One of the most challenging tasks in the rheology of liquid mixtures is to findquantitative relationship between flow and structure. Such a relationship should beplicit in a constitutive equation so that one can measure the evolution of the strucfrom knowledge of the macroscopic rheological material functions. An excellent recreview on the subject can be found in the work of Tucker and Moldenaers~2002!.
From a practical viewpoint, the ultimate task is to offer engineers a tractable theoical model that can be used to predict the final morphology under a given flow field~forinstance, at the exit of the die in the extrusion process! from knowledge of the on-line
a!Author to whom all correspondence should be addressed; electronic mail: [email protected]!Canada Research Chair on Polymer Physics and Nanomaterials.
rheological measurements. If such a model existed, then one could modify processconditions such as the screw rotation speed~average total shear rate! and modify themorphology of the outlet blend. However such an objective is very difficult to achieve fothe moment due to~i! the complexity of flow and heat transfer in the extruder and at theexit of the die,~ii ! commercial blends being concentrated mixtures with many additivesand~iii ! the components being viscoelastic. It is thus necessary to consider first mixtuof simple fluids such as Newtonian fluids and work out simple realistic models that acapable of quantitatively expressing the relationship between flow and structure in simflow field.
Predictive models of this kind exist for the linear regime such as small amplitudoscillatory shear flow@Palierne~1990!; Bousmina~1999!# and for the nonlinear regime atfirst order of the gradient of deformation@Doi and Ohta~1991!#. Doi–Ohta theory madea breakthrough in the field but it is only valid for a 1:1 mixture of two equidenseequiviscous Newtonian fluids. Their theory gives only a coarse grained picture of tstructure: the total surface area per unit volume and the average orientation of the inface. No information is given about discrete dispersed droplets. The absence of the inilength scale leads to a peculiar scaling law for stress,s, the first normal stress difference,N1 , and the total interfacial surface area per unit volume,Q, that vary linearly with shearrate,g. Introduction of a length scale into Doi–Ohta theory was made by Wagneret al.~1999! by modifying in an ad-hoc manner the relaxation term. Another extension of threlaxation term in the Doi–Ohta model was given by Almusallamet al. ~2000!, whoaveraged the anisotropy tensor over the droplet volume instead of the total volumHowever, it is difficult to calculate certain phenomenological parameters in these modewhich were determined empirically@Almusallamet al. ~2000!# or by fitting the experi-mental results or numerical simulations@Wagneret al. ~1999!#.
Time evolution of the structure with an initial length scale into an ellipsoidal shapwas proposed by Maffettone and Minale~1998! for dilute and semidilute mixtures of twoNewtonian fluids. Their model satisfies the volume preservation condition, but does ngive an expression for stress so then it becomes impossible to predict the variation ofstructure from rheological material functions. Recently Grmelaet al. ~2001! proposed ahybrid model that combined Doi–Ohta and Maffettone–Minale physics, but with morgeneral equations for volume preservation. The model recovers the Maffettone–Minmodel as a special case and additionally gives an explicit equation for stress.
In this article we work out simple equations that render the predictions of the modclearer in an engineering sense. The objective here is to show how the morphology ofdispersed phase can be extracted from rheological material functions and vice versa.approach used is illustrated in simple flows such as steady and transient shear flouniaxial elongational flow, and planar hyperbolic flow. Small and large amplitude oscilatory shear flows are discussed in a separate paper@Yu et al. ~2002!#.
II. MODEL
The model of Grmelaet al. ~2001! expresses in a quantitative manner the relationshipbetween flow and structure. The governing equations are general and satisfy conservalaws; they are compatible with thermodynamics and the volume preservation conditioThe morphology was treated both at the local level~a Doi–Ohta type picture! and at themacroscopic level~a Maffettone–Minale dispersion type picture! at given length scale.The model was derived in the spirit of GENERIC formalism@Grmela and O¨ ttinger~1997!; Ottinger and Grmela~1997!# that expresses, under isothermal conditions, the timeevolution of a set of variablesx as
1383RHEOLOGY AND MORPHOLOGY IN EMULSIONS
]x
]t5 L
dF
dx2
dC
d~dF/dx!, ~1!
