A STUDY ON POLYMER BLENDING MICRORHEOLOGY
A STUDY ON POLYMER BLENDING
MICRORHEOLOGY
P R O E F S C H R I F T ter verkrijging van de g r a a d van doctor in de t e c h n i s c h e w e t e n s c h a p p e n aan de Technische H o g e s c h o o l Delft op gezag van de rector m a g n i f i c u s , prof. dr J M. Dirken
voor een c o m m i s s i e a a n g e w e z e n door het college van d e k a n e n te v e r d e d i g e n op d i n s d a g 1 april 1986 te 1600 uur door J A C O B J A C O B U S E L M E N D O R P n a t u u r k u n d i g ingenieur g e b o r e n te Vlaardingen
krips repro meppel
Dit proefschrift is goedgekeurd door de promotor PROF. DR. IR. A. K. VAN DER VEGT.
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L. INTR0DUCTI01 '
2.
.1. Zero-interfacial tension syst< 9
,1.1. . . . ■ ■ ■ I : . . . . : ■ ■ ...'■■ ,3.2, . . . ' ■ ' ■ ■ ■ ■ .. . .. ' .' ■ ■ ■ . . . ■ ' ■■ 2.4. Coalesi ena -*! er blei 3. 41 4. DR01 LI '■' 43 U, 1 , 4.I.J. Generatioi 44 . . , . ■ ■ ■ 45
4.2, Expe r ime n tal resul 1 48
■i. J. / . "- ' 48 ï.2,2. 52 . . . '' 58 5. DROP "■ 65 3.1 . 65 5.2. Experimental part 69 5.3. Conclusions
THE DEFORMATION OF HCXi: "" NAL 75 FLOW
76 6 . 3 . 76 6 . 4 . 79 . . 81 83 7.1 . 84 T.I.I, '■■ 8 5 . . . 8 8 . . . 91 94 . . . 96 101 , s t a t e m e n t of t h e pr< >l*lem 101 104
influence on blend morphi : 110
. ■. . II' 3.5. ■ . '13 . . 114 117 119 LIST Of- : i
1. EN
I n i
in! are . . ■ . ■ ■ ■ ■ . ' .
the
Lending operai . i process of ;
I as yet. microrheolog
,. ■ ■ . ble for the foi i n I I i
it. In order it ion of morpli'
t polymer ess. In t he coin ■ ■ this study an insight, in thi riding process i
and Introduce thereby the studies on <':■•
hapters,
Since polymers art blended Ln the Liqui tate, the para that can Ln the bli nd Lng proc in thi i heo] ogical behai
components and the u . Die former i
■ ties, e the lattei
can be eo sed i nterfacial tensi n, dif fui ■: i m d maxima]
solubiLity.
'I he mosl important parametet E thes< indoubtedly the maximal
solubili i ...::' , Whei : Ls small, the system is inmiscil
ture wil] l"1 formed under all Condi 1 components are miscible, a single phase structure can b\
The miscibility oi po] ymei Li Liqi , general, goverm I tropic factors. The Lhe chains ' amount of possible conformations
tem and thereby the entropica] ee energj ol . Evidently, two Liquids are miscible when th< i i ent
single phase system is smaller than the free energy of the phase separated state, i.e. the free energy of mixing, AG, is negati< .
required but not sufficient, sinci ites can show than either fully phase or single phase states. According to Olabisi et al. (1) phase separation can only be Lnhil when the second derivative of Lhe free energy to concentration is po Li Lve(
9 G/9C > 0. From the well-known Flory-Huggins lattice theory and the more recent equation of state theories it can readily be explained that miscible polymer/polymer systems arc scarce.
As a first step in any mixing operation the two components arc brought to
gether. in low viscous systems the one is usually floating on the other whereas in the blending of molten polymers the components are initially
distributed as granules. When the size of the granules is much larger than the desired length scale, their size has to be reduced. This is achieved by
mechanical deformation of the two-phase system. This deformation is inevitably accompanied by an increase of interfacial area and, when an interfaeial tension is present, by an increase of the system's free energy. The
deformation will therefore by counteracted by the interfacial forces. The latter, however, being reciprocal to the local curvatures, are quite small in
the beginning of the blending operation where the size of the domains is still rather large.
Another consequence of the initially large length scale is the fact that
interdiffusion in miscible systems hardly promotes the homogeneity of the system (2). Therefore, miscible and inmiscible systems will show similar mixing behaviour in the initial stage of the blending process.
Under these conditions the liquids domains will be deformed along the streamlines of the flow field imposed on the blend by the blending device.
Spherical domains will become extended into highly extended particles, or, when two-dimensional flows are present, into lamellar structures. Since in a
successful blending operation the length scale of the blend is continuously shortened, a point will inevitably be reached where interfacial forces are no longer negligible. Threadlike bodies will be converted into lines of small
droplets and droplets small enough to resist the hydrodynamic, disruptive, forces will eventually be formed. In miscible systems on the other hand the
hydrodynamic processes will bring the length scale dovm to a level where diffusion processes can promote the homogeneity of the polymer blend and the
formation of a single phase.
It would be an oversimplification to assume that liquid domains would not collide during the blending operation. If colliding droplets are just small
enough to resist the hydrodynamic forces, a coalescence will yield a droplet
too large to do so and break-up will rapidly follow. Temporarily however, the
droplet size is larger and the time average of the droplet size can therefore
be expected to increase with increasing number of coalescences. The latter
dependence will even be more pronounced when droplets formed by a coalescence
collide and coalesce before break-up has set in, etc.. The thus formed "dispersion" is therefore dynamic by nature i.e. droplets are
continuously-breaking up and coalescing.
Break-up of droplets is a two-step process. First a droplet is stretched
out by the hydrodynamic interactions. When a critical stretch is exceeded,
the interfacial forces will "neck" the droplet and two daughterdroplets are
formed. When the size of the droplet is relatively large, this necking cannot
be completed before elongation into a highly extended body is a fact (see
Figure 11)
-O °
r~\ R>Rcrit.
R»Rcrit. b
Fig. 1-1. Differences in break-up mode for R > J ?c r i t (a) and R » «crit (b).
It can therefore be expected that when droplets are present which, by repeated
coalescences, are far above their critical size, size reduction is achieved via the formation of highly extended bodies and the subsequent interfacial tension driven break-up of them. Indeed, observation of polymer blend
morphologies can reveal extended bodies with periodic distortions, apparently frozen-in during their conversion from extended bodies into a line of
droplets (see Fig. 1-2). Above a certain length scale and concentration the extended bodies are so numerous that coalescence between those bodies will
take place even before break-up can be completed, thus forming a "dynamic" co-continuous network (see Fig. 1-3).
