A simple constitutive equation for immiscible blends

A simple constitutive equation for immiscible blends Wei Yua) and Chixing Zhou Advanced Rheology Institute, Department of Polymer Science and Engineering, Shanghai Jiao Tong University, Shanghai 200240, People’s Republic of China (Received 17 March 2006; final revision received 27 December 2006兲

Synopsis A simple rheological constitutive equation for immiscible blend is suggested in this work. The model is based on the ellipsoidal description of droplets. Droplet deformation is described by the ellipsoidal model, and droplet breakup/coalescence is considered to affect the droplet size only. The discrete breakup/coalescence process is approximated by a continuous dynamic equation. A simple rescaling method is suggested to integrate the volume-preserving ellipsoidal deformation model and the droplet size evolution dynamics. The predictions of the model are compared with experiments in start up of shear and shear rate sweep modes. The agreement is quite satisfactory on transient and steady rheological properties, as well as the morphology of droplets. © 2007 The Society of Rheology. 关DOI: 10.1122/1.2437206兴

I. INTRODUCTION Blending two different polymers is an easy way to obtain materials which can take advantages of the properties of both components. However, it is crucial to control the phase morphology of blends to optimize the blend’s properties. It is known that the morphology evolution of a blend depends on the type and strength of flow field, viscosity ratio, the interfacial tension and also the initial morphology. There are lots of experimental as well as theoretical research works on the morphology evolution under flow field, although most theoretical work focuses on simple homogeneous flow field, while experimental work investigates both simple homogeneous flows and very complex flows, such as the flow field in twin screw extruder. One of the difficulties in predicting the morphology under a flow field is the capability and accuracy of constitutive equation in describing the morphological and rheological properties. Two types of models have been suggested according to the description of morphology, i.e., models based on droplet morphology 关Maffettone and Minale 共1998兲; Jackson and Tucker 共2003兲; Yu and Bousmina 共2003兲兴 and models based on local coarse grained morphology 关Doi and Ohta 共1991兲; Lee and Park 共1994兲; Wagner et al. 共1999兲; Bousmina et al. 共2001兲; Grmela et al. 共2001兲; Peters et al. 共2001兲; Almusallam et al. 共2000, 2004兲兴. The models that can describe local coarse grained morphology use a statistical area tensor or interfacial anisotropic tensor to describe the complex interface. These models include the well-known Doi-Ohta model, and many improved versions of it. Although the models based on local coarse grained morphology are more powerful in

a兲

Author to whom correspondence should be addressed; electronic mail: [email protected]

© 2007 by The Society of Rheology, Inc. J. Rheol. 51共2兲, 179-194 March/April 共2007兲

