Theory for drop deformation in viscoelastic systems

Transcription

1 _ Theory for drop deformation in viscoelastic systems Wei Yu Department of Polymer Science and Engineering, Shanghai Jiao Tong University, Shanghai 224, People s Republic of China and Department of Chemical Engineering, CRASP (École Polyetchnique, Montréal), Laval University, Sainte-Foy G1K 7P4, Canada Mosto Bousmina a) Department of Chemical Engineering, CRASP (École Polyetchnique, Montréal), Laval University, Sainte-Foy G1K 7P4, Canada Chixing Zhou Department of Polymer Science and Engineering, Shanghai Jiao Tong University, Shanghai 224, People s Republic of China Charles L. Tucer III Department of Mechanical and Industrial Engineering, University of Illinois at Urbana Champaign, 126 West Green Street, Urbana, Illinois 6181 (Received 14 August 23; final revision received 2 December 23) Synopsis The deformation of a viscoelastic drop suspended in a homogeneous viscoelastic matrix is investigated. The drop and the matrix fluids are assumed to obey linear viscoelastic constitutive equations. Small-deformation analysis is carried out on the basis of the general solution for creeping flow around a spherical drop. The corresponding stress is calculated by extending Batchelor s approach to viscoelastic media for both small and large deformations. Good agreement was found between the model predictions and experimental results available in the literature. 24 The Society of Rheology. DOI: / I. INTRODUCTION The deformation and relaxation behavior of a Newtonian drop in a Newtonian matrix has been investigated extensively by experiments Taylor 1934; Bentley and Leal 1986; Guido and Villone 1998, numerical simulations Rallison and Acrivos 1978; Kennedy et al. 1994; Khayat et al and theoretical models Taylor 1934; Cox 1969; Franel and Acrivos 197; Maffettone and Minale 1998; Almusallam et al. 2; Wetzel and Tucer 21; Peters et al. 21; Wuet al. 22; Yuet al. 22a,b; Yu and Bousmina 23; Jacson and Tucer 23. The parameters that control the deformation of Newtonian drops are the viscosity ratio p d / m ( d and a Author to whom correspondence should be addressed; electronic mail: bousmina@gch.ulaval.ca 24 by The Society of Rheology, Inc. J. Rheol. 482, March/April /24/482/417/22/$

2 418 YU ET AL. m are the viscosity of the drops and matrix, respectively, the characteristic time of drop relaxation m R/ R is the radius of spherical drop and is the interfacial tension and the intensity of the flow field capillary number, Ca, where is the shear rate. For small and moderate deformation, the shape of a Newtonian drop can be accurately approximated by an ellipsoid. Several ellipsoidal drop models have been proposed, including the models of Maffettone and Minale MM 1998, Jacson and Tucer JT 23, Grmela et al. GBP 21, and Yu and Bousmina YB 23. For Newtonian systems with p 1 and i.e., affine deformation of the drop, the time evolution of the drop shape can be written in the compact form proposed by Wetzel and Tucer 21: DG Dt LT GG L, where G is a second-order tensor that describes the ellipsoidal shape of the drop. Points on the droplet surface satisfy G:xx 1, and L is the macroscopic velocity gradient tensor, given by 1 L ij u i x j. 2 D/Dt is the material derivative and the superscript T indicates a tensor transpose. When the drop has interfacial tension and/or a different viscosity from the matrix, the velocity at the interface will be different from the macroscopic velocity applied. This distinction is taen into account in some way by the models listed above. In recent, very explicit treatment, Yu and Bousmina 23 considered three velocities: i macroscopic velocity applied velocity, ii velocity induced in the matrix and iii velocity induced inside the drop. The velocities both inside and outside the drop were related to the macroscopic velocity by boundary integral equations. The tangential velocity was assumed to be continuous across the interface and the velocity gradient was taen to be homogeneous inside the drop and equal to the velocity gradient at the interface. The YB model gives nice predictions for both small and large droplet deformation in a variety of flow fields. However, it is restricted to dilute emulsions of Newtonian drops suspended in a homogeneous Newtonian matrix. Theoretical analyses of the deformation of drops in Newtonian systems give useful insight into the complex contribution of the interface to the overall response of a mixture for a given macroscopic external deformation or stress field. This interfacial contribution becomes even more complex if the components of the mixture also exhibit viscoelastic relaxation. Then both molecular relaxation of the fluids and nonuniform anisotropic interfacial deformation become important for describing drop deformation and the overall response of the mixture. Experimental wor by Elmendorp and Maalce 1986 showed that normal stress of the dispersed phase tended to stabilize the shape of the drop, while normal stress of the matrix had the opposite effect. The same findings were reported by Mighri et al. 1997, Tretheway and Leal investigated the deformation and relaxation of Newtonian drops in a Boger fluid Tretheway and Leal 21 and viscoelastic drops in a Newtonian liquid in planar extensional flow Tretheway and Leal Their results showed that, for a small capillary number, steady drop deformation in the viscoelastic system is the same as in a Newtonian system with the same viscosity ratio; the effect of viscoelasticity appears only at high capillary numbers. Various numerical methods have been employed to simulate drop deformation in viscoelastic systems, such as the finite difference method Ramaswamy and Leal 1999a,