Fig. 1-2. Varicose bodies of the disperse phase in a PP/PS blend, indicating thai the subsequent formation and break-up of fluid threads is a possible size reduction mechanism.
Pig. 1-3. Dispersion (a) and co-continuous (b) type morphologies.
When the blending process is stopped, the structure will be fixed by the
solidification of the polymers. Since inevitably a time will pass between the
cessation of the deformation of the blend and the solidification of its components, intcrfacial tension .induced motion can change the morphology
somewhat. Since no external forces are acting on the system these changes in morphology are called "spontaneous". Full minimization of the interfaci.al
area would only be completed when the phases have separated completely and the one is floating on the other. This will, however, not occur in molten
polymer systems since the low density differences and the high viscosities prevent gravity induced coalescence (the settling velocity is of the order of 1CT13 m/s). The spontaneous changes in morphology will therefore be limited to spherification of anisometric particles, break-up of liquid threads or, in the case of a co-continuous structure, a coarsening of the blend.
These interfacial tension induce'd changes in morphology have two
implications on the problem of polymer blending:
i ) when one is trying to tavlor-make a morphology by carefully choosing components and blending conditions, one has to realise that upon
cessation of the blending operation the spontaneous changes will
change the morphology formed in the blending operation itself.
ii) when one is trying to analyse polymer blending operations from the formed morphologies, as is quite common, one has to realise that inter
facial tension induced changes have occurred. The observed morphology will therefore differ from the morphology that resulted from the blending process itself, which makes it hard to obtain correlations
between the blending conditions and the observed morphology.
A striking example of the latter is found when cooling down a blend at different rates. Fig. 1-4 is a SEM micrograph of a PP/PS blend prepared on a two roller mill at 200°C and cooled down rapidly by quenching it in liquid
nitrogen. A co-continuous structure is observed. Fig. 1-5 is a SEM micrograph of the same blend but annealed for 10 minutes at 200°C before cooling down
slowly. The co-continuous network appears to have been converted into a dispersion type morphology. The rate of those spontaneous changes can be
estimated from microrheological considerations. For typical polymer melts it can be established (2) that an ellipsoidal droplet will convert into a spherical droplet in several seconds and that highly extended bodies will
convert into lines of small droplets within minutes. The coarsening rate of a co-continuous structure was treated by McMasters who showed that it can be up
Fig. 1-4. Micrograph oi a PP/PS blond, prepared on a two roller mil] at 200X and quenched during the blending process.
• » i ' ■ * ■ . * - ' ;
■ ^ ?
1'
*
Fig. 2-5. 77ie same falend as in Fig. 1-4, annealed for 10 minutes at 200rjC.
to 100 nra/s (3). Comparison of these characteristic times with the
solidification times in polymer blending processes, which can be up to minutes for thick-walled specimens, shows that these interfacial tension
induced morphological changes should be taken i.nto account in both predicting and analyzing polymer blend morphologies.
It is realised that the phenomenology of the blending process as given in this .introduction is far from complete. It is nevertheless possible to use it
to explain qualitatively some trends that are reported in literature. To make more quantitative predictions the successive phenomena like droplet burst,
thread break-up, coalescence etc. should he investigated for the complex rheological behaviour of polymer melts. In the next chapter we have tried to investigate to what, extend the present state of knowledge on the specific
phenomena can be applied to our problem. In the successive chapters some attempts are reported to make additions to the present state of knowledge in
order to improve its applicability to the problem of polymer blending.
REFERENCES
1. 01abisi, 0., L.M. Robeson and M ,T. Shaw, "Polymer/polymer miscibility" , Academic Press, New York, (1979).
2. F.lmendorp, J.J., in "Mixing in polymer processing", C. Rauwendaal ed. ,
Marcel Dekker, New York, (1986).
2. LITERATURE SURVEY
In this chapter a survey is given of the literature on the topics that were shown in chapter 1 to be of importance in the understanding of
morphology originsti on. Wo shaJ1 follow the chronological sequence of the blending process and discuss subsequently:
i ) deformation of two-phase systems under conditions where the
inter-iaci.al tension plays a negligible role;
ii ) the conversion of thread-like bodies into lines of small droplets by i nter facial tension induced motion;
iii) the stability of liquid droplets in flowing matrix liquids;
iv ) coalescence phenomena and their effect on blend morphology.
This survey wil 1 deal with papers on topics of various nature, some of
which are far outside the field dealt with in this study. In these cases only those facets that can be related directly to our problem will be quoted.
2,1. Zero interfacial tension systems
It was emphasized in the Introduction that miscible and inmiscible systems can show similar mixing behaviour in the initial stage of the blending
process. Under these conditions the deformation of the two-phase system is governed by hydrodynamic factors only, until a length scale is reached where processes such as diffusion or interfacial tension induced motion become
noticeable.
To treat some features of the deformation of heterogeneous systems, we
will make the basic simplification that the behaviour of domains in arbitrary flow fields can be constituted from their behaviour in simple shear fields on
the one hand and elongational fields on the other.
2.1.]. Simple shear flows
The most elementary flow in two-phase flow is the stratified flow, see
Fig. 2-1. Applying a shear stress on interface 2 will cause the components to flow with different shear rates. It can easily be shown, when the interface 1
2-1. A stratified two-phase system in simple shear flow.
cannot withstand tangential loads, that:
HiYi
L12 = ''l'l " n2Y2 '
in the absence of pressure gradi ent along the i nterfaces, while the
velocities of interface 1 and 2 are given by:
When the system Is not straLisfied but contains discrete particles of
second phase, the mathematical treatment is far more complex and the (2-1)
(2-2)
viscosity ratio is unity or smaller, the particles will deform along the streamlines of the matrix phase (1). An initially spherical particle will be
stretched out into an ellipsoid with length L, it can easily be shown that:
3g = A Y2 + i ) (2-3)
with
Y the total shear
R the droplet radius
The behaviour of systems with p >> 1 was treated first by Taylor in his
classical papers (2,3) on emulsion rheology and generalised to time dependent flows by Cox (4). ft was shown that, in spite of the fact that interfacial
forces are absent, the deformation of the droplet remains finite. Due to the rotation present in a shear field the particle will rotate and undergo a
periodically varying force (5). Cox (4) derived an expression for the deformation of such a particle:
D
L + 2p (sin (IJY':)) (2-4)
«ith L and B the length ami width of the droplet.
The finjteness of the deformation of such highly viscous particles might
explain the difficulty one encounters i.n trying to disperse a highly viscous
liquid in a lower viscous one.
