ARTICLE IN PRESS. Numerical investigation of the effect of insoluble surfactants on drop deformation and breakup in simple shear flow

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1 S (05) /FLA AID:11865 Vol. ( ) [+model] P.1 (1-26) YJCIS:m5+ v 1.50 Prn:10/01/2006; 14:37 yjcis11865 by:vik p. 1 Journal of Colloid and Interface Science ( ) Numerical investigation of the effect of insoluble surfactants on drop deformation and breakup in simple shear flow Ivan B. Bazhlekov, Patrick D. Anderson, Han E.H. Meijer Materials Technology, Dutch Polymer Institute, Eindhoven University of Technology, 5600 MB Eindhoven, The Netherlands Received 6 July 2005; accepted 11 December 2005 Abstract The effect of insoluble surfactants on drop deformation and breakup in simple shear flow is studied using a combination of a three-dimensional boundary-integral method and a finite-volume method to solve the coupled fluid dynamics and surfactant transport problem over the evolving interface. The interfacial tension depends nonlinearly on the surfactant concentration, and is described by the equation of state for the Langmuir isotherm. Results are presented over the entire range of the viscosity s ratio λ and the surface coverage x, as well as the capillary number Ca that spans from that for small deformation to values that are beyond the critical one Ca cr. The values of the elasticity number E, which reflects the sensitivity of the interfacial tension to the maximum surfactant concentration, are chosen in the interval 0.1 E 0.4 and a convection dominated regime of surfactant transport, where the influence of the surfactant on drop deformation is the most significant, is considered. For a better understanding of the processes involved, the effect of surfactants on the drop dynamics is decoupled into three surfactant related mechanisms (dilution, Marangoni stress and stretching) and their influence is separately investigated. The dependence of the critical capillary number Ca cr (λ) on the surface coverage is obtained and the boundaries between different modes of breakup (tip-streaming and drop fragmentation) in the (λ; x) plane are searched for. The numerical results indicate that at low capillary number, even with a trace amount of surfactant, the interface is immobilized, which has also been observed by previous studies. In addition, it is shown that for large Péclet numbers the use of the small deformation theory to measure the interfacial tension in the case where surfactants are present can introduce a significant error Elsevier Inc. All rights reserved. Keywords: Drop deformation; Drop breakup; Surfactant; Interfacial tension; Marangoni stress; Tip steaming; Shear flow; Boundary integral method 1. Introduction Often in multiphase systems it appears that drop dynamics depends more strongly on interfacial properties than on the rheology of the bulk fluids,see for instance [1] and [2]. In addition, a small amount of interfacially active materials, called surfactants, can reduce the interfacial tension several times [3,4], which makes surfactants economically attractive. According to Hu et al. [5], a reduction of the interfacial tension by only 3% reduces the critical capillary number for coalescence by a factor of 6, see also [6]. That is why surfactants are extensively used to stabilize the morphology of blends. * Corresponding author. Present address: Koolmeeshof 19, 5672 VP Nuenen, The Netherlands. address: i.bazhlekov@gmail.com (I.B. Bazhlekov). Two types of surfactants exist, soluble and insoluble. In the case of soluble surfactants the surface active material is initially added to one or more of the bulk phases [3,4,7]. Insoluble surfactants can be generated by a reaction on the interface, via the in situ formation of block copolymers [5,8,9]. From a mathematical point of view the main difference between the two types is that in the case of soluble surfactants an extra transport equation for the concentration of surfactants has to be solved in the bulk phases. In most of the theoretical studies where soluble surfactants are considered an assumption is made that the surfactant distribution in the bulk phase is in one of the two limiting regimes: diffusion-controlled or adsorption/desorption-controlled [10 12]. With these assumptions the surfactant distribution problem reduces to the interface only, which greatly simplifies the numerical procedure. The numerical method presented in this study can also be extended to deal with such models for soluble surfactants, however, given /$ see front matter 2005 Elsevier Inc. All rights reserved. doi: /j.jcis

2 S (05) /FLA AID:11865 Vol. ( ) [+model] P.2 (1-26) YJCIS:m5+ v 1.50 Prn:10/01/2006; 14:37 yjcis11865 by:vik p. 2 2 I.B. Bazhlekov et al. / Journal of Colloid and Interface Science ( ) the already large number of parameters, this is not done here. Another way to distinguish between mathematical models in use is in the type of the equation of state. The most simple, linear equation of state, has been widely used [13 15], however it is relevant only for small surfactant concentrations. The nonlinear equations of state are found to fit experimental data well in a broader range of surfactant concentrations [7,8,16].Themost popular among them are the equations of state that correspond to the Frumkin [7] and Langmuir [17] isotherm, respectively. The later, known as the Szyszkowski equation of state, has only two parameters, the elasticity number E and the surface coverage x, and has been used in a number of studies [9,18 20]. We adopt this model as well. A disadvantage of all equations of state discussed is that they predict physically unrealistic negative interfacial tensions when the surfactant concentration Γ approaches the saturation limit Γ. To overcome this problem an ad hoc restriction on the value of the interfacial tension has been used [20]. Regarding the numerical methods used, the vast majority of investigations has been conducted applying the boundaryintegral method in combination with a finite-difference method for the surfactant transport in the 2D axisymmetric case [12,13] and a finite-volume method in the three-dimensional case [14,21]. These methods have shown to be very efficient because they require discretization only on the interface. Despite, finite-element [9,20] and volume-of-fluid [15,22] methods have also been used successfully. Drop deformation and breakup have been investigated in different types of flows: in axisymmetric and planar extension as well as in simple shear flows. The most relevant with respect practical applications is simple shear flow. Indeed, symmetric drop deformation in an extensional flow is difficult to maintain even in an experimental setup [4], and special precautions have to be taken [5]. One of the most important issues in determining drop-size distributions in blends is to find critical flow conditions that reflect the boundaries between the occurrence of a stable drop shape and drop breakup, see for instance the insets in Fig. 1 at λ = 0.6. These conditions are often given in terms of a critical capillary number as a function of the viscosity ratio, Ca cr (λ), known as the Grace curve [23], see also [24].InFig. 1 the Grace curve for simple shear flow is shown in the case of a pure interface. This case have been extensively investigated and the different modes of drop deformation and breakup have been determined, see the insets in Fig. 1. At small viscosity ratios and sub-critical Ca the drop has pointed tips [25,26]. At large viscosity ratios and relatively small Ca the drop is aligned with the flow (the bottom right frame), while at large Ca drop tumbling is observed [27], see the inset for Ca =, λ = 10. At a capillary number just above the critical value drop fragmentation appears in two main parts with in between satellite drops. At Ca several times larger than Ca cr the drop undergoes a large deformation, towards a lamellar shape for small λ (λ = 0.1) [28], or a long thread for λ 1, which eventually breaks up in a number of small drops [29]. The presence of surfactants can significantly alter the drop deformation (D is defined as D = (L B)/(L + B), see Fig. 2b), and consequently the value of the critical capillary number. Surfactants lower the interfacial tension and, in general, promote drop deformation. When the drop is subjected to an external flow, the surfactant concentration becomes nonuniform due to the surfactant convection, introducing interfacial tension gradients, known as Marangoni stress. The Marangoni stress effects the drop deformation and the external flow as well Fig. 1. The critical capillary number for simple shear flow as a function of the viscosity ratio, the Grace curve [23]. Simulations using present method in the case of pure interfaces are also presented to show different modes of stable drop shapes as well as breakup.

