Phase behavior of flowing polymer mixtures

Fluid Phase Equilibria,

30 (1986)

367

367-380

Elsevier Science Publishers B.V., Amsterdam - Printed in The Netherlands

PHASE BEHAVIOR

OF FLOWING POLYMER MIXTURES

MAl-THEwTlRRELL

Department of Chemical Engineering and Materials Science University of Minnesota Minneapolis, MN 55455 USA

ABSTRACT Polymeric mixtures are frequently processed under flowing conditions. the flow modification

Observations

of

of the apparent phase behavior of polymer fluids have been made for thirty

years, but a coherent data base and theoretical interpretation are still lacking. An attempt to organize all the observations

systematically

is made here, along with a critical examination

status of the data, theory and technological

of the current

significance.

INTRODUCTION Mechanical stresses imposed on polymer mixtures can change the apparent phase behavior of polymer mixtures.

The adjective “apparent” is introduced as a signal that the effects to be

discussed in this article may not be on the same footing as “equilibrium” under quiescent conditions.

phase behavior observed

The extent of applicability of analyses based on equilibrium

thermo-

dynamics to flowing polymer systems is a significant and currently pertinent question. The phenomena

are certainly real...apparent

consolute points have been observed to shift

by as much as 30°C with application of modest mechanical action...but they remain to be organized into a coherent, widely-understood

pattern. Their practical significance

is great, as well. Multi-

component polymers are the norm in technology and they are frequently subjected to vigorous mechanical

action (pumping, mixing, processing, extruding, molding, etc.). The relevant phase

behavior, with its implications

for the performance of the material, is that produced by the

mechanical processing experienced by the polymer. The organization

of this article is to present first a survey of the important phenomena,

along with the techniques used to observe them. The attempts at theoretical description will then be discussed, finishing with a critical assessment of the current status important research areasand technological implications.

0378”3812/86/$03.50

0 1986 Elsevier Science Publishers B.V.

368 MANIFESTATIONS PHASE BEHAVIOR

OF FLOW-INDUCED

MODIFICATION

OF POLYMER

The earliest work in this line is that of Joly and coworkers (Barbu and Joly, 1953), and later Go and coworkers (Kondo, et al., 1969), who observed shear induced precipitation aggregation

of proteins, such as horse serum aIbumin,

and synthetic polypeptides.

and

Shear rates of

several hundred reciprocal seconds are capable of rendering a clear quiescent protein solution irreversibly into a cloudy mixture with visible inhomogeneity polymer in the native state adopts a specific configuration biological activity.

between the polymer and the environment.

phase transition analogous to thermal denaturation

where application

of thermal energy disrupts a configuration

With solutions of synthetic macromolecules, early, of observations the application

(e.g. a-helical)

associated with its

Shearing disrupts the interactions that stabilize the specific configuration,

creating new interactions shear-induced

(Tirrell, 1978). In these cases, the

of precipitation

It is fair to think of this

of protein (cooking of egg white)

irreversibly.

there are numerous citations, some quite

of microgels, fibers, aggregates and crystallites

of flow (Rangel-Nafaile,

induced by

et al., 1984; Eliassef, er al., 1955; Lodge, 1961; Peterlin

and Turner, 1965; Peterlin, er al., 1965; Steg and Katz, 1965; Frank, et al., 1971; and Fritzsche and Price, 1974). These are materials with random coil configurations change in interactions

between the polymer and the solvent environment

not SO obvious as with protein denaturation.

or the ability to interact, between

so that they aggregate in the observed ways. The coil stretches and elongates

along the principal axes of the deformation. with statistically random configurations, that, in homogeneous

A major difference between the flexible polymers,

and the biological polymers with specific configuration,

systems, the deformed configurations

former case so that the stored elastic energy is recoverable, in the mechanical

due to mechanical action is

What clearly does happen is that the configuration

changes under flow. This may, in turn, modify the interactions the macromolecules

in the quiescent state so the

is

readily revert to random coils in the and indeed manifests itself as elasticity

behavior of the fluid.

