Modes of dispersion of viscoelastic fluids in flow

Modes of Dispersion of Viscoelastic Fluids in Flow H. VANOENE Scientific Research Staff, Ford Motor Company Dearborn, Michigan 48121 Received July 22, 1971; accepted October 12, 1971 It is shown that when a viscoelastic mixture of molten polymers is extruded, "alloy" composites are produced as a result of the formation of two distinct modes of dispersion: stratification or droplet-fiber formation. Important parameters responsible for these effects are: particle size, interfaciM tension, and differences in the viscoelastic properties of the two phases. The formation of polymer spheres, ribbons, or fibers in a matrix can be predicted on the basis of a new theory which has been confirmed experimentally and provides a route to composites of controlled structure and properties. Explicit expressions are derived for the interfacial tension between two phases a and/3 in flow: 0

0

~/~ = v ~ + ~a~[(~2)~ -- (~2)~] and v~ = v ~ -- ~a~[(~)~ -- (~)~], 0 t h e interfacial where -/~. is the interfacial tension of a droplet of fluid i in matrix j; -/ij tension in absence of flow; ai, the droplet radius; and (e~)i the second normal stress function of fluid i which depends on molecular weight, molecular weight distribution, and shear stress. Since v~i >_ 0 for droplet formation to be possible, these equations predict that when (~2)~ > (~)¢, phase ~ will always form droplets/fibers in phase ft. Droplet for0 ^ mation of phase fl in phase ~ will be possible only when aa _< 6v~/[(¢2)~ - (~)~], a quantity of the order of 1-0.1 ~ for polymer melts. If the phase dimension of phase fl, therefore, is larger than 1 ~, as may be accomplished by incomplete mixing, phase/~ does not form droplets but stratifies. A further consequence is that in the submicron region, phase a will form single droplets; droplet formation of phase fl leads to composite droplets, i.e., droplets of ~ containing smaller droplets of a. Differences in viscosity, shear rate, extrusion temperature, and residence time in the capillary influence only the homogeneity of the dispersion and not the mode of dispersion.

INTRODUCTION

The type of dispersion achieved in flow when both phases are viscoelastic fluids, When a mixture of immiscible fluids is subsuch as polymer melts, is studied in this jected to flow, the various components of paper. The flow behavior of such fluids is this mixture will achieve some ultimate g e n e r a l l y described i n t e r m s of t h r e e funcshear-stable state of dispersion. Usually an tions; one f u n c t i o n a c c o u n t s for t h e d e p e n d e m u l s i o n is formed, i.e., a dispersion of droplets of one p h a s e i n a c o n t i n u o u s second p h a s e (1). T h e h y d r o d y n a m i c factors which i n f l u e n c e the f o r m a t i o n a n d d e s t r u c t i o n of fluid droplets i n shear flows h a v e b e e n s t u d ied i n g r e a t detail (2). I n m o s t cases, however, a t least one of t h e fluid phases is a N e w t o n i a n liquid.

ence of t h e viscosity o n t h e shear rate, two o t h e r f u n c t i o n s , t h e so-called n o r m M stress f u n c t i o n s , describe t h e pressure d i s t r i b u t i o n which arises i n flow f r o m t h e elastic p r o p e r ties, or d e f o r m a b i l i t y , of t h e fluid (3). One m a y expect t h a t t h e shape of droplets i n flow will be d e t e r m i n e d b y b o t h the shear-

Journal of Colloid and Interface Science, Vol. 40, No. 3, September 1972

448

C o p y r i g h t 0 1972 b y Academic Press, Inc. All rights of reproduction in a n y f o r m reserved.

D I S P E R S I O N OF V I S C O E L A S T I C F L U I D S

ing forces arising from the viscosity and the pressure distribution around the droplet. The state of dispersion may therefore, be influenced by both the viscosity as well as the elasticity of the fluids. Dispersions of viscoelastic fluids have not been studied either theoretically or experimentally in any great detail. The present study, therefore, serves two purposes: to provide basic experimental data on the phenomenon itself and secondly to provide some theoretical foundation to guide the interpretation of the results. In the following sections, the flow of properties of viscoelastic fluids through capillaries and the theory of hydrodynamic stability of fluid droplets will be reviewed. It will then be shown how stability requirements are to be modified when one or both phases are viscoelastic fluids. The simplest approach is to complete the expression for the interfaeial tension by deriving a term which accounts for the difference of the free energy of deformation in flow of the two phases. From this analysis some definite predictions can be made with respect to the type of dispersion realized in flow. It is found that, in general, the predictions made are indeed borne out by the experimental results. Finally, some of the implications of the results obtained in this work as they relate to the function of processing aids and the structures of polymer/polymer composites made by extrusion or injection molding will be discussed.

T(e0}=-pz-6.1 T(,r~) =

-

449

__ Jo f~ T 6.1 dr

~fr

[lc] [ld]

?3(r) = p~ + fo r 76.1 dr -- ~1 (0-2 -- 26-1)

[ie]

where z is the direction of flow, r the direction of the velocity gradient, 0 the direction normal to the plane (rz}, T(rz} the shear stress, f the specific driving force in the direction of flow, and ~ the isotropic pressure, which, as is seen from Eq. [le], depends on the radial position in the capillary. The so-called normal stress functions, 6.1 and 6.2, are defined by: T(rr} -- T(O0} = 6.1,

[2a]

W(00) = 6-2,

[25]

T(zz} -

where the circumflex denotes that the normal stress functions are taken to be functions of the shear stress. Equation [ld] shows that the normal stresses do not influence the magnitude of the shear stress. One may, therefore, infer that the theories of the hydrodynamic stability of the shape of a droplet as influenced by differences of viscosity and velocity gradient will retain their validity even for flow of viscoelastic fluids. The remaining relationships, however, imply that the forces acting in flow on a droplet dispersed in a viscoelastic fluid are very different from those acting in a Newtonian fluid. Not only do the components of the stress tensor differ in the three principal directions r, 0, and z, but the mean pressure THEORY itself has a radial dependence. Both factors A . POISEUILLE FLOW OF VISCOELASTIC influence the shape of a drop. In the ensuing discussion about droplet FLUIDS stability in flow, one has therefore to suppleThe stress tensor T associated with the ment the treatment of hydrodynamic staPoiseuille flow of a viscoelastic fluid may be bility by considering the elastic stability of written as follows (4): the deformed droplet. T(zz)