3 S (05) /FLA AID:11865 Vol. ( ) [+model] P.3 (1-26) YJCIS:m5+ v 1.50 Prn:10/01/2006; 14:37 yjcis11865 by:vik p. 3 I.B. Bazhlekov et al. / Journal of Colloid and Interface Science ( ) 3 (a) (b) Fig. 2. Streamlines, velocity and surfactant concentration for steady drop shapes at λ = 0.1: (a) no surfactant at Ca = 0.4; (b) surfactant at E = 0.2, x = 0.75 and Ca = In both cases the deformation is about D = (L B)/(L + B) = For a definition of the parameters see Section 2. (The streamlines at the same levels and the velocity field are shown in the x y plane.) (compare the pattern of the external flow for both frames in Fig. 2). In addition, the internal circulation in the drop phase, see Fig. 2a), also contributes to the surfactant distribution, which differs significantly from that in extensional flows. In the present work we offer an numerical investigation of the effect of surfactants on drop deformation and the critical capillary number in simple shear flow. Different surfactant related mechanisms are analyzed as well as different modes of breakup. 2. Mathematical formulation, numerical method and validation The governing equations given in this section are standard and can be found elsewhere [2,13,14,17,20,21], see also references therein. They are written below for completeness and definition of the parameters Mathematical model An initially spherical, with radius R, deformable drop of a Newtonian fluid with viscosity µ 1 in another immiscible Newtonian fluid with viscosity µ 2, is subjected to simple shear flow. All fluids are considered incompressible and gravity and inertial forces are negligible. In dimensionless terms the governing equations read i = 0, u i = 0, x Ω i, i = 1 (drop) and 2 (matrix), (1) where i = p i I + µ 1 /µ 2 ( u i + ( u i ) T ) is the stress tensor, I is the unit tensor, p i is the pressure, and u i is the velocity in the ith phase Ω i. The problem has only one viscosity ratio λ = µ 1 /µ 2. The boundary conditions at the interface S = Ω 1 Ω 2 are the stress balance and continuity of the velocity across the in-

4 S (05) /FLA AID:11865 Vol. ( ) [+model] P.4 (1-26) YJCIS:m5+ v 1.50 Prn:10/01/2006; 14:37 yjcis11865 by:vik p. 4 4 I.B. Bazhlekov et al. / Journal of Colloid and Interface Science ( ) terface Ca ( 1 (x) 2 (x) ) n(x) = f(x), u 1 (x) = u 2 (x), where n(x) is the unit vector normal at S. The capillary number is defined as Ca = RGµ/σ eq, where G is the shear rate and σ eq is the interfacial tension at uniform surfactant distribution on an equivalent spherical drop. In Eq. (2) the interfacial force f(x) will be specified later. The velocity at infinity is prescribed via a boundary condition which for simple shear flow is u (x) = (Y, 0, 0), x =, x = (X,Y,Z). For comparisons with previous results we also use axisymmetric and planar extensional flow, where u (x) = ( X, Y,2Z) and u (x) = (X, Y,0), respectively. The evolution of the interface S(x,t) is given by the kinematic condition dx = u(x,t), x S. (4) dt In the presence of surfactants on the interface the interfacial tension σ is not constant, but depends on the local surfactant concentration. Thus the interfacial force f(x) has two components: The capillary pressure, which is normal to the interface and proportional to the local curvature and interfacial tension and the Marangoni stress, the surface gradient of the interfacial tension. In dimensionless form the interfacial force reads f(x) = σ(x)2k(x)n(x) s (σ ), x S, where k(x) = 0.5 s n is the dimensionless mean curvature of the interface and σ = σ /σ eq is the dimensionless interfacial tension. In the present work the material parameters with indexes represent dimensional values, with an exception for σ av and Γ av. The nonindexed parameters are reserved for the corresponding dimensionless values. The surface gradient s is defined as: s = (I nn). Due to deformation of the drop, the interfacial velocity and surface diffusion, the concentration of the surfactant Γ on the interface will, in general, be nonuniform. The equation governing the surfactant concentration expresses the mass balance of the surfactant on the interface, known as the surfactant convection diffusion equation. In dimensionless terms it is (see for instance [13] and [17]) Γ + s (Γ u s ) 1 (6) t Pe 2 s Γ + 2k(x)Γ u n = 0, where Γ = Γ /Γ eq is the dimensionless surfactant concentration and Γ eq is the uniform concentration on the equivalent spherical drop. The tangential and normal component of the interfacial velocity are respectively u s = (I nn) u and u n = (n u)n. The surface Péclet number Pe = GR 2 /D s (D s is surface diffusivity), which is convenient to be expressed as (see [18]) Pe = CaΛσ 0 /σ eq, where Λ = σ 0 R/(µD s ) is a material parameter independent of the surfactant amount and the flow conditions. The equivalent interfacial tension σ eq is given by (2) (3) (5) (7): σ eq = σ (Γ eq ), in dimensionless terms it is σ(1) = 1, see Eq. (8); σ 0 = σ (0) is the interfacial tension for a clean interface. The convection diffusion equation in the form (6) applies for insoluble surfactants. In the case of soluble surfactants an extra term, taking into account the surfactant flux interfacebulk has to be added, see for instance [11]. In order to close the mathematical model (1) (6) we need a relation between the interfacial tension and the surfactant concentration, σ (Γ ), which is given by the equation of state Equation of state The equation of state gives a dependence of the interfacial tension on the surfactant concentration. Different equations of state have been used, for a discussion see [17]. One of the most popular is the equation of state that corresponds to the Langmuir adsorption isotherm, the so called Szyszkowski equation (often called Langmuir equation of state) σ (Γ ) = σ 0 + RT Γ ln ( 1 Γ /Γ ), where Γ is the maximum concentration for complete surface coverage in a monomolecular film, and RT is the product of the gas constant and the temperature. The Szyszkowski Eq. (7) has the advantage to be valid also for large surfactant concentrations in contrast to the linear equation of state. In dimensionless terms (7) becomes σ(γ)= 1 + E ln(1 xγ ) 1 + E ln(1 x), where E = RT Γ /σ 0 is the elasticity number and x = Γ eq /Γ is the surface coverage. With this definition of the elasticity number (based on σ 0 rather than σ eq ) E does not depend on the surfactant coverage x,see[9]. More complicated equations of state have also been used, for example the Frumkin isotherm or two-phase equation of state [17]. Their disadvantage, however, is a larger number of parameters. In a number of studies ([8] for systems of polymeric liquids and [30] for air/water interface) it has been reported that both Langmuir and Frumkin isotherms provide an equally good fit with experimental data. For water/oil interfaces Janssen et al. ([4,7,16]) have found that the Frumkin isotherm provides a better fit than the Langmuir one. A major disadvantage of all equations of state discussed above is that they predict a physically unrealistic negative values for the interfacial tension when Γ approaches Γ (Γ < Γ ). This is seen in Fig. 3 at an elasticity number E = 1: at a critical surfactant concentration, about 60% of the saturation limit Γ the interfacial tension becomes negative, see the solid line in Fig. 3. In contrast, the experimental measurements show that above some value of the surfactant concentration the interfacial tension remains almost constant, see for example [30] and [7]. Chang and Franses [30] discuss experimental data where the plateau for the interfacial tension at high surfactant concentration is about 50% of the corresponding value for clean interface. Their explanation is that Γ is a theoretical value and (7) (8)

5 S (05) /FLA AID:11865 Vol. ( ) [+model] P.5 (1-26) YJCIS:m5+ v 1.50 Prn:10/01/2006; 14:37 yjcis11865 by:vik p. 5 I.B. Bazhlekov et al. / Journal of Colloid and Interface Science ( ) 5 Fig. 3. The interfacial tension (the solid line) and a part of the Marangoni stress (the dashed line) as functions of the surfactant concentration given by the equation of state (7) at E = 1. in reality the maximum concentration is usually lower than Γ. This is, however, not predicted by the equations of state. A common opinion is that with increasing surfactant concentration the Marangoni stress also increases and since it opposes convection it prevents the interfacial tension from becoming negative. In order to check this, the Marangoni stress is written in a dimensionless form as ( s σ ) (σ /σ 0 ) /σ 0 = (Γ /Γ ) ( s Γ ) /Γ. While the second term in the right-hand side depends on the surfactant distribution, the first one is, at a given concentration, determined by the equation of state only. In Fig. 3 it is called Marangoni stress and is shown as a function of the dimensionless surfactant concentration Γ /Γ. At the critical surfactant concentration Γ 0.6Γ, when the interfacial tension becomes negative the Marangoni stress is insignificantly larger than that for much smaller Γ. Thus in the example discussed, E = 1, one cannot expect that the Marangoni stress alone can prevent the interfacial tension from going below zero. This conclusion is also supported by our simulations. To study how the critical surfactant concentration and the Marangoni stress depend on the elasticity number E, theyare shown in Fig. 4. It is seen that the critical surfactant concentration monotonically decreases with increasing E. The Marangoni stress initially decreases to a minimum at E 1, after which it increases monotonically, but for E>1intheinterval considered (E <5) has moderate values. For E<0.25 the critical surfactant concentration is close to the maximum one, and the Marangoni stress is significant in order to keep the surfactant concentration below the critical level, and consequently the interfacial tension positive Boundary integral formulation and numerical method For a given position of the interface S and surfactant concentration the solution of the mathematical model (1) (3) and (5) for the velocity u(x 0 ) at a given point x 0 S can be obtained by means of the boundary integral formulation [21] (λ + 1)u(x 0 ) = 2u (x 0 ) 1 f(x) G(x 0, x) ds(x) 4π λ 1 4π S S u(x) T(x 0, x) n(x) ds(x), where the integration is over the interfacial area S. The tensors G and T are the Stokeslet and stresslet, respectively: G(x 0, x) = I/r + ˆxˆx/r 3, T(x 0, x) = 6ˆxˆxˆx/r 5, where ˆx = x x 0,r= ˆx. The main advantage of a boundary integral method is that it involves integration only on the interfaces. The main problem, however, regarding the numerical implementation of (9), is due to the singularities of the kernels G and T at x = x 0.Theintegral in Eq. (9) that correspond to the capillary pressure (the first term of (5)) and the second integral in (9) are computed via a nonsingular contour integration on an adaptive triangular mesh [31]. The integration of the Marangoni stress (the second term of f(x)) is performed by standard surface integration, applying local mesh refinement at the singular point. Now, having the interfacial velocity (u s (x,t), and u n (x,t)) the position of the interface and the surfactant concentration at the next time instant t + t are obtained via Eqs. (4) and (6), respectively. The convection diffusion Eq. (6) is solved by a finite-volume method of second-order spatial accuracy in combination with Crank Nicolson time-integration scheme. Our method is similar to that of [14,21]. More information regarding the numerical (9)