The apparent irreversibility

of several of the observed phenomena

(Metzner and Wissbrun,

1984; Eliassef, et al., 1955; Lodge, 1961; Peterlin and Turner, 1965; Peterlin, ef al., 1965; Steg and Katz, 1965; Frank, et al., 1971; and Fritzsche and Price, 1974) is probably a kinetic effect due to the slow dynamics in entangled polymers. tion of a polymer-containing

Connection

fluid and the configurational

central issue in the development

between the global mechanical deformadeformation of the macromolecules

of polymer kinetic theory. An authoritative

the book by Bird and coworkers (Bird, et al., 1977). Generally reasonably

successful molecularly

is the

source on this point is

speaking, while there are some

based rheological constitutive equations, there is little

information pro or con on their ability to predict molecuIar deformation. suggest that the simplest models greatly over-predict

Those data that do exist

the molecular deformation

Our main attention here will be directed toward. observations

(Bird, et al., 1977).

of how reversible formations

of a second fluid phase are affected by flow. Several figures can be used to illustrate the phenomena.

Figure 1 is from the work of Rangel-Nafaile,

et al. (1984). One sees their major

increases in the range of the two phase region, that is, increases in the upper consolute temperature

369

with increases in shear stress. These observations visual observations

have been made, as have most in this area, by

of turbidity with the unaided eye. Instrumental

measurements

of turbidity or

light scattering have been made on occasion (Hashimoto, er al., 1986; Sondergaard and Lyngaae-Jorgensen, conditions.

1986). One prepares a solution and scans temperature under steady flow

The consolute curve is assumed to correspond to the cloud point curve. On some

occasions, a coincident

discontinuity

in the viscosity is seen and taken as corroborating

evidence of

phase separation (Wolf and Jend, 1979; Schmidt and Wolf, 1979).

30-

v

1000

0 2000

25-

0

4000

20-

15T,T = 0

10

I 0

I

T

I_

0.05

0.1

wPs Figure 1. Phase diagram of PS in dioctyl phthalate various stress levels. Curves drawn ?re schematic. (From Rangel-Nafaile, el al. , 1984)’ ”

Expansion

of the two phase region is not seen universally.

shows the opposite in the system:

polystyrene(PS)-terr-butyl

to be depressed in temperature, expanding the miscible region. systems of Figures 1 and 2 may behave differently.

acetate. The consolute curve appears It is not immediately

such as these are not unique to polymer-solvent

consolute curves (Sondergaard

and Lyngaae-Jorgensen,

and Carr (1983) on PS-polyvinylmethylether

clear why the

induction by shear outnumber

is found to be promoted (see also Rangel-Nafaile,

Observations

Figure 2 from Wolf (1980)

From this author’s recent review of the

literature, it seems fair to say only that reports of immiscibility where miscibility

at

those

et al., 1984). mixtures nor to upper

1986). Figure 3 shows data of Mazich

(PVME), which exhibit an LCST. Application

of

rather low shear rates can displace the consolute curve upward by about 7”C, in that case widening the zone of miscibility.

Phase diagram of PS in tert-butyl acetate (from Figure 2. Wolf, 1980). Solid line is quiescent consolute curve. Points. with error bars are cloud points at the indicated shear rates in reciprocal seconds. Shape of flow-modified consolute curve is thought by Wolf to be like that of Figure 4. T (OC)

::,“I

;“lESCEN,

0 SHEAR

120-

llO-

wPs Figure 3. Phase diagram of PS in PVME under quiescent and shearing conditions (from Mazich and Carr, 1983).

unique to shear flow. Ferguson, et al. (1986) have assembled a

Nor are such observations collection of observations undergoing

of structural changes which may be phase changes, in polymer solutions

pure, simple extensional

Rangel-Nataile,

et al. (1986).

MODELLING

FLOW-MODIFIED

flow. Observations

such as these have also been reviewed by

PHASE BEHAVIOR

Two generic modelling approaches have been deveoped to predict the modified phase er al.,

diagrams of the previous section (Rangel-Nafaile,

1984; Wolf, 1984). Each has been

directed one of the two broad categories of phenomena mentioned: miscibility promotion.

induction and

Implicit in all of the modelling efforts to date is the assumption that one is

dealing with phenomena under near-equilibrium phenomena

immiscibility

is to combine an equilibrium

conditions.