-

p,

-

-

T(rr} = -- p~ -- fo r_6.1r dr

-

J0

e,_

r

dr

[la] B. HYDRODYNAMIC [lb]

STABILITY OF DROPLETS

The deformation and hydrodynamic stability of viscous droplets in flow has been the subject of many experimental as well as

Journal of Colloid and Interface Science, Vol. 40, No. 3. September 1972

450

VANOENE

theoretical investigations. A detailed summary of this work has been given by Goldsmith and Mason (2). Briefly, the deformability of a Newtonian viscous drop is determined by two parameters: X, the ratio of the viscosity of the suspended fluid and that of the medium and l¢, the ratio of the interfacial tension to the product of the magnitude of the local shear stress and particle radius. According to Cox (5), the deformation D and orientation a of the deformed spheroid for arbitrary values of k and /~ are, in the limit of small deformation, given by the expressions: D =

5(19X + 16) [3a] 4(X -1- 1){(20/c) ~ -]- (19X)2}I/2

a = 1/~ _}_ 1/~ tan -~ (19X/20k),

[3b]

facial effects are dominant and X of the order of unity: D-

Furthermore, one may expect that large droplets will break up into mieron-submicron droplets, which because of the interfacial tension are shear stable (7). The orientation a of the deformed droplet in flow is determined by the relative magnitude of the interracial term k and the viscosity ratio X. When interfaeial effects are dominant, the droplets tend to be oriented at an angle of 45 ° with respect to the flow direction; when the viscosity term is dominant the droplets tend to be oriented in the flow direction. C. INFLtrENCE OF TEE FREE ENERGY OF

where

DEFOR~,IATIO N

D = ( L -- B ) / ( L

and

19X -}- 16 1 16X -t- 16 k

+ B)

[3c]

X = W/V0

[3d]

/c = 7/yoGa

[3el

L is the length and B is the height of the deformed spheroid; hi, the viscosity of the suspended fluid; no, the viscosity of the medium; G, the magnitude of the local shear rate; a, the particle radius and ~, the interfacial tension. The product n0G represents the local shear stress. The orientation angle a is the angle between the longer axis of the deformed spheroid and the direction of the velocity gradient. For small values of the parameter ]c the influence of interracial tension is negligible and the deformability becomes a function of the viscosity ratio only. To fix the orders of magnitude of the parameter k, let v0G be equal to a shear stress of the order of 105--10 ~ dyn/cm 2, typical for polymer melts , and let the interracial tension be of the order of 5 d y n / c m (6). Then for submicron droplets ]c ~-~ 5-50, or/c >> 1; for micron-sized droplets ]c ~ 1; for larger droplets k << 1. The viscosity parameter X is, in our work, of the order of 5 --> 0.2. The deformation will be small when X and /c are large. When inter-

ON

THE

~V[ODE

OF

DISPERSION The expressions for the stress tensor associated with Poiseuille flow show that the elasticity of the fluid modifies only the diagonal components of the stress tensor. For Weissenberg fluids (i,e., ~i = 0), these components are: T(rr) = T(O0) = - p ,

T
[4] -pz

+

a2

Hence, in flow there is a net tension in the flow direction. When flow ceases this tension relaxes, giving rise, for example, to dieswell. The molecular origin of this tension is the distortion of the isotropic segment distribution of the polymer molecule in shear flow (8). This distortion lowers the configurational entropy of the molecule which increases the free energy. The increase in free energy is recoverable when flow ceases or when the state of deformation of the molecule changes. The magnitude of the recoverable free energy depends on whether recovery takes place while keeping the boundaries of the fluid element constrained to the location it had in the streaming fluid (constrained recovery) or letting the boundaries be unspecified after recovery (free re-

Journal o.f Colloid and Interface Science, Vol. 40, No. 3, September 1972

DISPERSION OF VISCOELASTIC FLUIDS covery) (9). As observed by JaneschitzKriegl (10) the recoverable free energy for constrained recovery in shear flow, F~, is given by the expression Fg = 1/~ ( T r T -4- 3p,).

[5]

For a Weissenberg fluid in Poiseuille flow one obtains F~ = } ~ .

[61

In general, the recoverable free energy of deformation is a function of the state of flow; when the state of flow of a fluid element changes, its free energy of deformation will change accordingly. Even though constrained recovery is difficult to measure in a homogeneous polymer melt since it involves recovery in the flow channel after cessation of flow, in binary mixtures, or heterogeneous viscoelastic fluids, a change in the state of deformation of the composite fluid can be brought about by a certain mode of dispersion during flow. D.

THERMODYNAMICS TION I N FLow

OF DROPLET

FORMA-

In light of the above, one may ask under which circumstances droplet formation will take place in a mixture of two viscoelastic fluids. This question ~11 be treated on the basis of a new thermodynamic analysis of droplet formation: Consider two viscoelastic fluids a and and let

(~)~ > (~)~. It is shown in Appendix I that the boundary conditions for a droplet in equilibrium with its surroundings are given by the conditions : (i) d i r T = 0; (ii) T(rz) continuous through the interface; (iii) T r T continuous through the interface. The first boundary condition determines the stress tensor to within a constant; the seeend boundary condition arises from the fact that a liquid/liquid interface cannot support

451

a shear stress difference, since liquids flow under the influence of shear stress; the third boundary condition expresses the fact that the interfaeial tension shall be in equilibrium with the mean pressure on either side of the interface. From these conditions it is at once apparent that for two viscoelastic fluids condition (ii) and (iii) cannot be satisfied simultaneously since T r T is also a measure of the recoverable free energy of deformation. The thermodynamic question may, therefore, be stated as follows: Given two viscoelastic fluids a and ~, and (~2)~ > (~2)¢, can droplet formation lead to a net decrease in recoverable free energy of deformation? Let the interface between liquid a and be located at some radial position r in the capillary and let the shear stress at this location be given by T@z).Before droplet formation the fluid element of liquid ¢~ has a recoverable free energy of deformation given by 1/~(~2)~ and a mean pressure p~ }~(e2)~ • For a droplet of fluid ~ to form in the matrix of fluid a continuity of trace T requires that the mean pressure in phase become equal to that of fluid a and hence become equal to pz - }~(~2)~ • The mean recoverable free energy of deformation of the fluid element ~ will then be equal to ~ (~2) after droplet formation. Since (e2)~ > (~2)¢, the process of forming a droplet of fluid ~ in fluid a requires that the recoverable free energy of deformation in the fluid element ~ be larger after droplet formation than before this took place. Conversely, when during flow a droplet of phase a forms in a matrix of fluid ~, the recoverable free energy of deformation of the fluid element will be smaller after droplet formation by the amount: 1 ((~)~ - (~)~)'~ a 3,