3 DROP DEFORMATION IN VISCOELASTIC SYSTEMS b and the finite element method Hooper et al. 21. However, little is nown from a theoretical standpoint about drop deformation in viscoelastic systems. Using Palierne s model 199, 1991, Delaby et al. 1995, 1996 derived small-deformation equations to predict drop deformation in uniaxial elongational flow. Recently, Greco 22 presented a steady analytical solution for small drop deformation, where both the drop and the matrix obey the second-order fluid constitutive equation. In this paper, we propose a small-deformation theory for an emulsion of two viscoelastic fluids. The formulation is very general, although the linearized Oldroyd B constitutive equation will be considered here as a specific example to illustrate our approach. The first part of this paper illustrates our strategy for Newtonian systems, and shows that simple models such the MM model can be derived using the Stoes solution for flow around a spherical drop. The solution for the Newtonian system is thereafter applied to a mixture of viscoelastic media in the Laplace domain, where the relationship between the stress and the velocity field has the same form as in the Newtonian system a Stoes-type equation. Once solution of the velocity is found in the Laplace domain we use inverse Laplace transformation to return the solution to the time domain, and thus derive equations that describe both the evolution of the structure during flow and the corresponding stress, without any adjustable parameters. In contrast to the expression for the morphology tensor that will be wored in the small-deformation regime, the equation for stress will be given for a general case in both small and large deformation regimes. However, since stress is related to the morphology tensor, we will use here stress for only the small-deformation case. II. SMALL DEFORMATION ANALYSIS OF NEWTONIAN SYSTEMS Before presenting the solution in viscoelastic systems, we briefly review some important aspects of small-deformation theory in Newtonian systems. We consider only the limit of creeping flow, and the drop is assumed to be an ellipsoid that departs only slightly from a spherical shape. The shape of the drop can be described by Eq. 1 applied at its surface Wetzel and Tucer 21; Yu and Bousmina 23 DG Dt L I T GG L I, where L I is the velocity gradient tensor taen at the surface of the drop at the interface. At first order deformation, the surface of the drop can be given by Cox 1969; Franel and Acrivos 197 rx,t 1fx,t, r x x 1/2, 4 where is a small number and the function f is of the order of unity. All quantities will henceforth be expressed in dimensionless form. All velocities are divided by R, distances by R, time by 1, stress and pressure inside the drop by d, and stress and pressure outside the drop by m. The equations of momentum and mass conservation inside and outside the drop read, in dimensionless form, u*x,t, *x,t for r 1fx,t, 5 ūx,t, x,t for r 1fx,t, 6 where u*(x,t) and ū(x,t) are the velocities inside and outside the drop, and *(x,t) and (x,t) are the stress tensors inside and outside the drop, respectively. For Newtonian fluids, the constitutive equations can be expressed as 3

4 42 YU ET AL. *x,t p*x,tu*x,t T u*x,t, 7 x,t p x,tūx,t T ūx,t, where p*(x,t) and p (x,t) are the pressures inside and outside the drop. Equations 5 and 6 together with Eqs. 7 and 8 show that the velocity and the stress are related by the Stoes equation. The boundary conditions are given by Franel and Acrivos 197 ū e A A x as r, u*x,t ūx,t at r 1fx,t, K fx,t t ūx,t nx,t u*x,t nx,t at rx,t 1fx,t, 11 x,t nx,tnx,t x,t:nx,tnx,t p*x,t nx,tnx,t*x,t:nx,tnx,t at r 1fx,t, x,t:nx,tnx,tp*x,t:nx,tnx,t Ca R 1 R 2 at r 1fx,t, where e A and A are the rate-of-deformation tensor and the vorticity tensor of the undisturbed flow field, respectively, n is the unit vector normal to the surface that points outward, R 1 and R 2 are the principal radii of curvature, and K (r f ) 1. We consider here the case of a small capillary number ( Ca). The first term of the expansion, i.e., the zeroth-order solution, can be obtained by solving Eqs. 5 8 with boundary conditions given by Eqs This involves only the spherical harmonic solutions of second order Cox 1969: f F: 2 r xx r3 3F:xx r 2, 14 u* A x6s* x 5 14T* xr 2 2 7T*:xxx, 15 ū e A x A x 6S x 15S :xxx r 5 r 7 3T :xxx r 5, 16 F, S*, T*, S and T are symmetric, traceless second-order tensors. Using the boundary conditions Eqs and Eqs , wefind 6S* 3 5T* e A 3 5T, 6S* 3 7T* e A 9S 3 2T, p 16 7 T*12S* 2e A 48S 3T, 19 2e A 72S 9T p 3 7T*12S* 12F, 2 S*, T*, S and T can be solved from Eqs as a function of F Franel and Acrivos 197:

5 DROP DEFORMATION IN VISCOELASTIC SYSTEMS p19 S* 62p3 ea 2p319p16 F, T* 84 19p16 F, S p1 43p2 32p3 ea 2p319p16 F, T 1p1 32p3 ea 8 2p3 F, Inserting Eqs into Eq. 15 we obtain the velocity distribution inside the drop. Generally the velocity at the surface of the drop is not a simple linear function of the position. However, the velocity at the three points where the principal axes of the drop intersect the drop surface can be expressed by u* A x 5 12p1 2p3 ea x 19p162p3 F x. We approximate the velocity distribution at the surface of the drop by Eq. 25. The average velocity gradient at the interface is then given by L I A 5 12p1 2p3 ea 19p162p3 F. The above equation expresses the velocity gradient L I at the interface as a function of the tensor F, which needs to be determined. For small deformation, the ellipsoidal tensor G can be expanded at first order to G CaG, where G is a deviatoric tensor whose magnitude is O(1). Substituting Eq. 27 into the definition of the ellipsoid G:xx 1 and using r (x x) 1/2, we obtain r 1 Ca 2 G:xxOCa2. 28 Comparing Eq. 28 with the small-deformation expansion, Eqs. 4 and 14, gives G 6F. 29 Therefore, we can express the average interfacial velocity gradient from Eq. 26 as a function of G: L I A 5 2p1 2p3 ea 19p162p3 G. 3a G in Eq. 3a can be replaced by 1/Ca3G/tr(G). In fact, using Eq. 27 and noting that the deviatoric tensor G satisfies trg, we may write 1 Ca 3G trg Ca 1 3CaG trcag Ca 1 33CaG G, tr

6 422 YU ET AL. where tr denotes the trace of the tensor between large parens. Replacing G in Eq. 3a by 1/Ca(3G/trG) gives the following expression for L I : L I A 5 2p1 1 2p3 ea 19p162p3 Ca 3G tr G. 3b The symbol means that the equality is not unique and other expressions for G are possible. We have adopted the above approach because it easily leads to the MM model. In fact, by substituting Eq. 3b in the evolution equation for G Eq. 3, we obtain in a straightforward manner the dimensionless version of the MM model Maffettone and Minale 1998: Ca DG Dt A GG A 5 2p3 ea GG e A 4p1 3G G 2p319p16 tr G G. Maffettone and Minale s original development was a phenomenological way, so the present derivation provides a theoretical connection between their model and smalldeformation theory. In addition, the MM model is nown to match small-deformation theory when the ellipsoidal drop departs only slightly from a spherical shape, correspondence that supports the assumption of Eq. 26. III. SMALL-DEFORMATION ANALYSIS OF VISCOELASTIC SYSTEMS Viscoelastic polymers are characterized by a spectrum of relaxation times. For small deformation, the stress can be written in dimensionless form as x,t px,tp s u x,t T ux,t] x,t, 1 N, De x,t x,t p p ux,t T ux,t, 33 where s p, ps s /, p p p /, s is the viscosity contribution of the solvent, p is the th mode contribution to the overall viscosity of the polymer, and De is the Deborah or Weissenberg number, which is defined as the ratio of the time scale of fluid response to a characteristic time of the governing flow 1. is the ordinary time derivative of the extra stress tensor. Rigorously, the linear constitutive equations, Eqs. 32 and 33, are valid for oscillatory experiments under small amplitude of deformation and can also supply at least a first-order approximation of viscoelastic effects in steady flow with a small rate of deformation. The predictions of the present model for drop deformation in transient shear flow Figs. 1 and 2 show nice agreement with the numerical simulations of Toose The governing equations and boundary conditions are the same as in the Newtonian case Eqs. 5 and 6 and In Newtonian systems the velocity and the stress fields are governed by the Stoes equation, which can be solved the same way as in Sec. II. In contrast, for viscoelastic systems, substituting Eq. 32 into Eq. 5 shows that the governing equation contains an additional term compared to the Stoes equation. This renders the equation difficult to solve directly in the time domain t. To overcome the difficulty, we tae the Laplace transform of Eqs. 32 and 33. In the Laplace domain s, this generates an equation of Stoes form, which is easy to solve using the same procedure as that in Sec. II. Thereafter, we return the solution to the time domain to find the time variation of the velocity

7 DROP DEFORMATION IN VISCOELASTIC SYSTEMS 423 FIG. 1. Transient evolution of the longest semiaxis of the ellipsoid under uniaxial elongation flow. The viscosity ratios are p 1, p ds and p ms.5. field, and consequently the time variations of the velocity gradient and morphology tensor through the use of Eq. 3. In the Laplace domain s, the velocity and the stress fields are described by the following equations: with u*x,s, *x,s for r 1fx,s, 34 FIG. 2. Transient evolution of deformation parameter D for viscoelastic drop in the Newtonian matrix. The symbols are numerical simulations given by Toose 1997.