The deformation of liquid domains with intermediate values of viscosity
ratio p for systems with negligible interfacial tension has not been solved
yet. It involves undoubtedly high deformations, a problem unapproachable with the commonly applied perturbation theories (see section 2.4). Even numerical
solutions have not yet been given.
2.7.2. Extensional flows
The ease of a droplet dispersed in a matrix undergoing an extensional flow
differs from the above by the [act that rotation is absent. This implies that
a droplet of any viscosity ratio can be deformed to any extent, provided that;
the extensional field is applied for sufficient long time.
Fig. 2-2. Principle of the four roller apparatus generating a plane hyper bolic flow field.
For small deformations the problem was tackled by Kalb (6), who derived a relation giving the rate of extension of a droplet in a hyperbolic flow field,
as generated in a four roller apparatus (see Fig. 2-2):
1 dL 5 * rn Ks
2R r7t"
=2JTT3
ye
(2-
5)1 m
with
y = the rate of extension of the matrix.
e m
For large deformations Kalb derived an analytical expression for the ratio of extension rates of droplet and matrix phase:
(^ (in
(ffj) - 3/2) ♦ 1}
(2-6)with
A = L/2R the stretch of the droplet.
Graphic representation of this equation is shown in Fig.
2-Y
e_.t
Fig. 2-3. Ratio of droplet extension rate and matrix extension rate against total extension of the matrix, for various values of the viscosity ratio p,
It can be seen that even for large values of p the elongation rate ratio
approaches unity for large deformation. The latter implies that in a stratified system (see Fig. 2-4) undergoing extension the extensional rates
of the two phases are equal, independent of the viscosities of the two
phases. In conclusion we may say that for the dispersion of liquids at high
viscosity ratio and zero interfacial tension, elongational flows can be more
effective than shearing flows.
Direct application of the above to the problem of polymer blending is
inhibited by the fact that, in general, polymer melts exhibit non-Newtonian
properties that are not incorporated in the relations given above. The general idea, however, that, in the initial stage of the blending process,
the mixture, miscible or not, can be viewed upon as a zero interfacial
tension system, is independent of fluid rheology.
2.2. Capillary instabilities
It was shown by Elmendorp (7) that diffusion will promote the
homogeniza-tion of blends of miscible polymers only when the length scale of the blend
is reduced to the order of 10 nm. To estimate at what length scale inter
facial tension induced motion will occur in inmiscible systems we will
briefly discuss some fundamentals of the phenomenon of capillary
instabilities.
Slender bodies formed by the deformation of spherical domains will tend to reduce their interfacial area under the action of interfacial tension. They
will attain a more or less cylindrical shape which is subject to instabilities. The stability of such liquid threads surrounded by an inmiscible matrix has
been the topic of investigation in numerous studies. To categorize these theoretical and experimental studies would go beyond the scope of our study.
and Han (9). We will only give the theoretical background and cite references
that are in direct relation to the problem of polymer blending.
The break-up of liquid cylinders was firstly mentioned by Plateau (10) who observed the break-up of water f 1 owing flown from orifices. it was the
classical work of Lord Rayleigh (11) that gave a thorough theoretical description of this surface tension induced process, neglecting however the
viscosity of either phase. Tomotika (12) extended Rayleigh's treatment to Newtonian liquid threads embedded in a Newtonian matrix. He assumed the thread to exhibit a small axisymmetric sinusoidal distortion, as is shown in
Fig. 2-5:
RoTHR
Fig. 2-5. Definition of z, R , R, a, and X in a sinusoidal ly distorted cylinder
R(z) = R + a sin (~) (2-7)
with
R the average thread radius
a the distortion magnitude z a co-ordinate along the thread
_X the distortion wavelength
From conservation of volume it follows that the average radius R depends on a as follows:
2
(2-8)
with R the initial thread radius.
From calculation of the surface area of the sinusoidally distorted thread it
follows that, when A > 2TFRO the interfacial area decreases when the distortion amplitude increases as is shown in Fig. 2-0.
A-=3R
rt'Fig. 2-6. Interfacial area vs. distortion magnitude for various values of
A-For this reason a liquid thread is unstable to distortions with a wavelength
larger than the circumference of the thread. Tomotika's theory shows that,
when A > 2TTR the distortion must grow exponentially with time: — o (2-91 a = a exp (qt) where a = the distortion at t = 0 ■n CA,P) (2-10)
a = the interfacial tension H = the matrix viscosity
The value of ft (A..P) can be calculated from Tomokita's original equations 38-40 which are too lengthy to be reproduced here. The most striking fact of
which depends on the viscosity ratio p. Assuming to thread Lo contain all wavelengths with equal initial amplitudes at the time of creation, the
distortion at which Q (X,p) is maximal, X , will lead to thread break-up.
— " —m ' The value of the dominant wavenumber, X = 'Irll /X , and the dominant growth
m o —in
rate, Q (X ,p) are plotted against viscosity ratio p in Fig. 2-7. For p ■+ 0
tiie growth rate f unct ion Q (_Xj p ) approaches nn i t v , wh i cli deviates t: rom
earlier calculations of this graph (13) where a value ft (X ,0) = 0.3 was
found. The val idity of our resu 1 ts can be deduced f rom the fac t. that our
results are in agreement with Tomotika's original results.
Fig. 2-7. The wavenumber and growth rale of the dominant wavelength.
From equation (2-7) it appears that thread break-up is completed when a = R. According to equation (2-8) this will be the case when a = 0.81 R.
The time necessary to reach this distortion follows from equation (2-9):
- £n (-0.8 R
-)
(2-11)Apart from this axisymmetric distortion, asymmetric distortions can be present. They grow slowly, however, and can only be noticed when the
axi-symmetric mode is suppressed. Rumscheidt and Mason (13) were the first in a line of authors to study the lnterfacial tension driven break-up of fluid threads created by uni-axial extension of a fluid droplet in a four roller apparatus ( 14 , 15, Id , 1 7 ) . Since a four roller apparatus deforms droplets into non-rotationa3 symmetrical ellipsoids (18), it is questionable whether perfect cylinders where the starting point of their experiments. It was shown, however, that both growth rate and dominant distortion wavelength arc
predicted satisfactorily by Tomotika's relations, when Newtonian fluids are
used.
A complication arises when the matrix surrounding the thread is deforming.
When it is extending the thread will be extending at the same rate (see Fig.
2-3) and the wavelength of an initiated distortion will increase rapidly to a
region where the growth is small i.e. the wavelength will not remain dominant. It is known from the work of Mikami (19), Tomokita (20) and Flumerfelt (17)
that this renders the thinning filaments very stable, until at ultrathm
filaments sudden rupture occurs. In the case of a shearing matrix the thread
obtains a helicoil shape. It was shown by Grace (35) that both wavelength and
break-up time increase dramatically when a velocity gradient across the
thread is present.