6 S (05) /FLA AID:11865 Vol. ( ) [+model] P.6 (1-26) YJCIS:m5+ v 1.50 Prn:10/01/2006; 14:37 yjcis11865 by:vik p. 6 6 I.B. Bazhlekov et al. / Journal of Colloid and Interface Science ( ) Fig. 4. The critical surfactant concentration (the solid line) and the Marangoni stress (the dashed line) at σ = 0 as functions of the parameter E. Fig. 5. Comparisons of the present simulations with the numerical results of Velankar et al. (2004) at different values of the relative surfactant coverage (x = 0, 0.2, 0.95) for the length of the drop L, the drop width (W ) and breadth (B). As in [9] the curves for x = 0.2 and0.95 are shifted downward by 0.5 and 1, respectively, for clarity. method for the convection diffusion equation and its validation can be found in [32]. For most of the simulations presented here meshes of about 4000 triangular elements were used. In the cases of large drop deformation, or to test the convergence of the method, the number of elements was increased to about The time step t was in the range [10 3,10 5 ] depending on the parameters Ca and x. In all simulations presented in this work the conservation of the drop volume and surfactant amount were satisfied with an error less than 0.5%. A number of numerical tests and comparisons have been performed in our previous studies [31,32], to show the accuracy and the numerical stability of the method. In the following section additional comparisons, with experimental and numerical results, are discussed, regarding the effect of insoluble surfactants on drop deformation Comparisons with previous results The first comparison is with the numerical results of Velankar et al. [9], seefig. 5: an initially spherical drop is subjected to simple shear flow at Ca = 1 till a strain of 5, after which the flow stops and the drop relaxes back to a sphere. Fig. 5 shows a good agreement for the three values of x: pure interface x = 0, relatively small surfactant coverage x = 0.2 and high surfactant coverage x = 0.95.

7 S (05) /FLA AID:11865 Vol. ( ) [+model] P.7 (1-26) YJCIS:m5+ v 1.50 Prn:10/01/2006; 14:37 yjcis11865 by:vik p. 7 I.B. Bazhlekov et al. / Journal of Colloid and Interface Science ( ) 7 (a) (b) Fig. 6. Comparisons with the experimental results of Hu and Lips at: (a) λ = and (b) λ = 2.3 (the markers represent the experimental results and the lines correspond to our results). The second comparison discussed here is with the numerical results of Eggleton et al. [18] shown in their Fig. 3, regarding drop deformation in axisymmetric extensional flow. Our results are in very good agreement with those presented in [18] for surface coverage x For larger values of x, however, our results showed smaller deformations, up to 10 15% for x = and Ca We verified our results with a 2D axisymmetric version of our code and obtained excellent correspondence for all considered values of x. Inanattemptto find the reason for the disagreement at large x we compared the profiles of surfactant concentration, see Fig. 7 of Eggleton et al. The comparison showed that a very small difference 3 4% could exist with less surfactant in the results of Eggleton et al. Such insignificant differences, however, translate at large x (x = 0.996) to about 10 15% difference in the values of the interfacial tension and the capillary number. The last comparison is with the experimental results of Hu and Lips [8]. They studied the effect of insoluble surfactants on drop deformation in planar extensional flow using a fourroll mill apparatus. In Fig. 6 the results of Hu and Lips (see their Figs. 2a and 3a) are compared with our simulations. In the case of pure interfaces, x = 0, the agreement is very good, see the solid lines (our results) and the open squares (Hu and Lips results). In the presence of surfactants, although a qualitative correspondence could be found, the difference is disturbingly large. We checked for a possible error in the estimation of the parameters, varying the values of E, x and Λ, but that did not resolve the difference. A closer inspection of the results pre-

8 S (05) /FLA AID:11865 Vol. ( ) [+model] P.8 (1-26) YJCIS:m5+ v 1.50 Prn:10/01/2006; 14:37 yjcis11865 by:vik p. 8 8 I.B. Bazhlekov et al. / Journal of Colloid and Interface Science ( ) Fig. 7. The influence of the viscosity ratio λ and the surfactant on the drop deformation at small capillary number. The markers are our results: filled markers for pure interface, x = 0, and open markers for the case of surfactants. The small deformation theory is represented by solid lines. sented in Fig. 6 shows that the difference is significant even at small deformations, where the numerical results should be the most accurate. The interfacial tension in [8] has been estimated from the drop deformation using small deformation theory: D = Ca(19λ + 16)/(8λ + 8), see[33]. This idea had been used earlier by Hu et al. [5], who state that : This technique for measuring the interfacial tension is known to give reliable results, even when there is surfactant at the interface. To stay within the linear regime the measurements of the interfacial tension in [5,8] have been carried at a drop deformation around D = 0.1. A small deformation theory in the case of insoluble surfactants is presented by Stone and Leal [13] for small Péclet numbers. In this case the surface diffusion plays dominant role for the surfactant distribution. Thus, due to the dilution, the drop deformation D at given Ca is smaller than that for drop with pure interface, see Fig. 9 for Λ 1. In the case of small deformation the dilution is negligible as well as the effect of surfactant [14], see also Fig. 9. The latter agrees with the suggestion of Hu et al. [5]. In the case of large Péclet numbers, however, due to the strong influence of the Marangoni stress and stretching, the drop deformation is larger than that for pure interface. The difference can be significant even for small deformations, as it is shown by Stone and Leal, 1990 [13], about 20% at D = 0.1, see their Fig. 3c. Thus, one can see that the use of small deformation theory in the present case, large Péclet numbers, is not justified. In order to find out how the presence of insoluble surfactants at large Péclet numbers affects the deformation D, our results from Fig. 6 for deformations around D = 0.1 are compared with the small deformation theory in Fig. 7. It is seen that in the presence of surfactants the deformation depends also linearly on the capillary number. At the considered values of x the deformation is independent of the viscosity ratio λ and weakly depends on the surfactant coverage x. The disagreement of the results in the case of surfactants (open markers) with the linear theory is about 10% for λ = 2.3 and 25% for λ = 0.093, but the correspondence with the immobile curve, λ =, is relatively good. A similar trend at small deformations, in the case of axisymmetric extensional flow, is found in the theoretical studies of Milliken et al. [2] (see their Figs. 2 and 3). Milliken et al. have shown that the drop deformation in the presence of surfactants is unaffected by the viscosity ratio. For relatively weak surfactants (linear equation of state) their results are very close to the immobile one, and for larger decrease of the interfacial tension (the nonlinear equation of state) the deformation is larger. The numerical results of Eggleton et al. [18], see their Fig. 3, are also in support of this conclusion. Experimental evidence of the immobilization effect of the surfactants, even at very small amount, can be found in [34] in the case of an ascending bubble in a protein solution and in Janssen et al. [4] (see also references therein) in the case of planar extensional flow. Following the above discussion our conclusion is that in the presence of surfactants at large Péclet numbers the measurements of the interfacial tension based on the small deformation theory can lead to a significant error. The error is larger, up to 25% for small viscosity ratios, even at very small deformations, D There are, we believe, two effects which are responsible for this difference: First, the Marangoni stress, which even at a trace amount of surfactants completely blocks the interfacial motion, leading to immobilization of the drop. This increases the deformation to a level of large viscosity ratio drops (λ>10). Second, the tip stretching (see [8,17]) leads to an additional deformation in the drop tips, which (in the discussed range of E, x and Λ) is higher for larger surfactant coverage or/and smaller viscosity ratio. Good correspondence, see Fig. 8, is obtained if the results of Hu and Lips [8] are rescaled in such a way to fit with our simulations for small deformations, D 0.1.This,infact,isa