0148-6055/2007/51共2兲/179/16/$27.00

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describing complex morphology, especially for blends with co-continuous or irregular phase morphology, they are less widely used since little quantitative morphology comparisons have been made in the literature. This is partially attributed to the fact that several parameters in the model must be given in an ad hoc manner. This problem also restricts the capability of these models to be extended to viscoelastic system. The other widely used models are based on the assumption that the shape of droplets can be approximated by an ellipsoid, which is represented by an ellipsoidal shape tensor. These models include the Maffettone-Minale model 共1998兲 and also many extensions of it 关Jackson and Tucker 共2003兲; Yu and Bousmina 共2003兲兴. The ellipsoidal models have powerful predictability on the droplet deformation, relaxation and corresponding rheological properties. Moreover, the viscoelastic effects of components can be easily incorporated into the ellipsoidal model 关Maffettone and Greco 共2004兲; Yu et al. 共2004, 2005兲兴. However, the main problem with the ellipsoidal models is that only droplet deformation and relaxation can be predicted. Processes with interface breakup, such as droplets breakup and coalescence, cannot be described by these models. Some efforts have been made to include the interface breakup. Dressler and Edwards 共2004兲 considered the evolution of droplet numbers due to droplet breakup and coalescence. However, a single time scale for breakup and coalescence is used. Moreover, it is lack of quantitative comparison on morphology and rheology between model predictions and experiments, especially on a transient process under large step shear where the breakup process dominates and hysteresis phenomena where both breakup and coalescence process play a role. Another approach is to construct hybrid models. Zkiek et al. 共2004兲 and Almusallam et al. 共2003兲 established models based on local coarse grained morphology which includes the thread breakup under quiescent condition. Almusallam et al. 共2004兲 imposed a constraint on the interfacial relaxation term that preserves a frame-invariant quantity that approximates the volume of an ellipsoidal droplet, and the breakup of droplets is modeled by casting the Tomotika theory 共1935兲 in terms of anisotropic interface tensor. An ad hoc breakup function is chosen to correct the breakup behaviors under different limits. Peters et al. 共2001兲 developed a numerical algorithm that determines stress for a blend modeled as mixture of spherical and highly elongated droplets. Although some of the abovementioned models can make acceptable predictions under certain conditions, there are more or less arbitrary parameters that need to be specified manually. In this paper, we would suggest a simple model which is based on the ellipsoid assumption of droplet shape and includes the breakup and coalescence process. We try to make the model simple and fully predictive at the same time. Moreover, the predictions of the model under different conditions will be verified. II. MODEL We consider an immiscible blend with droplet morphology. The shape of dispersed droplet can be approximated by an ellipsoid. Therefore, the ellipsoidal shape tensor G is adopted to describe the droplet shape 关Wetzel and Tucker 共1999兲; Jackson and Tucker 共2003兲; Yu and Bousmina 共2003兲兴, i.e., any point xi on the droplet interface satisfies Gijxix j = 1. The ellipsoidal shape tensor G is a positive symmetric second order tensor. The eigenvalues of G are equal to the reciprocal of the square of the semiaxes of the ellipsoid, 共1 / L2 , 1 / B2 , 1 / W2兲. L, B, and W are the three semiaxes of the ellipsoidal droplet. It should be noticed that the shape tensor G is not dimensionless in its original ˜ = GR2, where R denotes the radius definition. Usually it can be made dimensionless by G of the undeformed spherical droplet 关Yu and Bousmina 共2003兲兴. In almost all the previous ellipsoidal models, a constraint of volume preservation is applied, e.g., detG = constant or

A SIMPLE CONSTITUTIVE EQUATION FOR BLENDS

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˜ = 1, which is a natural result from the constant volume fraction of dispersed droplet detG and constant number density of droplet. There will be no interface breakup under such constraint, which makes the classical ellipsoidal models suitable only for droplet deformation or relaxation. It is believed that the volume of a droplet is preserved during deformation and relaxation, and changes if breakup or coalescence happens. This means that the classical ellipsoidal models can still work for droplet deformation and relaxation, ˜ 关Wetzel and Tucker 共2001兲兴: which can be modeled by the dynamic equation of G ˜ DG ij ˜ +G ˜ L = 0, + LkiG kj ik kj Dt

共1兲

where D / Dt denotes a material derivative, Lij represents the velocity gradient tensor over the interface 共=⳵u j / ⳵xi兲. Lij can be taken from any classical ellipsoidal model, such as MM model 关Maffettone and Minale 共1998兲兴, JT model 关Jackson and Tucker 共2003兲兴 and YB model 关Yu and Bousmina 共2003兲兴 for Newtonian system, MG model 关Maffettone and Greco 共2004兲兴, YBZT model 关Yu et al. 共2004兲兴 and YZB model 关Yu et al. 共2005兲兴 for viscoelastic system. Although different models can be used to describe droplet deformation, the accuracy is different. Lij can be calculated from the velocity distribution across the interface, and it is definitely not equal to the velocity gradient of the applied flow. In this paper, the following expression is adopted 关Yu and Bousmina 共2003兲兴: ¯ ␣ + L␤ 兲R , Lij = ␻Aij + 共Bmnkl + Cmnkl兲RimRukeAuvRvlR jn + Rim共L mn mn jn

共2兲

eAij

is the rate of deformation tensor of applied flow field, Rij is the rotation tensor where which depends on the morphology of droplet, Bijkl and Cijkl are fourth order tensors which depend on the aspect ratio of an ellipsoidal droplet and related to the complete Eshelby tensor and alternate Eshelby tensor, ¯Laij and L␤ij are contribution from the interfacial tension which also depends the viscosity ratio and Eshelby tensors 关see Yu and Bousmina 共2003兲 for details兴. When there are droplet breakup and coalescence, both the volume of a single droplet and the number density of droplet will change, while the total volume of dispersed droplet is still a constant. This means first we need an additional dynamic equation to describe the evolution of number density of droplet 共or equivalently the dynamic equation of droplet radius兲. However, the volume 共or equivalently radius兲 of a droplet will change when such droplet undergoes breakup or coalescence. This makes it difficult to apply Eq. 共1兲 straightforward to droplet breakup or coalescence. An additional rescaling process for shape tensor should be taken. Since droplet breakup or coalescence is a discrete event, it is not straightforward to integrate it into a continuous model. In fact, the general approach is to describe the evolution of droplet radius R by a differential equation: R dR =− + rcoal , dt 3tbreakup