2.2.J. Viscoelastic systems
In spite of the good agreement between experiment and theory when Newtonian
fluids are used, the applicability of the above to the problem of polymer
blending Is limited in view of the complex rheological behaviour of polymeric
fluids.
The number of experimental and theoretical studies on capillary
instabilities in viscoelastic systems is limited and mostly confined to the break-up of jets in air (8,21,22,23). The general conclusion from these investigations is that the break-up of viscoelastic threads is delayed dramatically in the final part of the break-up process. To understand this remarkable phenomenon it must be realised that the deformation of the thread is, locally, mainly elongational and involves high stretches (18). It is a well-known fact ('Ik ) that vise, o el as tic f lui ds can exhibit extremely high
elongational viscosities at high stretches, slowing down the break-up
process. In addition to this, it was shown (8) that initial stresses, as can readily be present in a freshly created viscoelastic thread can increase its
break-up time many times when the relaxation time of the thread phase
competes with its break-up time.
2.3. Hydrod ynarnically induced burst of droplets
In the previous part of this chapter the stability of filaments was
discussed. It was shown that the growth rate of small disturbances (that can either be caused by thermal fluctuations (25) or flow instabilities (26)) is
reciprocal with the filament radius; thinner filaments will rupture faster than thicker ones.
Since, in a successful blending operation, the Length scale of the blend is continuously decreased, a stage will inevitably be reached where droplets
will break-up into two halves before elongation into a filament can be achieved. In this stage the influence of inLerfacial forces on the
deform-ability of droplets can no longer be neglected. Finally a situation will
ari.se where inLerfacial forces prevent the deformation to grow to such extent
that break-up can occur. It appears therefore that the droplet size in the final morphology is governed by the stability of a droplet against the
hydrodynamic forces induced by the deforming matrix fluid.
At all break-up processes, droplet break-up is a transient phenomenon by nature and prediction of the conditions leading to droplet burst inevitably
involves solution of the unsteady state equations of motion. Combined with the fact that droplet deformation is a moving boundary problem, the
mathematical complexity prevents any exact solution of the droplet deformation problem and the break-up phenomenon itself to be given. Perturbation analyses,
however, have lead to useful relations predicting the deformation of droplets in general flow fields. The strategy of this approach is that velocity
distribution, and corresponding stresses, within and outside a spherical droplet are calculated by solving the equations of motion with the appropriate
boundary conditions. These boundary conditions are determined by the fact that the flow field outside the droplet should, at large distances, match the undisturbed flow field. The velocity distributions at the droplet interface
should be such that the normal components are zero and the tangential components are equal. The differences in tangential stresses across the
interface should be equal to the interfacial tension gradient, which is zero when surfactants are absent.
The actual droplet shape can now be found by calculating the local
18
curvature of the interface by equating the di l:f erences of the normal components of the stress distribution to the pressure drop due to the inter
facial tension. Further steps can now be taken by repeating the procedure. Since the mathematical complexity increases exponentially with each step in
the perturbation procedure, only one or two steps have been reported. This inevitably bounds the applicability of these analyses to small deformations
only and will not yield useful relations in predicting droplet break-up.
To avoid this problem several authors attempted to carry out slender body analyses in which a perturbation is superimposed on a highly extended droplet shape (27,28,29). As will be discussed below in more detail, their analyses
proved useful in predicting the break-up of droplets.
Since the approaches in the various attempts to predict droplet deformation
and break-up are varying with the rheology of the fluids used and the geometry of the flow field assumed, they will be discussed in four sections
separately. Simultaneously the experimental results reported by various authors, applicable to the problem of polymer blending, will, be summarized.
2.3.1. Newtonian systems in simple shear flow v^ = Y.v- v = 0, vz= 0
The first experimental and theoretical study on droplet deformation was
carried out by Taylor (3). He considered the case of a Newtonian droplet
suspended in a Newtonian matrix and derived expressions for the deformation of droplets in a steady simple shear flow. From his study it appeared that
the behaviour of such droplets is governed mainly by two dimensionless groups:
i ) the Weber number v\ '{?Jo, being actually the ratio of shearing forces
and interfacial forces;
ii) the viscosity ratio p = ^ d ^ m '
Since his treatment is of first order and restricted to limiting values of
p and We, only asymptotic results for small deformations are obtained:
p » 1 ; We = 0(1) D = ~ (2-12a)
P - O ( l ) ; W e « l D = We C ^ ^ f ) ^ U b )
with L and B the length and width of the droplet respectively and the
deformation parameter D defined as D = L - B/L + B, as in Fig. 2-8.
Fig. 2-8. Definition of L. Bi end orientation angle 6 of a deformed droplet.
Bv equating the maximal pressure difference in the matrix fluid at the interface of the droplet to the Laplace pressure, 2o/R, Taylor derived a
relation predicting the conditions of break-up, neglecting the fact that the
droplet assumes a highly extended shape- upon burst:
- _ _ j ^ _ (_16p + 16, ( 7_1 3
Yc 2T] R U 9 p + 16J {" }
m r
with Y the shear rate at burst. In spite of the artificiality of equation
(2-13) it appeared to be valid in many studies (3,31,32,33,34,35).
Chaffey and Brenner (30) and Van Dam (36) extended Taylor's treatment to a
second order solution. Contrary to the expectation based on the fact that it
is a higher order solution, the predictions show, in general, worse agreement
with experiments than Taylor's prediction. A possible explanation for this
disappointing result may be that the perturbation analysis does not converge
for this specific problem.
An extension of Taylor's treatment to time-dependent flow fields was made
by Cox (4). This author derived relations predicting the time-dependent
droplet deformation limited to small deformations only. According to Cox the shape of a droplet is given by:
§ = 1 + cF t ~ = — - j /T1 , (2-14;
R pq dr 9r r r/K=l P q
with r the radial distance from the center of the droplet to the interface and
E F functionals that can be found bv solving the differential equations: pq
^-rr (cF ) = 7 7 — 17—7- i~7 '- ~ TT v f (.2.-1DJ
Dt ■ pq L9p 6 (p + 1 J pq We pq
where e are the components of the rate of strain tensor of the undisturbed pq
flow field. For a steady simple shear flow, solution of this equation yields:
_(19p
4 (p + 1}/((1C->P)' + (jfe> )
which equals Taylor's solution in the limits p
The orientation of the droplet is given by:
20
(2-16)
(2-17)
Although Cox's results are shown to agree with experimental data for small deformations, their applicability to the problem of droplet break-up is
limited since drop break-up involves deformations far beyond their region of
validity. To extend the theory of Cox to higher deformations Barthes Biesel
and Acrivos (37) carried out a second order perturbation analysis for
time-dependent flow fields. Their complex results predict droplet deformation
often worse than Cox's original results,
The analyses summarized above all consider the behaviour of isolated
droplets in a flowing matrix. In practical polymer blending conditions the disperse phase fraction is usually such that this assumption is no longer
valid. The influence of neighbour droplets on the deformation of a droplet was taken into account bv Choi and Schowalter (38). For moderately concentrated emulsions they showed that the neighbour interactions could
increase the deformation of a droplet up to 20%. Experimental verification of their predictions appear to be lacking. A fundamental assumption in all. the
analyses discussed above is that the difference in tangential stresses at the interface is zero i.e. pure interfaces are assumed. In order to incorporate
surface rheological effects Flumerfelt (,39) extended Cox's treatment to the deformation of droplets with well defined viscoelastic interfaces. His results
were verified by Phillips et al. (40). They show that Flumerfelts results can
be used to measure the surface-rheological properties of a mixture from the deformation of a droplet and the revolution time of the fluid within the
droplet.