whereF is the free energy,dF/dx is the conjugate variable ofx (d/dx is the functionalderivative!, L the Poisson bivector, andC(dF/dx) is a dissipation potential. If werestrict treatment to the case of droplet-type morphology, i.e., the state variablesx arechosen to be~v, M !, wherev is the velocity field andM is the positive second orderconformation tensor~we call it here the morphology tensor! characterizing the shape ofthe droplet, then Eq.~1! becomes
]M
]t5 ~V–M2M–V!1 f 2~D–M1M–D!2Bi~M !, ~2!
together with the standard time evolution for the velocity field in which the extra stresstensors is given by
sabs 5 2Mbg
dF
dMag1Labge
dF
dMge, ~3!
where D and V are the deformation rate and vorticity tensors defined asD 5 (¹v1¹vT)/2 andV 5 (¹v2¹vT)/2. Labge represents the fourth order tensor that makesthe coupling between the structure and the flow field. The tensorB is expressed by fourself-consistent equations (B( i ), i 5 1, 2, 3, 4! that ensure preservation of the volume(detM ) @Ait-Kadi et al. ~1999!; Grmelaet al. ~2001!#.
Bab(1) 5 L(b)$4
3 @2 23 I1I2FM1
1~I12 43 I2
2!FM21#Mab1 4
3 @I12FM1
1~2I1I223!FM21#~MM!ab
2 43 ~I1FM1
12I2FM21!~MMM!ab%, ~4!
Bab(2) 5 L(b)@2 2
3 ~I1FM112I2FM21
!Mab12~FM11I1FM21
!~MM!ab22FM21~MMM!ab#,
~5!
Bab(3) 5 L(b)$@2 8
3 ~ 13 I1
22I2!FM12 4
9 I1I2FM21#Mab2 4
3 ~I1FM12I2FM21
!~MM!ab
14FM1~MMM!ab%, ~6!
Bab(4) 5 L(b)$@ 2
9 ~7I1I222I13!FM1
1 23 ~I12 1
3 I12I21 2
3 I22!FM21
#Mab22~I2FM11FM21
!
3~MM!ab1 23 ~2I1FM1
1I2FM21!~MMM!ab%, ~7!
where
M1 5tr~M !
~detM !1/3, M21 5 ~ tr M21!~detM !1/3, I 1 5 tr M ,
I2 5 12 @~tr M !22tr~MM !#. ~8!
The physics in Eqs.~2!–~8! do not include breakup or coalescence. Only deformations upto critical deformation of breakup are considered. The family of the time evolutionequations@Eqs.~2!–~8!# is parameterized by the free energyF, by a parameterf 2 , and
suchnd,
al
ss.
y
t
1384 YU ET AL.
by the kinetic coefficientL (b) . 0. We consider here that the free energy depends onMonly through its dependence onM1 andM21 @Leonov~1976!#. Equation~2! then takesthe following form:
sabs 5 22 f2$FM1
~Mab2 13 I1dab!1FM21
@I1Mab2~MM!ab2 23 I2dab#%. ~9!
In the particular case in which the tensorB is given by Eq.~5! and the potentialF isgiven byF 5 *d3rf 5 *d3rK ln M21, the time evolution equation@Eq. ~2!# reduces tothe time evolution equation of Maffettone and Minale~1998!:
]M
]t5 f 2~M–D1D–M !1~V–M2M–V!2
f 1
t S M23I 3
I 2d D , ~10!
where f 1 /t 5 23 KL0 , t 5 hmR/G is a characteristic time,h i ( i 5 d,m) is the viscos-
ity of dispersed phase or matrix,R is the radius of initial undeformed drops,G is theinterfacial tension, andI i ( i 5 1, 2, 3) are the scalar invariants ofM . The factorK in thefree energy density depends on the viscosity ratiop 5 hd /hm , volume fraction ofdispersed phasew, and the ratio of interfacial tension and the initial radius,G/R @see Eq.~21!#. The extra stress tensor@Eq. ~9!# reduces in this particular case to
ss 52 f2K
I2~I1M2M–M2 2
3 I 2d!. ~11!
We recall that expression~11! for the extra stress tensor was not derived in the work ofMaffettone and Minale~1998!.
Equations~10! and ~11! give the time evolution of the droplet morphology and theassociated extra stress tensor. From an engineering viewpoint, the consequence ofrelations is that it is possible to extract the structure from stress measurements aconversely, stress can be determined fromin situ morphological observations.