[71

where a is the radius of the droplet. This net decrease in free energy of deformation favors the formation of droplets of phase a in phase

Journal of Colloid and l~terface Science, Vol. 40, No. 3, September I972

452

VANOENE

When this decrease in the recoverable free energy is formally taken into account as an additional contribution to the interracial free energy (i.e., the free energy of formation of the interface/unit area), one obtains for the interracial tension in flow 0

~.~

= ~.~

+ ~a[(~).

-

(~)~],

Is]

than Poiseuille flow since it requires only the evaluation of the trace of the stress tensor. To first order this can be evaluated readily by means of the procedure of Pipkin and Tanner (12). The interracial tension is a positive quantity, hence Eq. [8] is only satisfied when

(1) (~2). - (~s)s is positive where . o is the interracial free energy in the or (2) o absence of flow. Alternatively, one m a y consider the conIn other words, when these conditions are tinuity of Tr T to be equivalent to increassatisfied, a droplet of phase a may be formed ing the mean pressure from say _1/~ Tr T ( a ) in phase ft. to - - ~ Tr T(~). The work of formation of The interfaeial tension for a droplet of a droplet of radius a of phase a, W,~, is phase fi in phase a can be expressed as then given by ,~°

W,~ -- 47rL/~ 41r 31 + ~ - a .5[Tr T ( a ) -- Tr T(B)]

-

-

[9]

47ra-~ ( Po" -- Po~), 3

where -y,~ is the interfacial tension and (P0" - P0 ~) the pressure difference between phase a and ~, when the trace of the stress tensor is continuous through the interface. The interfacial tension with respect to the surface of tension, m a y be obtained from Kondo's (11) expression ~.~

-

3W

~a

~.

[10]

=

0

,.~

-

~a[(~).

-

(~2)~].

[11]

Hence, for a certain droplet size 3'~ may be zero. Only droplets smaller than this critical size are stable. An order of magnitude estimate of the elastic term can be obtained by noting that in the region of shear rates of interest (high shear rates) ~2 is of the same order of magnitude as the shear stress (3, 10). The quarttity (~2), - (~2)~ will, therefore, be also of this same order of magnitude, i.e., 105-108 d y n / c m 2. The hydrostatic interfacial tension for the polymer is of the order of ~ 5 d y n / c m (6). Hence, only for micron or submicron size droplets is 0

"Y.~ > 1/~a[(~2), -- (~2)~] When the viscoelastic terms are zero, substitution of Eq. [9] in Eq. [10] yields:

[12]

regardless of the sign of [(#2), - (~2)~]. For larger droplets the elastic term is dominant ½ (p. °8 o and the hydrostatic interfacial tension 0 7~ = a 0 -- P ) = ~ ' ~ , may be neglected. which is the expression for the interracial Even though the measurement of normal tension -y.~ 0 under hydrostatic conditions stress functions is difficult especially on (i.e., in absence of flow). Inclusion of the polymer melts at high shear rates, one can viscoelastic terms gives nevertheless obtain an estimate of the rela0 tive magnitude of normal stress functions from theory. For slow flows it has beer shown Hence the two methods of deriving an ex(13) that the second normal stress function pression for the interfacial tension are is proportional to the steady-state shear comequivalent. The mean pressure method may pliance Je : be more convenient than the direct free~2 = J~T(rz} 2, [13] energy method when dealing with flows other Journal of Colloid and Interface Science, Vol. 40, No. 3, September 1972

DISPERSION OF VISCOELASTIC FLUIDS

where J~ -

UT ~

\ ~

/

TM

[14]

and R, the gas constant; T, the temperature; p, the density; M~ the weight average tooleeular weight, M, and M~+~ the so-called z-average and z + 1 average molecular weight (10). Equation [13] is valid only at low shear stress (9, 13). Experimentally one finds that at high shear stress the normal stress function is proportional to the shear stress rather than the square of the shear stress. Nevertheless, the other implication of Eq. [13], i.e., that the deformability of a viscoelastic fluid is determined by the equilibrium compliance will probably be still valid at high shear stresses. In other words, the magnitude of the equilibrium compliance can be taken as a measure of the relative magnitude of the normal stress function (13b). When initially the domain size is larger than approximately 1 t~, the criterion for the formation of a droplet of an arbitrary phase a in phase ~ requires, therefore, that RT 1 (;fz ~+1~

] [15]

be positive. A number of definite predictions follow from this analysis. (a) If it is observed that phase ~ stratifies (i.e., does not form droplets) in phase a then phase a will form droplets in phase ~. Hence the extrudate structure observed in a mixture of 10 % a and 90 % ¢~ should be the opposite of that found in a mixture of 10 % and 90 % a. Such pairs of mixtures will be called complementary mixtures. (b) If a particular morphology is observed, it should not be influenced by the magnitude of the shear rate or by raising the temperature of extrusion, except for effects which can be attributed to the hydrodynamic stability of a particular mode of dispersion.