To give an extensive review of the experimental work reported in the
literature on droplet deformation and break-up would go bevond the scope of
this part; only references that can be brought in direct relation to the
problem of polymer blending will be cited.
Apart from Taylor's original work the first experimental studies on the deformation and break-up of isolated droplets are reported by Mason's group
(34,41). The transient and steady deformation and orientation appeared to agree quite well with Cox's predictions (4). It should be noted that this L - B good agreement is restricted to the deformation characteristic D = - jr. The
shape itself was predicted rather poorly as is shown in Fig. 2-9. Although
experimental and theoretical deformations seem the same the actual shapes are quite different. In this respect it is even more surprising to note that Cox's
results appear to be valid up to D - 0.5, where higher order theories fail.
PREDICTED SHAPE
OBSERVED SHAPE
Fig. 2-9. Predicted and observed shape of a droplet in simple shear flow R = 10~3 m, n = ] Pa.s, n = 0 Pa.s, 0 = 30 mN/m, Y = 10 s~}
m d
The break-up of droplets appears to he less predictable. Rumscheidt and
Mason (33) differentiated several break-up modes on basis of the viscosity
ratio p. It appeared that only at 0.2 < p < 1 the droplets disintegrated into two halves at shear rates agreeing with Taylor's criterion, equation (2-11).
At lower viscosity ratios the droplet attained a sigmoidal shape (see Fig.
2-10), and small droplets are shed of from the tops. At viscosity ratios
exceeding p = 3 break-up did not occur at all, regardless of the shear rate
applied. The latter can be understood from equation (2-16) which yields a
deformation of D < 0.4 for p > 3, which can be regarded as insufficient for break-up. The physical explanation of this limited deformation is the same as
the one given to explain the finiteness of the deformation of droplets with
zero interfaci.al tension at p < 3 (section 2.11).
o o o
o o o
Fig. 2-10. Sigmoidal shape of a droplet while tip-streaming.
Experimental data on the relation between critical shear rates (or better
Weber numbers) and viscosity ratio have been reported by several authors. The
record breaking range of 10 < p < 10 was reported to be studied by Grace
(35). His results are shown in Fig, 2-11. For p < 0.1 two sets of data were
reported. The horizontal curve represents size reduction via tip-streaming, where small droplets are shed off from the tops, while the upper curve
represents the actucal fracture of the main droplet into two daughter
droplets. The straight line dependence, on a log/log scale, allows us to fit
a power function to Grace's data for main fracture:
We„ = 0.16 p~0'6. (2-18)
which is in fair agreement
from a slender body analysis by Acrivos and Lo (39):
We„ = 0.2 p~2/3
rfith a relation that can be derived theoretically
(2-19) We. cr 1000 100 10 1 0.1 rotational shear_ (couette) irrotational shear (<troll)
lo'itf icTio'icTio' io" io' io" iö
V I S C O S I T Y R A T I O Tf|,Fio,. 2-11. Weber number at burst
■ simple shear,
viscosity ratio p. plane hyperbolic.
2.3.2. Newtonian systems in hyperbolic How, = "Vh
While in simple shear flow the velocity gradient is perpendicular to the direction of flow, it is aligned in the direction of flow in extensional
flows. This bounds the applicability of these types flow as extension and acceleration of the matrix are inseparable. It is for this reason that
droplets flowing through contractions experience extensional flow only for a short while i.e. the flow is instationary in the Lagrangian sense. To avoid
this experimental inconvenience Taylor used a four roller apparatus, generating a stationary hyperbolic flow field as long as the droplet remains
in the center of the flow field. He showed theoretically that droplet deformation and break-up can be predicted from equation (2—12b) and (2-13) by
replacing y by 2>' .
The generalized theory of Cox (4) yields the time-dependent response of
the deformation of a droplet to a step in extensional rate, for all values of
p:
D ( t
» ■
2Weh ( ü p ^ - i f »
1- « p <- wk-
h<*
h» <
2-
20>
with Weh = VhK / 0 .
The validity of equation (2-20) in predicting droplet deformation was
shown experimentally by various investigators (2,34,30). It is interesting to
note that equation (2-20) indicates a .large difference between shear and
extensional flows. As hyperbolic flows do not permit the droplet to rotate,
even highly viscous droplet can be deformed to any extent in hyperbolic flow.
Given time they will, according to equation (2-20), reach even higher deformations than low viscous droplets. The rate by which they attain their
equilibrium deformation is smaller however.
As in simple shear flow the most extensive work was carried out by Grace (35). The experimentally determined critical Weber number is plotted against
the viscosity ratio p in Fig. (2-11). From these results it appears that the anomalous behaviour above p = 3, as noticed in simple shear flow is absent.
2.3.3. Non-Newtonian systems in simple shear t low
The experimental and theoretical results on droplet behaviour have limited
applicability to polymer blending processes when they are limited to Newtonian
liquids. In shear flow, molten polymers and polymeric solutions inhibit, in general, shear rate dependent viscosities and normal stresses. Attempts have
been reported to predict droplet behaviour of non-Newtonian systems by
adjusting the Newtonian relations by substituting the shear rate dependent
viscosity and incorporating the elastic effects into the inter facial, tension. The latter was proposed by Van Oene (42) who derived, on thermodynamic grounds, a relation to account for the elastic effects:
V d = 0 +!( ( T1 1 -T2 2 ' d - ' h i " T2 2 'm' ( 2-2 1 1
It appeared, however, not possi ble to derive semi-empirical relations pre
dicting droplet deformation and break-up along this route (43).
