part properties and appearance.
Table 2.5. Parameters Effecting Heating Time
Parameters That Influence Fusion and Densification Times
1Heat transfer type (conduction, convection, radiation) Wall thickness of part
2 Oven temperature Oven temperature3 Resin melt temperature Heat flux4 Heat flux Mold surface to volume ratio5 Mold wall thickness Particle size of resin6 Mold surface to volume ratio Mold heat capacity7 Oven recovery time Resin melt temperature and heat of fusion
Parameters That Influence Induction Time
2 Literature Review 72
The heating cycle can be monitored by recording the temperatures of the oven, the
exterior and interior wall of the mold, and the air temperature inside the mold. These
data have a characteristic shape, unique to the rotational molding process. The oven
temperature is important because it provides a means of assessing the ability of the oven
to deliver heat. The thermal traces for the mold walls provide a measure of how heat is
received from the oven and transferred to the powder. The internal air temperature
provides perhaps the most interesting information. After the cycle begins, the internal air
temperature increases steadily (constant slope in plot) until it surpasses its tack
temperature [24]. This is sometimes referred to as the “kink” temperature and is
identified as the temperature at which the powder begins to melt and adhere to the mold
surface [24]. The heating rate slows as the cycle proceeds, indicating particle adhesion.
The plastic acts as a heat sink against the inner surface of the mold by absorbing energy
as it melts and retarding the rate of energy transfer to the cavity air. Once all the material
has melted, the temperature trace resumes a higher heating rate, often being very similar
to what was originally witnessed at the beginning of the cycle. There is generally a lag
between the internal mold thermal trace and the cavity air trace. This lag can be
increased further if there is either a large amount of powder or the powder has low
thermal conductivity [24].
Cooling
A thermal gradient remains after heating, with the mold’s external surface being
the hottest and the internal cavity air being the coolest parts of the system. The system is
typically cooled from the outside, which can be done by forced air, water mist, or water
2 Literature Review 73
shower. External cooling causes the maximum temperature to shift from the external
surface, through the mold and densified polymer, towards the mold cavity. This process
is referred to as thermal inversion. The rate of thermal inversion is dependent upon
relative thermal properties and the thickness of both the mold and polymer.
The rate of thermal inversion is important because it can adversely influence final
part properties by developing residual stresses [114]. Rapid cooling can freeze molecular
structure in a metastable state and eventually, chains move to increase stability, resulting
in warpage and distortion. High cooling rates can also reduce part density in
semicrystalline materials by increasing the amorphous content, which increases impact
strength and flexibility [114, 132]. In extreme cases the molding can develop surface
distortions and even collapse from the rapid change of internal pressure if a vent is not
present to equalize pressure [24]. The cavity may be pressurized to reduce shrinkage
ensure that the molding does not pull away from the mold because that dramatically
reduces the effectiveness of heat transfer. Therefore, water cooling must be applied
judiciously, typically only after thermal inversion and recrystallization has occurred. On
the other hand, slow cooling increases the crystalline content and density, while reducing
impact strength.
A number of techniques have been used to determine the most effective cooling
rate. Initially, cooling rates were selected by trial-and-error. Molders soon discovered
that it was a difficult and inefficient method to optimize cycle time and part performance.
An obvious addition to the selection process was the differential scanning calorimeter
2 Literature Review 74
(DSC). DSCs quickly and accurately provide valuable information about melting and
recrystallization by subjecting samples to various heating and cooling rates. As their
accuracy improves, process simulation has become a more common method of screening
process conditions. With the advent of portable multiplexed thermocouple platforms, the
most promising conditions can be quickly verified before being scaled to the larger, more
expensive equipment.
2.1.3.3 Vacuum and Pressure
Vacuum or additional pressure within the mold can produce a number of effects
depending on how it is applied. In general, introducing a small positive pressure (5 to 10
psi) during densification decreases both the frequency and size of entrapped bubbles [24].
Pressure compresses the trapped bubbles, acting to increase the solubility in the bulk
polymer by increasing the concentration gradient in the dissolving bubble and
accelerating bubble extinction. The majority of pinholes can be completely removed in
as little as thirty seconds [149]. It has been shown to increase properties like tensile
strength, modulus, and impact strength, but improvement may be limited to cosmetic
quality, which can be as important as mechanical properties. If pressure is applied before
bubble formation the air concentration is increased but as long as the pressure is
maintained there will be no net effect because there is no compressive force on the
bubbles (the pressure inside the bubbles is equivalent to that inside the mold). Of course
applying pressure prior to bubble formation then releasing it during densification will
cause bubbles to expand.
2 Literature Review 75
If a vacuum is introduced prior to bubble formation the effect is very similar to
applying pressure during densification. However, this method should produce better
results, especially in polymers with low gas permeability, because the concentration of air
is decreased [24]. By removing the air at the onset bubble extinction does not depend on
gas dissolution into the polymer. It is not necessary to hold the vacuum during the entire
heating cycle, just until bubbles form. At that point, releasing the vacuum will cause the
bubbles to disappear almost immediately. Since this process occurs while the mold
temperature is increasing, additional heating after bubble formation will cause bubbles to
grow. Applying a vacuum after bubble formation will cause bubble growth.
2.1.3.4 Mold Release Agent
Mold release agents are used because, even in the simplest geometries, moldings
stick to the mold surfaces. Mold release agents are designed to interfere with polymer
adhesion to the mold surface by reducing the mold’s surface energy [149]. Over 250
types of mold releases exist, varying from having to be applied each cycle to permanent
alteration of both internal and external surfaces.
Molten polymer will wet a mold surface if the surface energy of the mold is
greater than that of the polymer. Mold surface energies are typically an order of
magnitude greater than polymers so, in theory, they should spread easily without many
defects [3, 53]. In practice, bare metal surfaces are not as good as theory suggests
because an oxidative layer forms that reduces surface energy. Oxidation rates increase
with temperature and coincidentally, the polymer is exposed to the highest temperatures
2 Literature Review 76
during rotational molding at the mold/polymer interface [87]. In the most extreme cases,
like with polyethylene, oxidation causes the formation of chemical bonds between the
polymer and the mold surface. This escalates the sticking problem and introduces an
increased need for release agents. Unfortunately, mold release agents act by reducing
surface energy to impair adhesion and that reduction inherently, by reducing the
polymers affinity to wet the surface, increases the number of pinholes in the surface.
2.2 Thermotropic Liquid Crystalline Polymers
The purpose of this section is to review various aspects of main chain TLCP
rheology and mechanical properties that pertain to rotational molding. In fact, their
unique performance and rheological behavior inspired their selection for rotational
molding because they may extend the processing technique to higher performance
applications where conventional rotational molding polymers are insufficient.
The section begins with the review of TLCP mechanical properties in section
2.2.1. As some mechanical properties are dependent on the processing technique, it
should be stated that reported values might not be representative of what may be obtained
from rotational molding. This is especially true in properties that are strongly dependent
on the degree of molecular orientation. Section 2.2.2 focuses on various rheological
aspects of TLCPs.
2 Literature Review 77
2.2.1 Mechanical Properties
Strength and Modulus
Mechanical properties, especially tensile strength and modulus, depend upon the
degree of orientation achieved. This is limited by the fabrication method and geometry
of the manufactured item. A compression molded unoriented LCP has mechanical
properties similar to that of a conventional isotropic polymer [104]. Injection molding
imposes higher deformation rates that increase the degree of orientation, especially in the
skin, which makes the major contribution to stiffness [31]. Injection molded main chain
TLCPs show superior tensile moduli to those molded from conventional glass fiber
reinforced isotropic polymers. As the degree of orientation increases in the testing
direction, the mechanical properties approach those of main chain TLCP fibers [21, 130].
Simplistically, the layered structure of injection molded TLCP can be viewed as a
microcomposite composed of layers with varying directional orientation [104]. Each
layer contributes in an integral manner to the mechanical properties of the entire molding.
Modulus is dependent upon layer thickness and the direction and degree of orientation of
polymer chains in those layers. Each of these factors depends on the processing
conditions and the fluid’s rheological response to the imposed flow conditions. Stiffness
can also be varied through manipulation of the chemical composition. The flexural
modulus of injection molded tensile bars with varying HBA content was studied. As
HBA content increases and becomes great enough for the polymer to transition to a liquid
crystalline phase, the modulus monotonically increases [14]. Orientation was measured
as a function of depth through the moldings. Despite the obvious skin-core effects, it was
2 Literature Review 78
found that an increase in molecular orientation was mainly responsible for the increase in
modulus [13]. From this, it was concluded that simply creating a liquid crystal
mesophase is not sufficient to acquire a system with high, self-reinforcing modulus; the
mesophase must become sufficiently oriented. However, if this is done effectively,
TLCPs can exhibit exceptionally high strength and modulus. Moldings may retain these
properties at elevated temperatures (>200°C), which make them suitable for high
temperature applications.
TLCP articles possess some degree of anisotropy, a difference in properties when
tested parallel and perpendicular to the flow direction, which is reported as the anisotropy
ratio. The anisotropy ratio increases with the degree of orientation and is therefore
greatest in fibers. The anisotropy ratio for injection molded TLCP has been shown to fall
somewhere between 4:1 to 10:1, increasing with decreasing thickness as the proportion of
skin and core increases [118, 62, 29]. Introducing fillers tends to increase the anisotropy
ratio of conventional isotropic polymers, but disrupts molecular alignment and reduces
the ratio for TLCPs [118]. Although not typically desirable, this is one way of improving
the cross flow properties.
Dimensional Stability
TLCPs are used for high precision components that require close tolerances on
dimensional quantities. There is very little difference in molecular configuration, and
therefore a negligible density change between the melt and solid states for TLCPs. They
also typically possess little elastic recovery. This means that when molded, they exhibit
2 Literature Review 79
very low mold shrinkage and warpage when compared with isotropic polymers. Once
molded, they retain their molded dimensions well. One reason for this is that TLCPs
absorb very small quantities of water (typically less than 0.2% when immersed in water),
which makes the effects of swelling from moisture absorption negligible [104]. The
coefficient of linear thermal expansion is much lower for TLCPs than for conventional
polymers, even when conventional polymers have been reinforced with glass fibers [31].
Incidentally, they are quite similar to values for metals (TLCPs < 1, 1< Metals < 3
cm/cm/°Cx10-5). This similarity results in good integrity and minimal strain during
thermal cycling of components including both materials [104].
Barrier Properties and Chemical Resistance
TLCPs act as excellent barriers to gases, with permeability coefficients for He,
H2, Ar, N2, CO2 comparable to or smaller than those for polyacrylonitrile, one of the least
permeable polymers known [26]. See Table 2.6 for a comparison between Vectra A900
and PAN. They also have exceptionally low oxygen and water vapor permeabilities. The
low permeability seems to be more of a result of low gas solubility rather than low gas
diffusion, though it has been shown that diffusion can be greatly reduced by thermal
treatment in systems that are crystallizable [22]. It was calculated that crystalline content
would have to be near 90% or more to explain the solubility. However, crystallinity is
often below 20% for an unannealed sample so it seems that liquid crystalline order is
responsible [104].
2 Literature Review 80
Table 2.6. Comparison of Gas Transport Properties at 35°C of Vectra A900 and PAN
7. Rey, A.D., “Modelling the Wilhelmy Surface Tension for Nematic Liquid
Crystals,” Langmuir, 16, 845 (2000)
8. Ticona, Vectra Liquid Crystal Polymer Global Brochure (VC-7), Product
Literature (2000)
9. Baird, D.G., Collias, D.I., Polymer Processing: Principles and Design, John Wiley
& Sons, New York, (1998)
4 The Role of Transient Rheology in Polymeric Coalescence
136
4 The Role of Transient Rheology in Polymeric Coalescence
Preface
The work presented in this chapter partially addresses the first objective of this
research, as it represents the first step towards understanding the coalescence of TLCPs.
Specifically, the effect that transient rheology has on the coalescence of polymeric
materials. It was discovered during the review of the literature that a clear, consistent
understanding of the role that viscoelasticity plays in polymer coalescence was not
available but is required to understand the behavior observed for TLCPs. This chapter is
organized as a manuscript for future publication.
4 The Role of Transient Rheology in Polymeric Coalescence
137
The Role of Transient Rheology in Polymeric Coalescence
Eric Scribben†, Donald Baird†, and Peter Wapperom‡
†Department of Chemical Engineering, Virginia Polytechnic Institute and State University, Blacksburg, Va 24061
‡Department of Mathematics, Virginia Polytechnic Institute and State University, Blacksburg, Va 24061
4.1 Abstract
Polymeric coalescence is the process by which surface tension drives two small
drops to merge into a single uniform, homogeneous drop. In this work a coalescence
model, which equates the work of surface tension to the work done by viscous forces
while assuming biaxial extensional flow kinematics, is evaluated to determine its ability
to accurately predict the coalescence rates of three isotactic polypropylene resins with
increasing weight average molecular weight (M = 190k, 250k, 340k). There were three
variations of the model: one used a Newtonian constitutive model, another used the upper
convected Maxwell (UCM) constitutive model with the assumption of steady state stress
behavior, and the last used the UCM model with time dependent material functions.
Coalescence model predictions from each of these formulations were compared to
experimental data at 180°C. The surface tension was obtained by fitting the Bashforth
and Adams equation to a sessile drop profile and the constitutive equations were fit to
rheological data to obtain the viscosity and relaxation spectra. The coalescence rates
predicted by the Newtonian model were slower than those of the experimental data,
indicating that viscoelastic effects were significant even for the case of relatively low
Deborah numbers. The results from the steady state UCM model predict that
4 The Role of Transient Rheology in Polymeric Coalescence
138
viscoelasticity (as quantified by the relaxation time) acts to slow coalescence relative to
the Newtonian model and, therefore, was less accurate than the Newtonian model at
representing the experimental coalescence data. It was shown that the reduction in
coalescence rate with increasing relaxation time was due to the UCM model predicting
that the biaxial extensional viscosity approaches infinity at a critical extension rate. A
numerical algorithm was developed to treat the increased complexity that was introduced
by the transient UCM formulation. The solution demonstrated that coalescence was
accelerated by increasing the relaxation time, the opposite relationship of what was
predicted by the steady state UCM formulation. It was also found that the transient UCM
formulation was capable of quantitatively predicting the coalescence rates of the
polypropylenes at short times where viscoelasticity was important. This work illustrates
that the effective viscosity, over a range of times where the viscosity had not reached
steady state, was lower than the steady state value and lead to the acceleration of
coalescence.
4.2 Introduction
Coalescence refers to the process where, in an attempt to minimize surface area,
surface tension drives a collection of fluid drops to merge into a single, homogeneous
body. Coalescence begins when drops are brought into contact, and they instantaneously
deform in an elastic manner to create finite contact surfaces [1]. It is important to
emphasize that elastic contact scales with surface tension, creep compliance, and particle
radius but not time [2]. Elastic contact may also be referred to as neck formation,
4 The Role of Transient Rheology in Polymeric Coalescence
139
because it defines the formation of a bridge at the interface between particles.