453

(e) Changes in the molecular weight and the molecular weight distribution of the components can be used to verify the prediction that the phase ~4th largest normal stress functions will form droplets. E. CO~¢IBINED INFLUENCE OF DEFORMABILITY AND HYDRODYNAMIC STABILITY ON THE STATE OF DISPERSION

The theory of Cox (5), discussed previously, has shown that the relevant parameters governing the deformation of fluid droplets are the viscosity ratio, X, and the parameter k, the ratio of the interfacial tensicn and the shear stress. Since the viscosity is a rheologieal function, independent of the elastic properties of the material--the viscosity determines only the magnitude of the shear stress--the elastic properties can only have a direct influence through the parameter k. Cox, however, does not consider the deformation of viscoelastic droplets. Such droplets may appear hydrodynamically to be more rigid than a Newtonian droplet of the same viscosity (14). Hence, the analysis of the free-energy changes in droplet formation only provides a criterion for the process of droplet formation itself. The precise state of deformation of such droplets in flow would require a far more difficult analysis. Furthermore, there are more practical difficulties; the flow of the polymer melt may not be laminar at the shear rates of interest and second, the magnitude of the normal stress in flowing melts is rarely known. The "elastic turbulence" of the flow arises not so much from the flow through the capillary itself as from the passage through the entrance region to the capillary. It has been shown both theoretically as well as experimentally that the converging flow in the entrance region is always unstable (15). The instability in the capillary is, therefore, due to the persistence of this entrance turbulence. It is not known whether laminar flow can be expected in a sufficiently long capillary (16). It is known, however, that a typical elastic

Journal of Colloid and I~terface Science,

Vol. 40 No. 3, September1972

454

VANOENE

effect, such as die-swell, becomes independent of the length of the capillary for capillaries with an 1/d ratio of approximately 20 to 40. For this reason, rather long (//d of 33 and 66) capillaries should be employed so as to assure that Poiseuflle type equations retain some validity. In view of these quantitative limitations, the predictions made earlier with respect to the magnitude of the "elastic" contribution to the interracial tension will, therefore, have to be tested in rather qualitative fashion; especially when considering the ultimate degree of dispersion. The analysis shows the critical influence of particle size. As long as the size of the disperse phase is of the order of a few microns or larger, elastic effects will be dominant. When the particle size is of the order of a few microns or smaller, the absolute magnitude of the normal stress differences plays a decisive role in determining the absolute size of the disperse phase. The phase structure may either be preserved down to the 0.1 ~ level (e~ ~ 107 dyn/cm 2) or break down at the 1 ~ level (as ~ 106 dyn/cm~). The ultimate structure will, therefore, be difficult to predict or to achieve but will give some indication as to the importance of the viscoelastic terms. Moreover, as the ultimate degree of dispersion is attained, the bulk properties will tend to some average value, hence locally a droplet will not experience the stress field of the pure matrix but that of the average matrix. This particular aspect has not been taken into account in the previous analysis. EXPERIMENTAL METHODS A. EXTRUSION Extrusion of the mixtures was performed on a capillary rheometer (17). Capillaries used had an lid of 12, 33, and 66, respectively, with a diameter of 0.050 in. B . PREPARATION OF M I X T U R E S

Mixtures of polymers were prepared in two different ways. Relatively coarse mix-

tures were prepared by mixing the dry polymer powders (approx. 35 mesh), in the appropriate amount. These mixtures were compressed into pellets which could be easily loaded in the rheometer barrel. For a given run, the powder mixture was preheated for 5-10 min before actual extrusion took place. No special effort was made to preserve the extrudate morphology by rapid cooling (quenching) of the extrudate itself, The effect of such break-up of the extrudate structure during solidification would have been clearly evident upon subsequent microscopic examination. With the possible exception of extrusion at 250°C no evidence of a change in extrudate morphology during solidification was obtained. Even at 250°C the effect was minor. Mixtures for the investigation of the ultimate degree of dispersion were prepared by preblending the components using a Kenics mixer (18). This mixer is essentially a binary mixer. The incoming polymer stream is split and twisted by a bow tie-shaped element, subsequent elements repeat this process. The particular mixer used (see Fig. 1) could be fitted in the rheometer barrel and contained eight mixing elements; two passes through this mixer were found to be sufficient for our purposes. It was found convenient to extrude the output of the mixer through an inverted capillary so as to facilitate cutting the extrudate. The virtue of the Kenics-type mixer is that the degree of mixing depends only on the number of mixing elements employed and not on the magnitude of the shearing forces. Shear degradation is therefore minimized. C. MATERIALS USED

The following commercially polymers were used:

available

Polystyrene: Styron 666 (manufactured by the Dew Chemical Co.). Polyethylene: Marlex 6009 (manufactured by Phillips Petroleum). Ionomer: A polyethylene copolymer containing carboxyl groups (Na-salt), type:

Journal of Colloid and Interface Science, ¥ol. 40, No. 3, September 1972

DISPERSION OF VISCOELASTIC FLUIDS

455

molecular weight tail of the molecular weight distribution. This introduces unavoidably, rather larger errors in M= and M~+I. E. OPTICAL Optical

FIG. 1. T h e Kenics mixer a t t a c h m e n t for the I n s t r o n rheometer.

Surlyn A 1558 (manufactured by DuPont Chemical Co.). Polymethylmethacrylate (PMMA): Elvacite 2041, 2010 (manufactured by DuPont).

MICROSCOPY micrographs

were

taken

on

a

Leitz-Panphet microscope equipped with a Polaroid camera. In order to enhance optical contrast between the phases, micrographs were made with polarized light under phase contrast, by reflected or transmitted light as deemed appropriate. The polystyrene/polymethylmethaerylate samples were mounted in an epoxy resin (five parts, by weight, of Araldite CIBA 502 and one part hardener CIBA 956) and polished to a 0.3 ~ finish. These samples were studied in reflection. The Surlyn/Marlex mixtures could be easily microtomed and were examined in transmission using polarized light or phase contrast. When only the morphology of the disperse phase was of interest, the matrix was dissolved in an appropriate solvent, the disperse phase isolated and examined separately. F. ELECTRON~{ICROSCOPY

Microtomed sections of polystyrene/polymethylmethaerylate mixtures were prepared by mounting the extrudates in an epoxy resin. The epoxy resin used was either a D. GEL PER]~c[EATION CHROMATOGRAPHY mixture of seven parts, by weight, Araldite A Waters Associates Gel Permeation CIBA 502 and one part hardener CIBA 956, Chromatograph (gpc) operating at a flow or a mixture of six parts, by weight, Dow rate of approximately 1.0 ml/min was used. epoxy 332, three parts Dow epoxy 732 and Polystyrene was analyzed at 138°C in 1,2,4- one part hardener DEIt 20. Since in the trichlorobenzene solution, the polymethyl- latter mixture the hardener is sensitive to methacrylate samples at 50°C in toluene moisture, samples were embedded at 40°C. solution. The gpc columns were calibrated by No staining was found necessary as the coninjecting standard polystyrene samples. The trast between polystyrene (dark) and column calibration for PMMA was obtained PMMA (light) was sufficient to distinguish from the polystyrene calibration and a rela- the two phases. A Siemens model I electron tively narrow distribution ( M w / M ~ ~ 2.0) microscope was used. P M M A sample using the method of Zinbo RESULTS and Parsons (19). Average molecular weights were calculated directly from the gpc curves EXTRUDATE STRUCTURE OF COMPLEMENTARY ~/J[IXT URES without additional corrections. The values obtained for Mr and M,+I are limited by the In Fig. 2 sections normal to the flow experimental accuracy of detecting the high direction of the extrudates of 10/90 IonoJournal of Colloid and Interface Science,