At this stage, let us make three comments regarding Eqs.~10! and ~11!. First, therequirement of volume preservation imposes thatI 3 [ detM 5 constant. Then thescaled invariants can be rewritten asM1(I 1) 5 I 1I 3
21/3 and M21(I 2) 5 I 2I 322/3. Di-
mensional analyses show thatI 1 , I 2 , andI 3 represent the length, area, and volume of themicrostructure, respectively@Maffettone and Minale 1998#. Therefore,M1 can be re-garded as a function of the extension of the droplets, andM21 as a function of the areaof the droplets. To obtain the Maffettone–Minale expression@Eq. ~10!# from general Eq.~1!, the free energy densityf has to be given the following expressionf(M21)5 K ln M21, which is in fact a special case of the Leonov potentialf5 f(M1 ,M21). This means that the free energy density depends only on the interfaci
area of the droplets. This is only accurate for slight deformation of the droplet@Grmelaand Ait-Kadi 1998#.
The second comment is about the effect of nonaffine motion on the extra streEquation~3! shows that the extra stress has two contributions:~i! affine motion arisingfrom ss,affine 5 2dF/dM–M and ~ii ! nonaffine motion originating fromss,nonaffine
5 2L :dF/dM . The weighted contribution of the nonaffine motion can be evaluated b
ss,nonaffine5f 222
f 2ss, ~12!
where f 2 reflects the degree of the nonaffine motion contribution. For shear flow at firsorder of the deformation rate~small capillary numberCa 5 hmRg/G), f 2 can be ex-
e.
total
hat
e
. Tofirst
spec-was
the
logyuc-
ndtheared
1385RHEOLOGY AND MORPHOLOGY IN EMULSIONS
pressed asf 2 5 5/(2p13), which makes the contribution of nonaffine motion negativIf a second order capillary number is considered, thenf 2 takes the simple form off 25 5/(2p13)1 3Ca2/(216Ca2) @Maffettone and Minale~1998!#. In this case, the
contribution by nonaffine motion can become positive for largeCa. This means that thestress contribution that arises from nonaffine motion always tends to decrease theextra stress for a small deformation rate~small Ca), while it tends to increase it for alarge deformation rate~small p and largeCa).
In the third comment we note that a new family of volume preserving models tincludes and extends the families introduced by Maffettone and Minale~1998!, Ait-Kadiet al. ~1999!, and Grmelaet al. ~2001! has recently been introduced by Edwardset al.~2002!.
In this article we discuss the predictions of the model for the particular case wherf 1and f 2 take the simple form of
f1 540~p11!
~2p13!~19p116!
and
f2 55
2p13. ~13!
Equation~11! represents the extra stress contribution due only to the interfaceinclude the contribution of the components to the total extra stress, we use as aapproximation for Newtonian mixtures a simple mixing rule@Doi and Ohta~1991!#:
s 5 sdw1sm~12w!1ss; ~14!
sd and sm are the stress contributions of the dispersed phase and the matrix, retively. Another alternative is the use of the total extra stress in creeping flow thatgiven by Mellema and Willemse~1983!:
s 5 2hmD1sm1ss; ~15!
sm is referred to as the ‘‘viscosity ratio term’’@Lee and Park~1994!# that depends onpand on the structure. Equation~15! can be rewritten in a more compact form@Peterset al.~2001!# as
s 52~2p13!16~p21!w
2p1322~p21!whmD1
5
2p1322~p21!wss. ~16!
Equation~11! along with the expression for the contribution of the components tototal extra stress given by Eq.~14! or ~16! ~or other types of equations! gives the finalexpression for stress that has to be combined with the time evolution of morphogiven by Eq.~10!. This gives a dual relationship between the evolution of the microstrture and stress both under steady state and transient regimes.
In the following, predictions of the model will be evaluated in different flow fields aparticular discussion will be devoted to explicitly illustrate how one can connectmorphology to the stress and vice versa. The predictions of the model will be compto our own experimental data and to data in the literature.
en
rence
er-r
and
and
wsosityis
1386 YU ET AL.
III. SIMPLE SHEAR FLOW
Let us consider a Cartesian coordinates frame (x1 , x2 , andx3) such that the velocitydirection is taken alongx1 , the velocity gradient direction is alongx2 , and the vorticitydirection is alongx3 . For simple steady shear flow, the velocity gradient tensor is givby
¹v 5 S 0 g 0
0 0 0
0 0 0D . ~17!