Coalescence continues with the radial growth of the neck region. At this stage, the
particles retain their individuality because coalescence is not influenced by other contacts
from neighboring drops. In addition, there is no appreciable change in density of the
multi-particle structure [2]. The neck dimensions grow until a network of pores form and
eventually become entrapped bubbles. The process of eliminating the entrapped bubbles
is referred to as densification, and the densification rate depends upon gas permeability in
the molten material [2]. A complete description of the coalescence of multiple drops
must represent at least three stages: elastic contact, neck growth, and densification.
However, the neck growth process is the focus of this work because it embodies most of
the geometrical change experienced by a particle during coalescence.
Frenkel [3] first modeled viscous (Newtonian) coalescence by proposing an
expression for the coalescence rate of this system that was later corrected for continuity
by Eshelby [4]. They suggested that coalescence could be modeled by evaluating two
isolated particles, which they represented by the idealized system of spherical drops as
depicted in Fig. 4.1. In the figure ao is the initial particle radius, x is the neck radius, y is
the distance from the particle center to the contact plane, and θ is the angle between the
axial plane of symmetry and the line from the particle center to the neck radius.
4 The Role of Transient Rheology in Polymeric Coalescence
140
ao
y θ
a
x
aoao
y θ
a
x
y θ
a
x
Figure 4.1. Shape evolution during the coalescence of two spherical particles
Their model was derived from the mechanical energy balance by equating the work of
surface tension to the work done by the rheological stresses, which is referred to as
viscous dissipation for a purely viscous fluid [5],
∫∫∫=Γ− dVdtdS Dτ : (1)
where τ is the deviatoric stress tensor, D is one-half the rate of strain tensor,
( )TvvD ∇+∇=21 (2)
S is surface area, and V is the volume of both spheres. The presence of external stresses
and gravitational effects were neglected. The flow kinematics were assumed to be biaxial
stretching flow, with the components of the velocity gradient tensor given by,
−=∇
εε
ε
&
&
&
0002000
v (3)
4 The Role of Transient Rheology in Polymeric Coalescence
141
where ε& is the extension rate. The rheological stresses in Eq. 1 were assumed to be
described in terms of the Newtonian constitutive equation. They solved Eqs. 1 through 3
for the early stages of coalescence when the particle radius remained constant to give,
21
23
Γ=oo a
tax
η (4)
where t is time, Γ is the surface tension, and ηo is viscosity. Despite its simplicity by
assuming a Newtonian fluid and a constant particle radius, the Frenkel model defines a
relationship between material properties, particle size, and the rate of viscous
coalescence. It has been used as a qualitative benchmark for experimental data,
demonstrating good agreement between measured values of the dimensionless neck
radius (x/a) and the predicted exponent of ½ on time [6].
Pokluda et al. improved the accuracy of Eq. 4 by applying the conservation of
mass with constant density to account for the time dependence of the particle radius [7],
3/122 )
)](cos2[)](cos1[4()(
ttata o θθ −+
= (5)
where θ is referred to as the coalescence angle, θsin=ax , as shown in Fig. 1. This time
dependent radius was incorporated into the mechanical energy balance, Eq. 1, through the
definition for the surface area of the two spheres:
( )θπ cos14 2 += aS (7)
As outlined in reference [7] this system of equations can be rewritten as the following
homogeneous, first-order differential equation:
4 The Role of Transient Rheology in Polymeric Coalescence
142
( ) ( )( )[ ] ( )[ ] 3
43
521
35
cos1cos2sincos2
θθθθ
ηθ
+−Γ=
−
Kadtd
oo
(8)
where K1 arises from the definition of the extension rate,
dtdK
yvy θε 12
1 =∂∂
−=& (9)
and is defined by:
( ) ( )( )( )
−+
++−−=θθ
θθθθcos2cos1
cos1cos226
sin2
tan1K (10)
It should be added that K1 presented here is a slightly different form than was defined in
the original derivation, which was an approximation in the limit of small coalescence
angles. The K1 defined here is valid over all coalescence angles during the coalescence
process and is used because it represents an improvement in model accuracy.
In addition to the energy balance approach for modeling coalescence first
introduced by Frenkel [3], a complete flow description of coalescence was derived from
the equations for the conservation of mass and momentum, a constitutive equation
describing the development of stress with deformation, initial conditions, and boundary
conditions describing the movement of the free surface has been used [8]. The latter
approach provides greater resolution of stresses, and its complexity requires a numerical
simulation technique such as finite element method (FEM). This approach has been used
to explore numerous factors that pertain to the viscous coalescence problem
[8,9,10,11,12]. It was demonstrated that the complete flow description is capable of
accurately predicting the relative density for a packing of glass spheres, at least up to
densities where the kinetics are no longer controlled by neck growth [13]. When
4 The Role of Transient Rheology in Polymeric Coalescence
143
compared to the Newtonian model, Eq. 8, the coalescence predictions were in reasonable
agreement, which was not surprising because the predicted streamlines from the
numerical solution confirm the assumed biaxial extensional flow kinematics [7,8,11].
Bellehumeur et al. evaluated the Newtonian coalescence model, Eq. 8, for several
polyethylenes and found that the model over predicted the coalescence rates [6]. They
proposed a variation of the Newtonian coalescence model to incorporate viscoelasticity
by using the upper convected Maxwell (UCM) constitutive equation for the extra stress
tensor,
Dττ oηλ 2=+∇
(11)
where λ is the characteristic relaxation time, ∇τ is the convected derivative,
vττvττ ∇⋅−⋅∇−=∇
T
DtD (12)
and DtDτ is the substantial derivative. By assuming that the flow field is homogeneous,
and that at any instant the stresses are at steady state (though the stresses may change
with the extension rate throughout the process) the constitutive equation was simplified to
the following expression [5]:
[ ] Dvττvτ oT ηλ 2=∇⋅−⋅∇−+ (13)
After Eqs. 3, 5, 7, 9, and 13 were substituted into Eq. 1 and the integration was
performed, the model was rearranged into the following differential equation for the
steady state representation of coalescence:
4 The Role of Transient Rheology in Polymeric Coalescence
144
01282
21
1
2
1 =−
Γ+−+
dtd
KKaK
dtdK oo θηλθλ (14)
where K1 is defined as before, Eq. 10, and
( ) ( ) 35
34
35
2cos2cos1
sincos2θθ
θθ−+
=−
K (15)
Eq. 14 predicts that the coalescence rate decreases with an increase in the
relaxation time. The predicted behavior was used to suggest that viscoelasticity was
responsible for the observation that two propylene ethylene copolymers coalesced at a
slower rate than predicted by the Newtonian model [6]. This is interesting because
reported coalescence times for polytetrafluoroethylene (PTFE) and several acrylic resins
were significantly shorter than what was predicted by the Newtonian model [2,16].
Furthermore, the relaxation times were not measured experimentally. Instead, they were
adjusted to produce better agreement with data, and these values were unrealistically
large. Therefore, it was unclear if it was possible to accurately predict coalescence using
a viscoelastic constitutive equation with experimentally measured relaxation times.
The complete flow description with a FEM has also been extended to include the
viscoelastic representation of stresses. As with other cases where viscoelastic behavior
has been modeled with a FEM, the inability to achieve convergence with increasing
relaxation times limit the simulations to cases with relatively low Deborah numbers [14].
Although the complete flow description provides increased resolution of the stresses
relative to the energy balance approach, it appears that the flow kinematics are
predominately biaxial extension, as assumed in Frenkel’s Newtonian model [15].
4 The Role of Transient Rheology in Polymeric Coalescence
145
The objective of this work is to extend the approach devised by Frenkel to
describe the coalescence of two particles to the transient viscoelastic case by using the
upper convected Maxwell constitutive equation without the assumption of steady state
stress behavior. The ability of the model to predict coalescence is assessed by comparing
the measured coalescence values of three polymers with different relaxation times to the
predicted values. The significance of this work rests partially on the fact that all of the
material parameters in the UCM model are obtained directly from rheological data rather
than arbitrarily adjusted for the model to accurately predict coalescence data. Finally, the
results presented here are used to explain the predictions of a previous model presented
by others concerning the role of viscoelasticity in polymer coalescence.
4.3 Experimental
4.3.1 Materials
Isotactic polypropylenes (available from Sigma-Aldrich [CAS 9003-07-0]) of
three different molecular weights were selected for this study. Because the materials
possess different molecular weights, they are expected to demonstrate increasing
relaxation times, potentially allowing the evaluation of effect that the magnitude of the
relaxation time has on coalescence. The weight average molecular weight, the
polydispersity index, and the melt flow index are summarized in Table 4.1. The densities
4 The Role of Transient Rheology in Polymeric Coalescence
146
and the melt transitions for all samples were the same and were reported by the
manufacturer as 0.9 g/cm3 and 160-165°C respectively.
Table 4.1. Weight Average Molecular Weight, Polydispersity, and Melt Index
M (x10-3)
PDI (M/N)
MI (g/10min)
190 3.80 35.00 250 3.73 12.00 340 3.50 4.00
4.3.2 Surface Tension Measurement
The surface tension of each of the materials was determined by fitting the
Bashforth and Adams equation to the sessile drop profile of the molten polymer in an
inert atmosphere at 180°C [17]. Sessile drops were prepared by first extruding
polypropylene fibers with a Göttfert Rheograph 2001 capillary rheometer equipped with
a 0.5 mm capillary die (L/D = 10). Pellets, with a diameter of approximately 500 µm and
an aspect ratio of approximately one, were cut from the fibers. A single pellet was placed
vertically, standing on its cut end, on a glass slide in a Linkham hot stage set at 180°C,
where it was melted into a sessile drop. The sample was quenched and the glass slide
was rotated to allow a profile view of the drop. The sample was reheated to 180°C and a
digital image of the profile was recorded with an optical microscope equipped with a
miniDV camcorder. The accuracy of this technique (0.1% error) demands that the
particle radii must be small so that gravitational forces cannot influence the shape of the
profile. This was verified by calculating the Bond numbers Γ
= grBo2ρ (Bo190k=0.063,
4 The Role of Transient Rheology in Polymeric Coalescence
147
Bo250k=0.125, Bo340k=0.063) and supported by the observation that the profile shape did
not change when the glass slide was rotated.
4.3.3 Rheological Characterization
All rheological characterization was performed with a Rheometrics Mechanical
Spectrometer Model 800 (RMS-800). The instrument test geometry was a 25 mm
diameter cone and plate geometry with a 0.1 radian cone angle. All experiments were
performed in the presence of an inert nitrogen atmosphere to prevent thermo-oxidative
degradation. Test specimens were prepared by compression molding performs at 180°C
under nominal pressure and allowing them to slowly cool without applied pressure. This
method produces homogeneous samples with minimal residual stress. Reported
rheological results represent the average of at least three runs using different samples for
each run. Stress growth upon inception of steady shear flow experiments were conducted
to obtain single mode UCM model parameters. These experiments were performed at
shear rates in the zero shear viscosity limit at 180°C (steady values from 0.01 to 0.1 sec-1
are shown in Fig. 2). Small amplitude dynamic oscillatory shear measurements were
performed to determine parameters for the multimode UCM model. The tests
encompassed frequencies from 0.1 to 100 rad/sec at 180, 220, and 260°C. The
experiments performed at 220 and 260°C were then shifted to the coalescence
temperature (180°C, the master curves are shown in Fig. 4.2) by applying time-
temperature superposition to include frequencies lower than could be obtained by direct
measurement. The calculated errors for the stress growth and dynamic oscillatory
measurements were found to be ±10 and ±5%, respectively.
4 The Role of Transient Rheology in Polymeric Coalescence
148
0.01 0.1 1 10 100
100
1000
10000
100000
η, |k
| (P
a se
c)
g, ω (sec-1, rad sec-1)
Figure 4.2. Steady and complex shear viscosity master curves for polypropylene at 180°C. ( ) 190k, ( ) 250k, ( ) 340k. The open symbols represent small amplitude oscillatory shear measurements; filled symbols represent steady shear values.
4.3.4 Coalescence
The sample preparation for the coalescence experiments was identical to the
procedure used to generate sessile drops. Two of the pellets were placed vertically,
standing on their cut end, next to each other in the hot stage at 180°C. The hot stage was
capable of achieving a heating rate of 90°C per minute and could maintain the
temperature within 0.1°C, which assisted in providing nearly isothermal conditions. An
inert, nitrogen atmosphere was used to help eliminate thermo-oxidative degradation
throughout the experiment. Upon heating, the cylinders adopted a spherical shape before
coalescence began. When observed from above with the microscope, the system
geometry is identical to that shown in Fig. 4.1. The entire coalescence process was
recorded on high resolution video. Still images were extracted at prescribed intervals,
4 The Role of Transient Rheology in Polymeric Coalescence
149
and the neck and particle radii were measured using Scion Image, a digital image analysis
software available from Scion Corporation. Coalescence experiments were performed
both with and without a lubricant, in order to explore the importance of surface contact
with the glass slide, but no difference was observed. Each coalescence experiment was
conducted three times to ensure reproducibility, and the reported neck radius versus time
data is the average of the three runs. The standard deviation in the dimensionless neck
radius between the repeated runs was 0.01. A representative example of the recorded
images is shown in Fig. 4.3, where there is initially a finite contact area and the neck
radius increases with time until the two drops converge. In the example, the two particles
nearly reach a dimensionless neck radius of 1 within thirty seconds. Although the test is
stopped a few seconds later because the magnitude of the change in the dimensionless
neck radius becomes comparable to the magnitude of the error in the measurement, the
two particles do appear to continue to coalesce towards a single, nearly spherical drop.
Figure 4.3. Optical micrographs from the coalescence of 190k polypropylene drops
4 The Role of Transient Rheology in Polymeric Coalescence
150
4.4 Numerical Methods
4.4.1 Model Parameter Fitting
The single mode UCM model parameters were obtained from stress growth data
by minimizing the sum of the squared difference between the predicted and experimental
transient viscosity. To represent the transient response, fitting was performed at 0.5
second intervals from inception of flow until steady state was achieved. The single mode
UCM model fits to data at a shear rate of 0.01 sec-1 are shown in Fig. 4.4, where it is
shown that the model fits well at long times but not as well at short times. A summary of
the coalescence model parameters and calculated Deborah numbers (as defined for
coalescence,ooa
DeηλΓ= ) is provided in Table 2.
4 The Role of Transient Rheology in Polymeric Coalescence
151
0 2 4 6 8 10 12 14
1000
10000
h+ (Pa
sec)
t (sec)
Figure 4.4. Single mode UCM model fit to the transient shear viscosity from stress growth experiments at 180°C. The symbols represent the experimental data: ( ) 190k, ( ) 250k, ( ) 340k. The lines represent the single mode UCM fits to the data.
Table 4.2. Single Mode UCM Coalescence Model Parameters and Calculated Values for the Deborah Number at 180°C.