Vol. 40, No. 3, September 1972

456

VANOENE

FIG. 2. Morphology of complementary Marlex/Ionomer mixtures: (a) mixture of 10% Marlex and 90% Ionomer extruded at 190°C crosshead speed 0.2 in./min. (b) mixture of 90% Marlex and 10% Ionomer (extrusion conditions same as (a), both sections .k flow direction).

FIa. 3. Morphology of complementary Marlex/polystyrene mixtures : (a) mixture of 10% Marlex and 90% polystyrene extruded at 190°C, crosshead speed 0.5 in./min X 200; (b) mixture of 90% Marlex and 10% polystyrene (extrusion conditions same as (a), both sections II flow direction). Journal of Colloid and Interface Science, Vol. 40, No. 3 September 1972

D I S P E R S I O N OF V I S C O E L A S T I C F L U I D S

mer/Marlex and 90/10 Marlex Ienomer extruded at a rate of 0.2 in./min are compared. It is seen that the Ionomer phase forms droplets in a NIarlex phase, and that the Marlex stratifies in the Ionomer phase. In Fig. 3 the same effect is observed for complementary mixtures of Marlex and polystyrene. It is possible to dissolve the polystyrene phase ~dth toluene and isolate

457

the Marlex phase. As shown in Fig. 4, the ~iarlex phase is in the form of thin ribbons. The Marlex/polystyrene mixtures were studied at extrusion temperatures ranging from 170 to 250°C. No phase inversion with temperature was observed, those mixtures in which Marlex stratified at 170°C remained stratified. The polystyrene phase, in droplets at 170°C, formed a mixture of

FIG. 4. Marlex ribbons after dissolution of the polystyrene phase (crossed polaroids). Journal of Colloid and Interfacx Science,

Vol. 40, No. 3 September 1972

458

VANOENE

droplets and fibers at higher extrusion temperatures. Changing extrusion conditions by varying the length of the capillary or shear rate had no m~rked effect on extrudate morphology except that extrusion through a short capillary (1/d ~ 15) resulted in a coarser dispersion. The effect of flow instability was also more noticeable in this case. The particular extrudate morphology was found to be remarkably shear stable in the sense that a once extruded mixture could be reextruded without a change in extrudate structure. Reextrusion made the dispersions more homogeneous (Fig. 5). The load did not change on reextrusion. These results, therefore, confirm the general predictions made. Since neither shear rate, residence time in the capillary, nor temperature has a marked influence on the morphology of the extrudate, the various structures must arise from parameters that depend on the radial position in the capillary but are independent of the length of the capillary. These experimental observations by themselves point to the importance of the

deformability or "elasticity" of the phases, which in fully developed Poiseuille flow is only a function of radial position. INFLUENCE OF ~/[OLECULAR WEIGHT AND ~V[oLECULAR WEIGHT DISTRIBUTION

It was found convenient to use polymethylmethacrylate (PMMA)/polystyrene (PST) mixtures since P M M A samples of various molecular weights are readily available. The results of the gel permeation chromatography of the P M M A and polystyrene samples used are listed in Table I. The extrudates are compared in Fig. 6. The high molecular weight sample of P1V[MA forms large, irregular droplets. The primary effect in the mixture with low molecular weight P M M A is that of stratification. The PM~V[A sample with a molecular weight which almost matches that of the polystyrene matrix forms much smaller droplets. This P M M A sample was prepared by blending toluene solutions of the high and low molecular weights P M M A samples and subsequent freeze drying. Even though the Mw of this sample is somewhat smaller than that

FIG. 5. Influence of reextrusion of a mixture of 30% Marlex and 70% polystyrene: (a) extruded once; (b) extruded three times; both sections I[ flow direction. Journal of Colloid and Interface Science,

¥o1.40. No. 3, September 1972

459

DISPERSION OF VISCOELASTIC FLUIDS TABLE I MOLECULARWEIGHTDISTRIBUTIONOF POLYMETHYLMETHACRYLATESAMPLES AND POLYSTYRENE (STYRON666) B:C GEL PERMEATIONCHROMATOGRAPHY Mol. weight (10-6)

Elvacite 2010 Elvacite 2041 Mixture Sytron 666

M~

Mw

M,,

M~+I

0,069 0.24 0.087 0.088

0.130 0.54 0.255 0. 275

0. 200 1.06 0.75 0.62

.256 2.05 1.87 1.15

(M, Mz+I/ M~2)M w

0.4 4 5 2,6

~4 ~J

C FIG. 6. Influence of molecular weight and molecular weight distribution on extrudate morphology of a mixture of 30% P M M A and 70% polystyrene. (a) P M M A : Elvacite 2041 ~[ ~> M Styron; (b) P M M A : Mixture M~ ~.~ M~ Styron; (c) P M M A : Elvacite 2010 M < M Styron; extruded at T = 225°C, crosshead speed 0.5 in./min.

of the polystyrene mixture, the molecular weight distribution of the mixture is considerably broader (Table I). The definite conclusion is that the high molecular weight PMMA sample forms droplets in the polystyrene matrix, hence verifying the prediction that the phase with the largest second normal stress function forms droplets. All of the experimental results presented thus far were obtained on mixtures prepared from rather coarse powders ( ~ 3 5 mesh). INFLUENCE OF PARTICLE SIZE

Preblending the powder mixtures in a Kenics mixer (see Experimental section for details) results in a rather homogeneous composite. This particular method of mixing was found very satisfactory for compositions

which form either stratified or fibrous extrudates, much less satisfactory results were obtained when the second phase forms droplets. Extrusions of such a preblended mixture of 70% PST and 30% ~[arlex at 190°C results in a dispersion of PST droplets in a 5~[arlex phase (Fig. 7a, b), extrusion at 225°C yields a dispersion of PST fibers in Marlex (Fig. 7e, d). A similar transition is formed in Surlyn/5~[arlex mixtures (Fig. 8). If one interprets stratification as absence of droplet formation, these results are in agreement with those obtained on the coarser mixtures, shown in Figs. 2 and 3. For these particular mixtures, the ultimate mode of dispersion is droplets/fibers of the more elastic phase in a matrix of the less elastic phase.