共3兲

where the breakup process is represented by a characteristic breakup time tbreakup, and rcoal denotes the rate of coalescence. Actually, Eq. 共3兲 is originally derived from the dynamic equation of droplet number density, which assumes that the change of droplet number density due to breakup and coalescence is proportional to the droplet number density. Similar equations have been used in some previous work 关Peters et al. 共2001兲; Patlazhan and Lindt 共1996兲兴. The breakup time tbreakup is controlled by the capillary number Ca, which is defined as the ratio between shear stress ␩m␥˙ 共␩m is the viscosity of matrix, ␥˙ is the shear rate兲 and interfacial tension ⌫ / R 共⌫ is the interfacial tension兲. Although the

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breakup process is essentially complicated, and not all of the mechanisms have been completely understood, only two representative modes of breakup are considered in this work. When Ca is smaller than the critical value for breakup Cac, there will be actually no breakup, and tbreakup can be infinity. When Cac ⬍ Ca ⬍ ␬Cac, the breakup is dominated by necking 关Grace 共1982兲兴. The corresponding breakup time for necking is approximated by tbreakup ⬇ 85.3Cac p0.45␥˙ −1 关Grace 共1982兲兴, where p = ␩d / ␩m defines the viscosity ratio between droplet viscosity ␩d and matrix viscosity ␩m. The term ␬ is a flow dependent parameter, which is about 2 in simple shear flow and 5 in planar hyperbolic flow 关Janssen and Meijer 共1993, 1995兲兴. When Ca ⬎ ␬Cac, the droplet will be extended into a long thread and capillary instability will dominate the breakup process. Since the breakup process via capillary instability is rather complex to be modeled by either the ellipsoidal model or local coarse grained model, the breakup time tbreakup is used to describe the process, which is also experimentally shown to be tbreakup ⬇ zCa2/3 p0.5␥˙ −1 关Van Puyvelde 共2000兲兴 and z is taken as a constant here 共z = 26兲. The choice of critical capillary number Cac is important in determining the breakup time of necking. Cac can be taken from the experimental results of 关Grace 共1982兲兴. Another choice is the critical value predicted from the ellipsoidal model 关Maffettone and Minale 共1998兲; Jackson and Tucker 共2003兲兴. These two methods can give different values of Cac, and the value from the product of the smaller one and a factor 共0.85兲 is chosen in the present model to ensure convergence of the ellipsoidal model. For example, in the model blend I below, the viscosity ratio is 0.44. Cac from the Grace experiments is about 0.46. Cac from the ellipsoidal model 关for example, cf. Jackson and Tucker 共2003兲兴 is about 0.45. The Cac in the calculation is chosen to be 0.45*0.85= 0.39. The reason to multiply 0.85 is to increase the convergence of the ellipsoidal since the ellipsoidal model will predict infinite droplet deformation at Cac. The coalescence rate is taken from Janssen and Meijer 共1993, 1995兲 as rcoal =

R 4␾␥˙ J , 3 ␲

共4兲

where the probability of coalescence J is expressed as a product of collision probability Jcoll and drainage probability Jdrain,

冉

Jcoll = exp −

冊

␲ , 8␥˙ ␾tproc

冉

Jprob = exp −

冑3R 4hc

冊

pCa3/2 .

共5兲

The term tproc is the time of processing, ␾ is the volume fraction of droplets, and hc is the critical thickness of film breakup 关hc = 共AR / 8␲⌫兲1/3, A is the Hamaker constant兴; ␥˙ is used here just because the coalescence model is derived for shear flow. For general flow, it can be replaced by 冑2IIDij, where Dij is the rate of deformation tensor of applied flow field, and IIDij is the second invariant of tensor Dij. It should be stressed again that the breakup of a droplet 共or coalescence of two droplets兲 is a discrete event which happens at the time tbreakup 共or the corresponding coalescence time兲. These discrete events are modeled to be continuous by Eq. 共3兲. Therefore, the continuous approximation, Eq. 共3兲, is suitable to describe the continuous breakup 共and coalescence兲 process of droplets, and inappropriate for single breakup 共and coalescence兲 process of a droplet. It should be noticed that two different time scales are chosen in the present model for breakup and coalescence, respectively. The characteristic times of breakup are taken from the empirical results of experiments, while the characteristic time for coalescence is obtained from the lubrication theory of film drainage. For droplet breakup, the characteristic times depend on Cac or Ca, which are applicable for