Multimode UCM model parameters were obtained by simultaneously fitting Eqs.
16 and 17, using a nonlinear regression method to storage (G') and loss (G") modulus
data as outlined in Bird et al. [5]:
( ) ( )∑= +
=′N
k kj
jkkjG
12
2
1 λωωλη
ω (16)
4 The Role of Transient Rheology in Polymeric Coalescence
152
( ) ( )∑= +
=′′N
k kj
jkjG
121 λω
ωηω (17)
where ω is frequency and N is the number of modes. The optimum number of modes for
each sample was determined by adding modes until there was less than 10% reduction in
the error between the data and model. The multimode model fit for the 340k sample is
shown in Fig. 4.5 and is representative of the fits to the other samples. The multimode
UCM model parameters are summarized in Table 4.3.
0.1 1 10 1000.1
1
10
100
1000
10000
G'/
G" (
Pa)
ω (rad sec-1)
Figure 4.5. Multimode fit to the storage and loss moduli for the 340k sample at 180°C. ( ) G', ( ) G". The symbols represent the experimental data; the lines represent the multimode UCM fit to the data.
4 The Role of Transient Rheology in Polymeric Coalescence
153
Table 4.3. Multimode Upper Convected Maxwell Model Parameters at 180°C. k 1 2 3 4 5 6
To further evaluate the importance of accurately representing the transient
behavior, the energy based UCM model was reformulated without the steady state
assumption. The generalized upper convected Maxwell constitutive model is given by:
Dvττvτvττ kkkT
kkkk tηλ 2=
∇⋅−⋅∇−∇⋅+∂∂+ (18)
where the subscript k signifies the mode or the kth partial stress component of the total
stress, and the total stress is the sum of each of the modes. Assuming that there is a
homogeneous flow field, Eq. 18 is reduced to:
Dvττvττ kkkT
kkk tηλ 2=
∇⋅−⋅∇−∂∂+ (19)
for a single mode (k=1) and differs from Eq. 13 because the unsteady state stresses, t∂
∂τ ,
are included. This change increases the complexity of the problem because the
components of the extra stress tensor are not only differential equations, as shown in Eqs.
20 and 21, but each contains the extension rate, which is also defined as a differential
equation in Eq. 9:
−+=
λετ
λεητ 122
&&
xxxx
dtd
(20)
4 The Role of Transient Rheology in Polymeric Coalescence
154
++−=
λετ
λεητ 144
&&
yyyy
dtd
(21)
xxzz ττ = (22)
This requires the development of an alternative numerical approach to what was applied
to the steady state model. By assuming that the extension rate is constant over a given
time step, which is valid if discretization is sufficiently refined, and the initial condition
that the stresses are zero at t = 0, the differential equations for stress may be solved
analytically to give:
( )( )
( ) ( )
−
−+−
=−
−
−
−
−−
ελεητ
ελεητ
ελ
ελ
ελ
&
&
&
& &&&
212
212
1
1212121
n
n
xx
ttt
xx eee (23)
( )( )
( ) ( )
+
+++−=
−
−
+
+
+−
ελεητ
ελεητ
ελ
ελ
ελ
&
&
&
& &&&
414
414
1
1414141
n
n
yy
ttt
yy eee (24)
These equations for stress (Eqs. 23 and 24) may then be substituted into the energy
balance, Eq. 1, and integrated to give the following equation:
( ) ( ) ( ) 01sincos
cos2cos132 3
53
41
32
=−−+−Γ θθ
θθττ yyxxo Ka
(25)
Eqs. 9, 10, 23, 24, and 25 may be solved by determining the root, dtdθ , at a given time
and θ by using Müller’s method [18]. Convergence was achieved when there was less
than 1% difference in each predicted value of dimensionless coalescence between
successive reductions of the time step, dt. The accuracy of the numerical scheme was
evaluated by applying it to the steady state model and comparing the results to those
produced by the numerical integration of Eq. 15.
4 The Role of Transient Rheology in Polymeric Coalescence
155
4.5 Results and Discussion
4.5.1 Newtonian and Steady State UCM Coalescence Models
The results from the coalescence experiments are shown in Fig. 4.6 along with
predictions from the Newtonian and steady state UCM models using experimentally
measured parameters. The coalescence data are in qualitative agreement with what the
Frenkel model suggests; coalescence rates decrease with increasing viscosity. The
Newtonian model predicts slower coalescence rates than what were experimentally
measured. This result is consistent with the previously mentioned reports on PTFE and
acrylic resins in which it was concluded that the experimental coalescence was faster than
predicted by the Newtonian model [2,16].
4 The Role of Transient Rheology in Polymeric Coalescence
156
0 100 200 300 400 500 6000.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Dim
ensi
onle
ss N
eck
Rad
ius
(x/a
)
t (sec)
Figure 4.6. Experimental polypropylene coalescence data, the Newtonian, and steady state UCM model predictions at 180°C. The symbols represent the experimental data: ( ) 190k, ( ) 250k, ( ) 340k. The lines represent the model predictions: (—) Newtonian model, (----) steady state UCM model. The predictions from the steady state model obscured because they are nearly identical to the Newtonian results.
The inaccuracy of the Newtonian model at times longer than the characteristic
relaxation time, especially for the 190k and 250k samples, has discouraging implications
on the accuracy of viscoelastic model predictions at long times. Error between the
experimental data and the predicted Newtonian behavior at long times will also be
present in the UCM coalescence models because the UCM constitutive model predicts
Newtonian behavior at long times (this assumes that f < fc, where fc is the critical
extension rate at which the UCM model predicts the biaxial extensional viscosity to
approach infinity). Interestingly, the steady state UCM model predictions are essentially
identical to the Newtonian model predictions, overlapping not only at long times but also
at short times as shown in Fig. 4.6.
4 The Role of Transient Rheology in Polymeric Coalescence
157
Following the approach by Bellehumeur et al. [6], the relaxation time was
arbitrarily increased in an attempt to improve the accuracy of the predictions. As
illustrated in Fig. 4.7, the coalescence rates decrease at short times with large changes in
the relaxation time. However, this only decreases the accuracy of the predictions by
slowing coalescence rates relative to the predicted Newtonian limit. The facts that the
steady state UCM model predictions are the same as the Newtonian predictions for the
experimentally measured relaxation times, the magnitude of the relaxation times need to
be relatively large to observe a difference from the Newtonian model, and increasing the
relaxation time decreases accuracy suggest that the behavior predicted by the steady state
model may not be representative of the response of viscoelastic materials.
0 1000.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Dim
ensi
onle
ss N
eck
Rad
ius
(x/a
)
t (sec)
Figure 4.7. 340k polypropylene coalescence data, the Newtonian, and steady state UCM model predictions at short times. The symbols represent the experimental data: ( ) 340k. The lines represent the model predictions: (—) Newtonian, steady state UCM model with (----) λ =1.54 sec, (····) λ =200 sec, (─·─·) λ =400 sec.
4 The Role of Transient Rheology in Polymeric Coalescence
158
The reason that the coalescence rate of the steady state model is predicted to slow
as the relaxation time is increased can be determined by considering the magnitude of the
biaxial extension rates and the implications it has on the nature of the biaxial viscosity.
The evolution of the biaxial extension rates during coalescence, as predicted by the
steady state UCM model for the three evaluated relaxation times, are shown in Fig. 4.8.
The extension rates pass through a maximum then decay monotonically. The UCM
model predicts that the biaxial extensional viscosity approaches infinity at the critical
extension rate, λ
ε21=c& . It is reasonable to suspect that the enhanced viscosity, which
simulates a rate hardening behavior, excessively slows coalescence. The magnitude of
the extension rates for the large relaxation times of 200 and 400 sec do not exceed the
predicted critical extension rates of 0.0025 and 0.00125 sec-1, respectively, but they are
close enough to cause the biaxial viscosity to increase beyond 6ηo, as shown in Fig. 4.9.
The biaxial viscosity predicted for the experimentally measured relaxation time (λ = 1.54
sec) is practically identical to 6ηo and explains the similarity in coalescence rates for the
steady state UCM and Newtonian cases. By increasing the relaxation time, the biaxial
extensional viscosity significantly surpasses 6ηo leading to a decrease in the coalescence
rate relative to the Newtonian model. This explains why the steady state model predicts a
decrease in the coalescence rate as the relaxation time is increased.
4 The Role of Transient Rheology in Polymeric Coalescence
159
0 100 200 300 400 500 6000.0000
0.0005
0.0010
0.0015
0.0020
0.0025
f (s
ec-1)
t (sec)
Figure 4.8. Biaxial extension rate as predicted by the steady state UCM model during coalescence of the 340k sample where: (—) λ =1.54 sec, (----) λ =200 sec, (····) λ =400 sec.
4 The Role of Transient Rheology in Polymeric Coalescence
160
0 100 200 300 400 500 600
100000
h b (Pa
sec)
t (sec)
Figure 4.9. Biaxial extensional viscosity as predicted by the steady state UCM model during coalescence of the 340k sample where: (—) represents 6ηo that is predicted for Newtonian fluids, and (----) for λ =1.54 sec, (····) λ =200 sec, (─·─·) λ =400 sec.
4.5.2 Single Mode Transient UCM Model
Predictions were generated with the single mode transient model to ascertain the
role of the transient representation of material functions in accurately predicting the
coalescence rates of polymeric materials. Predictions for the 340k are presented with
data in Fig. 4.10. In addition to the Newtonian model predictions, which are included for
reference, the transient model predictions were produced with the experimentally
measured relaxation time and also, as before, two larger values. The transient model
predictions do not decrease the error that was observed at long times because the UCM
model predicts Newtonian behavior at long times as previously discussed. However, it
does appear that the transient model improves the accuracy at shorter times.
4 The Role of Transient Rheology in Polymeric Coalescence
161
0 25 50 75 200 400 6000.0
0.1
0.2
0.3
0.4
0.5
0.6
0.8
1.0
Dim
ensi
onle
ss N
eck
Rad
ius
(x/a
)
t (sec)
Figure 4.10. 340k polypropylene coalescence data, the Newtonian, and transient UCM model predictions. The symbols represent the experimental data: ( ) 340k. The lines represent the model predictions: (—) Newtonian, transient UCM model with (----) λ =1.54 sec, (····) λ =2.0 sec, (─·─·) λ =3.0 sec.
Focusing on the shorter times where viscoelastic influence is more relevant, it
becomes apparent that the transient UCM model predicts the reverse effect of relaxation
time on coalescence when compared to the steady state model. By using values for
relaxation time that are greater than the experimentally measured value, the coalescence
rate increases relative to the Newtonian model. This is in qualitative agreement with
what has been reported for the PTFE and acrylic studies where it was reported that
coalescence rates increase with increasing relaxation time [2,16]. The transient model
improves the accuracy of the coalescence predictions for these materials, producing
quantitative agreement within experimental error at short times. The transient model also
appears to be more sensitive than the steady state model to variations in the relaxation
4 The Role of Transient Rheology in Polymeric Coalescence
162
time and does not require unrealistically large values to affect the coalescence rate.
Furthermore, the biaxial extensional viscosity approaches but does not exceed 6ηo, thus
eliminating the rate hardening behavior predicted by the steady state model.
4.5.3 Multimode Transient UCM Model
The previous results, shown in Fig. 4.10, were based on calculations using a
single relaxation time. The multimode UCM model was used next to further examine the
importance of accurately representing the transient viscosity during coalescence. The
first step is to establish the significance of using a relaxation spectrum by evaluating the
shear stress growth predictions with model parameters obtained from the small amplitude
dynamic oscillatory shear data. These predictions are compared to the single mode fits
and experimental data in Fig. 4.11. The multimode model appears to be more accurate
than the single mode UCM model fits to the stress growth data, especially at short times,
and for the highest molecular weight sample. In general, it is expected that the
magnitude of the difference in predicted stress growth behavior between the single mode
and multimode will increase in materials possessing greater relaxation times or broader
relaxation spectra and, therefore, will become more important in the coalescence
predictions for those cases.
4 The Role of Transient Rheology in Polymeric Coalescence
163
0 2 4 6 8 10 12 14
1000
10000
h+ (Pa
sec)
t (sec)
Figure 4.11. Transient shear viscosity at 180°C, the single mode UCM model fits, and the multimode UCM model predictions. The symbols represent the experimental data: ( ) 190k, ( ) 250k, ( ) 340k. The lines represent the models: (—) single mode fits, (····) multimode predictions.
Now that the significance of using a relaxation spectrum has been identified, the
multimode coalescence model predictions are examined. The prediction for the 340k
sample, which is representative of what was observed in all samples, is shown along with
the Newtonian and single mode transient UCM results in Fig. 4.14. At short times (less
than 50 seconds) the multimode model exhibits a slight decrease in the coalescence rate
relative to the single mode model at short times. This reduction in coalescence rate
seems reasonable when we consider the single mode fit and multimode prediction of the
transient viscosity that was shown in Fig. 4.13. The multimode model predicts a higher
viscosity than the single mode at times less than approximately 2 seconds, which should
slow coalescence relative to the single mode. It is interesting that this seemingly minor
4 The Role of Transient Rheology in Polymeric Coalescence
164
difference at short times has such a dramatic effect on the magnitude of the coalescence
rate over much longer times. This emphasizes that slight differences in the representation
of the transient viscosity has a great influence on the early stages of coalescence and that
the predictions of the multimode model will be more significant with increasing
viscoelasticity (higher Deborah numbers).
0 1000.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Dim
ensi
onle
ss N
eck
Rad
ius
(x/a
)
t (sec)
Figure 4.12. 340k polypropylene coalescence data, the Newtonian, the single mode transient UCM model, and the multimode transient UCM model predictions at short times. The symbols represent the experimental data: ( ) 340k. The lines represent the model predictions: (—) Newtonian, (----) single mode transient UCM model, (····) multimode transient model.
4.6 Conclusions
4 The Role of Transient Rheology in Polymeric Coalescence
165
Three energy balance derived coalescence models, one with a Newtonian
constitutive equation while the other two utilized two different forms of the upper
convected Maxwell constitutive model, were evaluated to determine their ability to
predict coalescence rates of three polypropylenes with experimentally measured model
parameters. Rheological characterization of the selected polypropylenes showed that the
measured relaxation times were short relative to the coalescence times, and the Deborah
numbers were small. In spite of this, the experimental data indicated faster coalescence
rates than were predicted by the Newonian model. This suggests that modeling the
coalescence of viscoelastic materials requires the inclusion of viscoelasticity to accurately
predict coalescence rates, even for materials with small Deborah numbers. Predictions
from the steady state UCM coalescence model using the experimentally measured
parameters did not produce a significant change in the predicted coalescence rates. Upon
varying the relaxation time it was found that the model was insensitive to the magnitude
of the relaxation time and required unrealistically large values to observe a change in
behavior. The change that was observed did not improve the accuracy of the predictions
because increasing relaxation time slowed coalescence. It was determined that the source
of this behavior was that the UCM constitutive model predicts infinite viscosity at a
critical extension rate. By increasing the relaxation time the critical extension rate is
effectively lowered, causing the viscosity to increase beyond 6ηo and coalescence to slow
relative to the Newtonian case.