8 424 YU ET AL. and N *x,s p*x,s p ds 1 p dp De d s1 u*x,st u*x,s, 35 ūx,s, x,s for r 1fx,s, 36 M x,s p x,s p ms 1 p mp De ūx,st ūx,s; ms1 37 f (x,s) L f (x,t) represents the Laplace transform of the function f (x,t). Here a stress-free initial state is assumed, so Eqs. 35 and 37 can be reformulated as *x,s p *x,su*x,s T u*x,s, x,s p x,sūx,s T ūx,s, where *(x,s) ij *(x,s)/g ij d (s), p *(x,s) p*(x,s)/g d (s), g d (s) p ds N 1 (p dp/deds1) and ij (x,s) ij (x,s)/g m (s), p (x,s) p (x,s)/g m (s), M g m (s) p ms 1 (p mp /De ms1). The above equations show that, in the Laplace domain, the stress and the velocity Eqs. 34 and 36 together with Eqs. 38 and 39 have the form of the Stoes equation. Therefore, the velocity and the velocity gradient can be obtained by solving the above equation using the same route as that in the Newtonian case. We use the same expansion as in the Newtonian case, i.e., the shape of drop, velocity and stress are expanded to first order of Ca. Expansion with viscoelastic parameters was not performed here. This is because viscoelasticity only affects drop deformation for at least secondorder Ca. Greco 22 showed that, for steady deformation, viscoelastic expansion comes into play only for terms that are quadratic in Ca. This has been also confirmed experimentally by Tretheway and Leal 21. Therefore, our expansion to first order of Ca in the viscoelastic system can account for the Newtonian effect to first order and for the viscoelastic effect to zeroth order. Therefore, the time-dependent velocity inside and outside the drop can be written in the Laplace domain as u*x,s A s x6s*s x 5 7T*s xr 2 2 7T*s:xxx, 4 ūx,s e A s x A s x 6S s x r 5 15S s:xxx r 7 3T s:xxx r The velocity in the time domain can be obtained from Eqs. 4 and 41 by taing the inverse Laplace transform, u*x,t L 1 u*x,s, 42 ūx,t L 1 ūx,s. The total stress tensors inside and outside the drop tensors in the time domain are then given by 43 *x,t L 1 g d s*l 1 *x,s, x,t L 1 g m s*l 1 x,s, 44 45

9 DROP DEFORMATION IN VISCOELASTIC SYSTEMS 425 where f (t)*g(t) t f ()g(t)d indicates convolution of functions f (t) and g(t). Inserting Eqs into the boundary conditions, Eqs. 1 13, we obtain 6S*t 3 5T*t e A t 3 5T t, 6S*t 3 7T*t e A t9s t 3 2T t, e A t*l 1 g m sl 1 g m s*72s t9t tpl 1 g d s* 3 7T*t 12S*t 12Ft, 48 2e A t*l 1 g m sl 1 g m s*48s t3t t pl 1 g d s* 16 7 T*t12S*t. 49 With a spectrum of relaxation times, S*(t), T*(t), S (t) and T (t) can be solved formally by inverse Laplace transformation of the corresponding Laplace domain functions to the time domain. However, such inversion is analytically difficult when the number of relaxation modes exceeds two. Instead, to illustrate our strategy and to explore the predictions of our model, we now consider the simple case where both the matrix and the drop are characterized by a single relaxation time. Then, Eqs can be solved in the Laplace domain to obtain S*(s), T*(s), S (s) and T (s): S*s 5De d s1de m p ms s1 6a 1 s e A s 2De d s1de m s1a 4s Fs, a 1 sa 2 s 5 S s T*s 84De d s1de m s1 Fs, a 2 s a 5s 3a 1 s ea s 4De d s1de m s1a 3s Fs, a 1 sa 2 s T s 1a 5s 3a 1 s ea s 8De d s1de m s1 Fs. a 1 s 53 Expressions for coefficients a i (s) (i 1 5) can be found in the Appendix. Using the same approach as in the Newtonian case, we can obtain the average velocity gradient in the time domain as a function of G, L I t A 5p ms t e A 5p t 2p ds p3p ms 2p ds p3p ms 2 De d De m b 1 d 1 t e A e c 1 t dd 2 t e A e c 2 t d Ca p ds p3p ms 2 De d De m b 1 d 3 t 3G tr G e c 1 t d