9 S (05) /FLA AID:11865 Vol. ( ) [+model] P.9 (1-26) YJCIS:m5+ v 1.50 Prn:10/01/2006; 14:37 yjcis11865 by:vik p. 9 I.B. Bazhlekov et al. / Journal of Colloid and Interface Science ( ) 9 (a) (b) Fig. 8. The comparisons presented in Fig. 6, after the experimental results of Hu and Lips were rescaled: (a) λ = and (b) λ = 2.3. correction due to the effect of surfactants to the values of the equivalent interfacial tension initially calculated based on the small deformation theory. The good correspondence in Fig. 8, also for large deformations, indicates that only one correction step gives already a satisfactory prediction for the equivalent interfacial tension. 3. Results and discussion 3.1. Parameter values The mathematical problem (1) (8) posed in the previous section has the following dimensionless parameters: λ, Ca, E, x and Λ. In the present section we offer a relatively complete parametric investigation for λ, Ca and x. The viscosity ratio was set to λ =[0.001; 0.01; 0.1; 0.2; 0.3; 0.6; 1; 2; 3] and the surfactant coverage x =[0; 0.1; 0.25; 0.5; 0.75; 0.95; 0.975]. Themesh(λ,x) was further refined around the boundaries between different regions presented in Fig. 16. For each pair (λ,x) simulations were performed to find the steady drop shapes for different Ca. The capillary number was increased, starting from Ca = 0.1, with steps Ca = till its critical value Ca cr was reached, where at a further increase no steady drop shape exists. For a better resolution for Ca cr the step Ca was reduced when Ca approached Ca cr. It is important to bear in mind that the definition of the capillary number is based on the equivalent interfacial tension σ eq and varies with the surface coverage x. The value of the elasticity number was set to E = 0.2. It corresponds to the value used in previous numerical studies [9,18,35] and falls well in the range of practical two-phase

10 S (05) /FLA AID:11865 Vol. ( ) [+model] P.10 (1-26) YJCIS:m5+ v 1.50 Prn:10/01/2006; 14:37 yjcis11865 by:vik p I.B. Bazhlekov et al. / Journal of Colloid and Interface Science ( ) Fig. 9. The influence of the surface diffusion (parameter Λ) on the drop deformation at E = 0.2, x = 0.75 and λ = 0.1. system [8,18]. For large values of E the problem related with physically unrealistic negative interfacial tension is more likely to appear, see Fig. 4 and the discussion in Section 2.2. At E = 0.2 the limiting value of the surface coverage is x = 0.993, for larger x the equation of state predicts negative values for σ eq /σ 0. Results at twice smaller and twice larger values of the elasticity number, E = 0.1 and 0.4, are also discussed. The Péclet number Pe = ΛCaσ 0 /σ eq expresses the relative importance of the convection and the diffusion. To investigate the influence of the parameter Λ, simulations at different values were performed and the results are shown in Fig. 9. As it is seen, there are two limiting cases of the surfactant distribution: diffusion dominated Λ 0.1 and convection dominated Λ 100, the values which agree very well with those predicted in [20] in the case of axisymmetric extensional flow. In the diffusion controlled regime the surfactants are uniformly distributed with concentration Γ av and the interfacial tension is constant over the interface. Thus, the drop deformation at a given Ca is described well by that of a pure interface with interfacial tension σ(γ av ). In the more interesting, convection dominated regime, the influence of the surfactants on the drop deformation is stronger as is shown in Fig. 9. The results presented in the remainder of this study are devoted to this regime at Λ = This value corresponds well with the convection dominated regime for the whole range of λ and x considered here, and has been used extensively by previous authors (see for instance [18,19]). For easier comparison, in all figures where drops with a moderate deformation are presented the deformation is chosen to be about D = For the velocity field plots presented the velocity is scaled with the same factor, The reader should keep in mind that due to the projection on XY plane the interfacial velocity could be slightly changed. The starting positions for the stream lines in all figures shown are the same Surfactant related mechanisms of drop deformation: stretching, dilution and Marangoni stress There are three surfactant related mechanisms the combined effect of which alters the drop deformation from that of a drop with an uniform interfacial tension, well described by Pawar and Stebe [17] Surface dilution With an increase in drop deformation the interfacial area also increases. This leads to a decrease of the average surfactant concentration which in turn can lead to a increase of the average interfacial tension: S σ av = σ ds (10) A, where A is the total interfacial area. Hu and Lips [8] used a definition of the average interfacial tension based on the average surfactant concentration: σ HL av = σ(γ av), Γ av = S Γ ds A, (11) where σ(γ) is given by the equation of state. It is easy to see that while both definitions are equivalent for a linear dependence σ(γ) they differ for a nonlinear dependence. In general σav HL given by (11) overestimates σ av. For instance, in the case discussed by Hu and Lips (see their Ca = 0.13, D = 0.4) their definition predicts that σav HL is about 2.4% larger than the equilibrium interfacial tension. Our results, however, show that in this case the average interfacial tension is unchanged, σ av = 1. Because our prediction for A, respectively Γ av, agrees excellently with that of Hu and Lips we can conclude that the abovementioned difference is due to the different definitions of the average interfacial tension, Eqs. (10) and (11).

11 S (05) /FLA AID:11865 Vol. ( ) [+model] P.11 (1-26) YJCIS:m5+ v 1.50 Prn:10/01/2006; 14:37 yjcis11865 by:vik p. 11 I.B. Bazhlekov et al. / Journal of Colloid and Interface Science ( ) 11 Fig. 10. The Marangoni stress, s σ, for the case shown in Fig. 2b, E = 0.2, x = 0.75, λ = 0.1 andca = Marangoni stress Marangoni stress, the last term in Eq. (5), is determined by the product of the interfacial tension derivative with respect to the concentration and the concentration gradient s σ = σ (12) Γ sγ, where the interfacial tension derivative is σ Γ = xe ln(1 xγ ) (1 xγ )(1 + ln(1 x)). The Marangoni stress retards the convective flux and thus opposes the interface velocity. Depending on the initial coverage, x, one or both multipliers in Eq. (12) can have a significant contribution to the total Marangoni stress, see also [17]. For small x initially, at uniform surfactant distribution, the interfacial tension derivative is weak, as is the concentration gradient. This allows for the interface velocity to build up a significant gradient in surfactant concentration. As a result both parts of the Marangoni stress (12) become significant, at least locally at the drop tips. For large x (surface coverage close to the maximum value) the surface tension derivative is also large, even for small concentration gradients. Thus the Marangoni stress is significant and a nearly uniform surfactant concentration is maintained, which is the case in the dilution dominated regime. An important feature for extensional, axisymmetric or planar, flow at the final, steady, stage of the deformation is that the interface velocity is fully retarded by the Marangoni stress. Thus, even for a trace amount of surfactant the drop interface is fully immobilized, see for instance [4,20]. In contrast, for simple shear flow, due to the drop rotation [16], the interface velocity is significant even for high initial surface coverage, see Fig. 15b. Marangoni stress, however, can significantly alter the pattern of the interface velocity and consequently the velocity inside the drop. This is seen from a comparison between the two frames of Fig. 15. A typical vector field of the Marangoni stress is shown in Fig. 10 for relatively large surface coverage, x = It should be expected that Marangoni stress with such pattern retards the drop deformation, especially in the tip regions Stretching In previous studies [8,17], the stretching is considered as purely tip stretching: a higher deformation at the drop tips compared to that of drops with a constant interfacial tension. Pawar and Stebe (1996) [17] defined the difference between the local interfacial tension at the tips and σ eq (1 in dimensionless units) as a measure for tip stretching. Here we use a simple arithmetic to define the effect of surfactants on stretching. A starting point is, as in [8,17], that the total effect of surfactants on drop deformation can be decoupled in the three mechanisms D = D DMS = D P + dd D + dd M + dd S, (13) where D P is the deformation in the case of pure interface with interfacial tension σ eq (dimensionless 1), dd D and dd M denote the contributions due to dilution and Marangoni stress, respectively. The last term, dd S, is due to the stretching and has to be defined. Now, we decompose the interfacial force f(x), see Eq. (5), in a similar manner f(x) = f P (x) + df D (x) + df M (x) + df S (x), (14) where the terms correspond to those in Eq. (13). The first three terms in the r.h.s. of (14) are well defined: f P (x) = 2k(x)n(x), df D (x) = (σ av 1)2k(x)n(x), df M (x) = s σ. (15) Finally, replacing the expressions for these terms, (15), in (14) and comparing with (5) yields: df S (x) = (σ σ av )2k(x)n(x). (16) This mathematical expression for stretching shows that it is a result of the difference between the interfacial tension and the average one, not the equivalent value, σ eq, as it was suggested by [17]. Stretching can be divided schematically into tip and waist stretching. Tip stretching appears at the tips of the drops, because the interfacial tension σ there is smaller than the average