The viscoelastic coalescence model was solved without the steady state
approximation. This transient model was evaluated with a single and multiple modes.
4 The Role of Transient Rheology in Polymeric Coalescence
166
The single mode transient coalescence model demonstrated that the influence of
viscoelasticity is limited to short times and at long times it converges with the Newtonian
solution. This formulation was able to improve the model accuracy at short times by
predicting an increase in coalescence rate with an increase in relaxation time and in doing
so illustrates the importance of representing the transient viscosity. The model was also
more sensitive to the magnitude of the relaxation time, producing significant changes in
coalescence rates with only small changes in the magnitude of the relaxation time. While
the multimode transient UCM model more accurately represented the transient
rheological response it was unable improve the accuracy of the model predictions for
these materials but will likely be more important for fluids possessing a broader
relaxation spectrum.
4.7 Acknowledgements
This work was financially supported by a phase II SBIR grant from NASA, grant
number NAS-2S-4018-285, managed by Luna Innovations.
4 The Role of Transient Rheology in Polymeric Coalescence
167
4.8 References
1 K.L. Johnson, K. Kendall, and A.D. Roberts, Surface Energy and the Contact of
Elastic Solids, Proceedings of the Royal Society of London. Series A, 324, 1558
(1971) 301.
2 S. Mazur, Coalescence of Polymer Particles, in M. Narkis and N. Rosenzweig
(Eds.), Polymer Powder Technology, John Wiley & Sons, New York, 1995,
Chapter 8.
3 J. Frenkel, Viscous Flow of Crystalline Bodies Under the Action of Surface
Tension, Journal of Physics, (Moscow), 9, 5 (1945) 385.
4 J.D. Eshelby, Discussion in Paper by A.J. Shaler, Seminar on the Kinetics of
Sintering, Transactions of AIME, 185, 11 (1949) 806.
5 R.B. Bird, R.C. Armstrong, and O. Hassager, Dynamics of Polymeric Liquids,
Vol. 1., second edition, John Wiley & Sons, New York, 1987, Chapter 5.
6 C.T. Bellehumeur, M. Kontopoulou, and J. Vlachopoulos, The Role of
Viscoelasticity in Polymer Sintering, Rheologica Acta, 37, 3 (1998) 270.
7 O. Pokluda, C.T. Bellehumeur, and J. Vlachopoulos, Modification of Frenkel’s
Model for Sintering, AICHE Journal, 43, 12 (1997) 3253.
8 A. Jagota, P.R. Dawson, and J.T. Jenkins, An Anisotropic Continuum Model for
the Sintering and Compaction of Powder Packings, Mechanics of Materials, 7, 3
(1988) 255.
9 R.S. Garabedian and J.J. Helble, A Model for the Viscous Coalescence of
Amorphous Particles, Journal of Colloid and Interface Science, 234 (2001) 248.
4 The Role of Transient Rheology in Polymeric Coalescence
168
10 A. Jagota and P.R. Dawson, Micromechanical Modeling of Powder Compacts – I.
Unit Problems for Sintering and Traction Induced Deformation, Acta
Metallurgica, 36, 9 (1988) 2551.
11 J.I. Martinez-Herrera and J.J. Derby, Viscous Sintering of Spherical Particles via
Finite Element Analysis, Journal of the American Ceramic Society, 78, 3 (1995)
645.
12 H. Zhou and J.J. Derby, Three-Dimensional Finite-Element Analysis of Viscous
Sintering, Journal of the American Ceramic Society, 81, 3 (1998) 533.
13 A. Jagota, K.R. Mikeska, and R.K. Bordia, Isotropic Constitutive Model for
Sintering Particle Packings, Journal of the American Ceramic Society, 73, 8
(1990) 2266.
14 P.J. Doerpinghaus and D.G. Baird, Pressure Profiles along an Abrupt 4:1 Planar
Contraction, AICHE Journal, 49, 10 (2003) 2487.
15 R. Hooper, C.W. Macosko, and J.J. Derby, Assessing Flow-Based Finite Element
Model for the Sintering of Viscoelastic Particles, Chemical Engineering Science,
55 (2000) 5733.
16 S. Mazur and D.J. Plazek, Viscoelastic Effects in the Coalescence of Polymer
Particles, Progress in Organic Coatings, 24 (1994) 225.
17 J.F. Padday, in Matijević, E. (Ed.), Surface and Colloid Science, Vol.1, Wiley
Interscience, New York, 1969, Part II., 104.
18 IMSL Fortran Subroutines for Mathematical Applications, Math/Library, Vol. 1.,
Visual Numerics, 1997, Chapter 7, 846.
5 The Role of Transient Rheology in the Coalescence of Thermotropic Liquid Crystalline Polymers
169
5 The Role of Transient Rheology in the Coalescence of Thermotropic
Liquid Crystalline Polymers
Preface
The work presented in this chapter addresses the first objective of the research.
Specifically, the coalescence of TLCPs is investigated to determine if their behavior may
be explained solely by their viscoelastic character. This chapter is organized as a
manuscript for future publication.
5 The Role of Transient Rheology in the Coalescence of Thermotropic Liquid Crystalline Polymers
170
The Role of Transient Rheology in the Coalescence of Thermotropic Liquid Crystalline Polymers
Eric Scribben and Donald Baird
Department of Chemical Engineering , Virginia Polytechnic Institute and State University, Blacksburg, Va 24061
5.1 Abstract
Polymeric coalescence is the process by which surface tension drives two small
drops to merge into a single uniform, homogeneous drop. In this work a coalescence
model, which equates the work of surface tension to the work done by viscous forces
while assuming biaxial extensional flow kinematics, is evaluated to determine its ability
to accurately predict the coalescence rates of two TLCPs. Results from two variations of
the model (one used a Newtonian constitutive model and the other used the upper
convected Maxwell (UCM) constitutive model with the assumption of steady state stress
behavior) that had previously been reported were explained according to their predicted
viscosity behavior. The analysis suggested the importance of using the transient viscosity
in the model. The model employing unsteady stresses that was described elsewhere was
evaluated to determine its ability to accurately predict coalescence rates when using
experimentally quantified coalescence model parameters. This model represents a
qualitative improvement to the previous models because it predicted that coalescence
would occur at a higher rate than predicted by the Newtonian model, which was observed
experimentally. However, the model was unable to quantitatively predict the
experimental coalescence rates, as it over predicted the acceleration of coalescence that
5 The Role of Transient Rheology in the Coalescence of Thermotropic Liquid Crystalline Polymers
171
arises due to the transient viscosity. The origin of the disagreement is not known, but
could be due to the use of a constitutive relation designed for flexible chain polymers.
5.2 Introduction
Polymer coalescence is the basis of a number of polymer processing operations
such as: the fabrication of particulate performs, cold compression molding, dispersion
coating, powder coating, rotational molding, and selective laser sintering [1].
Coalescence refers to the process where, in an attempt to minimize surface area, surface
tension drives a collection of fluid drops to merge into a single, homogeneous body.
When drops are brought in contact, they instantaneously deform to create finite contact
surface [2]. This is also referred to as neck formation because it describes the formation
of a bridge at the interface between drops. Coalescence continues with the radial growth
of the neck radius, x, as is shown by the schematic in Fig. 5.1, where the drops are
represented by identical spheres.
5 The Role of Transient Rheology in the Coalescence of Thermotropic Liquid Crystalline Polymers
172
xx
Figure 5.1. Schematic of the geometric evolution of two coalescing spherical particles.
Thermotropic liquid crystalline polymers (TLCPs) exhibit a number of
mechanical and physical properties that are desirable for products manufactured by
techniques involving coalescence. TLCPs have demonstrated exceptionally high values
of tensile strength and modulus (strengths in excess of 1000 MPa and moduli near 100
GPa from fiber spinning and approaching 200 MPa and 20 GPa, respectively, in injection
molding) [3]. They retain their mechanical properties over a wide range of temperatures.
They have permeability coefficients for gases such as: He, H2, Ar, N2, CO2 that are
comparable to or less than those for polyacrylonitrile (PAN), one of the least permeable
polymers known [4]. They exhibit excellent resistance to acidic or basic environments
and a wide range of organic solvents [5]. Finally, they have low coefficients of linear
thermal expansion (CLTE), which are less than 1 cm/cm/°Cx10-5 [3].
5 The Role of Transient Rheology in the Coalescence of Thermotropic Liquid Crystalline Polymers
173
In our initial work, the coalescence behavior of two commercial TLCPs was
reported [6]. It was observed that coalescence was possible if the material exhibits a
well-defined zero shear viscosity. It was also demonstrated that TLCP coalescence rates
could not be predicted by either a Newtonian or a viscoelastic coalescence model. The
viscoelastic coalescence model incorporated viscoelastic effects by using the upper
convected Maxwell (UCM) constitutive equation to describe the extra stress tensor and is
referred to as steady state because it was assumed that the stresses, at any instant, were at
steady state. The reader is referred to the original papers for details on the complete
derivation of the Newtonian and the steady state UCM coalescence models [7,6]. The
cause for the inability of the models to accurately predict the coalescence rates of the
TLCPs was not known, but it was suggested that it could be due to the fact that the
transient viscosity was not represented in either of the evaluated coalescence models and
the time required for the viscosity to reach steady state was the same order of magnitude
as the time required for coalescence [6].
The UCM coalescence model was evaluated for several polypropylenes without
the steady state assumption to determine the effect that transient viscosity had on
predicting coalescence rates [9]. This model is hereafter referred to as the transient UCM
coalescence model because of its transient description of the rheological stresses. As
with both the Newtonian and steady state UCM coalescence models, the transient UCM
model predicted that the coalescence rate was proportional to the surface tension an
inversely proportional to viscosity. However, it was demonstrated the transient UCM
coalescence model was more sensitive to relaxation time than the steady state formulation
5 The Role of Transient Rheology in the Coalescence of Thermotropic Liquid Crystalline Polymers
174
and, in contrast to the steady state UCM model, the coalescence rate was accelerated by
increasing the relaxation time. The model also quantitatively predicted the coalescence
rates of the selected polypropylenes, using measured relaxation times, at short times
where the transient biaxial extensional viscosity had not reached steady state.
The objective of this work is to determine if the transient UCM coalescence
model can accurately predict coalescence rate of coalescing TLCPs. To accomplish this
objective, two TLCPs with markedly different rheological behavior are used. The
Newtonian and steady state UCM coalescence models are analyzed with respect to the
coalescence data to provide an explanation for their inablity to predict the measured
coalescence data. The UCM constitutive model is fit to transient shear viscosity data to
obtain UCM model parameters. Finally, predictions from the Newtonian, steady state
UCM, and the transient UCM coalescence models are compared to experimental data to
determine if the transient UCM model can improve the accuracy of the predictions.
5.3 Experimental
5.3.1 Materials
Two nematic TLCPs, Vectra A 950 and Vectra B 950, available from Ticona
(Summit, NJ), were selected for this evaluation. Both materials are randomly
copolymerized wholly aromatic copolyesters: Vectra A is composed of hydroxybenzoic
acid and hydroxynaphthoic acid, and Vectra B is composed of hydroxynaphthoic acid,
5 The Role of Transient Rheology in the Coalescence of Thermotropic Liquid Crystalline Polymers
175
terephthalic acid, and aminophenol. The chemical structure and mole fraction of each
monomer are shown in Fig. 5.2. The melt temperature, as determined by the peak value
from differential scanning calorimetry (DSC), is 279°C [10]. When the melt is cooled
quiescently, the crystallization temperature is 226°C, as defined by the beginning of the
exothermic crystallization peak measured by DSC. However, the resin may crystallize in
shear flow at higher temperatures, up to approximately 300°C [10]. The glass transition
temperature, as measured by the peak value from dynamic mechanical thermal analysis
(DMTA), is 147°C [11]. The nematic to isotropic transition temperature is unknown
because the degradation occurs at temperatures below the transition. The weight average
molecular weight and polydispersity index are thought to be around 30,000 and 2 [12].
Both materials were dried in accordance of manufacturer specifications, in vacuum oven
at 150°C for between 12 and 24 hours, before the measurement of surface tension,
rheological characterization, or coalescence experiments.
C O
O
p - Hydroxybenzoic Acid
C
O
O
Hydroxynaphthoic Acid 0.73 0.27
C O
O
p - Hydroxybenzoic Acid
C O
O
p - Hydroxybenzoic Acid
C O
O
C O
OO
p - Hydroxybenzoic Acid
C
O
O
Hydroxynaphthoic Acid
C
O
O
Hydroxynaphthoic Acid
C
O
OC
O
C
OO
OO
Hydroxynaphthoic Acid 0.73 0.27
Vectra A 950
C
O
O
Hydroxynaphthoic Acid
N
H
O
p - Aminophenol
C C
O O
Terephthalic Acid
0.6 0.2 0.2
C
O
O
Hydroxynaphthoic Acid
C
O
O
Hydroxynaphthoic Acid
C
O
OC
O
C
OO
OO
Hydroxynaphthoic Acid
N
H
O
p - Aminophenol
N
H
O
p - Aminophenol
N
H
ON
H
N
H
OO
p - Aminophenol
C C
O O
Terephthalic Acid
C C
O O
Terephthalic Acid
C C
O O
C C
OO OO
Terephthalic Acid
0.6 0.2 0.2
Vectra B 950
Figure 5.2. Chemical structure and composition of Vectra A 950 and Vectra B 950.
5 The Role of Transient Rheology in the Coalescence of Thermotropic Liquid Crystalline Polymers
176
5.3.2 Differential Scanning Calorimetry
Differential scanning calorimetry (DSC) was used to identify a potential
minimum temperature for the coalescence experiments by measuring the end of the melt
transition. The thermal analysis was performed with a heating rate of 10°C/ minute with
a Seiko Instruments SSC/5200 series auto cooling DSC-220C. The sample was exposed
to both a heating and a cooling cycle before the recorded measurement to ensure the
material had been properly dried and to impose a known thermal history. 320 and 330°C
were selected for the coalescence experiments because they were at least 10 to 15°
greater than the end of the melt transitions.
5.3.3 Surface Tension Measurement
The surface tension of each of the materials was determined by fitting the
Bashforth and Adams equation to the sessile drop profile of the molten polymer in an
inert atmosphere at 320 and 330°C [17]. This method was selected because it presents a
noninvasive means of measuring the surface tension of the TLCP as a melt with the
identical geometry, thermal history, and deformation history of the particles used in the
coalescence experiments. A single, 500 µm diameter sphere, identical to those used in
the coalescence experiments, was placed on a glass slide in the hot stage, where it was
melted into a sessile drop. A description of the procedure used to generate the spherical
particles is provided elsewhere [14]. The sample was quenched and the glass slide was
rotated to allow a profile view of the drop from above. The sample was reheated to the
test temperature and a digital image of the profile was recorded by an optical microscope
equipped with a miniDV camcorder. The Bashforth and Adams equation was fit to data
5 The Role of Transient Rheology in the Coalescence of Thermotropic Liquid Crystalline Polymers
177
points representing the profile shape that were extracted from the digital image of the
profile with Scion Image, an image analysis software available from Scion Corporation.