10 426 YU ET AL. d 4 t d 5 t 3G tr G e c 2 t 6 d 519p ds p16p ms 2 De d De m b 2 t 3G 3G tr G e c 3 t dd 6 2p ds pp ms 2p ds p3p ms 19p ds p16p ms 3Gt tr Gt. tr G e c 4 t d The coefficients b i (i 1,2), c i (i 1 4), and d i (i 1 6) are functions of the rheological properties of the viscoelastic components and are given in the Appendix. As mentioned above, substitution of G by 1/Ca(3G/trG) was used here to obtain Eq. 54. An integrodifferential equation is obtained by inserting Eq. 54 into Eq. 3, which can be rewritten as a set of simultaneous differential equations: 54 g i t G t L I T GG L I, 3G tr G c i g i, i 1 4, 55 where and g i t t 3G tr G e c i t d, i L I t A 5p ms t e A 5p t 2p ds p3p ms 2p ds p3p ms 2 De d De m b 1 d 1 t e A e c 1 t dd 2 t e A e c 2 t d Ca 1 2d 3 g 1 td 4 g 2 t 6d 5 g 3 td 6 g 4 t 52p ds p3p ms 2 De d De m b 1 519p ds p16p ms 2 De d De m b 2 2p ds pp ms 2p ds p3p ms 19p ds p16p ms 3Gt tr Gt. Equations together with the initial conditions G t and g i t can be solved numerically to obtain the transient shape of the drop in viscoelastic systems for arbitrary flow fields, without any fitting parameter. The effect of flow history on deformation of the drop is given by the integrals in Eq. 57. For example, in the case of start-up shear flow, the magnitude of the shear rate is given by 57

11 DROP DEFORMATION IN VISCOELASTIC SYSTEMS 427 t t t t. 58 (t) can be constant or time dependent, as in oscillatory shear flow. For the constant shear rate, (t), and steady shear flow (t ), the interfacial velocity gradient for a viscoelastic system, Eq. 57, reduces to the Newtonian case, Eq. 3b. This means that steady-state deformation of the drop in viscoelastic systems and Newtonian systems will be the same. This conclusion is obtained under the assumption of small deformation, which also implies that a steady shape exists. We thus predict that the viscoelastic character of the drop and matrix only influence the transient process of deformation, not the steady state. This will be discussed further below. IV. RHEOLOGY OF VISCOELASTIC IMMISCIBLE BLENDS The stress of a viscoelastic immiscible blend can be calculated once the velocity at the interface is nown. The expression for stress in Newtonian systems seen, e.g., Yu and Bousmina 23 is not applicable here, and one has to include relaxation of the components. This is not easy because even the term for stress arising from the contribution of components has to be modified. To solve the problem, we use the procedure proposed by Batchelor and rewor the equations for the viscoelastic case. The stress in a blend can be defined as an ensemble average of the stress in the system, which is equivalent to the volume average of stress over the whole domain Batchelor 197. Hence, the stress of the blend can be written as ij t 1 V V ijtdv 1 V VV ij tdv 1 V V ij tdv, 59 where V denotes the volume of the whole system, and V is the volume of the dispersed phase, which also includes the interfaces between the drop and the matrix. Using the Gauss theorem, Eq. 59 can be reformulated as follows: ij t 1 V VV ij m tdv 1 V S i m txj n ds 1 V V i t x x j dv. 6 The superscripts m and d denote the matrix and the dispersed drop, respectively. The last term of Eq. 6 will be identically zero in the absence of inertia effects and buoyancy forces Batchelor 197; Mellema and Willemse The second term can be obtained from the jump of normal stress at the interface Mellema and Willemse 1986: 1 m V itxj n ds 1 d S V itxj n ds S V S 1 3 ij n i n jds. 61 Returning to the multimode constitutive equations for the drop and matrix, we use Eqs. 32 and 33 to write the stress in either the drop or matrix phase as ij p ij u i, j u j,i ij, ijij pui, j u j,i. 62 The reason for retaining the time derivative of ij in this equation will become apparent in a moment. We then obtain from Eqs the general form for stress,

12 428 YU ET AL. ij t 1 V VV 1 V V M pt ij m u i, j tu j,i t m, m ij t dv 1 N pt ij d u i, j tu j,i t d, d ij t dv 1 V S 1 3 ij n i n j ds. 63 Here u i, j denotes u i, j u i /x j. Following Batchelor 197, we define the macroscopic average velocity gradient as U i x j 1 V V u i x j dv 1 V VV u i x j dv 1 V S u i n j ds. 64 Substituting Eq. 64 in Eq. 63, we have ij t ij V ptdv mu i,j tu j,i t d m u V V i tn j u j tn i ds S V S 1 3 ij n i n jds 1 M V 1 m VV ij m, tdv 1 N V 1 d, d ij tdv. V The above equation shows that stress for the viscoelastic blend can be divided into two parts: the first four terms and the last two terms. The first part has the same form as in the Newtonian system, but the velocity term that describes the interface is different since it is influenced by the viscoelasticity. The second part comes completely from the contribution of viscoelasticity. The last two terms can be calculated by performing direct and inverse Laplace transforms. For the drop phase, d d, ij tdv L V V 1L d d, ij tdv V V 1 L d dps u V V i,j su j,i ds1 sdv d t V V dp t dp d d 2 et/ d u i,ju j,i ddv dp u V i tn j u j tn i ds dp S d Similarly, for the matrix phase we obtain 1 V t e t/ d d S 65 u i n j u j n i ds. 66