Under steady shear conditions, the interfacial shear stress, first normal stress diffeand second normal stress difference, and viscosity are given by
s12s 5
2KCa f1f 22
3~Ca21 f 12!
, ~18!
N1s 5
4KCa2f22
3~Ca21f12!
, N2s 5 2 1
2 N1s , ~19!
hb 5~2p13!13~p21!w
2p1322~p21!whm1
10Kt f 1f 22
3~2p1322~p21!w!~Ca21 f 12!
, ~20!
where f 1 and f 2 are given by Eq.~13! andK is given by@Yu et al. ~2002!#
K 56G
5R
~p11!~2p13!w
5~p11!2~5p12!w. ~21!
For a 1:1 mixture of two equiviscous equidense Newtonian fluids with a complex intface, Doi–Ohta theory~1991! theory predicts the following relations for interfacial sheastress,ss, the difference in first and second normal stress,N1 and N2 , and for theviscosityhs:
s12s 5
Gg~12m!
3l~11m!2 , ~22!
N1 52Gg
l~11m!2Am~12m!
3, N2 5 2N1 , ~23!
hs 5~12m!G
3l~11m!2, ~24!
wherel andm are two phenomenological parameters related to relaxation of the sizeshape of the drops.
Clearly, Doi–Ohta theory predicts a linear scaling law for interfacial shear stressthe first normal stress difference with the shear rate (ss, N1 } g) and constant viscosity.In the present model with dilute dispersed ellipsoidal-type morphology, the scaling lawith shear rate for interfacial shear stress, in first normal stress difference, and viscare different. In fact, Eq.~19! shows that the interfacial first normal stress differenceconstant for shear rates higher than a certain critical value. For small values ofg, thedifference in interfacial first normal stress varies somewhat proportionally tog2 (Ca2)
ates
heis a
are
lend
1387RHEOLOGY AND MORPHOLOGY IN EMULSIONS
~Fig. 1!. Similar behavior was reported by Bousminaet al. ~2001!. The interfacial shearstress is proportional tog only at smallCa ~small g). The viscosity of the mixtureexhibits a Newtonian plateau at low shear rates and shear thinning at larger shear r~Fig. 1!.
The ratio ofN2 /N1 for the Doi–Ohta model is21, while it is21/2 in the predictionsof the present model. These differences originate from the peculiar topology of tstructure considered in the two approaches. In fact, in the present model the stressfunction of capillary numberCa, which includes information on the initial droplet size,whereas in the Doi–Ohta model such a length scale and consequently time scaleabsent.
Modifying the relaxation term, Wagneret al. ~1999! introduced a length scale intoDoi–Ohta theory and worked out the following expression for shear stress:
s12 5 hmgS 111
3l1
~22m!R2
81G2
hm2
w2l3 g21O~ g4!D , ~25!
FIG. 1. Dependence of the interfacial extra shear stress and the first normal stress difference for a model bsystem withhd 5 1000 Pa s,hm 5 2000 Pa s,w 5 0.1, G/R 5 2000 Pa.gc : Critical shear rate.
sely
iting
atress
milar
ch isthedeli.e.,s is
es,f
hat
tion,
1388 YU ET AL.
wherel is an unknown phenomenological parameter that was considered to be inverproportional to the volume fractionl 5 a/w, wherea takes various numerical valuesdepending on the system under consideration. The model is unable to recover the limcase given by Taylor~1932! unless the parameterl is taken as a function of the viscosityratio, which is not justified. It also predicts constant viscosity at low shear rate andparabolic increase for larger shear rates! The same can be said for the first normal sdifference.
Choi and Schowalter~CS! ~1975! ~the CS model! also predicted for droplet-typemorphology, the shear stress, and the first normal stress difference that scale in a simanner as the present model at low shear rate (s } g andN1 } g2):
s12CS 5 hmH 11w
5p12
2~p11!F11w
5~5p12!
4~p11!G J g, ~26!
N1CS 5
2
5
hmw
11Z2 F19p116
4~p11!S 11w
5~5p12!
4~p11!D G2
gCa, ~27!
N2CS 5 2
1
280
hmw
11Z2
~19p116!~29p2161p150!
4~p11!3
3F11w5~5p12!~41p21121p1188!
8~p11!~29p2161p150!G gCa, ~28!
Z 5~19p116!~2p13!
40~p11!CaF11w
5~19p116!
4~p11!~2p13!G .