The general approach to both class of

expression for the free energy of mixing with terms

aspiring to account for the flow modification. A. Modelling

of immiscibility

The fundamental

induction

by flow,

idea behind this class of models is that the free energy of the flowing

polymer system is raised by the elastic energy stored in the elongated polymer configurations. procedure adopted by several (Rangel-Nafaile,

The

et al., 1984; Vrahopoulou and McHugh, 1984;

1986) has been to write the following sort of expression for the free energy of mixing of n2 moles of stagnant polymer and nl moles of solvent to form a flowing solution: AGlvt = RT[nl Qn( 1-g) + n2Qn+ + x9( l-$)N]

The first term is the Flory-Huggins energy.

+ $tr

2

(1)

mixing term (Flory, 1953). The second is the stored elastic

In Equation (l), R is the gas constant, T is the absolute temperature, N is nl + mn2, with

m the ratio of polymer to solvent molar volumes, $ is the volume fraction of polymer, x is the interaction parameter, v is the molar volume of solvent and 2 is the deviatoric stress tensor. The Flory- Huggins description of the equilibrium stage of development

mixing is certainly an adequate description at this

of the field, though it clearly is inappropriate

of Figure 3, exhibiting LCST behavior.

for some systems, such as that

The physics of flow are embodied in the second term.

This particular form for the stored elastic energy is taken from the work of Marrucci (1972) and is derivable, as we shall show, from the statistical mechanics of stretching a random coil polymer (Bird, et al., 1977). The partition function for a random coil polymer, or more precisely the configurational probability

distribution

function for a random walk of n steps of length L, giving the

density for finding a vector r spanning the two ends of the walk, is given by a simple

Gaussian distribution: )3/2 e -3tr(rr)/2nl? P(r,n) = (- 3 27cnL2 where tr (rr) is the trace of the dyadic product of the end-to-end vector with itself. Clearly,

(2)

372

<

+r(rr)> =nL2

(3)

where the brackets represent an average over the distribution P(r,n) and P(r,n> satisfies the diffusion equation: aP -= an

L2 7 V2P

(4)

with: for r f 0

P(r,O) = 0

for all n > 0

(5)

The free energy associated with holding the two ends of Gaussian chain of Equation (2) at some fixed r is: GS = - kTfin P(r,n)

(6)

or, substituting

from Equation (2):

2

tr (rr) + constant.

Gs =

2

-

kT

(7)

nL2

Defining an effective spring constant for the Gaussian chain as:

H25

(8) nL2

we arrive at: GS = 4 H tr (rr)

+ constant.

The connection between the configuration be made by imagining

and the polymer contribution

to the deviatoric stress can

that the Gaussian chain really is a Hookean spring spanning r. This is the

physics of the linear elastic dumbbell model of polymer rheoIogy (Bird, er al., 1977). The spring, the chain, wilt have a tension, F, associated with a given r:

representing

,

F.dr=dGS

or, differentiating F=Hr

.

(10) according to Equations (9) and (10):

373

This tension will contribute to the deviatoric stress T across a surface oriented perpendicular

to unit

vector i according to:

(11) where vN is simply the volume of the system. the chosen direction,

The piece, i

and F is the force in that spring.

l

r, represents the projection of r along

Thus,

,

vNz =Hrr

(12)

and GS =

+%J (T)

(13)

which is precisely the last term in Equation (1). With assemblies of flexible polymer molecules, configurational

quantities such as rr must be replaced with averages over distribution

which are themselves

flow dependent.

Equation (13) holds nonetheless

functions

for the isolated Gaussian

chain. The book by Bird, et al. (1977) is the best source on issues related to this. This is a reasonable qualitative molecular interpretation to be quantitatively

of the extra term; it should not, however, be expected

accurate.

Clearly, stress raises the free energy. tiate with respect to compositions

To calculate the phase behavior one must differen-

in order to calculate the chemical potentials in the usual way. An

exception from the usual procedure is the maintenance stress, r12, constant.

This is conceptually

analysis in that 2t2 is, in general, composition empirically.

of some flow parameter, such as stress

straight forward but demanding

of experiment

and

dependent in a manner that is known at best

This, then, requires empirical interpolation

of the stresses in order to calculate the chemical potential.

formulae for the composition

dependence

For the linear, elastic dumbbell model

we are using, all the Zii are zero except Z11 the simple tensile stress along the flow direction (Bird, et al., 1977, p. 492) so that tr (z) can be related simply to the elasticity measured by the first normal stress difference:

(14) since x22 is zero. This is in turn related by linear viscoelastic theory (Graessley, 1974), for simple shear flow, to the observables

solution viscosity, 71, and the equilibrium

compliance,

Je, to give:

. where y is the shear rate. Equation (15) shows the rheological quantities to measure in order to determine the flow contribution

to the free energy.