A more rigorous treatment to the problem was given by Van Dam (44). This
author assumes the fluids to behave according to the second degree
Coleman-Nol] model. The relation between stresses and deformation rates are, according
to this model, given by:
I = nol * tt2 It I + ~{^V (2-22)
with
e the rate of strain tensor and o"= i[i + I|J ■ a^ = %ty .
Physically this model implies the viscosity nn, the first normal stress coefficient, >!-, , and the second normal stress coefficient, lp„, to be constant Van Dam succeeded in deriving analytics] expressions for the shape of a
droplet, in a simple shearing matrix through a first order perturbation
analysi s: r r i „ 19p + 16 n( 2 ) , , . ,,, Ö" - [1 - We ■ 7 5 ' -rr • P ( u ) • s i n 2tt + K 48 ( p + l j 2 {72\\i) * cos 2* 4- M0i * Y7 |}], ( 2 - 2 3 ) 1th P , ( u ) P <2 )( p ;
=
_
-i 0 ■ 3 ( l -- 3|j2) ■ M2) , and U = cos (9), while: M, = = [93^-p- (47 - 12a.) + 3 3 , B j (79 4- 26a.) + 1 1680 Cp + 1) (3d'P + 3m) -3,-3 (2 (425 p2 + 910 p + 434) - a (1225 P 2 +■ 2 1 2 0 p + 928)"} + Md m F ' ' -Hi ' - 32 f (775 p2 4- 1580 p + 892) - 2 ^ (575 p 4- 940 p 4- 476)}], [-432p2 (83 4-302a ) - 43,3 (73 4-277a.) + dr —m d m ~-Q 4- 3,-3 p }(S5p" 4- 544p + 508) 4- 0^ (515p 636p 932")} 4-+ 832 (2 (5p2 4- 32p + 32) 4- a (65p2 + 66p 4- 116)}], = \U 22'm,d = _Ï Ud w U.h B = 4 m,d 2nQ Y n0 ~ -" m, d m , dthe Weissenberg number with r, $ and 9 spherical co-ordinates.
It should be noted that this solution is of first order in We and 3 and upon substitution of ty-y = iK = 0 and 6 = 90°, reduces to the shape of the
equatorial plane in Newtonian systems:
| = 1 + We sin (2*) (2-24)
which is equivalent to Taylor's solution for simple shear .flow. Apart from being limited to small deformations the validity of these
relations is restricted to small values of the elasticity parameter 3
■ m,d
This prevents, a priori, the prediction of large influences of fluid
elasticity. The shapes of the equatorial planes of droplets in viscoelastic
matrices are shown in Fig. 2-12.
Fig. 2-22. Predicted shape of a viscoelastic droplet in a viscoelastic simple shearing matrix, We = 0.33 and X„„ - T = 0
1
e = e . =
ot
e = 0 , 3 , =
1.6
rod RI d
2 3 = 3 . = 2.6*6 =1.6, 3 . - 0
m d m a
It can be seen that, within the range of validity the elasticity exhibited by the droplet phase plays a minor role. The elasticity of the matrix phase was predicted to increase the orientation angle 6 considerably.
Experimental verification of these relations appears to be lacking and comparison to earlier experimental data is prohibited by the fact that Van Dam's results are restricted to relatively rare second order fluids. More
realistic model liquids, like the Boger fluids (45,46), should be applied to check Van Dam's results.
The first scouting experiments on droplet behaviour Ln viscoelastic
systems was carried out in Mason's group (21,47). These experiments were limited to the behaviour of Newtonian droplets in viscoelastic media. 1.1 was
shown that matrix elasticity tends to increase the orientation angle 6 considerably, which is in agreement with Van Dam's results.
The influence of a shear thinning behaviour of the matrix fluid lias been studied extensively by Flumerfelt (48). This author measured break-up shear
rates as a function of droplet size in a simple shearing matrix generated i.n a parallel band apparatus (see Fig. 2-13) and showed that his data could be fitted to the relation:
with C. and C„ empirical constants.
( 2 - 2 5 ;
^
Fig. 2-13. The parallel band apparatus generating a simple shear field
The parameter Xj is a measure for the rate of shear y at which the
viscosity starts deviating from its zero shear value and can, according to
Bird and Carreau's five parameter model be expressed like (49):
(2-26)
28
a' - ~ , n the power law index for large Y.
L,(G^') = the Zeta-Riemann function.
Calculating the lim' of equation (2-25) yields a value of the droplet size below which break-up appears to be impossible, regardless of the shear rate
applied: A o
R. = ^ - C. (2-27,
I 2nd l
Plotting the empirical constant C, against the viscosity ratio p yields a rather good correlation for the systems used.
The existence of such a lower limit in droplet size in shear thinning matrices can be explained from the triviality that at increasing shear rate
the matrix viscosity decreases and the viscosity ratio increases. When a viscosity ratio of p = 3 is reached drop break-up will cease in simple shear
flow. For a power law matrix fluid with n = kly| , the shear rate at which p = 3 can be calculated to be:
n, l/(n-l)
M
3= C^) (2-28)
yielding a limiting droplet size according to equation (2—13) :
n , 1 /( n - l ) , /, , .
R, - ^ - - ( f )
>i
Uln-
1](2-29)
d
Applying equation (2-29) to Flumerfelts results yields limiting values that
are in the same order of magnitude as his observed values. Quantitative agreement is neither expected nor found since the effect of fluid elasticity
is not incorporated in equation (2-29).
"*e \
2,3.4, Non-Newtonian systems in elongational flow v =—r- Xl v = ~-r- y; x 2 y 2 -v = Y z
z e
The behaviour of viscoelastic two-phase systems in extensional flows was rarely studied. It was reported by Flumerfelt (48) and Zana and Leal (50)
that, when Newtonian droplets are broken-up in hyperbolic flowing, shear-thinning matrices, no droplet size was detected below which dispersion was
impossible. The absence of such a lower limit can readily be understood from the fact that in extensional flows no upper limit in viscosity ratio at which
droplet break-up is achievable is present, as is emphasized in section 2.3.2
A rather extensive study on the deformation of droplets in viscoelastic systems flowing through a conical dye has been carried out by Chin and Han (51,52). Starting from the second degree Coleman-Nall model they derived an expression for the shape of the equatorial plane of a droplet via a first order perturbation theory:
f = 1 + We Z0 F, (u) + We ZQ [Z0 1 P, ClO + 20 2 f:,4 CP)] +
+ 3 We 2, P.-, (U) + Z„ P, (u) (2-30) m 1 2 2 4
with Z, , Z-., Z„, , Z..,-, complicated functions of the system parameters and P
1 2 01 02 " n and S . as in equation (2-23). Por the Newtonian constitutive equation this
m,d ' ' expression reduces to:
| = 1 + | We sin (240 (2-31)
implying that in this type of axisymmetri c flow the droplet deformation i.s equal to 1.5 times the deformation of droplets in simple shear flow with equal gradient.