12 S (05) /FLA AID:11865 Vol. ( ) [+model] P.12 (1-26) YJCIS:m5+ v 1.50 Prn:10/01/2006; 14:37 yjcis11865 by:vik p I.B. Bazhlekov et al. / Journal of Colloid and Interface Science ( ) value, see Fig. 2. In the main part of the drop (drop waist) the interfacial tension is, in general, larger than σ av. Mathematically this can be written as: df S (x) = df T (x) + df W (x), df T (x) = min(σ σ av, 0)2k(x)n(x), df W (x) = max(σ σ av, 0)2k(x)n(x). (17) For a comparison the definition of tip stretching of Pawar and Stebe [17] reads: df PS T (x) = min(σ 1, 0)2k(x)n(x), (18) which is based on the equivalent interfacial tension σ eq,1in dimensionless terms. A common perception is that with increasing the interfacial tension, locally the drop deformation D decreases. While this is true for an axisymmetric drop shape, in shear flow an opposite effect can happen. Indeed, an increase of the interfacial tension at the drop waist (the regions around the points O and O,asshowninFig. 2) results in a decrease of the breadth B of the drop and the drop deformation D increases. The reason of this is the nonuniform surfactant distribution along the waist of the drop. Indeed, the larger interfacial tension in the regions around the points O and O tends to decrease the curvature locally. This results in a further flattening of the interface in the mentioned regions of the already, due to the flow, slender drop (B <W). Thus W and L increase and B decreases, and consequently the drop deformation D = (L B)/(L+ B) increases. Similar is the situation of drop deformation in planar hyperbolic flow in the presence of surfactants. Below we evaluate the three surfactant related mechanisms using the following approach: from a given steady solution for the drop shape and the surfactant distribution as initial condition, a new simulation starts, where one or more terms from the interfacial force (14) are extracted. Thus by comparing different combinations of the terms (15), (16), estimates for the dilution, Marangoni stress and the stretching are obtained. Consider the following example: The contribution of the drop stretching dd S to the total deformation can be defined as dd S = D S D P, where D S is the deformation at f(x) = f P (x) + df S (x) and D P is the deformation of a pure drop interface, at f(x) = f P (x). Another possibility is: dd S = D MS D M, where D MS is obtained with f(x) = f P (x) + df M (x) + df S (x) and D M with f(x) = f P (x) + df M (x). In general, the above two constructions for dd S will provide two different values for the drop stretching. This is due to the fact that the final drop shape differs from the initial one, which introduces a deviation of the surfactant distribution from the desired, initial, one. This can affect the measurements of the stretching and Marangoni stress, because they depend on the surfactant distribution. However, it has no influence on the dilution, dd D, which is well defined by dd D = D D D P. Thus for measuring dd S and dd M we use the combinations at which the final drop shape is closest to the initial one. These appear to be dd M = D M D P and dd S = D MS D M.The total deformation D = D DMS is not used for measuring of dd S or dd M. It is reserved for an evaluation of the decoupling, the difference between the total effect of the surfactants dd DMS = D DMS D P and the sum of the effects of the three mechanisms, dd D + dd M + dd S. Using this approach we investigated the contributions of the different surfactant related mechanisms to the total drop deformation. Fig. 11 shows the deformation dd I (I stands for D, M and S) as a function of Ca at λ = 0.1 and E = 0.2 attwovalues of the surface coverage x. In the first case x = 0.75 (the surfactant distribution at Ca = can be seen in Fig. 2) the effect of dilution is negligible. For a surface coverage close to the saturation limit, x = 0.975, as it has to be expected, the dilution has a significant effect. In this case, see Fig. 11b, drop stretching cannot be measured beyond Ca = 0.3 because the drop breaks up when df D (x) or/and df M (x) are excluded from the total interfacial force. The tip stretching ddt PS, as defined in [17], is also shown (*). In the first case x = 0.75, see Fig. 11a, σ av is very close to unity and thus the definition of the tip stretching (17) is equivalent to that of Pawar and Stebe [17], see Eq. (18). Thus the difference between the stretching dd S and the tip stretching (*) is due to the waist stretching, which is neglected in [17] and [8]. For the case of the higher surface coverage, x = 0.975, tip stretching ddt PS as defined in [17], seeeq.(18), indicates no effect on drop deformation, see the asterisks in Fig. 11b. This is because the surfactant concentration is below unity (respectively σ>1, see Fig. 16) due to a significant dilution. Our results, however, show that in this case the drop stretching dd S is important for the total drop deformation, see Fig. 11b. In [8], the authors showed that for λ = Marangoni stress promotes drop deformation. Our results presented in Fig. 11, however, indicate the opposite, that Marangoni stress decreases the drop deformation. The latter conclusion is supported also by the patterns of the Marangoni stress presented in Fig. 10. In fact, some values of the parameters and the flow conditions are different, planar extensional flow was studied in [8] while simple shear flow is considered here. However, the main source of discrepancy, we think, is due to the approach used by Hu and Lips (2003) [8]. The way they measured tip stretching does not guarantee that the effect of Marangoni stress is excluded. This can affect the following estimate of the Marangoni stress as the residual of the tip stretching and dilution to the total contribution of surfactant. Finally, a comparison between the total effect of surfactant dd DMS and the sum of the contributions of the three mechanisms dd D + dd M + dd S can be a measure for the coupling between them. The small difference between these two curves, presented in Fig. 11 by open triangles (up- and down-wards), indicates that in the considered cases the three mechanisms are weakly coupled and that they indeed can be considered individually. It should be noted, however, that the excellent agreement between dd DMS and dd D + dd M + dd S in the first case, at x = 0.75, is because in this case the dilution is negligible. An advantage of the approach presented above is that the surfactant related mechanisms are fully decoupled at model level, Eqs. (15), (16). The disadvantage is that due to a deviation of the drop shape from the initial one, the surfactant distribution

13 S (05) /FLA AID:11865 Vol. ( ) [+model] P.13 (1-26) YJCIS:m5+ v 1.50 Prn:10/01/2006; 14:37 yjcis11865 by:vik p. 13 I.B. Bazhlekov et al. / Journal of Colloid and Interface Science ( ) 13 (a) (b) Fig. 11. Decoupling of the different surfactant related mechanisms that alter the drop deformation at λ = 0.1 and E = 0.2: (a) x = 0.75 negligible dilution; (b) x = The graphs represent the relative deformation dd to that of a drop with constant interfacial tension σ = 1. can differ from the desired (initial) and thus affects the estimates for Marangoni stress and drop stretching The effect of surfactants at small capillary numbers In order to have a preliminary idea about the effects of surfactants on drop deformation, the range of small deformations is considered below. At λ = 0.1 Fig. 12a shows similar behavior as in the case of planar extensional flow, see Fig. 7. For all values of x the deformation D depends linearly on the capillary number, provided that D<0.15. It is seen in Fig. 12a that for given Ca at a small viscosity ratio, λ = 0.1, with increasing surface coverage the deformation increases initially till x = 0.75, and after that it decreases. The initial increase of the deformation is due to the immobilizing effect of surfactants. Thus at x>0.25 the drop interface is immobile and the deformation is weakly dependent on a further increase of x. The slight increase of D when x increases in the interval x [0.25, 0.75] is due to additional drop stretching. At a further increase of x, x>0.75, the dilution becomes dominant leading to a decrease in drop deformation. This behavior can also be seen in Fig. 12b, where the deformation is shown as a function of the viscosity ratio at Ca = 0.1. It shows also that the deformation is almost independent of λ for λ<0.1, provided that x>0.1. With increasing the viscosity ratio, however, the influence of the surfactant coverage on the drop deformation decreases and at λ>10 it is negligible. The behavior for moderate and large deformations is similar and will be discussed further in more detail.