Journal of Colloid and Interface Science. Vol. 40, No. 3, September 1972

460

VANOENE

FIG. 7. Droplet/fiber transition in mixture of 70% Marlex and 30% polystyrene. (a) and (b) : extrusion at 190°C (sections _L and II flow direetion resp.) ; (e) and (d) : extrusion at 225°C (sections _L and l] flow direction resp.) Droplets and fibers are considered to be in the same morphological class, since for a stable Poiseuille flow the normal stress functions depend on the magnitude of the shear stress, which varies only with radial position. Hence the influence of the viseoelastic terms on the morphology of the dispersions should be seen most clearly on a cross seetion. I n two-dimensions therefore, the morphological criteria reduce to a circle (droplets/fibers) or an are (stratification). The morphology of preblended (30/70) c o m p l e m e n t a r y mixtures of Elvaeite 2010

and Styron 666, extruded at 225°C are compared in Fig. 9. Both extrudates are dispersions of small droplets of the 30% phase. Most of the discrete particles are smaller t h a n a micron; the difference in morphology of eomplementary mixtures has disappeared, both facts indicating t h a t the elastie t e r m contributing to the interfaeial tension is no longer dominant. Extrusion at 170°C of these same mixtures results in a size reduction of the dispersed phase of about a factor three (Fig. 10). A striking observation, however, is t h a t m a n y more eomposite

Journal of Colloid and Interface Sciel co, Vol. 44),No. 3, September 1972

I;ISPEIISION OF VISCOELASTIC FLUIDS

461

FIG. 8. Fiber morphology in a mixture of 70% Ionomer and 30% Marlex: (a) section ± flow direction;

(b) section I[ flow direction; crossed polars, the fringes show the orientation of the interstitial Marlex phase.

FIG. 9. Ultimate dispersions of Elvacite 2010/Polystyrene dispersions extruded at 225°C, crosshead speed 0.5 in./min; (a) 30% Styron 666/70% Elvacite 2010; (b) 30% Elvacite 2010/70% Styron 666. droplets are formed in the 70 % Styron/30 % Elvacite 2010 mixture t h a n in its complem e n t a r y mixture. T h e dispersions obtained are not as homogeneous as Figs. 9 and 10 would indi-

eate. Usually a more lamellar structure is found in the wall region (Fig. 11). In coarse mixtures, structures reminiscent of vortices are dearly evident (Fig. 12). Effects such as these, due primarily to the inherent flow

Journal of Colloid and Interface Science, Vol. 40, No. 3, September 1972

462

VANOENE

FIG. 10. Mixtures same as those of Fig. 9, extrusion temperature 170°C, crosshead speed 0.5 in./min; (a) 30% Styron 666/70% Elvaeite 2010; (b) 30% Elvacite 2010/70% Styron 666.

FIG. 11. Lamellar wall texture in ultimate dispersions: (a) 70% Etvacite 2010/30% Styron 666; (b) 70% Styron/30% Elvaeite 2010, extrusion temp. 225°C, crosshead speed 0.5 in./min.

instability of the mixed melts, will be discussed separately in a subsequent repo~t. DISCUSSION

The experimental results obtained confirm in detail the predictions made on the basis of the analysis of the free-energy changes which accompany droplet formation during the shear flow of a mixture of viscoelastic Journal of Colloid and Interface Science,

fluids. The surprising stability with respect to reextrusion of stratified extrudates carl be attributed to the fact that droplet formation of the disperse phase would result in an increase in free energy of that phase. On reducing the size of the droplets to within the submicron range, droplet formation becomes possible. The observations that in ultimate-type

¥oI.'40,No. 3, September1972

DISPERSION OF VISCOELASTIC FLUIDS

463

FIG. 12. Typical examples of the effect of flow instability in coarse mixtures, sections [[ flow direction: (a) 30% Marlex/70% polystyrene; (b) 70% Marlex/30% polystyrene, extrusion temperature 190°C, crosshead speed 5 in./min. dispersions more composite droplets are formed by the less elastic phase (see Figs. 9 and 10) can be accounted for by minimizing the total surface free-energy F: F = ~A~ivij > 0 where A ij is the surface area of droplet i in matrix j. In the mixture as a whole, as shown in Appendix II, this criterion leads to the following condition: (i) for composite droplets of phase ~ in phase ~aO

phase a and 5, respectively; 0 0 and ¢0 are the volume fractions of a and ~ in the composite droplet; and A#2 the difference in normal stress functions. These expressions are minimized by for Eq. [16]: 0

[18] 1 > ~ 4 A~2 for Eq. [17]:

0

4)80 = 0,

i.e., no composite droplets.

[19]

[161 1 2 (b~A~2 > 0 (ii) for composite droplets of phase a in phase =

~¢o

o

[17] 1

+ ~ 4~ A~ > 0 where ~

and ¢~ are the volume fractions of

In a sense the formation of composite droplets is similar to the phenomenon of "engulfing" observed by Torza and Mason (20) in three-phase systems. Because of the dependence of the interfacial tension on particle radius, a two-phase system of viscoelastic fluids in flow is essentially similar to a multiphase s y s t e m of Newtonian liquids. In binary mixtures of Ne~¢onian fluids no composite droplets will be formed. Alternatively, one could attribute the