13 DROP DEFORMATION IN VISCOELASTIC SYSTEMS 429 m m, ij tdv mpui,j tu V j,i t mp V V U i,j U j,i d mp m u i tn j u j tn i ds mp S 1 V Substituting Eqs. 66 and 67 in Eq. 65, we have t e t/ m d S ij t ij V ptdv msu i,j tu j,i t ds ms V V V S 1 3 ij n i n jds 1 V t 1 N dp et/ d d m t e t/ m u i n j u j n i ds. S u i tn j u j tn i ds M mp 1 m e t/ m M mp t u i n j u j n i dsd S 1 m e t/ m Ui,j U j,i d. This can be simplified using the notation employed by Yu and Bousmina 23 to find A ij t p ij 2 ms e ijt2ds ms e Iij t L c A ij tr A 1 3 ij M N mp 2 1 m dp 1 et/ d d t t e t/ m A e ijd M 1 mp m e t/ m e Iij d. 69 Here A ij n i n j ds/v is the area tensor as defined by Wetzel and Tucer 1999, L c is the local characteristic length scale defined as the ratio between the volume and the surface area of the ellipsoidal drop, and e I(ij) (t) is the rate-of-deformation tensor at the interface e I(ij) (t) L I(ij) (t)l I( ji) (t)/2. In contrast to the expression for the morphology tensor that was wored in the small and moderate deformation regimes, the equation for stress Eq. 69 is general and it is not restricted by the magnitude of deformation of the drop. It can be applied to any ind of deformation, small or large, transient or dynamic, and requires only that the linear viscoelastic constitutive relations used here are accurate. Once the velocity gradient at the interface is nown the third and last terms, the stress of the blend can be calculated regardless of the magnitude of the macroscopic deformation applied. The relaxation spectrum of the components is also taen into account in the expression of stress. If the relaxation times of the constituents vanish, then Eq. 69 reverts bac to the stress expression for the Newtonian case given by Yu and Bousmina 23. V. MODEL PREDICTIONS AND DISCUSSION As stated earlier, Eq. 57 reduces to Eq. 3 when t, which means that for small drop deformation, viscoelasticity has no influence on the steady drop shape or the

14 43 YU ET AL. steady velocity at the interface. This phenomenon has been observed experimentally by Tretheway and Leal 1999, 21. The steady deformation of a drop in a viscoelastic system is the same as in the Newtonian system with the same viscosity ratio for small capillary numbers. Such a regime of capillary number can be regarded as the linear regime for viscoelastic drop deformation. Deviations appear only when Ca is larger than a certain critical value, which depends on the relaxation times of the viscoelastic constituents. Figure 1 shows the transient evolution of the longest semiaxis L of the drop at constant capillary number and various relaxation times various Deborah numbers. The predictions of the model show that, at either very short times or in the steady state long time, the drop deformation does not depend on the relaclsxation times of the constituents. Only the transient regime of drop deformation is affected by the fluid elasticity. The elasticity of the drop and that of the matrix have different effects on the transient evolution of the drop shape. An increase in drop elasticity increases the initial rate of deformation, whereas an increase in matrix elasticity decreases the initial deformation rate. This phenomenon is ascribed to the memory effect of polymeric fluid. These predictions are consistent with the numerical results of Hooper et al. 21. We have not compared the predictions of the present model with the numerical results of Hooper et al. 21, because the capillary numbers they used exceed the linear regime of viscoelastic drop deformation. The transient deformation history predicted by the present model is compared with the numerical results of Toose 1997 for three viscoelastic drops with De d.33 and various viscosity ratios suspended in a Newtonian matrix subjected to uniaxial elongational flow Fig. 2. The results are plotted in terms of the Taylor deformation parameter D (LB)/(LB) L and B are the longest and shortest semiaxes of drop, respectively. The capillary number is.25, which is small enough to lie within the linear regime. As shown in Fig. 2, the predictions of the present model agree very well with the numerical results of Toose The relaxation process can be described by Eqs , with the shear rate given by t t, t t t r, t t r. 7 Figure 3 shows the relaxation of L/R after steady deformation in uniaxial elongational flow. The capillary number for the steady flow is.5. Several systems with the same viscosity ratio, but with different relaxation times, are compared. As in the deformation process shown in Fig. 2, the relaxation process is also affected by the elasticity of the components. The elasticity of the pure components extends the time for the drop to retract into its spherical equilibrium shape. This is mainly due to the time required to release the elastic energy stored by the stretched chains in the direction of flow. However, elasticity in the different constituents influences the retraction process differently. An increase in the elasticity of the drop speeds up the initial retraction rate, while an increase in the elasticity of the matrix slows down the initial retraction rate. This is because, during the retraction process, the relaxation of chains inside the drop will speed up retraction of the drop shape, and the relaxation of macromolecular chains within the matrix will prevent the drop shape from retracting. At both very short times and very long times, the relaxation of the deformed drop does not depend upon the elasticity of the components.