The CS model predicts linear dependence of shear stress on the shear rate, whisimilar to the Doi–Ohta model and the present model at low shear rate but, similar toDoi–Ohta model, the CS model predicts constant viscosity. The failure of the CS moto predict the shear-thinning viscosity is due to linear expansion of the droplet shape,small deformation constraint. Additionally, the effect of droplet size on the shear stresnot considered by the CS model.
Formally, droplet size is not important in steady shear flow at small shear ratwhereas it starts to play a role at large shear rates. In fact, using the expressions of 1 ,f 2 , andK given by Eqs.~13! and~21! and expanding Eq.~20! with respect to the volumefraction gives the following expression for the viscosity:
hb
hm5 11wF5~p21!
2p131
5
2p13
160~p11!~19p116!
1600~p11!21~19p116!2~2p13!2Ca2G1O~w2!.
~29!
The above equation shows that the viscosity is function of the capillary number tinvolves the size of the droplets. However, for small shear rates~small Ca), the dropletsize disappears in the expression of viscosity, and Eq.~29! reduces to Taylor’s model~1932, 1934!:
hb
hm5 11w
5p12
2~p11!, ~30!
which are also the leading terms in the CS model@Eq. ~25!#. For a dilute suspension ofundeformable rigid spheres, the present model also recovers Einstein’s equa
er.
do
ant,sin
re
er.
1389RHEOLOGY AND MORPHOLOGY IN EMULSIONS
hb 5 hm(112.5w) ~1906, 1911!. A similar analysis can be applied to the normal stressdifference. For small shear rate and low volume fraction, the CS model gives
N1CS 5
~19p116!2hmgCaw
40~p11!2
and
N2CS 5 2
~19p116!~29p2161p150!hmgCaw
1120~p11!3 , ~31!
while the present model@Eqs.~19! and ~16!# gives
N1 5~2p13!~19p116!2hmgCaw
40~p11!2~2p1322~p21!w!
and
N2 5 2 12 N1. ~32!
It is obvious from Eqs.~31! and ~32! that the predictions ofN1 are the same for thepresent model and the CS model under a vanishing volume fraction and capillary numb
Figure 2 compares the predictions of the interfacial area, the semiaxesL, B, andW ofthe ellipsoid, and the orientation angle in steady state with experimental results of Guiand Greco~2001!. L andB are considered in the shear plane (L . B), whereasW istaken along direction of the vorticity. At low shear rate, the droplets first deform slightlyand rotate at 45°. Upon an increase in shear rate, the deformation becomes dominwith slight variation in rotation. At large shear rates, the orientation no longer increaseand the droplets stretch into long ellipsoids. This can be seen more clearly in Fig. 3,which the relative variation ofL, u, and Q are plotted against the shear rate. Thisbehavior was also described theoretically by Bousminaet al. ~2001!.
The model predictions for shear viscosity and the first normal stress difference acompared with our experimental results@El Mesri ~2002!# in Fig. 4. Two PDMS/PIBsystems withp 5 0.49, but with different volume fractions of 0.05 and 0.1, were used.
FIG. 2. Orientation angle, semiaxes, and surface area of an ellipsoid as the function of the capillary numbScattered dots are experimental results of Guido and Greco~2001!. Lines are predictions of the present model.
ntaltheuresel
earzutid-
rens
tressgle
m
1390 YU ET AL.
Excellent agreement is found between the theoretical predictions and the experimeresults for the first normal stress difference. For the viscosity, the difference betweenmodel predictions and the experimental results is less than 10%. The model also captthe slight shear thinning behavior exhibited by the mixture with 10% of PIB. The modpredictions are also compared to the experimental results of Vinckieret al. ~1996! in Fig.5 and to those of Grizzutiet al. ~2000! in Fig. 6. Figures 5 and 6 report the predictions upto the critical shear rate, for which only some experimental data from Vinckieret al.wereavailable. Once again, the model nicely describes the variation of viscosity with the shrate and the slight shear thinning that appeared in the experimental results of Grizet al. ~2000!. We should stress here that the predictions were obtained without any ajustable parameter.
A. Quantitative relationships between rheology and structure
We will now show that is possible to extract some information about the structufrom rheological material functions and, conversely, that rheological material functiocan be computed fromin situ morphological observations.
1. Orientation angle of the droplets
The orientation angle of a dispersed ellipsoid under steady shear flow is given by
u 5 12 arctan~f1/Ca!. ~33!