Journal of Colloid and Interface Science, Vol. 40, No. 3, September 1972

464

VANOENE

oresence of composite particles to a change in the mode of hydrodynamic breakup. Usually hydrodynamic breakup of a particular phase structure gives rise to single droplets. It is not inconceivable, however, that for hydrodynamic disturbances of a given wavelength, the interfacial tension of the phase with smaller ~2 is still negative, hence pressure differences can be accommodated by forming a phase structure with negative curvature. During hydrodynamic breakdown, the phase of lower as would then "engulf" the phase of larger a2. The droplet-fiber transition observed in the polystyrene/Marlex mixtures as the extrusion temperature increases is not readily explained. At 225°C the viscosity of Styron 666 is almost equal to the viscosity of the Marlex phase. Hence, hydrodynamically, the phases are less distinct. If both the viscosity ratio and the interracial parameters are small, both phases will be deformed as the continuum. It is then possible that droplets drawn out into fibers in the entrance region of the capillary remain as fibers in the capillary. The stability of these fibers would be enhanced by the fact that if the fiber were to break up into droplets of the same radius as the fiber, the total surface area would be increased by a factor: 3n/(1 + 2n), assureing that originally the fiber had two spherical caps on either end. Another possibility is that during the coalescence of two droplets of different size, the intermediate stage, which resembles a "pointed" droplet or a droplet with a conical cap, is drawn out into a fiber by a mechanism discussed by Taylor (21). In this case the viscosities of the two phases need not be equal. The droplet/fiber transition was found in both coarse and fine mixtures of polystyrene/Marlex and Ionomer/1Viarlex mixtures. If the stability of the fibers is primarily due to the fact that breakup would result in an increase in interracial area, the droplet/ fiber transition should be considered as a general phenomenon, on a par with stratification.

The present theory should also apply to the case of melt blending of homopolymers (~/o = 0) and the state of dispersion of a mixture of a viscoelastic fluid and a nonelastic fluid (a~)~ = 0. When 7,~ 0 = 0, the morphology of the extrudate is completely determined by the elastic terms. A low molecular weight polymer stratifies in the high molecular weight polymer, conversely the high molecular weight polymer forms droplets/fibers. In flow, no fractionation need occur, since on a given streamline the high molecular weight component would form a separate phase rather than move across the streamlines into a region of lower shear stress. A nonelastic fluid (e.g., an incompatible plasticizer or processing aid) stratifies in the viscoelastic phase and will be excluded from the polymer phase; if present in sufficient concentration, such a phase can diffuse rapidly through the sample even after solidification of the composite. When the additive has a low viscosity, it may on phasing out migrate to the wall layer during flow, reducing the overall viscosity of the mixture considerably. In other words, the additive may act as a slip agent. In general, the dominant influence of the "elastic" terms on composite morphology have been demonstrated. As these elastic terms are increasing functions of the shear stress it is not necessarily required to employ "high shearing forces" to obtain good mixing. Low shearing conditions such as occur in a Kenics-type mixer are usually sufficient, especially when the components stratify. When the difference in elasticity is large and the disperse phase forms droplets neither low shear nor high shear mixing is satisfactory, since the droplets resist deformation. In such a case it is better to control the motions of the matrix than the forces. A liquid droplet may still be drawn into a fiber by an elongation-type flow, subsequent shearing may cause the fiber break up. It is interesting to note that in mill-rolling one does indeed subject the material to the combined action of shear and elongation flow.

Journal of Colloid and Interface Science, Vol. 40 No. 3, September 1972

465

DISPERSION OF VISCOELASTIC FLUIDS ACKNOWLED OMENTS It is a pleasure to acknowledge the stimulating interest of Dr. S. Newman, who first suggested this general problem, and discussions with Dr. A. J. Chompff, who also carefully read the manuscript. The assistance of Messrs. Y. F. Chang and D. J. Piacentini with the rheological measurement is greatly appreciated. Similarly, thanks are due to Mr. L. Bartosiewicz for his assistance with the optical microscopy, to Mr. K. Plummer, who carried out the electron microscopy and to Dr. M. Zinbo for performing the gel permeation chromatography analysis.

sure difference required to balance this tension. For liquid/liquid dispersions, the interfacial stress tensor will still be diagonal at equilibrium, since a liquid yields (flows) in shear. Hence, the boundary conditions for liquid/liquid droplets are very similar to the hydrostatic case. These boundary conditions are: (i) div T = 0 (ii) T~s= 0

APPENDIX

EQUILIBRIUM BOUNDARY CONDITIONS FOR FLUID DROPLETS

The interfacial tension may be derived from mechanical as well as thermodynamic consideration (22-24). The proof of the equivalence of these two methods usually (24) employs only the boundary condition for mechanical equilibrium: (i) div W = 0.

[I-1]

When the stress tensor is diagonal, i.e., when there are no shear stresses in the interface, the stress tensor may be written as T = Tt(~

for

i~j.

I

+ ~¢~) + T~nn

[I-2]

A third boundary condition is still required, since conditions (i) and (ii) determine the stress tensor up to a constant: the undetermined hydrostatic pressure. Across the interface, however, one has to require that these undetermined pressures are equal. This condition, as formulated by Buff, (22) m a y be stated as follows: (iii) f T r ( T -

G~l) dv = 0,

where T is the interfacial stress tensor, and t~ = [1 -- A(k)]W(a) -t- A(X)T(~); A(X) = 0

for

X < X°;

A(X) -- 1 for where Tt is the component in the plane (12), the interface, and T~ the component normal to this plane. When applied to a spherical interface, the boundary condition given by Eq. [I-1] yields the differential equation OT~ = ( T t Oh

Tn)J(X),

[2-3]

where X is a distance parameter and J(X) the radius of curvature of the plane. Integration of [I-3], after elimination of the higher order curvature terms, gives the final, defining equation for the interracial tension P~ -- P~ = v(X°)J(X°),

[I-4]

where 7(X °) is the interfacial tension of the surface loeate at distance X°, the so-called surface of tension, J(X °) the radius of curvature of this surface, and P~ - P~ the pres-

[I-5]

X > X°,

and X° the location of the interface. Hence, the trace of the stress should be continuous through the interface. The physical significance of these conditions becomes clear when considering the specific example of the stresses in a spherical membrane of radius r, subjected to radial forces only. Since the surface of tension may be regarded as an elastic membrane of zero thickness, there is a one-to-one correspondence between the interracial stress tensor and the mechanical stress tensor. The membrane stresses are (25) : T,~ = A + B / r 8 [I-6] Tt

=

A

-

B / 2 r 3,

where A and B are constants and r the radius of curvature. Hence, Tr T = T~ -t-

Journal of Colloid and Interface Science,

Vol. 40, No. 3, September I972

466

VANOENE

2Tt = 3A. The equilibrium boundary conditions for liquid/liquid droplets are therefore:

(i) div T = 0 (ii) To' = 0

for ij

[I-7]

where n~ : No. of composite a droplets, n¢ : No. of ~ droplets inside the a droplets. Therefore, F

= 3

I

4vaJ

(iii) Tr T continuous through the interface.