15 DROP DEFORMATION IN VISCOELASTIC SYSTEMS 431 FIG. 3. Relaxation of the longest semiaxis of the ellipsoid after steady deformation in uniaxial elongational flow. The viscosity ratios are p 1, p ds, p ms.5 and Ca.5. Predictions of drop retraction with small initial deformation are compared with experimental results in Fig. 4. The symbols are experimental results of Tretheway and Leal 21 for a PDMS Newtonian drop in a Boger fluid matrix. The Newtonian model of Yu and Bousmina 23 is also shown in Fig. 4 for comparison. Clearly the Newtonian model agrees with the experimental results only at very short times. At long times the Newtonian model underestimates the magnitude of shape relaxation of the drop, whereas the new viscoelastic model nicely describes the time evolution of the normalized deformation during the entire relaxation process. FIG. 4. Drop relaxation of a 1 sq m drop in a Newtonian-PIB/PB system for small initial deformation. Symbols are experimental results of Tretheway and Leal 21. The solid line is the prediction of the present model.

16 432 YU ET AL. FIG. 5. Drop deformation in small amplitude oscillatory shear flow, sin(t), of viscoelastic systems with d m 1 Pa s, p ds.5, p ms.5, 1 mn/m, R 5 m,.1 and.1 Hz. Horizontal lines are the maximum amplitude predictions of the Palierne model. The fact that the retraction process depends on the viscoelasticity of the blend constituents will have an important influence on the determination of interfacial tension in polymer blends by the deformed drop retraction method DDRM Xing et al. 2. If the maximum relaxation time of the constituents is larger than the characteristic relaxation time of the interface, as is the case in Fig. 4 where / De/Ca O(1)], the retraction process of the deformed drop will be greatly influenced by relaxation of the macromolecular chains. This will introduce appreciable errors in determination of the interfacial tension using Newtonian models. This difficulty can be avoided if the macromolecules are relaxed completely before drop retraction, which is possible in the drop retraction process after thread breaup Mo et al. 2; Xing et al. 2. Another linear aspect of drop deformation dynamics is drop behavior under smallamplitude oscillatory shear flow SAOS. Although steady drop deformation in a viscoelastic system is the same as in the Newtonian system for small capillary numbers, viscoelastic drops show a different maximum deformation in SAOS. Figure 5 illustrates the oscillatory drop deformation in a viscoelastic system under sinusoidal strain, sin(t), with.1,.1 Hz, d m 1 Pa s, p ds.5, p ms.5, 1 mn/m, and R 5 m. For drop deformation in SAOS, Palierne s model 199, 1991 expresses the amplitude of L/R by L R 1A, 71 where A 5 2D G m *19G d *16G m *, D 2G d *3G m *19G d *16G m *4/RG d *G m *. 72

17 DROP DEFORMATION IN VISCOELASTIC SYSTEMS 433 FIG. 6. Storage modulus and loss modulus of viscoelastic blends with d m 1 Pa s, p ds.5, p ms.5, 1 mn/m, R 5 m and.1. The symbols are predictions of the Palierne model, and the lines are predictions of the present model. Here G d * and G m * are the complex moduli of the drop and the matrix, respectively. They can be expressed for a single-mode linear viscoelastic model as G d * dip ds ip dp 1i d, G m * mip ms ip mp 1i m. 73 The maxima of L/R predicted by Palierne s model Eq. 71 are also shown in Fig. 5. The predictions of Palierne s model are reported here only as horizontal lines that describe the maximum deformation in steady state oscillation. The predictions of the present model are fully transient, but settle to steady state oscillation after a number of periods. The amplitudes of the oscillations for different viscoelastic systems predicted by the present model are in excellent agreement with Palierne s linear model. Figure 5 also shows that larger De d causes larger maximum deformation of the drop, and larger De m causes smaller maximum deformation. Such different deformation behavior of the viscoelastic system will affect the linear rheology of viscoelastic emulsions. The linear rheological properties of an emulsion of two viscoelastic media can now be calculated from Eq. 69 when the deformation history of inclusions is nown. The dynamic moduli G and G, calculated from our model in SAOS, are compared to those calculated using the Palierne model 199, 1991 in Fig. 6. The following parameters were used: d m 1 Pa s, p ds.5, p ms.5, 1 mn/m, R 5 m and.1. The predictions of the present model and the predictions of Palierne s model are exactly the same for the various conditions considered in Fig. 6. We should emphasize that there are no adjustable parameters, nor is any ind of mixing rule used in the present model see Eq. 69. The predicted rheological material functions, (t), 1 (t) and 2 (t), for the startup of shear flow and for relaxation at cessation of flow are shown in Figs Small elasticity of the drop has little effect on these material functions. However, high drop elasticity will cause a more rapid increase, and subsequently rapid relaxation of (t),