374

Ver Sh-ate and Philippoff (1974) examined 3% solutions of PS in decalin at T = 300°K for which x G 0.5 and J, z lo3 cm*/dyne.

Comparing

the Flory-Huggins

piece of Equation (1) with

equation (15) at a shear stress z12 of 3 x lo3 dynes/cm2 where they observed incipient turbidity one finds that both terms are of order 10m4cal/cm3. PSldecalin at 3% will normally phase separate at about 288’K under quiescent conditions, perturbation

so that this level of deformation

has produced a

of the phase diagram of about 12°K.

Rangel-Nafaile,

er al. (1984) have gone farther to calculate explicitly the effects of stress

on the critical consolute point and on the binodal curve. To calculate the former they need, as mentioned above, to make some empirical statement about the dependence of stress on composition. W)

They assume it to be parabolic at fixed shear stress:

= a(@ - +max)2 + b$ + tWmax

introducing

(16)

empirical parameters a and b. The maximum comes from the fact that the normal stress

is small at small $, grows with 0, then comes down at higher Q since, to keep z12 constant as . concentration increases, one must also diminish y. This brings down ‘tl 1, since it depends (cf. Equations (14) and (15)) quadratically (Rangel-Nafaile,

on ;I. Data show that this form is reasonable

er al., 1984) if empirical.

Determining

the critical point from the condition that the first two derivatives of the

chemical potential with respect to Q are zero shows that, subject to the above assumptions, critical composition,

Qc, is

JUKhQnged

the

by the application of stress while the change in the critical

temperature, expressed as change in the effective interaction parameter may be expressed as: Ax = x(i) - x(O) (17) av =2RT It is customary,

following Flory and Krigbaum

(Flory, 1953; Flory and Krigbaum,

1950), to

express x as: x= l/2 +w(WT-

1)

(18)

where v is the entropic contribution

to x , and 9 is the Flory “theta” temperature that can be thought

of as a ratio of the enthalpic to the entropic contributions

to the interaction parameter.

temperature is where the second virial coefficient is zero and corresponds Equation (17) gives the shift in the critical temperature corresponding

The “theta”

to x = l/2 (Flory, 1953).

to a change in x as:

AT = T,(Y) - T,(O) (19)

For PS in decalin at 300”K, and using the data: v = 390 cm3/mo1, 0 = 288OK and w = 1.48, one finds from Equations

(17) and (19):

AT=-

1.5x 10m6a

(20)

where a is the coefficient of the assumed quadratic dependence Rangel-Nafaile, tion of composition.

of z1 1 on 9 in Equation (16).

et al. (1984) have fit normal stress data at constant shear stress as a func-

For 212 = 2000 dynes/cm2, corresponding

to one of the curves of Figure 1,

one finds a = 3 x lo6 so that the predicted shift of the critical temperature is 4.5X.

The observed

shift in Figure 1 is of order 15 to 20°K. (The precise critical point under shear has not been determined).

Other data also conform to these analytical prediction which an accuracy of a factor of 3 to

4. With the limited data, it is difficult to assess the prediction that the critical composition change.

does not

However, the general shape of the cloud point curve appears distored under shear making

it likely that there is a shift in Qc. Rangel-Nafaile, numerical theory, incorporating diction is better, furnishing

et al. (1984) also developed a more complete

numerical differentiation

of the stress-concentration

accuracy to within about 3°K over all the cloud point curves measured.

This represents generally the state-of-the art in modelling immiscibility Some improvements (Vrahopoulou

induction by flow.

in the molecular modelling of the stretching of the polymer coil under flow

and McHugh,1984),

and of the treatment of polydispersity

McHugh, 1986) have been suggested. extensional

data. The pre-

Manucci

flow field on the isotropic-nematic

(Vrahopoulou

and

and Ciferri have considered the effect of an phase transition for liquid crystalline polymers

(1977). B. Modelling

of miscibility

promotion

by flow.