The accuracy of this technique (0.1%) demands that the particle radii must be small so
that gravitational forces cannot influence the shape of the profile. The absence of
gravitational forces was verified by calculating the Bond number Γ
= grBo2ρ (0.027 for
Vectra A 950 and 0.030 for Vectra B 950) and was supported by the observation that the
profile shape did not change upon rotating the glass slide.
5.3.4 Rheological Characterization
All rheological characterization was performed with a Rheometrics Mechanical
Spectrometer Model 800 (RMS-800). The instrument test geometry was a 25 mm
diameter cone and plate with a 0.1 radian cone angle. The magnitude of the complex
viscosity, |η*|, versus angular frequency, ω, and shear viscosity, η, versus shear rate, g,
data were measured in the presence of an inert nitrogen atmosphere to prevent thermo-
oxidative degradation. Test specimens were prepared by compression molding preforms
at 320°C under nominal pressure and allowing them to quiescently cool without applied
pressure. This method produces homogeneous samples with minimal residual stress that
were the dimensions desired for the test geometry. Reported rheological results represent
the average of at least three runs using different samples for each run. Small amplitude
dynamic oscillatory shear measurements were performed for angular frequencies between
0.1 to 100 rad/sec at 10% strain for both 320 and 330°C. The steady shear viscosity was
measured at low shear rates by recording the steady state value of a stress growth upon
5 The Role of Transient Rheology in the Coalescence of Thermotropic Liquid Crystalline Polymers
178
inception of steady shear flow experiment. The stress growth experiments were
performed because of the lengthy times required to obtain data at low angular frequencies
from dynamic oscillatory measurements. Transient viscosity values at shear rates less
than 0.01 sec-1 were thought to be the most pertinent to the coalescence process and were
used to obtain parameters for the UCM constitutive model by minimizing the sum of the
squared difference between the predicted and experimental viscosity values. To represent
the transient response, fitting was performed at 0.5 second intervals from inception of
flow until steady state was achieved.
Two procedures were used during rheological characterization because the
measured rheological response of TLCPs can be strongly dependent upon thermal and
deformation history. The first procedure was devised to introduce reproducible shear and
thermal histories to minimize variation in rheological data and was used for both the
stress growth and the dynamic oscillatory measurements. The cone and plate preform
was placed in the rheometer and the cone was brought to 0.05 mm from the plate while
the sample was heated to 340°C. Once the temperature reached 340°C, a steady shear
deformation was applied at a shear rate of 0.1 sec-1 for 10 seconds. After the preshear
was complete the sample was cooled to the test temperature where it was given five
minutes to reach a stress free state before beginning the test. The 340°C preheat
temperature was selected because the sample would not relax to a stress-free state within
the allotted time if lower temperatures were used. The second procedure was designed to
mimic the thermal and deformation histories of the samples used in the coalescence
experiments. This procedure was used only for repeating the stress growth experiments.
5 The Role of Transient Rheology in the Coalescence of Thermotropic Liquid Crystalline Polymers
179
The cone and plate perform was placed in the rheometer and the cone was brought to
0.05 mm from the plate while the sample was heated directly to the test temperature.
Once the set point was reached, the test was performed.
5.3.5 Coalescence Experiments
Coalescence measurements were conducted to determine the coalescence rates for
the TLCP particles at two temperatures, 320 and 330°C. Two spherical particles with a
diameter of 500 µm were placed in contact inside a Linkam THM 600 hot stage set at one
of two operating temperatures, as identified by the DSC and shear viscosity
measurements. The tests were performed in an inert, nitrogen atmosphere to assist in
eliminating thermo-oxidative degradation during the experiment. The heating rate for the
coalescence experiments was 90°C per minute and the test temperature was maintained at
the set point to within 0.1°C, which provided nearly isothermal conditions. The
coalescence process was observed in the hot stage with a Zeiss Axioskop equipped with a
color CCD camera. The video feed was recorded to high resolution digital video. The
coalescence between the two particles was identical to that shown in Fig. 5.1. Still
images from the digital video were extracted at prescribed intervals, and the neck and
particle radii were measured using Scion Image, a digital image analysis software
available from Scion Corporation. Each coalescence experiment was conducted three
times to ensure reproducibility, and the reported neck radius versus time data is the
average of the three runs.
5 The Role of Transient Rheology in the Coalescence of Thermotropic Liquid Crystalline Polymers
180
5.4 Results and Discussion
5.4.1 Rheological Characterization
The results from steady shear and small amplitude dynamic oscillatory
measurements are shown in Figures 5.3 and 5.4. Vectra A 950 at 320°C did not exhibit a
zero shear viscosity over the measured range of deformation rates tested. The measured
steady shear viscosity at low deformation rates exhibited yield-like flow behavior. A
well defined zero shear viscosity was observed for Vectra B 950 at 320°C for shear rates
below 1×10-2 sec-1 and the magnitude of the viscosity was approximately an order of
magnitude less than was measured for Vectra A 950. At 330°C, both materials exhibited
a zero shear viscosity at shear rates below approximately 1×10-2 sec-1. This represented a
dramatic change in the shape of the shear flow curve for Vectra A 950 relative to the
measurement at 320°C. At 330°C, the magnitude of the zero shear viscosity of Vectra A
950 was approximately an order of magnitude greater than the zero shear viscosity of
Vectra B 950.
5 The Role of Transient Rheology in the Coalescence of Thermotropic Liquid Crystalline Polymers
181
10-3 10-2 10-1 100 101 102100
101
102
103
104
h, |k
| (Pa
sec)
g, w (sec-1), (rad sec-1)
Figure 5.3. Steady and complex shear viscosity master curves for Vectra A 950 ( ) and Vectra B 950 ( ) at 320°C. The open symbols represent small amplitude oscillatory shear measurements, filled symbols represent steady shear values.
5 The Role of Transient Rheology in the Coalescence of Thermotropic Liquid Crystalline Polymers
182
10-3 10-2 10-1 100 101 102100
101
102
103
104
h, |k
| (Pa
sec)
g, w (sec-1), (rad sec-1)
Figure 5.4. Steady and complex shear viscosity master curves for Vectra A 950 ( ) and Vectra B 950 ( ) at 330°C. The open symbols represent small amplitude oscillatory shear measurements, filled symbols represent steady shear values.
The transient shear viscosity, of Vectra A 950 at 330°C and Vectra B 950 at both
temperatures, was within experimental error for the two test procedures. A representative
example of this is shown by the data for Vectra A 950 in Fig. 5.5. This result illustrated
that thermal and shear histories had no affect on the transient shear viscosity in the limit
of low shear rates at the test temperatures used for the coalescence experiments.
5 The Role of Transient Rheology in the Coalescence of Thermotropic Liquid Crystalline Polymers
183
0 200 400 600 800 1000
1000
η+ (Pa
sec)
Time (sec)
Figure 5.5. Comparison of the transient shear viscosity for Vectra A 950 measured at a shear rate of 1×10-2 sec-1 for the two different thermal and deformation histories. The data was measured using the prescribed pretest shear and thermal histories: ( ) at 320°C and ( ) at 330°C. The samples measured without the prescribed pretest shear and thermal histories: ( ) at 320°C and ( ) at 330°C. The lines represent the UCM fits to the data.
A representative example of the UCM model fits to the transient viscosity data at
a shear rate of 0.01 sec-1 is shown for Vectra A 950 in Fig. 5.5. The UCM model, despite
slight inaccuracy at short times, could represent the transient shear viscosity data at the
test conditions. A summary of the coalescence model parameters and calculated Deborah
numbers (as defined for coalescence,ooa
DeηλΓ= ) is provided in Table 5.1. Because a
true zero shear viscosity was not observed for Vectra A 950 at 320°C, the value shown
was approximated by fitting to transient shear viscosity at a shear rate of 1×10-2 sec-1.
5 The Role of Transient Rheology in the Coalescence of Thermotropic Liquid Crystalline Polymers
184
Table 5.1. UCM coalescence model parameters and calculated values for the Deborah Number.
A representative example of the micrographs recorded during coalescence is
shown in Fig. 5.6 for Vectra B 950, where there is initially a finite contact area and the
neck radius increases with time until the two drops converge. In the example, the two
particles nearly reached a dimensionless neck radius of 1 within thirty seconds. Although
the test was stopped a few seconds later because the magnitude of the change in the
dimensionless neck radius becomes comparable to the magnitude of the error in the
measurement, the two particles did appear to continue to coalesce towards a single,
nearly spherical drop.
5 The Role of Transient Rheology in the Coalescence of Thermotropic Liquid Crystalline Polymers
185
Figure 5.6. Optical micrographs from the coalescence experiments of Vectra B 950 at 320°C.
Coalescence data is shown in Fig. 5.7 for Vectra A 950 and Vectra B 950 at both
temperatures. The coalescence rate for Vectra A 950 at 320°C was slower than for
Vectra A 950 at 330°C and Vectra B 950 at both temperatures. It was also observed that
Vectra A 950 failed completely coalesce at 320°C, which may be explained by the
presence of the yield-like shear viscosity behavior at low shear rates at that temperature.
The yield-like behavior may be the explained by the presence of residual crystallites.
Wilson et al. [12] showed that residual crystallites were present in the melt at 320°C, but
were completely melted by 330°C for a series of Vectra A 950 thin films. If residual
crystallites were present in Vectra A 950 at 320°C, they may act as physical cross links,
effectively increasing the viscosity and, hence, slowing coalescence.
5 The Role of Transient Rheology in the Coalescence of Thermotropic Liquid Crystalline Polymers
186
0 10 20 300.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Dim
ensi
onle
ss N
eck
Rad
ius
(x/a
)
Time (sec)
Figure 5.7. TLCP coalescence data, where ( ) represents Vectra A 950 and ( ) represents Vectra B 950. Open symbols are used for the data at 320°C and filled symbols represent the data at 330°C.
The qualitative behavior of the coalescence data at the three conditions where
coalescence occurred cannot be explained by the differences in the measured rheological
data according to the relationship defined by the Newtonian coalescence model. The
model indicated that the coalescence rate is proportional to surface tension and inversely
proportional to viscosity. The surface tension of each TLCP was independent of
temperature and was approximately equal for both materials. Therefore, the driving force
for coalescence was equivalent in all cases and the measured differences in the
coalescence rate may be attributed to relative differences in the shear viscosity.
Assuming a constant surface tension, the Newtonian model predicts that the coalescence
rate decreases by increasing the viscosity. The magnitude of the shear viscosity for
Vectra B 950 was slightly greater at 330°C than at 320°C, but, in contrast to the
5 The Role of Transient Rheology in the Coalescence of Thermotropic Liquid Crystalline Polymers
187
Newtonian model predictions, coalescence occurred faster at 330°C. In addition, the zero
shear viscosity of Vectra A 950 at 330°C was approximately an order of magnitude
greater than for Vectra B 950 at either temperature, yet Vectra A 950 coalesced faster.
The steady state UCM coalescence model is also incapable of explaining the
qualitative behavior of the coalescence data. This model was derived by substituting the
UCM constitutive model into the Newtonian derivation to introduce the effects of
viscoelasticity. As a first approximation the steady state assumption was imposed, which
effectively eliminates the time dependence of the biaxial extensional viscosity, as is
shown by Eq. 1.
ελη
εληη
&& 414
212
++
−= oo
b (1)
The same biaxial extensional viscosity is predicted, at small values of λ or f, for
both the Newtonian and the steady state UCM coalescence models, ηb= 6ηo. Under these
conditions, the steady state UCM coalescence model is expected to predict nearly the
same coalescence rate as the Newtonian coalescence model, which was shown could not
qualitatively explain the experimental data. As the biaxial extension rate is increased the
biaxial extensional viscosity passes through a slight minimum before beginning its
increase towards infinity. Increasing viscosity by will only act to further reduce the
coalescence rate relative to the Newtonian model. This effect does not assist in
explaining the relative differences in the coalescence data because, as reported in Table
5 The Role of Transient Rheology in the Coalescence of Thermotropic Liquid Crystalline Polymers
188
5.1, Vectra A 950 at 330°C has a longer relaxation time than those measured for Vectra B
950 and, thus would be predicted to coalesce more slowly.
Unlike the Newtonian and steady state coalescence models, the transient UCM
coalescence model has the capacity to predict the experimental coalescence rates. In the
limit of small λ and f, the transient UCM coalescence model predicts the same behavior
as the Newtonian coalescence model, increasing viscosity slows coalescence. The time
required for the biaxial extensional viscosity to approach the steady state value, 6ηo,
increases as the relaxation time is increased. This transient response, as predicted by Eq.
2, effectively reduces the viscosity at times shorter than the characteristic relaxation time
and reducing the viscosity accelerates coalescence. Depending on the relative
magnitudes of the relaxation times, it is possible for the fluid with the greater steady state
viscosity to coalesce faster.
( ) ( )
( ) ( )
+
+=
−
−=
+
+−−
+
−=
+
−
+
+−
−
−−
ελεητ
ελεητ
εελεη
εεληη
ελ
ελ
ελ
ελ
ελ
ελ
&
&
&
&
&&
&
&&
&&
&&&&
414
212
414
212
41
22
21
11
224141
112121
yy
t
xx
t
tttt
b
eCeC
Cee
Cee
(2)
Although the transient UCM coalescence model has the capacity to correctly
predict qualitative differences in coalescence rates of TLCPs, their rheological behavior
can be more complicated than what is observed for a conventional isotropic viscoelastic
fluid. In some cases the steady shear viscosity is not described by a Newtonian plateau in
the limit of small shear rates and shear thinning at increased shear rates. Instead, the
5 The Role of Transient Rheology in the Coalescence of Thermotropic Liquid Crystalline Polymers
189
measured viscosity exhibits a yield-like behavior that is referred to as a three-region flow
curve, where the magnitude of the steady shear viscosity is related to different stages of
the destruction of polydomain liquid crystalline texture into a single nematic phase by the
imposed shear deformation [15]. Also the rheological stresses can be unique functions of
the deformation rate, strain, time, and may also be strongly dependent on the sample’s
thermal and deformation histories. Furthermore, there have been reports of oscillations
of the shear stresses during start-up of steady shear flow [16]. There have also been
reports of interesting behavior in the normal stress differences, such as negative values at
steady state or local minima before reaching steady state.