18 434 YU ET AL. FIG. 7. Effect of viscoelasticity on the transient shear viscosity of immiscible blends during startup and at cessation of flow. 1 (t) and 2 (t) compared to the Newtonian system. The effect of matrix elasticity is quite different: Matrix elasticity slows down the change of stress both in startup and relaxation. The elastic properties 1 (t) and 2 (t) of the blend presented in Figs. 8 and 9 come from a contribution by the interface. VI. CONCLUSIONS We have proposed a new perturbation theory for drop deformation in emulsions of two viscoelastic liquids. Two major results were obtained: i the deformation of the drop was calculated within the linear viscoelastic regime, and ii a general expression for stress FIG. 8. Effect of viscoelasticity on the transient first normal stress coefficient of immiscible blends during startup and at cessation of flow.

19 DROP DEFORMATION IN VISCOELASTIC SYSTEMS 435 FIG. 9. Effect of viscoelasticity on the transient second normal stress coefficient of immiscible blends during startup and at cessation of flow. was developed that covered both small and large deformations. We have demonstrated that the ellipsoidal model of Maffettone and Minale 1998 can be obtained from zerothorder perturbation analysis of a spherical Newtonian drop. A similar perturbation analysis was then applied to the viscoelastic system. When the constituents are viscoelastic, the stress and the velocity fields cannot be described by the classical Stoes equations used in the Newtonian case. However, by using a Laplace transform, the governing equations were given a Stoes-type form in the Laplace domain. This allowed calculation of the velocity field in the Laplace domain using the same route used to solve the Newtonian problem. Thereafter, the results were transformed bac into the time domain through the inverse Laplace transform. The velocity gradient tensor and the morphology tensor were then calculated for an arbitrary flow field in the linear regime. The stress was calculated by extending Batchelor s analysis to viscoelastic media. The resulting expression for stress is general within the framewor of the linear viscoelastic constitutive equation, but does not require any mixing rule. It is directly connected to the morphology through the interfacial velocity gradient tensor and the area tensor. The proposed model for both morphology and stress does not contain any fitting parameter. The predictions of the model show that viscoelasticity has no effect on steady drop deformation for small capillary numbers. The elasticity of the drop tends to increase the initial deformation and the relaxation rate, while elasticity of the matrix has the opposite effect. An increase in the elasticity of any constituent of the emulsion extends the time required to reach equilibrium, both in deformation and relaxation. All these predictions are consistent with experimental results and numerical simulations reported in the literature. ACKNOWLEDGMENTS This wor was financially supported by the Natural Sciences and Engineering Research Council of Canada NSERC, the Canada Research Chair on Polymer Physics and Nanomaterials, the Natural Science Foundation of China, Grant Nos and , and National Science Foundation Grant No. DMI

20 436 YU ET AL. APPENDIX The coefficients in Eqs are listed below. a 1 s 2p3sDe d 2p ds p3de m 2p3p ms s 2 2p ds p3p ms De d De m, a 2 s 19p16sDe d 19p ds p16de m 19p16p ms s 2 19p ds p16p ms De d De m, a 3 s 3p2sDe d 3p ds p2de m 3p2p ms s 2 3p ds p2p ms De d De m, a 4 s 16p19sDe d 16p ds p19de m 16p19p ms s 2 16p ds p19p ms De d De m, a 5 s p1sde d p ds p1de m pp ms s 2 p ds pp ms De d De m, b 1 De d 2p ds p3de m 2p3p ms 2 4De d De m 2p32p ds p3p ms, b 2 De d 19p ds p16de m 19p16p ms 2 4De d De m 19p1619p ds p16p ms, c 1 De d2p ds p3de m 2p3p ms b 1, 2De d De m 2p ds p3p ms c 2 De d2p ds p3de m 2p3p ms b 1, 2De d De m 2p ds p3p ms c 3 De d19p ds p16de m 19p16p ms b 2, 2De d De m 19p ds p16p ms c 4 De d19p ds p16de m 19p16p ms b 2, 2De d De m 19p ds p16p ms d 1 p ds De d2pds p31p ms p ms De m2p3pms 1p ds De d De m 2p dsp2pms 2 p ms 36p ms p ds 2p3p msb1 p ms De m 1p ds p ds De d 1p ms, d 2 p ds De d2pds p31p ms p ms De m2p3pms 1p ds De d De m 2p dsp2 2 p ms p ms 36p ms p ds 2p3p msb1 p ms De m 1p ds p ds De d 1p ms, 2 2 d 3 32p ds p31p ms De d2p2p3pms 1p ds De m4p 2 2 p ds2pds p92p 2 12p ms 336pp ms 4p3p msded De m b 1 31p ms De d 2p1p ds De m, 2 2 d 4 32p ds p31p ms De d2p2p3pms 1p ds De m4p 2 2 p ds2pds p92p 2 12p ms 336pp ms 4p3p msded De m b 1 31p ms De d 2p1p ds De m,

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