While the theories for immiscibility macromolecular promotion

configurations

in the homogeneous

begin with consideration

fundamental

induction focus on stored elastic energy in the single-phase region, the theories of miscibility

of the already-demixed

solution (Wolf, 1980; 1984). The

idea behind this class of models is that stress applied to the two-phase mixture will

break up droplets of the discontinuous

suspended phase, tending to re-homogenize

the system.

The theory is more mechanical than thermodynamic. Stripped to the essentials, the assertion of this class of models is that droplets of the discontinuous

phase will break up when the stress applied to the droplet from the deformation

(PI)

of the medium exceeds the sum of the stress due to interfacial tension (PZ) and the stress due to the droplet elasticity (Pg) which are working to maintain the integrity of the droplet. Wolf argues for particular mathematical

forms for these terms. Homogeneity

is said (Wolf, 1980) to be restored

when the size of the droplets is reduced to R

the size of the individual polymer coils. g’ Elasticity is put into the models via a term much like Equation (15), where it is related to

the equilibrium

compliance of the material in the droplets. Elasticity is playing a different role in

these theories from that which it plays in the immiscibility configurational

induction theories. No mention of the

elasticity of individual polymer coils is made in the miscibility promotion models.

Since it is quite conjectural,

we will not reproduce Wolfs complete model here. Several

qualitative comments suffice to put it in perspective. the models of miscibility

promotion.

Interfacial tension plays the dominant role in

Recalling the expression for the stress due to interfacial

376 tension o: p =E 2 r

(21)

where r is the dropIet size, we see that the force maintaining

the integrity of the phase-separated

dropIets will increase as they become smaller and, of course, as CJincreases. vanishes at the critical point, one anticipates the major manifestations homogenization

to be seen near Tc and that such homogenization

Since tension

of this stress-induced

to become increasingly

difficult

away from T,. Figure 4 illustrates the anticipated shape of the consolute curve under stress. Wolf has suggested (1980) that the data points in Figure 2 for the system PS-rerr-butylacetate

fall on such

a curve. He has coined the term culyric point (Wolf, 1984) for the minimum in Figure 4, where, in principle, an equilibrium

among three fluid phases should exist, a situation somewhat analogous to

a eutectic point. Presently available data are inadequate to assess this point.

TPc) 197

0.1 wPs Phase diagram of PS in decalin. Quiescent Figure 4. data is along upper curve. Lower curve is theory of Wolf for 5000 s-1 (from Wolf, 1980; 1984).

It is important to realize that the models of this category are not necessarily in conflict with the immiscibility

induction models, although some of language used in recent papers on the subject

suggests that they are (Rangel-Nafaile,

er al., 1984; Wolf, 1980). They treat different aspects of

377

the phase separation and it is possible that the effects included in each dictate the outcome under certain circumstances. Perhaps better nomenclature uration models (for immiscibility

for the two categories of models would be molecular config-

induction) and interfacial tension models (for miscibility promo-

tion). The former (Rangel-Nafaile,

CTal., 1984) explicitly assume that interfacial tension is zero but

neglect the droplet break up aspects of the experimental molecular configurations

observations;

the latter do not consider

(Wolf, 1980; 1984).

SURVEY OF CURRENT

STATUS AND SUGGESTIONS

FOR FUTURE WORK

Data. What is most sorely needed in this area is more data on stress-modified diagrams.

While anecdotal information

it is inadequate for a thorough fundamental polymer molecular weight, polymer-solvent ionic interactions

and polydispersity

phase

on precipitation and cloud points is useful and provocative understanding

of the phenomena.

and/or polymer-polymer

Systematic studies of

interaction parameter, polar or

of molecular weight ought to be undertaken,

time gathering the auxiliary data necessary for theoretical interpretation,

while at the same

such as composition-

dependent rheological (viscous and elastic) and interfacial tension data. Control of extraneous effects such as viscous heating is essential. Especially useful would be data on individual phase compositions or binodal temperatures. circumstances,

Fluorescence

rather than cloud points

spectroscopy may be a useful tool here since, under some

it permits the determination

of individual phase compositions

without physically

isolating them (Torkelson, et al., 1984). These are needed in order to assess properly questions about the shape of the consolute curves under flow. Deformations

other than simple shear should be studied since flows such as simple

extension are more effective at stretching (Bird, et al., 1977) and have some advantages in theoretical interpretation Theory. enormous.