47r a T ]

+ -~

A P P E N D I X II

=3

The total surface free energy F of a dispersion is given by

n~ a .

E~ [1 1]] 4~+

~+

--

0

3 n O ao 3

4/

1 + ~ 4, ~

where A~ is the surface area of droplet i in matrix j and ~'i~, the interfacial tension of droplet i in matrix j. For the mixture as a whole

Hence F~i. when 4o° = 0: No composite droplets. Case I I : Composite droplets of phase fl in phase a.

minimum.

Case I: Composite droplets of phase a in a matrix ft.

•

=

0

A~2

o "Y-0

F = ~ A~-,

F > 0,

o

@

+ a~ : radius of droplet a

@

(2

0 *~- = ~-0 - ~ a0(A~2) •

The volume fractions are 4o = 4o° - ~ o 4. = 1+

~,o_

~o

0

a0 : radius of droplet/3 The volume fractions of phase a and/3 are: 4. = 4.

-

Therefore, 0

0 + n . 4 v a , 2 [7-0 + 1/~ a.(A~2)]

4~°

40 = 1 -- ~o + ~oo, where ~o: volume fraction of phase a, calculated on the basis of the radius of the composite droplets. 4o°: volume fraction of phase fl within the a droplets. Hence, 2

2

F = no4~rao [~.o -- 1/~ ao(A~2)]

= 3 no

a°Z - t - n ~ - ~ - J as

1 ['41r +2L-3-n"aJ

=314

+[

~/~0

4~r 37 -- ~ n Oa0.j (A~2)

+114.o;

o oo

0 2

0

+ n~4~ao [~-o -- 1/~ ao(A~2)] ' Journal of Colloid and Interface Acience, Vol. 40, No. 3, S e p t e m b e r 1972

1 2 4o Az:

DISPERSION OF VISCOELASTIC FLUIDS Hence~ Fmin when ~o

o

>

1

o

^

the condition ~ o = 0 yields limiting size of stable ~ droplet:

a~ < 6y°~ =

A$'2

when ¢ 0 ~ O: composite droplets. REFERENCES 1. RICHARDSON, E. G., i n "Flow Properties of

Disperse Systems" (J. J. Hermans, ed.), Chap. II. North-Holland, Amsterdam, 1953. 2. GOLDSMITH, H. L., AND MASON, S. G., i n

"Rheology" (F. R. Eirich, ed.), Vol. 4, Chap. II. Academic Press, New York, ]967. 3. e.g., COLEMAN, B. D., MARKOVITZ, H , AND

NOLL, W., "Viscometric Flow of NonNewtonian Fluids." Springer-Verlag, Berlin, 1966. 4. TRUESDELL,C., AND NOLL, W., Handbuch der Physik Vol. III/3, Section 117 p. 470. Springer-Verlag, Berlin, 1965. 5. Cox, tg. G., J. Fluid Mech. 37, 601 (1969). 6. Wu, S., J. Phys. Chem. 74, 632 (1970). 7. RUIVfSCHEIDT, F. D., AND MASON, S. G., or. Colloid Sci. 16, 238 (1961). 8. FIXMAN, M., J. Chem. Phys. 45, 793 (1966); SEGAWA, W., J. Phys. Soc. Japan 25, 1519 (1968); GIESEKUS, H., in "Second-Order

Effects in Elasticity, Plasticity and Fluid Dynamics" (IV[. Reiner and D. Abir, eds.), p. 553. MacMillan, New York 1964. 9. LODGE, A. S., "Elastic Liquids," Chap. 7. Academic Press, London 1964.

467

10. JANESCHITZ-KRIEGL,H., Advan. Polymer Sci. 6, 170 (1969). 11. KONDO, S. J., dr. Chem. Phys. 25, 662 (1956). 12. TANNER, Pt. I., .4.ND PIPKIN, 2~k. C., Trans. Rheol. Soc. 13, 471 (1969). 13. a) COLEM.~.N, B. D., AND MARKOVITZ,H., dr. Appl. Phys. 35, 1 (1964). b) WALES, J. L. S., Pure and Appl. Chem. (IUPAC) 9.0, 331 (1969). 14. RUMSCHEIDT,F. D., ANDMASON, S. G., J. Colloid Sci. 16, 210 (1961); BARTOK, W., AND MASON, S. G., J. Colloid Sci. 13, 293 (1958). 15. •IESEKUS, H., Rheol. Acta 8, 4ll (1969); ADAMS, E. B., WHITEHEAD, J. C., AND BOGUE, D. C., A.I.Ch.E.J. 11, 1026 (1965). 16. BHATNAGAR,R. K., AND GIESEKUS, H., Rheol. Acta 9, 53 (1970). 17. MERZ, E. H., AND COLWELL, R. E., A S T M Bull. No. 232, 63, 1958 as modified by the

Instron Company. 18. Kenics "Static Mixer", U.S. Pat. 3,286, 992, manufactured by Kenics Corporation, One Southside Road, Danvers, MA 01923. 19. ZINBO, M., AND PARSONS, J. L., Ford Motor Company Tech. Rep. SR 70-94. 20. TORZA, S., AND MASON, S. G., dr. Colloid Interface Sci. 33, 67 (1970).

21. TAYLOR, G. I., Proc. llth International Congress Applied Mechanics (H. Gortler Ed.), p. 790. Springer-Verlag, Berlin, 1966. 22. BUFF, F. P., J. Chem. Phys. 23, 419 (1955). 23. ONO, S., AND KONDO, S., in "Handbuch der Physik" (S. Flugge, ed.), 10, p. 134. SpringerVerlag (1960). 24. MELROSE,J., Ind. Eng. Chem. 60, 53 (1968). 25. PRESCOTT,J., "Applied Elasticity," Chap. 12, Sec. 213, p. 328. Longmans, Green, London, 1924.

Journal of Colloid and Interfave Science, Vol. 40, No. 3. September1972