Therefore, Eqs.~18! and ~19! readily give the following expression that relates the ori-entation angle to the ratio of shear stress and the difference in first normal stress:
s12s /N1
s 5 12 tan~2u!. ~34!
It is then possible to extract the orientation angle from the measurements of shear sand the first normal stress difference. Note that, at low shear rate, the orientation anscales asg21. The above equation is identical to the one given by Jansseuneet al.~2000!. However, variation of the orientation angle with the shear rate is different frothe affine deformation model proposed by Jansseuneet al. ~2001!. This is because the
FIG. 3. Relative change of semiaxesL, surface areaQ, and orientation angleu with Ca.
en
of
1391RHEOLOGY AND MORPHOLOGY IN EMULSIONS
present model expresses stress as a function of the viscosity ratio, which was not takinto account by Jansseuneet al. ~2001!.
2. Deformation of the droplets
To express the shear stress and the first normal stress difference as a functiondroplet deformation, we define two deformation parameters (D2,LB andD2,WB), whichare slightly different from the classical deformation parameter defined by Taylor~1932,1934!:
D2,LB 5L22B2
L21B2 , ~35!
FIG. 4. Comparisons of the model predictions of the shear viscosity and the first normal stress difference withthe experimental results@El Mesri ~2002!#. Solid lines are predictions of the present model.
e
rst
1392 YU ET AL.
D2,WB 5W22B2
W21B2 . ~36!
From Eqs.~18!, ~19!, and~35!, we obtain the following expressions for steady interfacialextra stress and the first normal stress difference as a function ofD2,LB andu:
s12s 5 2
3 KD2,LB2 tan~2u!, ~37!
N1s 5 4
3 KD2,LB2 . ~38!
Equation~38! shows that the deformation term can be extracted from knowledge of thfirst normal stress difference~and vice versa!. Knowing D2,LB it is then possible to
FIG. 5. Comparisons of the theoretical predictions with the experimental results of the shear viscosity and finormal stress difference. Scattered dots are experimental results of Vinckieret al. ~1996!. Solid lines arepredictions of the present model.
yple
ons
heartress
ery
eatly
sient
cat-
1393RHEOLOGY AND MORPHOLOGY IN EMULSIONS
extract the orientation angle from knowledge of the shear stress~and vice versa! usingEq. ~37!. ParameterK is given by Eq.~21!.
The deformation parameterD2,LB plays an important role in determining the steadshear rheological material functions for immiscible blends. This is expected since simshear flow is a special two-dimensional flow. The steady rheological material functishould only be dependent on the morphology in the shear plane, i.e., onL, B, and theorientation angle. Although morphology information for the vorticity direction, i.e.,W,does not affect the steady state rheology, it does influence the transient evolution of sstress. Figure 7 shows the evolution of interfacial shear stress, the first normal sdifference, and the two deformation parameters (D2,LB andD2,WB) for various values ofCa. An interesting point to be noted is that the evolution of interfacial shear stress is vsimilar to that of the deformation parameterD2,WB. This is particularly true for largeshear rates, where boths12
s andD2,WB show overshooting, which is not seen in the curvof D2,LB . This means that the behavior of transient interfacial shear stress is greinfluenced by semiaxisW. For small shear rates,W is nearly 1 and the interfacial stressis linear with the shear rate. For large shear rates,W starts to deviate from 1 and theinterfacial stress starts to show some nonlinear behavior, such as overshooting in transtress and shear-thinning phenomena. The role ofW in controlling the linear behavior isalso found in large amplitude oscillatory shear flow@Yu et al. ~2002!#.
IV. UNIAXIAL ELONGATIONAL FLOW
For uniaxial elongational flow, the velocity gradient tensor is given by
¹v 5 S e 0 0
0 2 e/2 0
0 0 2 e/2D . ~39!
The steady elongational stress takes the following expression:
FIG. 6. Comparisons of the theoretical predictions with the experimental results of the shear viscosity. Stered dots are experimental results of Grizzutiet al. ~2000!. Solid lines are predictions of the present model.
-i–
1394 YU ET AL.
ss 5 s11s 2s22
s 52KCa f2
2
f 15 hm
3~19p116!w
2@5~p11!2~5p12!w#e. ~40!
The interfacial contribution to the steady elongational stress is proportional to the elongational rate applied. Therefore, the steady elongation viscosity is constant. The DoOhta model also predicts constant elongational viscosity:
ss 5G~213m2Am~415m!!