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deformed droplet. For droplet coalescence, the characteristic time is obtained for almost spherical droplet, which is only an approximation for the coalescence between deformed droplets. Now the problem turns to how to integrate the volume preservation dynamic Eq. 共1兲 with the dynamic evolution Eq. 共3兲 of droplet size. It is known that at time ti, the shape ˜ and orientation angle ␪ of droplet, e.g., dimensionless length of droplet axes ˜Li, ˜Bi, W i i ˜ . This can be and ␸i, can be determined completely from the dimensionless shape tensor G ˜ . At the next time readily done by calculating the eigenvalues and eigenvectors of G ti+1 = ti + ⌬t, the droplet radius becomes Ri+1 according to Eq. 共3兲 due to possible breakup or coalescence. It can be assumed that during the breakup 共or coalescence兲, only the maximum length of droplet changes, all other length of axes and orientation remain ˜ is also changed from unchanged. Keeping in mind that the scaling parameter of tensor G 2 2 Ri to Ri+1, we suggest the following rescaling to consider the volume variation of droplet:

冉冊

˜L = ˜L Ri+1 i+1 i Ri

2

,

˜B = ˜B i+1 i

冉冊

Ri , Ri+1

˜ =W ˜ W i+1 i

冉冊

Ri , Ri+1

␪i+1 = ␪i,

␸i+1 = ␸i . 共6兲

It should be noticed that the lengths of droplet axes are scaled at different droplet ˜ at time radius at different times, ˜Li = Li / Ri and ˜Li+1 = Li+1 / Ri+1. The droplet shape tensor G ˜ ˜ ˜ ti+1 = ti + ⌬t can be reconstructed by the droplet shape parameters Li+1, Bi+1, Wi+1, ␪i+1 and ␸i+1. It should be emphasized that no assumption has been made here on the number of droplets produced by breakup 共or coalescence兲. To model the real immiscible blends, the droplets’ size distribution should be included since all of the above equations tackle only single droplet problem. It will be illustrated below that the droplet size distribution has a great effect on the transient rheological properties of immiscible blends. Either discrete or continuous droplet size distribution can be used in the model. We assume that initially the number density ␾n of droplets can be described by a log-normal distribution function:

␾n =

1

冑2␲␴R e

ln共R/Rc兲2/2␴2

共7兲

The number average radius Rn and volume average radius RV can be calculated from 2 2 Eq. 共7兲 as Rn = 兰⬁0 ␾nRdR = Rce−␴ /2 and RV = 冑3 兰⬁0 ␾nR3dR = Rce−3␴ /2. The polydispersity of 2 droplet size can be defined as PI ⬅ RV / Rn = e␴ . Therefore, the droplet size distribution Eq.共7兲 can be completely determined by the parameters ␴2 = ln PI = ln共RV / Rn兲 and Rc = RV共PI兲−3/2. In fact, Eq. 共7兲 is only used to get the initial droplet size distribution in this work. The droplet size distribution will evolve in time under a flow field. In the calculation, droplet size distribution is discretized into several subpopulations with the number density ␾n,i, from which the volume fraction ␾i of each subpopulation can be readily calculated and ␾ = 兺i␾i. For each subpopulation, the droplet deformation and breakup can be calculated according to the above model, but modeling on coalescence becomes a big problem. In the present model, a simplified treatment is taken. For each subpopulation of droplet size, the coalescence is restricted between droplet with same size, but the volume fraction used is the total volume fraction ␾, not the volume fraction ␾i of subpopulation. Such simplified treatment is of course not accurate since coalescence between different droplets is neglected. However, the cumbersome population balance can be avoided. The morphology of immiscible blends can be calculated by integrating Eqs. 共1兲–共7兲 using Runge-Kutta method. The corresponding rheological properties of the blends can be computed by the stress expression for Newtonian blends 关Yu and Bousmina 共2003兲兴 or

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TABLE I. Model blends definitions and properties.