Despite the complex behavior exhibited by some TLCPs that suggests a complex
constitutive equation may need to be considered, it is possible that the rheological
behavior of TLCPs during coalescence can be sufficiently described by a more
conventional viscoelastic constitutive model. The dimension of liquid crystalline
domains for the materials evaluated in this study is on the order of 1 µm, while the
particle diameter is approximately 500 µm [17]. The particles were prepared in such a
manner that the liquid crystalline domains were not preferentially oriented, a description
of the process is provided elsewhere [14]. Therefore, without preferential orientation of
the domains, a random distribution of orientation exists and the particle is effectively
isotropic on the 500 µm scale. The magnitudes of the deformation rates and strains are
small during the coalescence of two particles. The deformation rates predicted by the
Newtonian model, using a value for the viscosity that is comparable to the zero shear
viscosity of the TLCPs used in this work, exhibit a maximum value during coalescence
5 The Role of Transient Rheology in the Coalescence of Thermotropic Liquid Crystalline Polymers
190
that is less than 1×10-2 sec-1. Also, the predicted strain for two identical spherical
particles is limited to one strain unit. Possible influence of negative values of the normal
stress differences should not be factor because the flow kinematics for coalescence are
biaxial extension. In addition, the transient biaxial extensional viscosity can be
approximated by the transient shear viscosity, using the relationship, ηb+ = 6ηo
+, which
was verified for these materials for deformation rates (g, f) up to 0.1 sec-1 [18].
5.4.3 Transient UCM Coalescence Model Predictions
Predictions from the transient UCM, Newtonian, and steady state coalescence
models are shown for Vectra A 950 at 330°C in Fig. 5.8. Similar results were observed
for Vectra A 950 and Vectra B 950 at 320°C. As shown in the figure, the Newtonian
coalescence model predictions began to deviate from the experimental data within the
first few seconds and, afterwards, under predicted the coalescence rates. The steady state
UCM coalescence model predicted slower coalescence than was predicted by the
Newtonian model and, in doing so, was less accurate than the Newtonian model. The
transient UCM coalescence model predicted greater coalescence rates than the Newtonian
model, which was in qualitative agreement with the experimental data. Unfortunately,
the model grossly over predicted the coalescence rate, which was nearly instantaneous
and was almost identical for the three experimental conditions. Also, the transient UCM
coalescence model was unable to correctly predict the order of coalescence times for
Vectra A 950 at 330°C and Vectra B at 320 and 330°C. The experimental coalescence
times were tVA330°C < tVB330°C < tVB320°C, as was shown in Fig. 5.7 and the model predicted
tVA330°C < tVB320°C < tVB330°C. Although the predicted results are not shown, the predicted
5 The Role of Transient Rheology in the Coalescence of Thermotropic Liquid Crystalline Polymers
191
order of coalescence times was consistent with the calculated Deborah number, the
samples with larger Deborah numbers coalesced faster.
0 10 20 300.0
0.2
0.4
0.6
0.8
1.0
D
imen
sion
less
Nec
k R
adiu
s (x
/a)
Time (sec)
Figure 5.8. Vectra A 950 coalescence data at 330°C and predictions from the Newtonian, steady state UCM, and transient UCM coalescence models. The symbols represent the experimental data ( ) and the lines represent the coalescence model predictions: Newtonian (—), steady state UCM (----), and the transient UCM (····).
5.5 Conclusions
The coalescence of two TLCPs was studied by comparing experimentally
measured values of the dimensionless neck radius with predicted values from the
Newtonian, steady state, and the transient UCM coalescence models. Several significant
conclusions can be drawn from this work. The first is that the behavior of the shear
viscosity at low shear rates can be used to identify conditions where successful
5 The Role of Transient Rheology in the Coalescence of Thermotropic Liquid Crystalline Polymers
192
coalescence will occur. It was shown that the measured rheological behavior, specifically
the transient viscosity in the limit of small shear rates, can be accurately represented by
the UCM constitutive model. The biaxial extensional viscosities, as predicted by the
Newtonian and steady state coalescence models, were analyzed to explain why those
models were incapable of qualitatively predicting the experimental coalescence data.
Finally, it was shown that the transient UCM coalescence model predicted faster
coalescence than the Newtonian model, which was in qualitative agreement with the
experimental data, but much faster.
5.6 Acknowledgements
This work was financially supported by a phase II SBIR grant from NASA, grant
number NAS-2S-4018-285, managed by Luna Innovations.
5 The Role of Transient Rheology in the Coalescence of Thermotropic Liquid Crystalline Polymers
The dynamic angle of repose of the generated powder was measured. An
example of an image used during this measurement is shown in Fig. 6.7 for sieve number
30. Powder flow occurred in a steady state fashion and the dynamic angle of repose was
found to be relatively constant (32-35°) for all of the sizes over the tested range of
rotation rates (approximately 1 to 10 rpm). Powders commonly used in rotational
6 The Rotational Molding of a Thermotropic Liquid Crystalline Polymer
217
molding have dynamic angles of repose between 25 and 50° [7]. The fact that the
powder had the ability to flow was a tremendous improvement over the particles
generated by cryogenic grinding.
Figure 6.7. Dynamic angle of repose of a sample from sieve number 30.
The evaluation of the powder flow characteristics for cryogenically ground pellets
and the spherical particles produced several important results. The poor performance of
milled TLCP pellets in the selected tests suggested that an alternative approach was
necessary to produce a powder that was acceptable for rotational molding. A technique
was devised that produced spherical particles with a range of particle sizes that are
commonly used in rotational molding. The apparent density was much higher for the
spherical powder than for the milled pellets and steady state powder flow was observed in
a horizontal rotating cylinder. These observations and measurements suggest the
6 The Rotational Molding of a Thermotropic Liquid Crystalline Polymer
218
generated powder should possess acceptable powder flow characteristics for the
rotational molding process.
6.4.2 Coalescence
DSC was used to identify the end of the melt transition, and the measured DSC
thermogram is shown in Fig. 6.8. The peak of the melt transition occurred at 284°C and
the melt transition was complete by 310°C, both are represented as stars in the figure. In
an attempt to ensure that complete melting would occur, 320 and 330°C were selected for
the coalescence experiments.
0 50 100 150 200 250 300 350
260
280
300
320
340
360
380
Endo
ther
m (µ
W/m
g)
Temperature (°C)
Figure 6.8. DSC thermogram of Vectra B 950 with the peak and end of the melt transition represented by stars.
6 The Rotational Molding of a Thermotropic Liquid Crystalline Polymer
219
The surface tension and shear viscosity were measured at 320 and 330°C, to
verify their relative magnitudes and the absence of a three-region viscosity behavior. The
surface tension was measured as 0.029 ± 0.002 J/m2 and was independent of temperature
and the surrounding atmospheric composition. This demonstrates that the driving force
for coalescence is constant for the two temperatures. The shear flow curves at the two
temperatures in nitrogen are shown in Fig. 6.9. As shown in the figure, the material does
not exhibit three-region viscosity behavior at either temperature, but, instead, a well
defined zero shear viscosity that is similar in magnitude at both two temperatures. With
the driving force and the resistance to flow equal at the two temperatures, coalescence
was expected to progress at similar rates.
10-3 10-2 10-1 100 101 102100
101
102
103
104
h, |k
| (Pa
sec)
g, w (sec-1), (rad sec-1)
Figure 6.9. The magnitude of the complex viscosity versus frequency are represented as for 320°C and for 330°C. The shear viscosity versus shear rate is represented as
for 320°C for 330°C, error bars represent deviation in the measurements.
6 The Rotational Molding of a Thermotropic Liquid Crystalline Polymer
220
In addition to the complex and steady shear viscosity values, two stress growth
upon inception of steady shear flow experiments were conducted in the zero shear limit
(0.01 sec-1) at 320°C, one in the presence of nitrogen the other in air, the results are
shown in Fig. 6.10. The transient viscosity of the sample measured in nitrogen reaches a
steady state value while the sample that was measured in the presence of air increases in
an unbounded fashion. It is doubtful that this behavior is the result of recrystallization
during shear because it did not occur in the sample measured in nitrogen at the same
temperature. However, a possible explanation is that the molecular weight was
increasing, as has been reported for a similar wholly aromatic TLCP, Vectra A 950. It
was shown, for Vectra A, that the melt was polymerized by interchain transesterification
that begins at temperatures approximately 35°C above the melt temperature [24].
Polymerization increases the molecular weight by one liquid crystal molecule forming a
convalent bond with one of its neighbors. Regardless of the reason for the increase in
viscosity, the measured behavior suggests that the presence of an inert atmosphere will be
essential to obtaining complete coalescence.
6 The Rotational Molding of a Thermotropic Liquid Crystalline Polymer
221
0 100 200 300 400 500 6000
1000
2000
3000
4000
5000
6000
7000
h+ (Pa
sec)
Time (sec)
Figure 6.10. Transient shear viscosity from stress growth experiments at a shear rate of 0.01 sec-1 and 320°C. The symbol represents the test conducted in the presence of nitrogen and is in the presence of air.
Coalescence experiments were carried out to confirm that the conditions
identified by rheological characterization were appropriate. A representative example of
the images recorded during the coalescence experiments is shown in Fig. 6.11. Initially,
there was a finite contact area and the neck radius increased with time until the two drops
converged. In the figure, the two particles nearly reached a dimensionless neck radius,
ax , of 1 within 20 seconds. Although the test was stopped shortly thereafter because the
magnitude of the change in the dimensionless neck radius becomes comparable to the
magnitude of the error in the measurement, the two particles do appear to continue to
coalesce towards a single, nearly spherical drop.
6 The Rotational Molding of a Thermotropic Liquid Crystalline Polymer
222
Figure 6.11. Optical micrographs from the coalescence experiments of Vectra B 950 in nitrogen at 320°C.
The coalescence experiments confirm the observations made from rheological
characterization and the measurement of surface tension. The results from all three sets
of experiments, 320 and 330°C in nitrogen and 320°C in air, are shown in Fig. 6.12. For
the coalescence experiment at 320°C in the presence of air, the two particles began to
coalesce at nearly the same rate as the samples at the other conditions but stopped
prematurely at a dimensionless neck radius of approximately 0.6. This result was
anticipated by the large increase in viscosity at low shear rates in the presence of air. In
nitrogen, the coalescence rate at 330°C was marginally faster than was measured at
320°C. This was in agreement with what was anticipated from the similarity in the
relative magnitudes of the surface tension and viscosity at the two temperatures. Because
there was no advantage in using the higher temperature, 320°C was selected for the
single-axis rotational molding experiments.
6 The Rotational Molding of a Thermotropic Liquid Crystalline Polymer
223
0 5 10 15 20 250.0
0.2
0.4
0.6
0.8
1.0
Dim
ensi
onle
ss N
eck
Rad
ius
(x/a
)
Time (sec)
Figure 6.12. Results from the coalescence experiments, where 320°C in nitrogen is represented as , 320°C in air is , and 330°C in nitrogen is .
6.4.3 Densification
Several sizes and distributions of the spherical powder were created to evaluate
the effect that particle size and distribution had on densification. The apparent density of
each sample is shown in Table 6.3. As was observed for the individual particle sizes, the
apparent density of the distributions increased as the mass average particle size was
decreases. In fact, the apparent density of all of the samples, except D4, increased as the
particle size decreased. Although distribution D4 has a larger average particle size than
sample S4, the apparent density was greater, demonstrating that a small mass fraction of
fine particles significantly increased the apparent density.
6 The Rotational Molding of a Thermotropic Liquid Crystalline Polymer
224
Table 6.3. The apparent density of the samples used in the densification study.
Several interesting results were shown by the measured tensile properties. The
tensile modulus was near 1 GPa for all samples except S1. It is possible that the results
6 The Rotational Molding of a Thermotropic Liquid Crystalline Polymer
226
for S1 were not representative because of the low average density in that sample. The
ultimate tensile strength of samples S1 through S4 increased with the average density,
indicating that strength was dependent on the extent of densification. The values for
strength were normalized for differences in their average density to determine if the
relationship between strength and average density was solely due to variation in the
extent of densification or if there was also a change in adhesion between particles.
Because the strength of the normalized results was not all equal it was concluded that
adhesion had been increased. A clear relationship between tensile strength and average
density was not observed for the sample distributions. Sample D1 delivered the highest
tensile strength, but D4 possessed the highest average density. Perhaps the small fraction
of extremely fine powders in distribution D4 actually reduces the strength. If this were
true, it could be rationalized by the distribution possessing a greater number of particles,
and, therefore particle interfaces.
The two trials that were used to evaluate the possibility of increasing the strength
by introducing the oxidative effect were unsuccessful. The results are shown in Table 6.5
with the results for the sample molded in nitrogen. It was found that the strength of the
samples that were exposed to air was less than the sample molded in nitrogen. Not only
does this imply that the speculated molecular weight increase by transesterification was
not occurring, it also demonstrates the importance of supplying an inert atmosphere
throughout the entire molding process and not only during the initial stages when
coalescence occurs.
6 The Rotational Molding of a Thermotropic Liquid Crystalline Polymer
227
Table 6.5. Comparison of the tensile strength and modulus of sample S3 when molded in the presence of air.
Cycle 40 min in N2 20 min N2/ 20 min air 40 min N2/ 20 min air Strength (MPa) 10.08 9.49 9.83 Modulus (GPa) 1.140 1.109 1.126
The results from the densification experiments identify that gas removal
represents a major obstacle to rotational molding TLCPs. It was not possible to
completely densify the bulk powder as evaluated in static coalescence. The average
density of the eight selected samples increased as the apparent density was increased.
The particle sizes and size distributions did not affect the tensile modulus, but did
influence strength, with an increase in the ultimate tensile strength as the particle size was
decreased. By normalizing the tensile strength for differences in the average density, it
was discovered that the increase in strength could not be accounted for solely by the
increase in density and, thus, adhesion between particles was improved. Finally, the
tensile strength was decreased by exposing the sample to air during the heating cycle.
Unfortunately, this implies that the molded product cannot be strengthened by this
technique, but it does demonstrate the importance of supplying an inert atmosphere
throughout the entire molding cycle.
6.4.4 Single Axis Rotational Molding
The conditions identified from evaluating the powder characteristics, coalescence,
and densification were tested in a single-axis rotational mold to determine if they could
be translated to the rotational molding of the bulk powder. The powder successfully
rotationally molded using the previously identified conditions. Pictures of the internal
6 The Rotational Molding of a Thermotropic Liquid Crystalline Polymer
228
and external surfaces of the rotationally molded sample are shown in Figs. 6.14 and 6.15.
The internal surface of the molded cylinder was smooth, but the external surface
contained a fair amount of surface pores. The surface pores did not extend all the way
through the sample wall, which demonstrates that sufficient coalescence was achieved
during rotational molding.
Figure 6.14. Internal surface of the rotationally molded sample D4 in the 1.59 cm diameter cylindrical mold.
6 The Rotational Molding of a Thermotropic Liquid Crystalline Polymer
229
Figure 6.15. External surface of rotationally molded sample D4, where the width image is 50.8 mm.