Like in simple shear flow, the relations are only capable of predicting small influences of fluid elasticity. In the region of applicability good agreement with experimental values was reported.
In c one 1 us ion we can say that extensional flows appear more effective in dispersion processes than simple shear flows. The dependence of the critical Weber number on the viscosity ratio is less pronounced while no upper limit in viscosity ratio at which break-up is possible is observed. Furthermore it is a well-known fact that elongational viscosities are less sensitive to deformation rate than shear viscosities. This favours the dispersion of droplets at high deformation rates.
The effectiveness of elongational flows is often concluded from the fact
that droplet behaviour is equal when y = 2y, = 1.5 Y , which is essentially wrong. Efficiencies should be compared on basis of the power P required to deform or break-up a droplet. Since q = 0.25 T), = 0.33 n (36), the powers
generating equal droplet behaviour relate to eachother as P = P, '= fP . s , h e This implies that both simple shear and hyperbolic flows are TT times as efficient as axisymmetric elongational flow.
2.4. Coa1escence
As was summarized l.n the previous par t, the flow induced break-up of liquid domains was studied extensively with the aim to relate the rheological
properties of components to the result of a mixing operation. It should be realized, however, that only at extremely low disperse phase concentrations
collisions of droplets can he avoided and their influence on blend morphology neglected. When the number of collisions resul, t trig in a coalescence is in the
same order of magnitude as the number of breakages, it is evident that the coalescence phenomenon has an effect on the time average length scale of the blend.
■ Supposing for a start that droplets whose sizes are above the maximal stable droplet size, break-up at a total shear of the matrix v , It can then be calculated that the number of breakages per unit volume and time can be
expressed as:
N, = r*- - : (2-32)
with C the disperse phase volume fraction.
From Smoluwchofski 's work we can obtain the number of collisions per unit volume and time (54):
fr2
N = y r y (2-33)
T T I T
From equations (2-32) and (2-33) we learn that the disperse phase fraction at which these number rates are equal can be expressed by:
C = 2 -b=c 8 Y
According to Grace (35) the critical shear is 30 at least, yielding a C,_ of 0.012. At higher disperse phase fractions the number of collision exceeds
the number of breakages.
This short introduction is an extreme in simplification. It allows us, however to conclude that the phenomenon of shear induced coalescence might be
an important parameter in governing the morphology resulting from a blending process. In order to introduce the general way of quantifying the influence of coalescence we will discuss some papers In the H e l d of emulsion
this topic in polymer technology manyfold.
The relation between droplet size and disperse phase fraction, in agitated vessels, was reported by various authors who fitted their data to various empirical relationships. The simplest of these is:
R = A (1 4- B'°C) W e "0 - 6 (2-35)
with 2 „
We„ - the tank Weber number = - ^ d = blade diameter
p - density
N = stirrer frequency
where A varied between 0.06 and 1 and B between 3.75 and 9 (55,56,57,58), indicating that the droplet size can increase considerably upon an increase in disperse phase fraction. A review of the more complicated empirical relations between droplet size and disperse phase fraction can be found in literature (58).
The unknown character of both the flow field and the coalescence process itself prevent any exact solution of the balance between coalescences and
breakages. Therefore stochastic approaches have received considerable attention. Since the continuous break-up and coalescence processes are responsible for the mass transfer in flowing two-phase systems, a good description of the process is indispensible for optimizing industrial. processes, such as extraction etc..
The distribution of both droplet size and solute cocentration are found by
solving che "population balance equation" of which various kinds are proposed in literature (59,60,61). In these integro-differential equations the break
up and coalescence behaviour must be substituted. This is in general done via probability distributions: N, (v), the number rate of breakages of droplets of volume v; 3(v,v') the number probability density distribution of
volume v of daughter droplets formed on the breaking of a droplet of volume v'; V(v) the average number of droplets of volume v; N (v,v'), the number
rate at which a droplet of volume v collides with a droplet of volume v' per drop of volume v' and P (v,v') the probability that a collision between
droplets of volumes v and v' results in a coalescence. Theoretical modelling of these functions were carried out by Coulaloglou and Tavlarides (58,62).
Experimental determination of the constants that complete the model can be done by chemical ways (following the distribution of reagents) (61,62,63,64,
32
65,66,67,68), physical methods (spreading of a dye, levelling off of density differences) (59,68,69,70), determination of the response of droplet size to a stepwise change in mixing intensity (71,72) and other methods (73,74).
Knowing the probability functions, the population balance equation can be solved to give both the final and the transient distribution of droplet size
and solute concentration. Interested readers are referred to specific literature on stochastic modelling of two phase mixing (75,76,77,78,79).
Although, in principle, a similar approach can be used in trying to solve
the yet untackled problem of predicting droplet sizes in polymer blending operations, little effort has been spent along this line. The only attempt.
undertaken sofar was reported by Tokita (80,81), who modelled coalescence and break-up phenomena on energetic grounds and showed that this could explain qualitatively the relation betv/een disperse phase fraction and particle size
in EPDM/NR blends.
2.5. Polymer blend morphologies
It would be an enormous task to review all literature on polymer blending and, therefore, we refer to several text books on this topic (82,83,84). It
would not have been within the scope of this study anyway, since no clear correlation appears to be obtainable, neither between the theoretical
micro-rheological considerations on the one hand and the more technological papers on the other, nor amongst the various papers themselves. Large discrepancies
between practical conditions and basic assumptions, required to render to mathematics of the theoretical approaches tractable, inhibits the former,
while the latter is thwarted by the fact that neither the materials used nor the blending conditions applied are comparable from one paper to the other.
We will therefore limit this survey to some qualitative trends that are
generally accepted to be independent of the exact nature of the components or the specific features of the blending process.
One of these generalties is the fact that the average domain size
increases with increasing disperse phase fraction. This was ascribed to the phenomenon of coalcescence by Tokita (80,81), as was mentioned above.
Further increase in disperse phase fraction can lead to a situation where the disperse domains are in continuous contact, thus forming a co-continuous
network. Still further increase of Lhe disperse phase fraction can result in
a reversal of phases where the ini tial con I. Lnuous phase becomes the disperse phase. The problem of phase i nversion was stud Led extensive]y by Mjroshn Lkov et al. (85) and Avgaropoulos et al. (86). These authors report that the
region where one of the components can be the continuous phase extends when its viscosity increases, indicating that in a binary mixture the least
viscous component forms the conti nuous phase over a Larger composi t.j on region than the more viscous one. This can be explained in terms of a better
deformabi1ity of the disperse phase enabling it Co deform Lnto highly extended bodies which, upon coalescence, form the continous phase. Fig. (2-14) shows graphically the phase continuity as a function of composition
and viscosity ratio for a blend of polvhutadiene (PBD ) and ethylene-propylene rubber (EPDM).
i
TORQUE ^
RATIO 2
1
EPDM -8
PBD 'X
0 (125 0.50 0.75 1.0
WEIGHT FRACTION PBD
Fig. 2-14. Phase continuity of a EPDM/PBO blend (after reference 86).