No. I II III

Model blends 10% PIB/PDMS 5% PIB/PDMS 10% PIB/PDMS 10% PIB/PDMS 30% PIB/PDMS

Viscosity ratio

Interfacial tension 共mN/m兲

Initial radius 共␮m兲

References

0.441 0.927

2.3 2.8

9.5 ¯

Vinckier et al. 共1997兲 Minale et al. 共1998兲

1.6a

3.0

¯

Grizzuti et al. 共2000兲

a

Zero-shear viscosity at 9 ° C.

viscoelastic blends 关Yu et al. 共2004, 2005兲兴. In the following predictions and comparison, the stress expression for Newtonian blends will be used 关Yu and Bousmina 共2003兲兴:

␶ = 2␩meA + 2共␩d − ␩m兲␾e −

冉

冊

1 ⌫␾ A − ␦ , Lc trA 3

共8兲

where eA is the deformation rate tensor of applied shear flow, e is the deformation rate tensor on the interface, Lc is the local characteristic length scale defined as the ratio between the volume and the surface area of the ellipsoid 共Lc ⬅ Vd / Sd兲. A is the area tensor, which can be converted from the ellipsoid morphology tensor G 关Wetzel and Tucker 共1999兲兴. Equation 共8兲 means that the stress of the blends only depends on the droplet shape, and the effect of droplet breakup or coalescence on the stress are not included. The detailed calculation on e can be found in the literature 关Yu and Bousmina 共2003兲兴. When the droplet size distribution is considered, the stress ␶i of each subpopulation with volume fraction ␾i is calculated by Eq. 共8兲. The total stress is obtained by a linear addition ␶ = 兺i␶i␾i. A comment on the present approach is that theoretically both Newtonian blends and viscoelastic blends can be described, once the breakup time and coalescence mechanism for viscoelastic system are known. Even the breakup and coalescence conditions are not quantified at present, the dynamics of droplet breakup and coalescence for Newtonian system can also be integrated with the viscoelastic droplet deformation to give a first order approximation for viscoelastic system. III. MODEL PREDICTIONS AND COMPARISONS The above suggested model can be used to predict both the morphological information and the rheological properties of an immiscible blend without a fitting parameter. In fact, the model consists of two parts, the droplet deformation and the breakup/coalescence of droplet. Although both the Newtonian model and the viscoelastic model can be used for droplet deformation, the breakup/coalescence process is modeled only from the experimental results of the Newtonian system. It is possible that the viscoelasticity of both components will affect the process of breakup/coalescence. However, the effect of viscoelasticity on droplet breakup or coalescence is neglected in the present model. Such effect could readily be integrated into the model once the influence is known. Therefore, the model is applied only to Newtonian blends here to verify its validity. The ellipsoidal model for droplet deformation is taken to be Yu-Bousmina model 共2003兲. The rheological properties and morphological information of three model blends are predicted and compared with experiments. The definition, properties, and references of three model blends are listed in Table I. Two model Newtonian fluids are used, poly-


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FIG. 1. Transient first normal stress difference of model blend I subjected to different shear rate. The solid lines are the model predictions. The scatter symbols are the experimental data from Vinckier et al. 共1997兲.

isobutylene 共PIB兲 and polydimethylsiloxane 共PDMS兲. Since the model predictions on the process of droplet deformation and relaxation are well compared with the experiments 关for example, cf. Yu and Bousmina 共2003兲兴, it is more interested in flow field that breakup or coalescence could happen and dominate the process. The first comparison is made on model blend I with the viscosity ratio 0.44. The initial volume average radius of droplets is 9.5 ␮m. The blends were subjected to different shear rate which ranged from 1.0 to 5.0 s−1. The excess first normal stress differences are compared first due to its high sensitivity to the morphology evolution. The experimental data are taken from Vinckier et al. 共1997兲 and scaled by the initial value of N1,excess, which is taken to be 16 Pa. To consider the effect of initial droplet size distribution, a log-normal distribution, Eq. 共7兲, is used and the polydispersity of droplet radius is taken as PI = 1.05. The model predictions as well as experimental data are shown in Fig. 1. It should be stressed that all the parameters in the model calculations are taken from the experiments except the polydispersity of droplet size. The predicted N1,excess shows a quick increase at short time, followed by an overshooting, and an equilibrium value at long time. The overshooting of N1,excess is well predicted by the model, and the predicted equilibrium values at long time are consistent with the experiments. The agreement here seems to be better than the model of Peters et al. 共2001兲. To show more clearly the contribution from droplets with different size, the components of N1,excess are shown in Fig. 2. The droplet size distribution is discretized into ten components, and the discrete distribution is shown in the inset of Fig. 2. It is clearly seen that the overshooting in N1,excess of blend is attributed to the overshooting in N1,excess of droplets with different size. The N1,excess for droplet with monodispersed radius shows an overshooting, and a sharp decrease corresponding to the end of breakup 共as shown in Fig. 3兲. However, the overshooting of N1,excess is not so sharp which is smoothed out by the breakup of droplets with different size. Figure 3 shows the relationship between the droplet deformation, droplet radius, and N1,excess for a monodispersed blend. The droplet first undergoes a fast deformation, and starts to break up. The dimensionless length of the droplet long axes ˜L = L / R continues to increase as the droplet breakup 共droplet radius decreases兲. There are two factors that influence the dimensionless length ˜L of axes. On the one hand, flow field tends to stretch the droplet, which increases