The results from the measured average density, ultimate tensile strength, and
tensile modulus for the rotationally molded cylinder are shown in Table 6.6. The results
for the same distribution from the densification study are included for comparison.
Surprisingly, all quantities were increased relative to what had been measured for static
coalescence. The increase in density suggests that further improvement may be possible
in the rotational molding experiment. Although the tensile modulus was over twice that
measured in the densification study, it was still only a fraction of the 20 GPa that can be
obtained with proper molecular alignment and implies that further increase may still be
possible. The measured increase in the tensile strength was quite significant because it is
very close to the nominal strength (17.9 MPa) required for industrial cross-linked high
density polyethylene tanks [25]. This was sufficient to pressurize the molded sample to
1.59 MPa before rupture occurred (50 gram sample with 3.8 mm wall thickness).
6 The Rotational Molding of a Thermotropic Liquid Crystalline Polymer
230
Table 6.6. The average density, tensile strength, and tensile modulus from the rotationally molded distribution D4 compared the results for the distribution from the densification study.
0.012 2.022 ± 0.013
0.14 17.49 0.16±Ultimate Tensile Strength (MPa)
Tensile Modulus (GPa)
1218 ±
10.51 ±
0.930 ±
Sample D 4 Results from the Densification Study
Sample D 4 Results from Rotational Molding
Average Density (kg/m3)
8 1288 ± 7
When comparing the results from the densification study with the results from
rotational molding, it appears that the dynamics introduced by rotation increased the
average density. The increase must occur by reducing the amount of gas that initially
gets trapped during coalescence because it was demonstrated in the densification study
that once gas is encapsulated, it cannot be removed. Understanding the mechanism that
controls the relationship between rotation and the amount of gas that is encapsulated
during coalescence may lead to a method to optimize the rotation rate.
To explain the increased density the static densification and dynamic rotational
molding processes were reconsidered. The encapsulation of bubbles in the static scenario
may be described as shown in Fig. 6.16. The powder is heated by conduction from the
mold wall. As the powder melts, coalescence occurs and a network of connected
particles is formed. Eventually the network collapses, and bubbles are encapsulated,
which are removed by dissolving and diffusing through the surrounding melt. However,
the permeability of a gas in the TLCP is extremely low so any bubbles that are formed
remain in the melt, as in stage two of the figure.
6 The Rotational Molding of a Thermotropic Liquid Crystalline Polymer
231
Figure 6.16. Comparison of bubble formation in static bulk powder and in rotational molding.
In the case of the rotating system, as in the static example, the powder is heated
by conduction from the mold wall. As the powder melts, coalescence occurs and a
network of connected particles begins to form. However, in the rotational molding case,
the coalescing particles are attached to the rotating mold wall and a layer of particles is
removed from the tumbling powder bed. Assuming the layer of particles is only on
particle thick, a three-dimensional network does not exist and, therefore, cannot collapse
and encapsulate gas. The validity of the layer thickness assumption depends upon the
rate of conduction from the molten layer to the powder bed and the amount of time a
particular position on the mold surface stays in the powder bed, which is proportional to
the rotation rate. The optimal rotation rate should be slow enough for a single layer of
particles to coalesce in one rotation. If rotation is too slow, more that one particle will be
6 The Rotational Molding of a Thermotropic Liquid Crystalline Polymer
232
attached to the mold wall as it passes through the powder bed. If it is too fast, the layer
will not completely coalesce before the next layer of particles attach.
The single axis rotational molding experiment was repeated to test the proposed
explanation of the observed difference in average density between the static and dynamic
cases. The rotation rate used in this experiment was determined by using the coalescence
time measured in the coalescence experiments. The particles coalesced at 320°C within
approximately 20 seconds. To promote the formation of a single particle layer, the
rotation rate was set at 20 seconds per revolution or 3 rpm.
Several of the measured properties were improved by reducing the rotation rate.
The density of the sample that was rotationally molded at 3 rpm was increased to 1392 ±
6 kg/m3. This is a significant improvement because it is essentially the material density
and, therefore, complete densification was achieved by the reduction in rotation rate.
Unfortunately, an increase comparable to what was measured in density was not
measured in the ultimate tensile strength and tensile modulus. The measured values were
17.63 ± 0.019 MPa and 2.010 ± 0.014 GPa for strength and modulus, respectively, which
were within the experimental error of the sample molded at 10 rpm. The surface pores in
the exterior surface were reduced as shown in Fig. 6.18. Although some surface pores
still exist, this represents a dramatic improvement over what had been found at 10 rpm,
shown in Fig. 6.17.
6 The Rotational Molding of a Thermotropic Liquid Crystalline Polymer
233
Figure 6.17. Exterior surface of the cylinder rotationally molded at 3 rpm, where the width image is 50.8 mm.
6.5 Conclusions
A commercially available TLCP, Vectra B 950, was evaluated for use in
rotational molding by separately investigating the phenomenological steps of powder
flow, coalescence, and densification and applying the identified conditions to rotational
molding. Several important conclusions can be drawn from this work. The available
pellets are not acceptable for use in rotational molding and they cannot be ground into a
freely flowing powder by conventional means. It is possible to convert the pellets into an
freely flowing powder by the described melt blending process. The particles can be
rotationally molded at 320°C in nitrogen, as was identified by the coalescence
experiments. Densification by dissolution and diffusion was not possible. However, the
dynamics of rotational molding could be used to achieve complete densification and
improve the surface appearance. Although the ultimate tensile strength was only a
6 The Rotational Molding of a Thermotropic Liquid Crystalline Polymer
234
fraction of what can be obtained by this material it still exhibited a value that was
approximately equal to the requirements for cross-linked high density polyethylene.
6.6 Future Work
Although this work has answered several questions about the rotational molding
of TLCPs in a single axis rotational mold, further assessment of the performance such as:
the structural integrity during thermal cycling, the mechanical performance at cryogenic
conditions, and permeability to a variety of gases, would be invaluable in evaluating the
molded structure for potential use as a cryogenic storage vessel. It would also be
beneficial to rotational mold the material in a larger, possibly more complex, mold with
biaxial rotation. Such an investigation would address the ability to effectively fill corners
and allow for the evaluation of their mechanical performance. Finally, an investigation
should be performed to address the possibility of increasing the limited strength values by
altering the TLCP chemical structure with the addition of telechelic ionomeric groups to
increase the interfacial adhesion between particles.
6.7 Acknowledgements
This work was financially supported by a phase II SBIR grant from NASA, grant
number NAS-2S-4018-285, managed by Luna Innovations.
6 The Rotational Molding of a Thermotropic Liquid Crystalline Polymer
235
6.8 References
1 MacDonald, W.A., Chapter 8, “Thermotropic Main Chain Liquid Crystal
Polymers,” in Liquid Crystal Polyners: From Structures to Applications, edited by
! upper convected maxwell particle coalescence program implicit none integer :: i,j,ierror integer :: nroots,itmax,info(1) real :: ao,visc,gam,t,x real :: F(1000,4),tf,dx,dt,sol,K,rate,st1,st2,tvisc real :: errabs,errel,eps,eta,fcn common ao,visc,gam ,t,x external ZREAL, fcn ! solution matrix F(:,:) = 0 ! define parameters ao = 2.50E-4 visc = 2301.1 gam = 0.028 ! set initial conditions t = 0.0001*visc*ao/gam
x = asin((3./2.*gam*t/(visc/ao)**0.5)
! enter initial conditions into solution matrix F(1,1) = t F(1,2) = sin(x) ! end time tf = 1. ! set time step (adjust for convergence) dt = 0.00001 ! initial guess at solution (dx = dtheta/dt)
Appendix D. Programs for the Neck Growth Model
321
dx = 1.0 ! initialize counters ! i = 2 because 1 is initial conditions ! j is for data reduction loop i = 2 j = 1 ! define zreal parameters errabs = 1.0E-5 errel = 1.0E-5 eps = 1.0E-5 eta = 1.0E-2 nroots = 1 itmax = 100 ! open file to write to open (unit=10, file='output.ecs', status='new', iostat=ierror) ! write initial conditions to file write(10,*) F(1,1),F(1,2) ! solve at each time step do while (t < tf) ! find root of expression call ZREAL(fcn, errabs, errel, eps, eta, nroots, itmax, dx, sol, info) dx = sol t = t + dt ! data reduction if (j == 100) then F(i,1) = t F(i,2) = sin(x + sol*dt) ! calculate extension rate for output file K=tan(x)/2.-sin(x)/6.*((2.*(2.-cos(x))+(1.+cos(x)))/(1.+cos(x))/ (2.-cos(x))) rate = K*dx ! calculate stresses for output file st1=2.*visc*rate st2=4.*visc*rate ! calculate viscosity for output file
Appendix D. Programs for the Neck Growth Model
322
tvisc = (st1+st2)/rate F(i,3) = rate F(i,4) = tvisc ! write solution to file write(10,*) F(i,1),F(i,2),F(i,3), F(i,4) i = i + 1 j = 0 end if j = j +1 x = x + sol*dt end do end program !- supporting function ----------------------------------------- real function fcn(dx) implicit none real :: ao,visc,gam,t,x,dx real :: K,T1,T2 common ao,visc,gam,t,x ! for small angles (use only for angle approximation) ! K = sin(x)/(1+cos(x))/(2-cos(x)) ! full expression K = tan(x)/2.-sin(x)/6.*((2.*(2.-cos(x))+(1.+cos(x)))/(1.+cos(x))/(2.-cos(x))) ! 1st normal stress (growth) Tau11 T1=2.*visc*K*dx ! 2cnd normal stress (growth) Tau22 T2=-4.*visc*Kdx ! define energy balance equation fcn=2.**(2./3.)*ao*K/3./gam*(T1-T2)*(1.+cos(x))**(4./3.)* (2.-cos(x))**(5./3.)/cos(x)/sin(x)-1. return end
Appendix D. Programs for the Neck Growth Model
323
D.2 Single Mode Steady State upper convected Coalescence Model
! upper convected maxwell particle coalescence program implicit none integer :: i,j,ierror integer :: nroots,itmax,info(1) real :: ao,visc,gam,lam,t,x real :: F(1000,4),tf,dx,dt,sol,K,rate,st1,st2,tvisc real :: errabs,errel,eps,eta,fcn common ao,visc,gam,lam,t,x external ZREAL, fcn ! solution matrix F(:,:) = 0 ! define parameters ao = 2.50E-4 visc = 2301.1 gam = 0.028 lam = 50.80 ! set initial conditions t = 0.0001*visc*ao/gam
x = asin((3./2.*gam*t/(visc/ao)**0.5)
! enter initial conditions into solution matrix F(1,1) = t F(1,2) = sin(x) ! end time tf = 1. ! set time step (adjust for convergence) dt = 0.00001
Appendix D. Programs for the Neck Growth Model
324
! initial guess at solution (dx = dtheta/dt) dx = 1. ! initialize counters ! i = 2 because 1 is initial conditions ! j is for data reduction loop i = 2 j = 1 ! define zreal parameters errabs = 1.0E-5 errel = 1.0E-5 eps = 1.0E-5 eta = 1.0E-2 nroots = 1 itmax = 100 ! open file to write to open (unit=10, file='output.ecs', status='new', iostat=ierror) ! write initial conditions to file write(10,*) F(1,1),F(1,2) ! solve at each time step do while (t < tf) ! find root of expression call ZREAL(fcn, errabs, errel, eps, eta, nroots, itmax, dx, sol, info) dx = sol t = t + dt ! data reduction if (j == 100) then F(i,1) = t F(i,2) = sin(x + sol*dt) ! calculate extension rate for output file K=tan(x)/2.-sin(x)/6.*((2.*(2.-cos(x))+(1.+cos(x)))/(1.+cos(x))/ (2.-cos(x))) rate = K*dx ! calculate stresses for output file st1=2.*visc/(1.-2.*lam*rate)*rate st2=4.*visc/(1.+4.*lam*rate)*rate
Appendix D. Programs for the Neck Growth Model
325
! calculate viscosity for output file tvisc = (st1+st2)/rate F(i,3) = rate F(i,4) = tvisc ! write solution to file write(10,*) F(i,1),F(i,2),F(i,3), F(i,4) i = i + 1 j = 0 end if j = j +1 x = x + sol*dt end do end program !- supporting function ----------------------------------------- real function fcn(dx) implicit none real :: ao,visc,gam,lam,t,x,dx real :: K,T1,T2 common ao,visc,gam,lam,t,x ! for small angles (use only for angle approximation) ! K = sin(x)/(1+cos(x))/(2-cos(x)) ! full expression K = tan(x)/2.-sin(x)/6.*((2.*(2.-cos(x))+(1.+cos(x)))/(1.+cos(x))/(2.-cos(x))) ! 1st normal stress (growth) Tau11 T1=2.*visc*K*dx/(1.-2.*lam*K*dx) ! 2cnd normal stress (growth) Tau22 T2=-4.*visc*K*dx/(1.+4.*lam*K*dx) ! define energy balance equation fcn=2.**(2./3.)*ao*K/3./gam*(T1-T2)*(1.+cos(x))**(4./3.)* (2.-cos(x))**(5./3.)/cos(x)/sin(x)-1. return end
Appendix D. Programs for the Neck Growth Model
326
D.3 Single Mode upper convected Maxwell Coalescence Model
! upper convected maxwell particle coalescence program ! this is the program for the transient upper convected maxwell coalescence model with 1 ! mode. implicit none integer :: i,j,ierror integer :: nroots,itmax,info(1) real :: ao,visc,gam,lam,t,x,C11,C22 real :: F(1000,6),tf,dx,dt,sol,K,tvisc real :: errabs,errel,eps,eta,fcn,fcn1,T1,T2 common ao,visc,gam,lam,t,x,C11,C22,T1,T2 external ZREAL, fcn, fcn1 ! solution matrix F(:,:) = 0 ! define parameters ao = 2.74E-4 visc = 11128.9 gam = 0.028 lam = 1.54 ! set initial conditions t = 0.0001*visc*ao/gam x = asin((3./2.*gam*t/visc/ao)**0.5) ! enter initial conditions into solution matrix F(1,1) = t F(1,2) = sin(x) ! end time tf = 650. ! set time step (adjust for convergence) dt = 0.001 ! initial guess at solution (dx = dtheta/dt) dx = 1. ! initialize counters ! i = 2 because 1 is initial conditions
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! j is for data reduction loop i = 2 j = 1 ! define zreal parameters errabs = 1.0E-5 errel = 1.0E-5 eps = 1.0E-5 eta = 1.0E-2 nroots = 1 itmax = 1000 ! open file to write to open (unit=10, file='output.ecs', status='new', iostat=ierror) ! write initial conditions to file write(10,*) F(1,1),F(1,2) do while (t < tf) ! find root of expression if (i == 2) then call ZREAL(fcn, errabs, errel, eps, eta, nroots, itmax, dx, sol, info) dx = sol t = t + dt F(i,1) = t F(i,2) = sin(x + sol*dt) ! calculate extension rate and viscosity for file K = tan(x)/2.-sin(x)/6.*((2.*(2.-cos(x))+ (1.+cos(x)))/(1.+cos(x))/(2.-cos(x))) tvisc = (T1-T2)/K/dx !biaxial extensional viscosity F(i,3) = K*dx !biaxial extension rate F(i,4) = tvisc F(i,5) = T1 !normal stress 11 F(i,6) = T2 !normal stress 22 ! write solution to file write(10,*) F(i,1),F(i,2),F(i,3),F(i,4),F(i,5),F(i,6) else if (i > 2) then call ZREAL(fcn1, errabs, errel, eps, eta, nroots, itmax, dx, sol, info) dx = sol t = t + dt K = tan(x)/2.-sin(x)/6.*((2.*(2.-cos(x))+ (1.+cos(x)))/(1.+cos(x))/(2.-cos(x))) ! data reduction if (j == 100) then F(i,1) = t
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F(i,2) = sin(x + sol*dt) ! calculate extension rate and viscosity for file tvisc = (T1-T2)/K/dx !biaxial extensional viscosity F(i,3) = K*dx !biaxial extension rate F(i,4) = tvisc F(i,5) = T1 !normal stress 11 F(i,6) = T2 !normal stress 22 ! write solution to file write(10,*) F(i,1),F(i,2),F(i,3),F(i,4),F(i,5),F(i,6) j = 0 end if end if ! calculate stress constants for next time step C11 = exp((1./lam-2.*K*dx)*t)*(T1-2.*visc*K*dx/(1.-2.*lam*K*dx)) C22 = exp((1./lam+4.*K*dx)*t)*(T2+4.*visc*K*dx/(1.-2.*lam*K*dx)) ! step forward x (theta) x = x + sol*dt ! advance counters i = i + 1 j = j + 1 end do end program !- supporting function ----------------------------------------- ! fcn is a function that is called by zreal to find dtheta/dt. It is only used for the first time ! step to initiate the program by using integration constants for the stress expressions that ! are determined from the conditions: t=0 Tau11 = Tau22 = 0. !------------------------------------------------------------------- real function fcn(dx) implicit none real :: ao,visc,gam,lam,t,x,dx real :: K,C11,C22,T1,T2 common ao,visc,gam,lam,t,x,C11,C22,T1,T2 ! for small angles (approximation) ! K = sin(x)/(1+cos(x))/(2-cos(x)) ! full expression K = tan(x)/2.-sin(x)/6.*((2.*(2.-cos(x))+(1.+cos(x)))/(1.+cos(x))/(2.-cos(x))) ! 1st normal stress (growth) Tau11 T1 = 2.*visc*K*dx/(1.-2.*lam*K*dx)*exp(-(1./lam-2.*K*dx)*t)* (exp((1./lam-2.*K*dx)*t)-1.)