14 S (05) /FLA AID:11865 Vol. ( ) [+model] P.14 (1-26) YJCIS:m5+ v 1.50 Prn:10/01/2006; 14:37 yjcis11865 by:vik p I.B. Bazhlekov et al. / Journal of Colloid and Interface Science ( ) (a) (b) Fig. 12. The influence of the surfactant coverage x on the drop deformation as a function of: (a) the capillary number (0.025 Ca 0.1) at λ = 0.1; (b) the viscosity ratio (0.001 λ 10) at Ca = The influence of the viscosity ratio In the case of a pure interface it is well known, see for instance [27,36], that at the limiting cases of small and large viscosity ratios the viscosity of the drop stabilizes its shape. This is also seen from the Grace curve presented in Fig. 1. For a small viscosity ratio and a relatively large Ca, pointed drop tips with high curvature are formed, see the inset in Fig. 1 at λ = 0. Such a shape plays an important role for the stability of the drop. In the presence of surfactants, their concentration increases at the tips of the drop locally, see Fig. 13a, leading to a decrease in interfacial tension. This results in a stretching of the tips and, if the capillary number is sufficiently large, tipstreaming appears, see [3,16]. At a high viscosity ratio the internal circulation in the drop (see Fig. 2a), plays a stabilizing role in addition to the interfacial tension, see also [27,37]. The relatively high viscous stress of the internal circulation also aligns the drop in direction to the external flow, see [27] and the inset in Fig. 1 at λ = 3. This decreases the viscous stresses from the external flow and once more favors a smaller drop deformation. If surfactants are present on the interface of a drop at high viscosity ratios, the interface velocity, which is coupled with the drop circulation, tends to decrease the surfactant concentration at the tip regions. In addition, the drop orientation leads to a smaller contribution of the external flow on the advection of surfactants towards the tips, further reducing the surfactant concentration gradient.

15 S (05) /FLA AID:11865 Vol. ( ) [+model] P.15 (1-26) YJCIS:m5+ v 1.50 Prn:10/01/2006; 14:37 yjcis11865 by:vik p. 15 I.B. Bazhlekov et al. / Journal of Colloid and Interface Science ( ) 15 (a) (b) Fig. 13. The distribution of the surfactant on the drop interface for E = 0.2andx = 0.5at:(a)λ = 0.1, Ca = 0.3; (b) λ = 3, Ca = In both cases the deformation is about D = 0.43 (the streamlines are in the plane x y). A comparison between the surfactant distribution at low, λ = 0.1, and high, λ = 3, viscosity ratios is shown in Fig. 13. It shows that, even for a more than double capillary number, Ca = 0.65, the surfactants for λ = 3 are about two times more uniformly distributed than that in the case of λ = 0.1. In summary, at small viscosity ratios the mechanisms related to high surfactant gradients are dominant, mainly tip stretching. In contrast, at large viscosity ratios dilution can play a significant role at high surface coverage. This can be quantitatively seen in Fig. 14, where below a given viscosity ratio, which depends on x, the mode of breakup is tip-streaming (marked curves). Fig. 14 also shows that in the range of small viscosity ratios, λ 0.1, the drop deformation is independent of a further decrease in λ. For instance, at all considered values of x the differences between the curves for λ = 0.1 and λ = are within 5%. An explanation is that, at a given x, with decreasing λ the Marangoni stress becomes dominant for the immobilization of the interface and a further decrease of the drop viscosity has no effect The influence of the surface coverage As was discussed in previous sections the effect of surfactants on drop deformation is a result of the combined action of three mechanisms: dilution, drop stretching and Marangoni stress. While the first two directly affect the interfacial tension, the Marangoni stress alters the interface velocity. This is seen in

16 S (05) /FLA AID:11865 Vol. ( ) [+model] P.16 (1-26) YJCIS:m5+ v 1.50 Prn:10/01/2006; 14:37 yjcis11865 by:vik p I.B. Bazhlekov et al. / Journal of Colloid and Interface Science ( ) (a) (b) Fig. 14. The influence of the viscosity ratio on the drop deformation at different values of the surface coverage: (a) x = 0.25; (b) x = 0.5; (c) x = 0.75; (d) x = The marked curves correspond to tip-streaming mode of breakup. Fig. 15, where the interface velocity at λ = 0.1 is shown for the two limiting cases of no surfactants, x = 0, and surfactants at concentration close to the maximum level x = By comparing these two velocity fields with the Marangoni stress from Fig. 10 it can be concluded that the Marangoni stress directly affects the interface velocity. Fig. 15 indicates that in the case of no surfactant, x = 0, the interface velocity is directed towards the tips of the drop. Thus for a relatively small surface coverage, negligible Marangoni stress, the interfacial velocity will convect the surfactants to the tip regions, leading to a highly nonuniform surfactant distribution. At high surface coverage, the Marangoni stress significantly alters the interface velocity to a pattern similar to that of Fig. 15b, leading to more uniformly distributed surfactants. This can be seen in Fig. 16 for two values of x = 0.5 and In the first case, Fig. 16a the interfacial tension in the tip regions is more than twice smaller than the equivalent value σ eq.atx = 0.975, Fig. 16b, the same deformation D = 0.43 is achieved for almost twice higher capillary number, Ca = This is due to the strong effect of the Marangoni stress, that keeps the interfacial tension higher than 1 and leads to significant dilution. It is also seen in Fig. 16 that the Marangoni stress affects the external flow around the drop. At large x a boundary layer around the drop is formed, see also Fig. 2 at x = For smaller x the external flow is affected locally around the drop tips, where the Marangoni stress is significant, compare Fig. 16a with Fig. 2a.

17 S (05) /FLA AID:11865 Vol. ( ) [+model] P.17 (1-26) YJCIS:m5+ v 1.50 Prn:10/01/2006; 14:37 yjcis11865 by:vik p. 17 I.B. Bazhlekov et al. / Journal of Colloid and Interface Science ( ) 17 (c) (d) Fig. 14. (continued) It should be mentioned, however, that the influence of the surface coverage x decreases with increasing viscosity ratio. As it was already shown in the case of a high viscosity ratio, the viscous stresses of the internal drop circulation become dominant, suppressing the influence of surfactants. Thus at high viscosity ratio the surfactants have a contribution to the drop deformation only at high x via dilution and possibly via Marangoni stress. It should be reminded that the direct effect of the surfactants on the interfacial tension is automatically taken in Ca via the equivalent interfacial tension. Quantitative information about the influence of the surface coverage on the drop deformation is presented in Fig. 17. At a small viscosity ratio, λ = 0.1, the deformation is nonlinear with x. The drop deformation increases initially with increasing x due to drop stretching and is maximal at x = 0.75, see the inset in Fig. 17a. With a further increase of x, Marangoni stress and dilution, respectively, become dominant, leading to a decrease in deformation. It was shown in the previous section that the results are independent of λ for λ 0.1, provided that x Thus the results in Fig. 17a forx 0.25 are valid also for smaller λ, λ 0.1. Figs. 17c and 17d indicate that for λ 1 the influence of surface coverage on drop deformation is negligible, provided that x 0.6. Thus, there exists a region at high viscosity ratio and relatively small x, where the surface coverage has no influence on the drop deformation. This region is shown in Fig. 19 by the dash-dotted line.