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FIG. 2. The contribution of different droplet subpopulations to N1s at ␥˙ = 5.0 s−1. The inset shows the initial droplet size distribution used in the calculation. The solid dots are the experimental data from Vinckier et al. 共1997兲.

droplet length L. The ability of droplet deformation depends on the capillary number. The larger Ca, the larger droplet deformation. On the other hand, droplet breakup happens simultaneously during droplet deformation; the droplet radius R and capillary number decreases, which tends to decrease the droplet deformation. Therefore, the dimensionless length ˜L increases at the startup of shear. After some shear strain, the breakup continues the deformation process which causes ˜L to decrease. The droplet will break up successively until the capillary number becomes smaller than the critical value. After that, the droplet size and droplet deformation reach a steady state. Coalescence between droplets can also happen in the late stage when the volume fraction is large and droplet size is small enough. The breakup process for droplets with different size is almost the same, except that the larger droplet takes a longer time to finish the breakup. Correspondingly, N1,excess increases almost linearly with strain at the beginning, and shows a sharp decrease to an equilibrium value. During the breakup process, when the capillary number is larger

FIG. 3. The relationship between droplet deformation and first normal stress difference under a shear rate ␥˙ = 5.0 s−1.


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FIG. 4. Comparison with the prediction of Dressler-Edwards 共DE兲 model. The parameters in DE model are ␥˙ = 5.0 s−1, 冑G⌫ = 93 Pa, ␭s = 1.7 s, ␭c / ␭s = 0.025, ␭c␭s / ␭2n = 1, p = −1.26, k = 2, ␪ = 0.001.

than the critical, the droplet would undergo large deformation to breakup. When Ca becomes smaller than Cac, the equilibrium deformation is rather small as compared with the deformation during breakup. So, there would be a sharp transition of droplet deformation at the end of droplet breakup. It is known that N1,excess is strongly related to the droplet deformation in steady state 关Yu et al. 共2002兲兴. It is expected that there will be a sharp transition of N1,excess which corresponds to the end of droplet breakup. Therefore, it can be inferred from the components’ contribution of N1,excess in Fig. 2 that smaller droplets finish breakup first, and then the larger ones. This is consistent with our understanding of the droplet breakup process. A comparison with the prediction of Dressler-Edwards model 关DE model, Dressler and Edwards 共2004兲兴 is added in Fig. 4. The parameters used in the DE model are also listed. The prediction of the DE model on N1,excess also shows an overshoot, and then decreases to a steady value, but the overshoot is not consistent with the experimental data. Correspondingly, the droplet long axis increases first and then decreases to a steady value, which is quite similar to the present prediction 共Fig. 3兲. However, the prediction on droplet size distribution is completely different. The prediction of the DE model on the representative average droplet size 共trS / 3兲 shows a pronounced overshoot and reaches a steady state after strain about 300. The number density also shows an overshoot and decreases to a steady state. From these predictions, it is hard to explain the initial breakup process 共overshoot in the number density兲 with the increase of average droplet size 共overshoot in trS / 3兲. However, the present model predicts a monotonous decrease of droplet size due to breakup 共Fig. 3 R ⬃ strain兲, and a shift of droplet size distribution to smaller droplet size 共see Fig. 7兲. On the other hand, it is difficult to conclude from the predictions of the DE model the breakup process of different droplets. While from the present model, it can be inferred from tbreakup about the breakup process of different droplets: when Cac ⬍ Ca⬍ ␬Cac, tbreakup is independent of droplet size which means the large droplet and the small droplet start to break up at the same time scale; when Ca ⬎ ␬Cac, we have tbreaking ⬀ Ca2/3 which means the large droplet takes a long time to start breaking up. From Figs. 2 and 3, it is also conclusive that smaller droplets finish the breakup 共droplet size below critical value兲 earlier than larger droplets. This information is