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! 2cnd normal stress (growth) Tau22 T2 = -4.*visc*K*dx/(1.+4.*lam*K*dx)*exp(-(1./lam+4.*K*dx)*t)* (exp((1./lam+4.*K*dx)*t)-1.) ! define energy balance fcn1 = 2.**(2./3.)*ao*K/3./gam*(T1-T2)*(1.+cos(x))**(4./3.)* (2.-cos(x))**(5./3.)/cos(x)/sin(x)-1. return end !- supporting function ----------------------------------------- ! fcn1 is a function that is called by zreal to find dtheta/dt. It is used for all time steps ! after the first iteration. The integration constant for the stress expressions are !determined using Tau11 & Tau22 from the previous time step. !------------------------------------------------------------------- real function fcn1(dx) implicit none real :: ao,visc,gam,lam,t,x,dx real :: K,C11,C22,T1,T2 common ao,visc,gam,lam,t,x,C11,C22,T1,T2 ! for small angles (approximation) ! K = sin(x)/(1+cos(x))/(2-cos(x)) ! full expression K = tan(x)/2-sin(x)/6.*((2.*(2.-cos(x))+(1.+cos(x)))/(1.+cos(x))/(2.-cos(x))) ! 1st normal stress (growth) Tau11 T1 = exp(-(1./lam-2.*K*dx)*t)*(2.*visc*K*dx/(1.-2.*lam*K*dx)* exp((1./lam-2.*K*dx)*t)+C11) ! 2cnd normal stress (growth) Tau22 T2 = exp(-(1./lam+4.*K*dx)*t)*(-4.*visc*K*dx/(1.+4.*lam*K*dx)* exp((1./lam+4.*K*dx)*t)+C22) ! define energy balance fcn = 2.**(2./3.)*ao*K/3./gam*(T1-T2)*(1.+cos(x))**(4./3.)* (2.-cos(x))**(5./3.)/cos(x)/sin(x)-1. return end
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D.4 Multi-Mode upper convected Maxwell Coalescence Model
! upper convected maxwell particle coalescence program ! this is the program for the transient upper convected maxwell coalescence model with 1 ! mode. implicit none integer :: i,j,n,modes,ierror integer :: nroots,itmax,info(1) ! set # of modes parameter (modes = 5) integer, dimension(1:modes,1:2) :: L real :: ao,visc(modes),gam,lam(modes),t,x real :: C11,C22,T1(1:modes),T2(1:modes) real :: F(1000,6),tf,dx,dt,sol real :: errabs,errel,eps,eta,fcn common n,ao,visc,gam,lam,t,x,L,C11,C22,T1,T2 external ZREAL, fcn ! set n = modes n = 5 L(:,:) = 0 F(:,:) = 0 ! define parameters ao = 2.74E-4 visc(1) = 1084.603559 visc(2) = 2677.551217 visc(3) = 3778.27556 visc(4) = 3259.505989 visc(5) = 1274.613484 gam = 0.02832 lam(1) = 0.02243756 lam(2) = 0.162908159 lam(3) = 1.010652684 lam(4) = 6.48507642 lam(5) = 49.20180938 ! set initial conditions t = 0.0001*(sum(visc))*ao/gam x = asin((3./2.*gam*t/(sum(visc))/ao)**0.5)
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! enter initial conditions into solution matrix F(1,1) = t F(1,2) = sin(x) ! end time tf = 650. ! set time step (adjust for convergence) dt = 0.001 ! initial guess at solution dx = 1. ! initialize counters ! i = 2 because 1 is initial conditions ! j is for data reduction loop i = 2 j = 1 errabs = 1.0E-5 errel = 1.0E-5 eps = 1.0E-5 eta = 1.0E-2 nroots = 1 itmax = 1000 ! open file to write to open (unit=10, file='output.ecs', status='new', iostat=ierror) write(10,*) F(1,1),F(1,2) do while (t < tf) ! find root of expression if (i == 2) then call ZREAL(fcn, errabs, errel, eps, eta, nroots, itmax, dx, sol, info) dx = sol t = t + dt F(i,1) = t F(i,2) = sin(x + sol*dt) ! calculate extension rate and viscosity for file K = tan(x)/2.-sin(x)/6.*((2.*(2.-cos(x))+(1.+cos(x)))/ (1.+cos(x))/(2.-cos(x))) tvisc = (sum(T1)-sum(T2))/K/dx !biaxial extensional viscosity F(i,3) = K*dx !biaxial extension rate F(i,4) = tvisc F(i,5) = sum(T1) !normal stress 11
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F(i,6) = sum(T2) !normal stress 22 ! write solution to file write(10,*) F(i,1),F(i,2),F(i,3),F(i,4),F(i,5),F(i,6) else if (i > 2) call ZREAL(fcn1, errabs, errel, eps, eta, nroots, itmax, dx, sol, info) dx = sol t = t + dt K = tan(x)/2.-sin(x)/6.*((2.*(2.-cos(x))+(1.+cos(x)))/ (1.+cos(x))/(2.-cos(x))) if (j == 1000) then F(i,1) = t F(i,2) = sin(x + sol*dt) ! calculate extension rate and viscosity for file ! biaxial extensional viscosity tvisc = (sum(T1)-sum(T2))/K/dx F(i,3) = K*dx !biaxial extension rate F(i,4) = tvisc F(i,5) = sum(T1) !normal stress 11 F(i,6) = sum(T2) !normal stress 22 ! write data to file write(10,*) F(i,1),F(i,2),F(i,3),F(i,4),F(i,5),F(i,6) j = 0 end if end if ! calculate stress constants for next time step C11 = exp((1./lam-2.*K*dx)*t)*(sum(T1)-2.*visc*K*dx/ (1.-2.*lam*K*dx)) C22 = exp((1./lam+4.*K*dx)*t)*(sum(T2)+4.*visc*K*dx/ (1.-2.*lam*K*dx)) ! step forward x (theta) x = x + sol*dt ! advance counters i = i + 1 j = j + 1 end do end program !- supporting function ----------------------------------------- ! fcn is a function that is called by zreal to find dtheta/dt. It is only used for the first time ! step to initiate the program by using integration constants for the stress expressions that ! are determined from the conditions: t=0 Tau11 = Tau22 = 0. !------------------------------------------------------------------- real function fcn(dx)
Appendix D. Programs for the Neck Growth Model
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implicit none integer :: i,j,n,modes parameter(modes = 5) integer, dimension(1:modes,1:2) :: L real :: ao,visc(modes),gam,lam(modes),t,x,dx real :: K,C11,C22,T1(1:modes),T2(1:modes) common n,ao,visc,gam,lam,t,x,L,C11,C22,T1,T2 ! for small angles (approximation) ! K = sin(x)/(1+cos(x))/(2-cos(x)) ! full expression K = tan(x)/2.-sin(x)/6.*((2.*(2.-cos(x))+(1.+cos(x)))/(1.+cos(x))/(2.-cos(x))) i = 1 j = 1 ! multimode 1st normal stress (Tau11) ! loop to determine each mode's contribution to stress ! conditions to simplify the stress equation once steady state has been reached do while (i <= n) if (L(i,1) == 1) then T1(i) = 2.*visc(i)*K*dx/(1.-2.*lam(i)*K*dx) !steady state else if (exp(-(1./lam(i)-2.*K*dx)*t)*(exp((1./lam(i)-2.*K*dx)*t)-1.) == 1.) ` then T1(i) = 2.*visc(i)*K*dx/(1.-2.*lam(i)*K*dx) L(i,1) = 1 else T1(i) = 2.*visc(i)*K*dx/(1.-2.*lam(i)*K*dx)* exp(-(1./lam(i)-2.*K*dx)*t)*(exp((1./lam(i)-2.*K*dx)*t)-1.) end if i = i+1 end do ! multimode 2cnd normal stress (Tau22) do while (j <= n) if (L(j,2) == 1) then T2(j) = -4.*visc(j)*K*dx/(1.+4.*lam(j)*K*dx) else if (exp(-(1./lam(j)+4.*K*dx)*t)*(exp((1./lam(j)+4.*K*dx)*t)-1.) == 1.) then T2(j) = -4.*visc(j)*K*dx/(1.+4.*lam(j)*K*dx) L(j,2) = 1 else
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T2(j) = -4.*visc(j)*K*dx/(1.+4.*lam(j)*K*dx)* exp(-(1./lam(j)+4.*K*dx)*t)*(exp((1./lam(j)+4.*K*dx)*t)-1.) end if j = j+1 end do ! define energy balance fcn = 2.**(2./3.)*ao*K/3./gam*(sum(T1)-sum(T2))*(1.+cos(x))**(4./3.)* (2.-cos(x))**(5./3.)/cos(x)/sin(x)-1. return end !- supporting function ----------------------------------------- ! fcn1 is a function that is called by zreal to find dtheta/dt. It is used for all time steps ! after the first iteration. The integration constant for the stress expressions are !determined using Tau11 & Tau22 from the previous time step. !------------------------------------------------------------------- real function fcn1(dx) implicit none integer :: i,j,n,modes parameter(modes = 5) integer, dimension(1:modes,1:2) :: L real :: ao,visc(modes),gam,lam(modes),t,x,dx,C11,C22 real :: K,C11,C22,T1(1:modes),T2(1:modes) common n,ao,visc,gam,lam,t,x,L,C11,C22,T1,T2 ! for small angles (approximation) ! K = sin(x)/(1+cos(x))/(2-cos(x)) ! full expression K = tan(x)/2.-sin(x)/6.*((2.*(2.-cos(x))+(1.+cos(x)))/(1.+cos(x))/(2.-cos(x))) i = 1 j = 1 ! loop to determine each mode's contribution to stress ! conditions to simplify the stress equation once steady state has been reached ! multimode 1st normal stress (Tau11) do while (i <= n) if (L(i,1) == 1) then T1(i) = 2.*visc(i)*K*dx/(1.-2.*lam(i)*K*dx) !steady state else if (exp(-(1./lam(i)-2.*K*dx)*t)*(exp((1./lam(i)-2.*K*dx)*t)+C11) == 1.) then T1(i) = 2.*visc(i)*K*dx/(1.-2.*lam(i)*K*dx) !steady state
Appendix D. Programs for the Neck Growth Model
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L(i,1) = 1 else T1(i) = exp(-(1./lam(i)-2.*K*dx)*t)*(2.*visc(i)*K*dx/ (1.-2.*lam(i)*K*dx)*exp((1./lam(i)-2.*K*dx)*t)+C11)!transient end if i = i+1 end do ! multimode 2cnd normal stress (Tau22) do while (j <= n) if (L(j,2) == 1) then T2(j) = -4.*visc(j)*K*dx/(1.+4.*lam(j)*K*dx) !steady state else if (exp(-(1./lam(j)+4.*K*dx)*t)*(exp((1./lam(j)+4.*K*dx)*t)+C22.) == 1.) then T2(j) = -4.*visc(j)*K*dx/(1.+4.*lam(j)*K*dx) !steady state L(j,2) = 1 else T2(j) = exp(-(1./lam(j)+4.*K*dx)*t)*(-4.*visc(j)*K*dx/ (1.+4.*lam(j)*K*dx)*exp((1./lam(j)+4.*K*dx)*t)+C22)!transient end if j = j+1 end do ! define energy balance fcn = 2.**(2./3.)*ao*K/3./gam*(sum(T1)-sum(T2))*(1.+cos(x))**(4./3.)* (2.-cos(x))**(5./3.)/cos(x)/sin(x)-1. return end
Technology, Plastics Design Library, William Andrew Publishing, Norwich, New York, 2002.
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9. Sawyer, L., Shepherd, J., Kaslusky, A., Knudsen, R., Tech Spotlight: Unfilled liquid crystalline polymers, [Online], Available: http://www.ticona-us.com/literature/documents/LCP_Article_01_351res72dpi.PDF, June, 2001.
10. Wissbrun, K.F., “Rheology of Rod-Like Polymers in the Liquid Crystalline State,” J. Rheol., 25, 6, 1981, pp. 619-662.
11. High Density Crosslinked Polyethylene (HDXLPE) Storage Tanks, [Online], Available: http://www.polyprocessing.com/updates/GenSpecrev2-HDXLPE.pdf
Key Words Rotational molding, LCP, mechanical properties