(see below).

The number of theoretical questions and opportunities

A few can be enumerated

raised by this work is

as especially interesting.

1. Applicability ofequilibrium-rype &m-y.

The theoretical basis for the thermodynamics

of elastic fluids in a steady state under the influence of an external field, rather than at static equilibrium,

was developed by Coleman (1964). The question arises, however, of the nature of

the external field imposed by various deformation experiences.

If the flow field v is irrotational so

that:

vxv=o

,

(22)

then v is derivable from a potential, B: v=-VB The frictional forces exerted by the flow on the macromolecule derivable from a potential, so that as far as the macromolecule

(23) are proportional

to v and thus also

is concerned it is acted upon by an

378

external field of conservative

force (Kramers, 1946). Equilibrium-type

theories should be

appropriate under these circumstances. Simple shear flow is not irrotational, however, and therefore may not be amenable to a thermodynamic

treatment.

This point suggests simple extension, or some other experimentally

achievable potential flow, as a better theoretical and experimental target. Marrucci and Ciferri (1977) have used this in their study of liquid crystalline polymers. Hanley and coworkers (Evans and Hanley, 1981; Hanley and Evans, 1982; Hanley, et al., 1983; Evans, et al., 1984; Rainwater, et al., 1985; Romig, 1986) have observed distortions from the equilibrium pair distribution function using nonequilibrium

molecular dynamics simulations.

What they see is qualitatively similar to experimental observations by Ackerson and Clark (1983) in light scattering from sheared colloidal dispersions. distribution

Quite generally, such distortions in molecular

functions are expected to produce thermodynamic

have formulated a thermodynamics

effects. Hanley and Evans (1982)

for a system under shear. It includes Maxwell’s relations,

treating shear rate as a state variable, that could be subjected to experimental test. 2. Simple additivity field Flory-Huggins

of mixing

and stretching

contributions

to the free energy.

The mean

lattice model assumes random mixing of polymer segments and solvent, with

no correlations other than connectivity

in the placement of segments (Flory, 1953). Stretching the

coils may do more than add the stress-dependent

term of Equation (1). It may also modify the

entropy and enthalpy of mixing terms since, even in the mean field spirit, the average interactions among stretched coils may be different from interactions among random coils. 3. Unified models encompassing

all phenomena.

Even with the present, inadequate data

base it is reasonable to begin theoretical consideration of models that include both configurational elasticity and inter-facial tension, in order to ascertain the circumstances under which one or the other may dominate.

There are better models for configurational

deformation (Bird, et al., 1977)

and for drop break up (Levich, 1962) than have been applied thus far, so that the estimates of what may be the dominant effect can be sharpened without necessarily undertaking a comprehensive modelling effort on the entire flow-modified 4. Connections

phase behavior.

with otherflow-modified

critical phenomena.

de Gennes (1979) has

exploited the picture of a polymer coil as a critical object, in the sense that it experiences correlated fluctuations

in the densities of its segments.

Boyer,1979);

Beysens and Gbadamassi,1980)

Beysens and coworkers (Beysens, Gbadamassi

have demonstrated changes in critical behavior, and

anisotropy of critical fluctuations, in low molecular weight fluid mixtures. framework encompassing mentioned

and

the polymer and small-molecule

Pursuit of a theoretical

phenomena may be useful. As

above, Hartley and coworkers are moving in this direction.

Technological

Importance.

The implications

of this work are significant

wherever a

structured, multiphase polymer product is being made under flowing conditions (Ferguson, ei al., 1986).

Crystallization

deformation

(McHugh and Forrest, 1975; Mackley and Keller,1973)

occurs under strong

conditions in fiber spinning, film blowing and other processing operations.

Polymers

in porous media (Aubert and Tirrell, 1980), as in membrane permeation or enhanced oil recovery can be strongly stretched. Block copolymers (Alward, et a[., 1986) and poiymer alloys are

379

frequently processed by extrusion. arrangement

Not only the phase diagrams, but also the very structure or

of phases can be strongly perturbed by flow.

Equilibrium fluid mechanics

thermodynamics

provides the jumping off point but proper inclusion of the

is a must.

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