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FIG. 5. The mechanism of droplet breakup during startup of shear.

inclusive from the DE model since the time scale ␭n in their model is taken as a fitting parameter and independent of droplet size. Therefore, it is believed that the present predictions are more reasonable. According to the comparison between the evolution of morphology and N1,excess, the mechanism of droplet breakup during the start of step shear is shown in Fig. 5. At the initial stage, there is a sudden increase of N1,excess, which corresponds to the stretching of the droplet. The overshooting regime includes the breakup of the smaller droplet and deformation of the larger droplet, the maximum point of N1,excess corresponds to the end of breakup of the droplet with radius RV, and the transition to equilibrium value denotes the end of breakup of the largest droplet. In the equilibrium regime, the droplet undergoes steady deformation and possible coalescence. The morphological evolution, which corresponds to the transient normal stress difference, is a little different from the one suggested by Mewis et al. 共2000兲. According to the mechanism of Mewis et al. 共2000兲, the transient increases of N1,excess before the maximum is attributed to the droplet deformation, and droplet begins to break up at the maximum N1,excess. In our model, however, the time for pure droplet deformation is quite limited, and the droplet starts to break up after a quick initial stretch. There are two reasons for the difference. The first one comes from the treatment of breakup in the present model. In fact, the breakup of droplet is a discrete event which takes a period of time tbreakup. However, such process is modeled as a continuous process by the dynamic equation, Eq. 共3兲. Second, the mechanism of Mewis et al. 共2000兲 does not seem precise for the breakup of droplets with different size. In their mechanism, both the maximum point and the inflection point of N1,excess are breakup points, although it is not clearly pointed out in the corresponding droplet size. The breakup process of droplets and the transient normal stress difference are correlated more clearly in the present model. The end of breakup of RV droplet and largest droplet can be readily inferred from Figs. 2 and 3. Moreover, it can be easily demonstrated the droplet breakup begins before the maximum point in N1,excess curve. For example, in Fig. 2 of the present paper, when we consider the breakup of a droplet with a radius of 6 ␮m, the breakup strain can be estimated to be tbreakup␥˙ ⬇ 26Ca2/3 p0.5 ⬇ 70, which is much smaller than the peak in N1,excess. Since the overshooting in N1,excess is ascribed to the droplet breakup, the droplet size distribution should affect the overshooting. This effect is shown in Fig. 6. Three distributions with same RV but different PI are predicted from the model. As PI increases, the extent of overshooting in N1,excess decreases, but the time to reach equilibrium becomes


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FIG. 6. The effect of droplet size distribution on the first normal stress difference.

longer. In all the calculation results shown in Fig. 6, the droplet size distributions are discretized into ten subpopulations with the same radius. The predictions on N1,excess for large PI seems a little wavy, which is due to insufficient number of sub-population in discretization of droplet size distribution. The evolution of droplet size distribution during the step shear can also be predicted by the present model, which is shown in Fig. 7. The distribution shifts to small droplet size and the shape of distribution changes as well. After complete breakup, the droplet size in every subpopulation is equal to the critical droplet size, which means that the droplet size becomes almost monodispersed. The excess shear stress is compared in Fig. 8. The excess shear stress prediction is not as good as the first normal stress difference, although the model prediction shows the same trend as the experimental data. Such problem exists in other models 关Peters et al.

FIG. 7. Evolution of the droplet size distribution during start up of shear flow.

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FIG. 8. The transient interfacial shear stress from present model and experiments 关Vinckier et al. 1997兴. The shear rate is ␥˙ = 3.0 s−1.