18 S (05) /FLA AID:11865 Vol. ( ) [+model] P.18 (1-26) YJCIS:m5+ v 1.50 Prn:10/01/2006; 14:37 yjcis11865 by:vik p I.B. Bazhlekov et al. / Journal of Colloid and Interface Science ( ) (a) (b) Fig. 15. The interfacial velocity for λ = 0.1 at:(a) x = 0 (no surfactant), Ca = 0.4; (b) x = 0.975, Ca = Modes of drop breakage An important question regarding the drop-size distribution in a suspension is: At what conditions drops will remain compact and when they will breakup? In the case of a drop with constant interfacial tension the answer is given by the dependence Ca cr (λ), known as the Grace curve [23], see also Fig. 1. The presence of surfactants can significantly alter drop deformation and breakup and, consequently, the critical capillary number, see for instance [4]. The influence of the surface coverage on the critical capillary number is shown in Fig. 18. It is seen that for small λ the presence of surfactants on the drop interface leads to a decrease of Ca cr due to the tip-streaming, see also Fig. 17. This is the case even for an extremely small surface coverage, note that at x = 0.1 the decrease of the interfacial tension is only about 2%, σ 0 /σ eq = The previously discussed effect of independence of drop deformation on viscosity ratio at small λ is expressed here as a plateau part of Ca cr (λ) for all values of x considered. The height of the plateau has a similar dependence on surface coverage as drop deformation: with increasing surface coverage the critical capillary number initially decreases, till x = 0.75, after which it increases (see the inset in Fig. 18). This behavior agrees well with previous results for axisymmetric [19] and planar [5] extensional flows. Our prediction of the dependence of Ca cr on σ 0 /σ eq agrees well with the experimental results of de Bruijn (1993) presented in his Table 1. Indeed, Fig. 17, see also Fig. 19, indicates that for λ 0.1 tip-streaming appears for 0.1 <x 0.95, which in terms of the relative interfacial tension is 1.02 <σ 0 /σ eq 2.5. Taking into account that in the experiments of de Bruijn (1993) [3] σ 0 = 30 mn/m, the above interval for σ eq becomes 29 mn/m σ eq 12 mn/m, which corresponds well to the experimental data in [3]. For the value of x at which Ca cr is minimal this approach translates our x = 0.75 (σ 0 /σ eq = 1.38) to a value of σ eq = 21.7 mn/m. The corresponding value measured by de Bruijn (1993) is 22 mn/m. With increasing viscosity ratio the effect of surfactant on Ca cr becomes negligible, provided that x<0.75, see Fig. 17. At large x and λ the critical capillary number increases, see Fig. 18, due to the effect of dilution. It should be reminded that Ca, respectively Ca cr, is based on σ eq, which depends on the surface coverage x.

19 S (05) /FLA AID:11865 Vol. ( ) [+model] P.19 (1-26) YJCIS:m5+ v 1.50 Prn:10/01/2006; 14:37 yjcis11865 by:vik p. 19 I.B. Bazhlekov et al. / Journal of Colloid and Interface Science ( ) 19 (a) (b) Fig. 16. The distribution of the surfactant on the drop interface for λ = 0.1 ande = 0.2 at:(a)x = 0.5, Ca = 0.3; (b) x = 0.975, Ca = (the streamlines are in the plane x y). Finally, the results are summarized in Fig. 19, where regions of different modes of breakup are shown by the solid lines. As it is well known for large viscosity ratios in the absence of surfactants, no breakup can appear, independently of the capillary number. In the presence of surfactants this region is broadened at large x due to the stabilizing effect of the Marangoni stress and dilution. Drop fragmentation appears at intermediate viscosity ratios for all x or at small viscosity ratio in combination with small x 0.1 or large x The tip-streaming mode, see Fig. 19, is typical for small λ and intermediate surface coverage 0.1 <x<0.95. The regions where drop deformation is independent of the viscosity ratio (dashed line) or surface coverage (dash-dotted line) are also shown in Fig. 19. The application of the present results is limited to the case of initially spherical drop subjected to a homogeneous shear flow with slowly increased shear rate. As in the case of pure interfaces the critical conditions for drop breakup can depend on the initial drop shape and/or nonhomogeneous shear flow Tip-streaming/dropping Simulations of tip-streaming are facing a number of difficulties, see [19]. They involve tip regions of extremely high curvature and surfactant concentrations approaching the saturation limit. That is why present numerical simulations are limited to 2D axisymmetric flows, see for instance [19]. Experimental results exist for planar extensional [4], and simple

20 S (05) /FLA AID:11865 Vol. ( ) [+model] P.20 (1-26) YJCIS:m5+ v 1.50 Prn:10/01/2006; 14:37 yjcis11865 by:vik p I.B. Bazhlekov et al. / Journal of Colloid and Interface Science ( ) (a) (b) Fig. 17. The influence of the surface coverage on the drop deformation at different viscosity ratio: (a) λ = 0.1; (b) λ = 0.3; (c) λ = 1; (d) λ = 2. The marked curves correspond to tip-streaming mode of breakup. shear flow [3,16]. In these studies drop-phase soluble surfactants have been considered. Below we investigate the process at λ = 0.1, E = 0.2 and x = 0.5 for different capillary numbers. Fig. 16 (see the dash-dotted line) suggests that they are applicable in the whole range λ 0.2, regarding the drop deformation and the critical capillary number. An open question is whether the size of the emitted drops is also independent of the viscosity ratio. Fig. 20 shows a typical drop shape at the moment when the daughter drops are emitted. The process of tip stretching and daughter drop formation is shown in Fig. 21. The main elements of the tip-streaming process are described elsewhere [3, 16,19]: with increasing capillary number the surfactant concentration at the drop tips increases and consequently the interfacial tension decreases. When Ca exceeds the critical value, Ca cr, the interfacial forces at the tips become almost zero, Fig. 21a, and cannot longer resist the viscous deformation forces. Thus the tip is stretched further, Fig. 21b. With progressing stretching, however, the interfacial area increases and the surfactant is locally diluted leading to an increase of the interfacial tension, Fig. 21c. This promotes capillary breakup of tiny drops, about 25 times smaller than the main drop. The equivalent surfactant concentration of the daughter drop is about 97% of the saturation limit with an equivalent interfacial tension about 1/3 of that of the main drop. This is in good qualitative agreement with the experimental observation in simple shear flow [3,7], as well as with the simulations of Eggleton et al. [19].

21 S (05) /FLA AID:11865 Vol. ( ) [+model] P.21 (1-26) YJCIS:m5+ v 1.50 Prn:10/01/2006; 14:37 yjcis11865 by:vik p. 21 I.B. Bazhlekov et al. / Journal of Colloid and Interface Science ( ) 21 (c) (d) Fig. 17. (continued) Now the dependence of the thread and daughter drop size on the capillary number is investigated. Janssen, Boon and Agterof (1994) found that the size of the daughter drop increases with increasing Ca, referring to tip-dropping. Similar behavior is predicted by our simulations as shown in Figs. 22 and 23. Fig. 22 shows that at larger capillary number the main drop undergoes a larger deformation. The zoomed tip region, presented in Fig. 23, indicates that the size of the daughter drop increases with Ca. The thread size initially, for values 0.4 Ca 0.5, increases and is then almost unaffected by a further increase of Ca. The equivalent interfacial tension of the daughter drop also increases with Ca from about σ eq = 0.4 atca = 0.4 to almost twice this value at Ca = The effect of the elasticity number The effect of the elasticity number E on drop deformation has been studied by Eggleton et al. [11,18]. They have shown, as should be expected, that for a given surface coverage the higher the value of E, the stronger the effect of surfactants. Here we do not investigate the effect of the elasticity number in the common fashion, i.e. at fixed other parameters. The idea is to find parameters which in combination with E determine the main behavior of the drop. If this is possible, then we can extend the validity of the results presented for values of E different from 0.2. A natural choice for such a combination is the ratio between the interfacial tensions of a clean interface to the equivalent one, σ 0 /σ eq. This ratio includes both parameters E

22 S (05) /FLA AID:11865 Vol. ( ) [+model] P.22 (1-26) YJCIS:m5+ v 1.50 Prn:10/01/2006; 14:37 yjcis11865 by:vik p I.B. Bazhlekov et al. / Journal of Colloid and Interface Science ( ) Fig. 18. The critical capillary number as a function of the viscosity ratio for different surfactant coverage. Fig. 19. The solid lines bound to the different regions of drop breakup (fragmentation or tip-streaming/dropping). The boundaries of surfactant dominated (viscosity ratio has no influence) and drop viscosity dominated (surface coverage has no influence) regions are given by the dashed and dash-dotted lines, respectively. Fig. 20. Tip streaming at E = 0.2, x = 0.5, λ = 0.1 andca = The critical capillary number is about Ca cr =

23 S (05) /FLA AID:11865 Vol. ( ) [+model] P.23 (1-26) YJCIS:m5+ v 1.50 Prn:10/01/2006; 14:37 yjcis11865 by:vik p. 23 I.B. Bazhlekov et al. / Journal of Colloid and Interface Science ( ) 23 Fig. 21. The evolution of the tip for the case shown in Fig. 20. Fig. 22. Tip dropping at E = 0.2, x = 0.5, λ = 0.1: Ca = 0.4 top; Ca = 0.6 bottom. Fig. 23. The zoomed tip region for three different capillary numbers. The frames show the tip regions just before dropping. The first and the third frames correspond to Fig. 22. (The three frames are scaled equally.)