2l~11m!2 e. ~41!
Steady interfacial stress can be related to deformationD2,LB by
ss 54Kf2D2,LB
31D2,LB. ~42!
FIG. 7. Evolution of interfacial stress, first normal stress difference, and deformation,D2,LB , D2,WB over timein simple shear flow under differentCa.
,l
1395RHEOLOGY AND MORPHOLOGY IN EMULSIONS
The deformation parameterD2,LB can be extracted from Eq.~42! in a similar manner tothat in simple shear flow. The dependences of interfacial elongational stress, viscosityand semiaxes on the elongation rate are shown in Fig. 8. The interfacial elongationaviscosity remains constant before droplet breakup. The evolution of interfacial elonga-tional stress,L, and interfacial areaQ are shown in Fig. 9. Clearly, both elongationalstress andL vary in a similar manner in time and the rheological behavior remains linearfor all elongation rates.
V. PLANAR HYPERBOLIC FLOW
For planar hyperbolic flow, the velocity gradient tensor is given by
¹v 5 S e 0 0
0 2 e 0
0 0 0D . ~43!
FIG. 8. Steady interfacial stress and viscosity in uniaxial elongational flow.
FIG. 9. Evolution of stress, semiaxesL, and surface areaQ over time in uniaxial elongation flow.
1396 YU ET AL.
Steady stress takes the following expression:
ss 5 s11s 2s22
s 58KCa f2
2
3 f 15 hm
2~19p116!w
5~p11!2~5p12!we. ~44!
The interfacial contribution of extra stress is similar to that found for uniaxial elonga-tional flow. The only difference is the constant coefficient. The Doi–Ohta model alsogives the linear variation of elongational stress with the deformation rate:
ss 54G~2m11!
3l~m11!2e. ~45!
FIG. 10. Steady interfacial stress and viscosity planar hyperbolic flow.
FIG. 11. Evolution of stress, semiaxesL, and surface areaQ over time in planar hyperbolic flow.
elon-
w is
dualxture
ofogi-.
Re-ande
1397RHEOLOGY AND MORPHOLOGY IN EMULSIONS
Steady interfacial stress can be related to the deformation of droplets as
ss 54
3Kf2D2,LB . ~46!
The dependences of interfacial elongational stress, viscosity, and semiaxes on thegation rate are shown in Fig. 10. The evolution of interfacial stress,L, and interfacial areaQ are shown in Fig. 11. Behavior similar to that observed in uniaxial elongational flofound.
VI. CONCLUDING REMARKS
The main result of this work is that the present proposed model gives a directrelationship between the microstructure and rheological material functions for a miof two Newtonian liquids. From an engineering viewpoint, this allows the extractionmorphological information from rheological material functions and, conversely, rheolcal material function functions can be extracted fromin situ morphological observationsSuch a relationship is valid for both transient and steady state regimes.
ACKNOWLEDGMENTS
This work was financially supported by the Natural Sciences and Engineeringsearch Council of Canada~NSERC!, Canada Research Chair on Polymer Physics,Nanomaterials~for one of the authors M. B.!, and the Fonds pour la Formation dChercheurs et l’Aide de la Recherche du Que´bec ~FCAR! funds.
NOMENCLATURE
Scalars
Ca capillary number(5 hmRg/G);D2,LB deformation parameters(L22B2)/(L21B2);D2,WB deformation parameters (W22B2)/(W21B2);I i ( i 5 1, 2, 3! scalar invariant ofM ;N1 first normal stress difference;N2 second normal stress difference;R radius of undeformed drops;p viscosity ratio (hd /hm);Q total interfacial surface area per unit volume;e elongation rate;w volume fraction of dispersed phase;g shear rate;h i ( i 5 d,m,b) viscosity of dispersed phase, matrix, or blend;l, m phenomenological parameters of the Doi–Ohta model;u orientation angle;F free energy;G interfacial tension.
Vectors and tensorsv velocity vector;D deformation rate tensor;M second order conformation tensor;V vorticity tensor;
ts
l
s:
es,’’
-
,’’
c
sults
ol-
nds
-
m
e
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1398 YU ET AL.
d identity tensor;s stress tensor;s i ( i 5 d,m) stress contribution of the dispersed phase or matrix;sm viscosity ratio contribution of the stress tensor;ss interfacial stress tensor.
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