An interesting application of the latter arises when two polymers are blended that show a distinct difference in viscosity-temperature or viscosity-shear
rate relationship. A change in either temperature or shear rate will change the viscosity ratio and can change the morphology from a dispersion type into
a co-continuous type or might even induce phase inversion. When the exact relation between phase continuity and viscosity ratio is known, the process conditions can be a powerful tool to control the phase continuity of the
polymer blend.
For constant disperse phase fraction the droplet size is reported to
depend on the viscosity ratio. Starita (1) reported that for p smaller than unity a fine structure was obtained, while for large viscosity ratios coarser
structures are observed. This observation was extended by Kufeznev's group (85,87,88) who observed a minimum in droplet size for p = 0.3. These results were explained by noting that for p = 0.3 a minimum value for the Rayleigh
wavelength (see section 2.2) is predicted. This explanation cannot he generally valid since the burst of droplets in two halves appears to be most
effective at p = 0.3 too. Apart from this the dependence of dominant Rayleigh wavelength on viscosity ratio is far less strong than the observed relation
between droplet size and viscosity ratio as is shown in Fig. 2-15.
Fig. 2-15. Droplet size vs. viscosity ratio for a blend of PMMA and PS (after reference 88). The dashed line represents the viscosity ratio dependence of the size of droplets originated from the break-up of equally sized cylinders, 'lhe absolute value of the latter was chosen such that the minima of the two curves coincide.
When composition and viscosity ratio are kept constant, the droplet size
An Increase in shear rate or energy dissipation results in a decrease in average droplet size.
When composition, viscosity ratio and shear rate are kept constant, the droplet size is reported to depend on the difference in surface tension (90). It should he mentioned that the correlation should be made with the inter-facial tension rather than with the differences in the surface tension. The authors di.d not measure the interfacial tension directly. The general result
of their work indicated that an increase in surface tension differences resulted in an increase in droplet size. The latter might he one of the reasons to explain the compatibilizing effect of the addition of block-copolymers to a polymer mixture. This was shown by Noolandi (91) to decrease the interfacial tension considerable. The effect of the addition of block-copolymers to a polymer mixture was studied extensively by Ueikens' group (92,93). The result of these studies indicate that the particle size is reduced by the addition of the block-copolymer of either component of the mixture.
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SCOPE OF THE PRESENT STUDY
The scope of the present study is to obtain some understanding of
morphology origination in polymer blending processes. A problem like this can be, and has been, approached by carrying out systematic experiments,
considering the blender as a black box and attempting to correlate the output of the blender (the blend morphology) to its input variables (properties of
the system, its components and the blending conditions). In some cases these
studj.es have been given a theoretical basis by svmplil'ving the geometrical
features of the blending device or the rheological behaviour of the components. The gap, however, with fundamental, theoretical, work on the
specific phenomena occurring in flowing two phase systems is still large. We can try to bridge this gap from both sides: either by extending the available
fundamental knowledge on phenomena that are relevant to the problem, towards the conditions of the polymer blending process, or by symplifying the
blending conditions towards the assumptions of the fundamental studies. It is only when these approaches have merged that a full understanding of the
problem is within reach.
The aim of this study is to initiate this approach and attempt to quantify
the polymer blending process as far as possible. it may be obvious from the literature review that the influence of fluid elasticity, exhibited by
almost any polymer melt, on the behaviour of droplets is not fully known as yet. This influence will be investigated by systematic variation of fluid
elasticity (chapter A ) . The behaviour of droplets in extensional flowing matrices has only been established for the stationary case. Chapter 5 will
discuss some scouting experiments on instationary flows. The behaviour of droplets from the ultimate in viscoelasticity, viz. solid elastic materials,
will be discussed shortly in chapter 6. The break-up of liquid threads which can be a transient droplet shape upon burst under certain conditions will be
discussed from the Newtonian case via viscoelastic threads towards molten polymer systems.
The current thinking on droplet coalescence will be used to investigate the influence on polymer blend morphology in chapter 8.
It will be shown that on one point the two approaches have merged and the gap was bridged. It appeared possible to predict droplet size in polymer
blends by using low disperse phase concentrations and a highly simplified blending geometry.
DROPLET DEFORMATION AND BREAK-UP IN SIMPLE SHEAR FLOW
11 was mentioned in chapter 2 that the colli siun of' disperse domains and
their influence on the formed morphology can be neglected at volume concen trations well below 1%. Under these conditions the deformation and break-up
of droplets are solely determined by the balance between the cohesi ve Lnter-facial forces and the disruptive hydrodvnamic forces. Although it might seem
that these low concentrations are seldom encountered in the blending of polymers it should be realised that additives like anti-oxidants, stabilizers
and pigments are sometimes introduced as jnmiscible liquids at these low concentrations. Apart from this, knowledge on the "isolated droplet size" is needed to estimate whether phenomena such as coalescence have been present
and can therefore be of help in obtaining a first insight in the phenomenolo gy of the blending process at higher concentration.
As was pointed out in the literature survey, droplet burst Is induced when a critical shear stress is exceeded. Alternatively, when a two-phase
system is subjected to deformation the liquid domains will be reduced to a size where the inter facial forces can just balance the hydrodynamic forces.
It would therefore seem obvious to Limit ourselves to the problem of break up. However, since no break-up is possible without proceeding deformation of
the droplet, additional information might be obtained from the deformation and orientation of droplets in simple shear flow. In view o I the apparent
lack of theoretical results on viscoelasti.c systems one Is compelled to turn to experimental approaches to establish the influence of fluid elasticity on
droplet behaviour.
The strategy chosen was to study the response of droplet behaviour on a systematic variation oi the first normal stress difference, a strong measure
of the fluid elasticity. A prerequisite in this approach is that, when fluids of different elasticities are compared, the other properties that can
influence droplet behaviour are equal. This condition excludes the use of polymer melts, since their elasticities are coupled on their viscous
behaviour, viz. via the molecular weight. An additional degree of freedom is obtained by choosing polymer solutions as model liquids. By dissoIving
decreasing concentrations of increasing molecular weight material series of liquids can be made with comparable viscous behaviour and different levels of