24 S (05) /FLA AID:11865 Vol. ( ) [+model] P.24 (1-26) YJCIS:m5+ v 1.50 Prn:10/01/2006; 14:37 yjcis11865 by:vik p I.B. Bazhlekov et al. / Journal of Colloid and Interface Science ( ) Fig. 24. Comparison of the drop deformation at different values of the elasticity number. and x, and has also been used as an estimate of the strength of the surfactant in experimental work, where E and x cannot be measured. The ratio σ 0 /σ eq has been used by previous authors as a base for comparison with experimental results, see for instance [19]. Eggleton, Tsai and Stebe (2001) in their Fig. 2 show relatively good agreement between their numerical results and the experimental data of Hu, Pine and Leal [5] regarding the critical capillary number. Because the elasticity number in the experiments had not been known, σ eq /σ 0 was used as parameter. Above we also used σ 0 /σ eq to compare our results with the experimental measurements of de Bruijn [3]. To have an idea how reliable this approach is, we performed simulations at different values of the elasticity number. A small viscosity ratio was used, where the effect of surfactants is the strongest. The results for drop deformation were compared, see Fig. 24, for values of E and x at which σ 0 /σ eq is the same. The comparisons between the marked and unmarked lines in Fig. 24 show a reasonable agreement between the results for E = 0.2 and E = 0.4. At larger elasticity numbers, E>0.4, the differences become larger. Similar is the trend for smaller values E<0.2. For a twice smaller elasticity number, E = 0.1, the difference with the results for E = 0.2 is similar to that shown in Fig. 24 for E = 0.4. Bearing in mind that for higher viscosity ratios the effect of surfactants is smaller, we can conclude that the results presented here for E = 0.2 can be applied in a broader interval, E [0.1, 0.4], ifx is chosen in a way to keep σ 0 /σ eq unchanged The effect of surfactants at high capillary number An intuitive expectation is that the higher the capillary number, the smaller the effect of surfactants. Indeed, surfactants affect interfacial forces which at high capillary numbers are dominated by viscous forces. This is in agreement with the experimental observation of Janssen, Boon and Agterof (1994) regarding the dependence of the mode of breakup on the capillary number. Our simulations show that already for a capillary number about 3 4 times larger than the critical value, the surfactant has a negligible effect. This is shown in the case of lamellae formation, typical for small viscosity ratios and large Ca, see[28]. The simulation presented in the inset in Fig. 1 for Ca = 5 and λ = 0.1 was repeated in the case of surfactants at E = 0.2 and x = 0.5, see Fig. 25. The differences between the two cases, regarding drop deformation, are barely noticeable which confirms the above conclusion of no influence of the surfactants at high capillary numbers. It is seen from the surfactant concentration distribution shown in Fig. 25, that the external flow is more important for stretching the drop than for convection of surfactants. Thus the maximum concentration is much below the saturation level, Γ, and tip-streaming is not observed. Similar is the situation for intermediate viscosity ratio (λ in the range of unity). The effect of dilution on drop deformation for the case presented in Fig. 25 is negligible, also for higher x. Indeed, dilution is important in the flat regions of the interface where, however, the interface curvature is negligible and thus the total contribution to the interfacial forces is small. At high viscosity ratios, as it was shown for capillary numbers Ca <Ca cr, the effect of surfactants on drop deformation is in general small and it is even smaller for higher Ca. For clean interfaces at high Ca and λ the drop behaves in a tumblinglike motion, see [27] and the inset in Fig. 1 for Ca = and λ = 10. Our investigations for λ>5 show that surfactants have no effect on drop behavior. In this case the viscous forces from the drop phase play a dominant role on the interface velocity and prevent large surfactant concentration gradients. Thus the contribution of the Marangoni stress and drop stretching is negligible. The same is the situation with dilution, which is negligible due to the small deformations and small interfacial tension gradients.

25 S (05) /FLA AID:11865 Vol. ( ) [+model] P.25 (1-26) YJCIS:m5+ v 1.50 Prn:10/01/2006; 14:37 yjcis11865 by:vik p. 25 I.B. Bazhlekov et al. / Journal of Colloid and Interface Science ( ) 25 Fig. 25. The formation of lamellae (flattened drop) at large capillary number, Ca = 5, and small viscosity ratio, λ = 0.1, in the presence of insoluble surfactant at E = 0.2 andx = 0.5; t = Conclusions The presence of surfactants on a drop interface reduces the interfacial tension as given by the equation of state. Most of the models in use assume that the concentration of surfactants on the interface is limited by Marangoni stresses. Indeed, in the models the Marangoni stress becomes infinite at Γ,however, they predict a physically unrealistic negative interfacial tension at concentrations before Γ reaches Γ. Another upper bound exists for the concentration of surfactant molecules on the interface, which is determined by the finite dimension of the molecules and the intermolecular forces between them [18].An incorporation of such bound in the models can lead to a more realistic prediction of the interfacial tension at high surfactant concentrations. In the experimental studies [5,8] the small deformation theory for pure interfaces has been used to measure the interfacial tension in planar extensional flows. While this is justified in the case of small Péclet numbers, it is shown that in the presence of surfactants at Pe 1, due to a immobilization of the interface the drop deformation can differ significantly from that for pure interface, especially for small λ and/or relatively high x. Fora more accurate estimate of the interfacial tension in these cases we offer a predictor corrector procedure: The predictions based on experimental measurements for the drop deformation and small deformation theory are corrected using numerical simulations. Results presented here indicate that one correction step already gives a satisfactory accuracy. Depending on the Péclet number, Pe = ΛCaσ 0 /σ eq,different regimes of surfactant distribution are distinguished: diffusion controlled at Λ<0.1, convection dominated Λ>100 and transitional. In the first two regimes the drop behavior is independent of Pe and in this study the convection dominated surfactant distribution at Λ = 1000 is investigated. At small capillary numbers in simple shear flows, as well as in planar extensional flows, the drop deformation depends linearly on Ca. At a given Ca the deformation initially increases with increasing surface coverage x till x = 0.75 and at further increase of x, D decreases, see Fig. 12. The drop deformation is independent on the viscosity ratio for small λ, λ<0.1, provided that x>0.1. With increasing the viscosity ratio, the effect of surfactants decreases. These trends are also seen at intermediate drop deformations. For a better understanding of the influence of surfactants, the additional surface tension is decoupled into: dilution, Marangoni stress and stretching. We show that in addition to tip stretching [8,17], waist stretching also plays a significant role and has to be considered in the total drop stretching. Dilution and drop stretching are a direct result of the change of the interfacial tension which affects the normal interfacial forces, see Eqs. (15), (16). The Marangoni stress being tangential to the interface opposes the interfacial velocity leading to immobilization of the interface. The interface can be immobilized locally as well as globally, depending on the amount of surfactant, x. While in extensional flows the immobilization is related to complete blockage of the interface velocity, in simple shear flows the interface is immobilized in a broader sense: The effect on the interface velocity is similar to that of high viscosity drop (conveyor-belt like motion, see Fig. 15b). The Grace curve, Ca cr (λ, x) in the presence of surfactants is shown to have a plateau region (viscosity ratio independent) at small λ, seefig. 18, which at some λ joins that for the pure interface (surface coverage independent), provided that x<0.75. At Ca just above the critical value, different types of drop breakup are observed, see Fig. 19: Tip streaming at small λ (λ <0.5) and intermediate x (0.1 x 0.95). Stable drop shape, as for pure interface for λ>3.5, which is broader at higher values of x. And drop fragmentation in between. The size of drops emitted during the tip-streaming increases with Ca, as well as their equivalent interfacial tension. At large values of the capillary number, Ca several times larger than Ca cr, there is no effect of surfactants on the drop deformation. Finally comparisons between results at different elasticity numbers indicate that the predictions at a given value of the parameter σ eq /σ 0 (E, x) provide a good first approximation for a range of the elasticity numbers around E. Thisallowsfora broader application of the results obtained at a given value of the elasticity number.