共2001兲兴. However, the excess shear stress only account for a small part of the total shear stress, about 10% in this case, which means the model predictions on the total shear stress or the shear viscosity of blends should not deviate that much. The effects of droplet breakup on the rheological properties of blend can have been illustrated for model blend I. The effect of droplet coalescence on the rheological properties is not so significant, and usually it needs a long time for coalescence to take place. One of the important effects of coalescence is the hysteresis phenomena, i.e., the morphology 共and also rheology兲 of blends under a specific shear condition would be different if the shear history is different. For example, Minale et al. 共1998兲 showed that the droplet size would be different when a blend is subjected to shear rate sweep under a step-up and a step-down manner. The droplet radius under a step-up shear and followed by a stepdown shear is shown in Fig. 9. The predictions by the present model are also shown. In these calculations, droplet size distribution is not considered. During the step-up shear, the droplet size is controlled by the deformation and breakup process, and droplet coalescence dominates the step-down shear process. Both the experiments and model predictions show that the droplet size in the step-down shear is in general smaller than that in the step-up shear. The agreement between the experiments and the model predictions is acceptable despite the fact that the breakup and coalescence mechanism is considered in a very simple manner. The corresponding shear viscosity of blend II is predicted and shown in Fig. 10. Similarly, the shear viscosity shows hysteresis. The shear viscosity during the step-up shear is lower than that during the step-down shear, which is ascribed to the difference in droplet size. A typical phenomena is that during the step-up shear, the shear viscosity decreases first and a plateau appears. The plateau in the shear viscosity is attributed to the breakup of droplet. In blends with certain droplet size distribution, the plateau is not so significant and looks like a shoulder instead. Figures 11 and 12 show the comparisons for model blend III. The main difference between blend III and I, II is that one component 关particle desorption mass spectrometry 共PDMS兲兴 of blend III exhibits evident shear thinning. The viscosity ratio increases with the shear rate. The experimental droplet radius shown in Fig. 10 is taken from Grizzuti et al. 共2000兲 in measurements by step down in shear rate. The predicted droplet radiuses


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FIG. 9. Morphology hysteresis of model blend II produced from step-up and step-down shear. The experimental data are from Minale et al. 共1998兲.

include results from step-up and step-down shear. During the step-up shear, the initial radius is set sufficiently large so that the equilibrium droplet radius under smallest shear rate is determined by droplet breakup. During the step-down shear, the initial radius is set sufficiently small so that the equilibrium droplet radius under largest shear rate is determined by droplet coalescence. It is seen that during the step-up shear, the droplet radius is almost constant at high shear rate since the droplet is hard to break up due to the high viscosity ratio. Although the experimental droplet radius is measured from the step-down shear experiments, its value is closer to the model prediction from step-up shear. The droplet size from step-up shear is the same for blends with different concentrations. This is due to the fact that the effect of droplet interactions in the breakup of droplet is not considered in the model. Concentration has some effect on the equilibrium droplet size in

FIG. 10. The corresponding viscosity hysteresis in blend II.

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FIG. 11. Droplet size of model blend III as a function of shear rate.

the step-down shear, where the coalescence dominates the process. The comparisons on shear viscosity are shown in Fig. 12. For blend with low volume fraction 共PIB 10%兲, there is a small difference between the step-up and step-down shear, and the model predictions are quite close to the experiments. For intermediate concentration 共PIB 30%兲, the predictions show different shear viscosity in step-up and step-down shear. The experimental results are also close to the model prediction with step-up shear under small shear rate. Three regimes, shear thinning at small shear rate, shoulder at intermediate shear rate and shear thinning again at high shear rate, are clearly shown. The deviation between model prediction and experiments starts to increase at high shear rate, which is ascribed to the hydrodynamic interaction between droplets at high volume fraction and high shear rate.

FIG. 12. The shear viscosity of model blend III.


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IV. CONCLUSIONS We suggest a rheological constitutive equation to describe the evolution of morphology and rheology of immiscible under a specified flow field. The model is based on the ellipsoidal description of droplet, and the droplet deformation/relaxation is described by any ellipsoidal model. The breakup/coalescence process is assumed to only affect droplet size, and modeled by a dynamic equation of droplet radius. The conservation of droplet volume during deformation and variation of droplet volume due to breakup or coalescence are unified by a simple mapping approach. The complete model is rather simple and easy to apply. The model can predict quite well the overshooting in the first normal stress difference in the transient startup of shear. Moreover, the morphology and rheology hysteresis under different shear history can be well predicted.

ACKNOWLEDGMENT This work was financially supported by the national Natural Science Foundation of China Grant Nos. 20474039, 10590355, 50390095 and 20490220.

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